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G e n e ra l re p o r t /D is c u s s io n s e s s io n 9 : S e le c t io n o f d e s ig n p a ra m e te rs fo r u n d e rg ro u n d

c o n s tru c t io n

R ap po rt d e s p é c ia lis te s /S é a n c e d e d is c u s s io n 9 : C h o ix d e s p a ram è tre s p o u r c o n s tru c t io n s s o u te r ra in e s

A .O .UR IEL, Uriel & A soc iados S .A ., M adrid, Spain

C .SAGASETA , Univers ity o f Cantabria , Santander, Spain

S YNOP S I S : Th e Re p o r t i s ma i n l y d e v o t e d t o p a r a me t r i c s e n s i t i v i t y a n a l y s i s . 2 - D t u n n e l s t a b i l i t y i n c l a y i s s h o wn t o b e c o n t r o l l e d b y t h e s t a b i l i t y r a t i o . Th e l o a d f a c t o r s h o u l d b e u s e d i n s t e a d f o r h e a d i n g s t a b i l i t y a n a l y s i s . Gr o u n d mo v e me n t s i n d u c e d b y t u n n e l l i n g h a v e r e c e i v e d g r e a t a t t e n t i o n . Gr o u n d l o s s i s s h o wn t o b e t h e o v e r r i d i n g p a r a me t e r . A n e w f o r mu l a t o e s t i ma t e i t i s p r o p o s e d wh i c h a c c o u n t s f o r t h e H/ D r a t i o . A r e c e n t l y a v a i l a b l e a n a l y t i c a l s o l u t i o n f o r g r o u n d d i s p l a c e me n t s i s d e s c r i b e d a n d e x t e n d e d t o t h r e e n e w c a s e s : n o n - h y d r o s t a t i c e x i s t i n g s t r e s s e s , e l a s t i c a n i s o t r o p y a n d c o mp r e s s i b i l i t y . Th e e f f e c t o f t h e H/ D r a t i o o n t h e s e t t l e me n t t r o u g h i s s e e n t o b e r e l e v a n t , wh e r e a s t h e i n f l u e n c e o f a n i s o t r o p y s e e ms t o b e s ma l l . Th e l a t e r a l e a r t h p r e s s u r e c o e f f i c i e n t h a s a n i mp o r t a n t e f f e c t o n b o t h t h e s e t t l e me n t p r o f i l e a n d t h e h o r i z o n t a l d i s p l a c e me n t s . S o i l c o mp r e s

s i b i l i t y s h o ws a c o n s i d e r a b l e i n f l u e n c e o n t h e r a t i o b e t we e n g r o u n d l o s s a n d s e t t l e me n t v o l u me .

1 I NTRODUCT I ON

T we n t y e x a c t y e a r s h a d e l a p s e d wh e n t h e Ri o Co n

f e r e n c e wa s h e l d s i n c e t h a t Au g u s t i n 1 9 6 9 wh e n P e c k p r e s e n t e d h i s S t a t e - o f - t h e - Ar t Re p o r t : ' De e p E x c a v a t i o n a n d Tu n n e l l i n g ' a t t h e Mé x i c o Co n f e

r e n c e . Gr e a t a d v a n c e s h a v e t a k e n p l a c e i n u n d e r

g r o u n d d e s i g n a n d c o n s t r u c t i o n s i n c e t h e n , p a r

t i c u l a r l y i n t h e a n a l y s i s o f t h e r e l a t i v e r e l e

v a n c e o f t h e p a r a me t e r s g o v e r n i n g s o i l b e h a v i o u r . I n s p i t e o f t h e t i me p a s s e d , P e c k ' s S t a t e - o f - t h e - Ar t Re p o r t r e ma i n s f u l l y f r e s h a n d v a l i d . Th e Ge n e r a l Re p o r t e r s a i m t o p a y a mo d e s t h o ma g e t o P e c k ' s c o n t r i b u t i o n i n i t s t we n t i e t h a n i v e r s a r y .

P e c k ( 1 9 6 9 ) g i v e s t h e t h r e e b a s i c r e q u i r e me n t s f o r a s a t i s f a c t o r y t u n n e l : Th e f i r s t o n e i s t h a t ' i t s h o u l d b e a b l e t o b e b u i l t ' . I t r e f e r s ma i n l y t o s t a b i l i t y o f t h e e x c a v a t i o n . Th e s e c o n d r e

q u i r e me n t i s t h a t ' c o n s t r u c t i o n o f t h e t u n n e l s h o u l d n o t e x c e s s i v e l y d a ma g e a d j a c e n t o r o v e r - l y i n g b u i l d i n g s , s t r e e t s o r u t i l i t i e s ' . I t i s r e l a t e d t o g r o u n d mo v e me n t s i n d u c e d b y t u n n e l

l i n g . Th e t h i r d r e q u i r e me n t i s t h a t ' i t s h o u l d b e c a p a b l e o f wi t h s t a n d i n g d u r i n g i t s l i f e t i me a l l t h e i n f l u e n c e s t o wh i c h i t ma y b e s u b j e c t e d ' . I t c a n b e r e l a t e d t o l i n i n g d e s i g n .

2 S COP E AND GUI DELI NES

Th i s Re p o r t c o v e r s b o r e d t u n n e l s . Cu t - a n d - c o v e r c o n s t r u c t i o n wa s d e a l t wi t h a t t h e Di s c u s s i o n S e s s i o n a n d h a s b e e n l e f t o u t o f t h e s c o p e o f t h i s Re p o r t . I t i s n o t i n t e n d e d t o b e a s t a t e - o f - t h e - a r t r e p o r t b u t a f r a me wo r k f o r p a r a me t r i c s e n s i t i v i t y a n a l y s i s .

Th e t h r e e r e q u i r e me n t s g i v e n a b o v e f o r m t h e g u i d e l i n e s o f t h i s Re p o r t . F i r s t l y , s t a b i l i t y o f t h e e x c a v a t i o n i s d i s c u s s e d . Th e n , t h e ma i n p a r t o f t h e Re p o r t i s f o c u s s e d o n g r o u n d mo v e me n t s c a u s e d b y s h a l l o w t u n n e l s ( t h o s e wh i c h c a n i n d u c e g r o u n d mo v e me n t s l a r g e e n o u g h t o a f f e c t t h e s t r u c t u r e s b u i l t o n g r o u n d s u r f a c e ) . F i n a l l y , s o me b a s i c i d e a s o n l i n i n g d e s i g n a r e s u mma r i z e d a n d a p a p e r o n t h i s t o p i c s u b mi t t e d t o t h e Co n f e r e n c e i s b r i e f l y c o mme n t e d .

3 GROUND I DEALI ZATI ON. S OI L P ARAMETERS

Th e c o mp l e x i t y o f mo s t g e o t e c h n i c a l p r o b l e ms ma k e s i t n e c e s s a r y t o a d o p t s o me s i mp l i f y i n g a s s u mp t i o n s r e g a r d i n g g e o me t r y , i n i t i a l s t a t e a n d g r o u n d p r o p e r t i e s . Un l e s s n u me r i c a l me t h o d s a r e a p p l i e d g r o u n d g e o me t r y i d e a l i z a t i o n s h o u l d b e v e r y s i mp l e , wi t h a f e w l a y e r s s e p a r a t e d b y h o r i

z o n t a l l i n e s i n c l u d i n g t h e wa t e r t a b l e . Ea c h g r o u n d l a y e r h a s t o b e i d e n t i f i e d b y me a n s o f t h e e x i s t i n g s t r e s s e s a n d p o r e p r e s s u r e s , t h e b u l k d e n s i t y , t h e v o i d r a t i o , a n d t h e g e o me c h a n i c a l p a r a me t e r s d e s c r i b i n g s o i l b e h a v i o u r .

3 . 1 Ex i s t i n g s t r e s s e s

Wi t h r e s p e c t t o i n i t i a l l y e x i s t i n g s t r e s s e s , t h e f u n d a me n t a l p a r a me t e r s a r e t h e u n i t we i g h t o f t h e s o i l a n d t h e e f f e c t i v e l a t e r a l e a r t h p r e s s u r e c o e f f i c i e n t , K0 . Th e s e , t o g e t h e r wi t h t h e h e i g h t

o f t h e g r o u n d wa t e r t a b l e a b o v e t h e p o i n t u n d e r c o n s i d e r a t i o n , a l l o w t h e d e t e r mi n a t i o n o f h o r i

z o n t a l a n d v e r t i c a l t o t a l a n d e f f e c t i v e s t r e s s e s a n d p o r e p r e BBu r e . F o r t o t a l s t r e s s a n a l y s e s t h e c o r r e s p o n d i n g l a t e r a l e a r t h p r e s s u r e c o e f f i c i e n t Kt c a n b e u s e d . I t i s r e l a t e d t o K0 a n d t h e p o r e

p r e s s u r e a s f o l l o ws :

R _ £ho_ _ O, ho+Uo = Ko +Up / o ' y o = K„ +r u

* ° »o ° ' vo+ u o l + Uo / o ' v o 1 + r u

No t e t h a t r u i s n o t t h e p o r e p r e s s u r e p a r a me t e r

i n t r o d u c e d b y Bi s h o p ( 1 9 5 5 ) f o r s l o p e s t a b i l i t y a n a l y s i s s i n c e t h e e f f e c t i v e v e r t i c a l s t r e s s i s u s e d i n s t e a d o f t h e t o t a l v e r t i c a l s t r e s s .

Th e p r e v i o u s s t r e s s h i s t o r y p l a y s a n i mp o r t a n t r o l e i n s o i l b e h a v i o u r a n d s h o u l d b e a c c o u n t e d f o r . Th e s i mp l e s t wa y t o t a k e i t i n t o a c c o u n t i s t h r o u g h t h e o v e r - c o n s o l i d a t i o n r a t i o wh i c h r e

l a t e s t h e c u r r e n t v e r t i c a l e f f e c t i v e s t r e s s t o t h e p a s t ma x i mu m o n e :

OCR = o ' v , „ / o ' v 0 ( 2)

2521

Th e OCR c a n b e e a s i l y o b t a i n e d b y me a n s o f o e d o - me t e r t e a t s wh e r e a s e s t i ma t i o n o f K0 r e q u i r e s t h e

me a s u r e me n t o f t h e ' i n s i t u ' h o r i z o n t a l s t r e s s .

I n o r d e r t o a v o i d s u c h a d i f f i c u l t me a s u r e me n t , S c h mi d t ( 1 9 6 6 ) p r o p o s e d a n e mp i r i c a l e x p r e s s i o n f o r K0 a s a f u n c t i o n o f t h e e f f e c t i v e l a t e r a l

e a r t h p r e s s u r e c o e f f i c i e n t f o r n o r ma l l y c o n s o l i

d a t e d c l a y s Koi k a n d t h e OCR:

K0 = K0„c . OCRn ( 3)

wh e r e n c a n b e g i v e n t h e v a l u e 0 . 4 .

3 . 2 S h e a r s t r e n g t h

N = Ovo/ Su ( 8 )

Th e l a t e r a l e a r t h p r e s s u r e c o e f f i c i e n t i n t e r ms o f t o t a l s t r e s s , Kt , t h e o v e r l o a d f a c t o r , N, a n d

t h e i n i t i a l s h e a r s t r e s s r a t i o , a „ , a r e i n t e r r e

l a t e d t h r o u g h t h e f o l l o wi n g s i mp l e e q u a t i o n s :

Kt = 1 - 2 a 0/ N ( 9)

A 0 = N( l - Kt ) / 2 ( 1 0 )

N = 2A0/ ( 1 - Kt ) ( 1 1 )

Ot he r us e f ul pa r a me t e r i s t he ' over l oad f act or ' :

Re g a r d i n g s o i l s t r e n g t h , t h e s i mp l e s t p a r a me t e r s t o c o n s i d e r a r e t h e e f f e c t i v e f r i c t i o n a n g l e , * ' , a n d c o h e s i o n , c ' ( f o r d r a i n e d o r l o n g t e r m c o n d i

t i o n s ) , a n d t h e u n d r a i n e d s h e a r s t r e n g t h , s u ( f o r u n d r a i n e d o r s h o r t t e r m c o n d i t i o n s ) .

La d d a n d F o o t t ( 1 9 7 4 ) f o u n d e x p e r i me n t a l l y t h e r a t i o i n t e r ms o f t h e s a me r a t i o f o r n o r

ma l l y c o n s o l i d a t e d c l a y s a n d t h e OCR:

bu / o ' v0 = ( s u / a ' v 0 ) „c . OCRA ( 4 )

wh e r e A c a n b e g i v e n t h e v a l u e 0 . 8 . Th e s a me f o r mu l a wa s t h e o r e t i c a l l y o b t a i n e d b y Wr o t h a n d Ho u l s b y ( 1 9 8 5 ) .

I t i s a l s o v e r y u s e f u l t o i n t r o d u c e a p a r a me t e r wh i c h a c c o u n t s f o r t h e i n i t i a l s h e a r s t r e n g t h mo b i l i z a t i o n ( b e f o r e t u n n e l e x c a v a t i o n ) . Th e mo s t a p p r o p r i a t e i s t h e ' i n i t i a l s h e a r s t r e s s r a t i o ' b e t we e n t h e d e v i a t o r s t r e s s a t t h e t u n n e l d e p t h a n d t wi c e t h e u n d r a i n e d s h e a r s t r e n g t h :

= ̂^vo- ^ho) / ^ Su ( 5)

S e v e r a l a u t h o r s h a v e a l r e a d y ma d e u s e o f i t , wi t h d i f f e r e n t p u r p o s e s t h o u g h u s i n g d i f f e r e n t d e s i g

n a t i o n s . Th i s i s a n e v i d e n c e o n h o w i n t u i t i v e i t i s . I t wa s f i r s t i n t r o d u c e d b y Da v i s a n d P o u l o s

( 1 9 6 8 ) a s a me a s u r e o f t h e i n i t i a l s a f e t y f a c t o r f o r f o u n d a t i o n s a n d wa s d e n o t e d b y f . Da v i s a n d P o u l o s u s e d t h i s p a r a me t e r t o e s t i ma t e t h e r e l a

t i v e i mp o r t a n c e o f t h e i mme d i a t e s e t t l e me n t wi t h r e s p e c t t o t h e d e l a y e d o n e . D' Ap p o l o n i a ( 1 9 7 1 ) r e d i s c o v e r e d t h i s p a r a me t e r a s t h e d e g r e e o f i n i t i a l p l a s t i f i c a t i o n f o r s t u d y i n g t h e h e a v e o f t h e b o t t o m o f a n e x c a v a t i o n . D' Ap p o l o n i a d e s i g

n a t e d i t a s t h e ' s h e a r s t r e s s r a t i o ' , f . S a g a s e t a ( 1 9 8 4 ) r e d i s c o v e r e d a g a i n t h e s a me p a r a me t e r f o r t h e a n a l y s i s o f t h e c a v i t y e x p a n s i o n , g i v i n g i t t h e d e n o mi n a t i o n o f ' i n i t i a l s t r e s s r a t i o ' , 6 0 . Wr o t h a n d Ho u l s b y ( 1 9 8 5 ) u s e d t h i s p a r a me t e r a g a i n wi t h t h e d e s i g n a t i o n o f 1 i n i t i a l s t r e s s p a r a me t e r ' , f , a d e n o mi n a t i o n wh i c h i s a n h y b r i d b e t we e n t h e o n e s a d o p t e d b y D' Ap p o l o n i a a n d S a g a

s e t a , a n d p r o p o s e d t o b e u s e d i n g e n e r a l f o r c l a y s . Ho u l s b y a n d Te h ( 1 9 8 8 ) u s e d t h i s p a r a me t e r o n c e mo r e a t t h e I n t . S y mp . o n P e n e t r a t i o n T e s t

i n g f o r t h e a n a l y s i s o f t h e CP T r e s u l t s a n d d e n o

t e d i t b y a . At t h e s a me s y mp o s i u m, Wr o t h ( 1 9 8 8 ) r e f e r r e d t o i t a s t h e ' i n i t i a l s h e a r s t r e s s r a

t i o ' o r ' h o r i z o n t a l s t r e s s i n d e x ' ,

Wr o t h a n d Ho u l s b y ( 1 9 8 5 ) g a v e a r e l a t i o n s h i p b e t we e n a „ a n d t h e OCR, b a s e d o n e q u a t i o n s ( 3 ) a n d ( 4) :

a „ = ( a ' v0- a ' h 0 ) / 2 s u = ( l - K0 ) / ( 2 s u / o ' v0 ) ( 6 )

_ 1-K, hc . 0CR"

0 ‘ 2( bu / o ' v0),,c . OCRa

3 . 3 De f o r ma b i l i t y

De f o r ma t i o n p a r a me t e r s a r e u s u a l l y d e f i n e d a s s u

mi n g e l a s t i c s o i l b e h a v i o u r . I n t e r ms o f e f f e c

t i v e s t r e s s e s a n d f o r t h e c a s e o f i s o t r o p y t h e p a r a me t e r s a r e t wo : t h e mo d u l u s o f e l a s t i c i t y , E' ( E ' >0 ) , a n d t h e P o i s s o n ' s r a t i o , p ' ( 0 < p ' < 0 . 5 ) . Th e s h e a r mo d u l u s , G = E ' / 2 ( l + p ' ) , i s o f t e n u s e d i n s t e a d o f t h e mo d u l u s o f e l a s t i c i t y . F o r u n d r a i

n e d c o n d i t i o n s a n d t o t a l s t r e s s e s t h e i s o t r o p i c p a r a me t e r s r e d u c e t o o n e , t h e u n d r a i n e d mo d u l u s , Eu , wh i c h c a n b e e a s i l y r e l a t e d t o t h e e f f e c t i v e

e l a s t i c p a r a me t e r s : Eu = 3 E ' / 2 ( l + p ' ) . P o i s s o n ' s

r a t i o e q u a l s o n e h a l f f o r t h e c o n d i t i o n o f z e r o v o l u me c h a n g e . Th e s h e a r mo d u l u s i s : G = Ey/ 3 .

Ne v e r t h e l e s s , ma n y s e d i me n t a r y s o f t s o i l s e x h i

b i t c r o s s - a n i s o t r o p y , wi t h t h e v e r t i c a l l i n e a t a n y p o i n t a s t h e a x i s o f r a d i a l s y mme t r y . Th i s k i n d o f a n i s o t r o p y i s d e s c r i b e d b y me a n s o f f i v e i n d e p e n d e n t p a r a me t e r s . I n t e r ms o f e f f e c t i v e s t r e s s e s , t h e y a r e : t h e e l a s t i c mo d u l u s i n t h e h o r i z o n t a l a n d v e r t i c a l d i r e c t i o n s , E' h a n d E ' v ,

r e s p e c t i v e l y ; t h e P o i s s o n ' s r a t i o s ( f o r t h e

e f f e c t o f h o r i z o n t a l s t r e s s o n h o r i z o n t a l s t r a i n ) a n d p' yf t ( f o r t h e e f f e c t o f v e r t i c a l s t r e s s o n

h o r i z o n t a l s t r a i n ) , a n d t h e i n d e p e n d e n t s h e a r mo d u l u s , G^ . F o r t h e p u r p o s e o f p a r a me t r i c s e n

s i t i v i t y a n a l y s e s , t wo u s e f u l r a t i o s a r e :

n ' = E' h / E' v g ' = G^ / E' , , ( 1 2 a , b )

Th e s t r e s s - s t r a i n e q u a t i o n s f o r c r o s s - a n i s o t r o p i c s o i l s c a n b e f o u n d s o me wh e r e a n d a r e o mi t t e d h e r e . Th e c r o s s - a n i s o t r o p i c e l a s t i c p a r a me t e r s c a n n o t t a k e a n y a r b i t r a r y v a l u e s . Th e l i mi t s we r e t h e o r e t i c a l l y e s t a b l i s h e d b y He a r mo n ( 1 9 6 1 ) , b a s e d o n s t r a i n e n e r g y c o n s i d e r a t i o n s . P i c k e r i n g ( 1 9 7 0 ) g a v e a c l e a r d e s c r i p t i o n o f t h e t h e o r e t i

c a l b o u n d i n g v a l u e s o f t h e e l a s t i c p a r a me t e r s .

E ' h , E ' v a n d G ^ s h o u l d b e p o s i t i v e a n d s o n ' a n d

g ' . Th e a b s o l u t e l i mi t s f o r p *„j, a r e : - l i p ' ^ i l .

Th e e n v e l o p e f o r a l l o wa b l e v a l u e s o f a n d p' , ^, i s :

n ' ( 1 —P ' h h ) i 2 P ' h v J ( 1 3 )

Ur i e l a n d Ca ñ i z o ( 1 9 7 1 ) s h o we d t h a t t h e b o u n d i n g s u r f a c e g i v e n b y e q u a t i o n ( 1 3 ) i s t h e c o n e r e p r e

s e n t e d i n F i g u r e 1. I t s v e r t e x i s p o i n t B a n d i t s a x i s l i e s o n p l a n e p ' hh=0 a n d i s i n c l i n e d a t 4 5 * .

Th e c r o s s s e c t i o n o f t h e c o n e i s a n e l l i p s e wh o s e ma j o r d i a me t e r i s t wi c e i t s mi n o r d i a me t e r . Th e s e a u t h o r s p r o p o s e d a n o t h e r u s e f u l r a t i o t o b e t t e r e x p r e s s t h e d e g r e e o f s o i l a n i s o t r o p y :

2 5 2 2

n =

(JLhh

F i g u r e 1. Bo u n d s f o r a n i s o t r o p i c p a r a me t e r s .

r 1 = ( 1 4 )

wh i c h i s t h e c o mp r e s s i b i l i t y r a t i o u n d e r u n i a x i a l s t r e s s i n t h e p r i n c i p a l d i r e c t i o n s . Th e r a t i o r 1 i s a l s o t h e d e f o r ma b i l i t y r a t i o i n t h e v e r t i c a l a n d h o r i z o n t a l d i r e c t i o n u n d e r h y d r o s t a t i c c o m

p r e s s i o n . E v e r y a n i s o t r o p i c s o i l i s d i l a t a n t ( e x c e p t t h o s e l y i n g o n t h e p l a n e ABC o f F i g u r e 1, f o r wh i c h r ' = l ) i n t h e s e n s e t h a t a s o i l c a n b e d e f i n e d a s d i l a t a n t wh e n u n d e r g o e s v o l u me c h a n g e a s a r e s u l t o f s h e a r i n g s t r e s s e s wi t h o u t v a r i a

t i o n i n t h e h y d r o s t a t i c s t r e s s . Th e v o l u me t r i c s t r a i n i s :

3e oct = ( l - 2 p 1 hv/ n 1 ) ( 1 - r 1 / a 1 ) &o 1 V/ E 1 v =

( n ' - 2 p ' ) ( r ' - a ' ) 3ao'

1 - a 1( 1 5 )

wh e r e a' = - a o 1 „ / ( 4 0 ' h+Aa ' x) a n d o ' h , o' j d e n o t e

h o r i z o n t a l s t r e s s e s i n p e r p e n d i c u l a r d i r e c t i o n s . F o r a ' = 1 ( i . e . , n o v a r i a t i o n i n t h e h y d r o s t a t i c s t r e s s ) , t h e s i g n o f t h e v o l u me c h a n g e ( d i l a t i o n o r c o n t r a c t i o n ) d e p e n d s o n t h e p o s i t i o n o f t h e s o i l wi t h r e s p e c t t o t h e p l a n e ABC. S o i l s l y i n g a b o v e t h a t p l a n e u n d e r g o v o l u me c h a n g e o f t h e s a me s i g n a s a o ' v .

Two i n t e r e s t i n g l i n e s o f p l a n e ABC a r e AB a n d BC. Bo t h s t r a i g h t l i n e s b e l o n g t o t h e b o u n d i n g c o n e . L i n e AB i s t h e l o c u s f o r i n c o mp r e s s i b l e ma t e r i a l s . S i n c e l i n e AB i s t h e i n t e r s e c t i o n o f t h e p l a n e s l - p ' h h - p ' h v = 0 a n d n ' _ 2 H, hv = 0/ t h e r e

f o r e r ' i s u n d e t e r mi n e d a n d i s e q u a l t o z e r o

i r r e s p e c t i v e o f t h e s t r e s s c h a n g e . S o i l s l y i n g o n l i n e AB d o n o t u n d e r g o v o l u me t r i c s t r a i n .

Th e s h e a r s t r a i n u n d e r p r i n c i p a l s t r e s s i n c r e

me n t s AO' y, AO' h= AOl 1 i s :

- 2 a , ( P ’ hv+ n ' ) + ( p ' hh- 2 p ' hv- l )

* 0« = - - - - - - - - - - - 7- r - : - - - - - - - - - - - - - - : r: — d 6 )l + 2 a ' E \

Th e d i s t o r t i o n u n d e r h y d r o s t a t i c p r e s s u r e o i s :

t oct = 2 o ( p ' llh- i i ' l w+ n 1- l ) / 3 E ' ll ( 1 6 a )

Ni c e e x a mp l e s o f d i s t o r t i o n u n d e r h y d r o s t a t i c

p r e s s u r e c a n b e f o u n d i n Al l i r o t e t a l ( 1 9 7 7 ) .

S i n c e l i n e BC i s t h e i n t e r s e c t i o n o f t h e p l a n e s P ' h v + n ' = 0 a n d P ' m T 2 !1 ' h v - 1 = ° > h e n c e Yoc t i s

e q u a l t o z e r o a n d , t h e r e f o r e , s o i l s l y i n g o n l i n e BC d o n o t u n d e r g o s h e a r s t r a i n .

I t i s i n t e r e s t i n g t o n o t e t h a t , a p a r t f r o m t h e l a r g e n u mb e r o f p a r a me t e r s t o h a n d l e , t h e me a s u

r e me n t o f t h e i n d e p e n d e n t s h e a r mo d u l u s i s v e r y d i f f i c u l t , a s s h o wn b y Ur i e l ( 1 9 7 3 ) .

E f f e c t i v e s t r e s s e s c o n t r o l s o i l d e f o r ma t i o n . He n c e , a s s u mi n g t h e s o i l s k e l e t o n b e h a v e s a s a l i n e a r e l a s t i c c r o s s - a n i s o t r o p i c ma t e r i a l , t h e f i v e e l a s t i c p a r a me t e r s c o mp l e t e l y d e f i n e t h e p r o b l e m. Un d e r s a t u r a t e d a n d u n d r a i n e d c o n d i t i o n s ( a s s u mi n g i n c o mp r e s s i b i l i t y o f b o t h wa t e r a n d s o i l p a r t i c l e s ) t h e s t r e s s - s t r a i n r e l a t i o n s h i p s i n t e r ms o f t o t a l s t r e s s a r e a l s o c r o s s - a n i s o t r o - p i c ( Ur i e l , 1 9 7 0 ) . Th e g e n e r a l e x p r e s s i o n f o r t h e c h a n g e i n p o r e wa t e r p r e s s u r e d u e t o t o t a l s t r e s s i n c r e me n t s Ao h , ¿ o 1 a n d a o v i s :

( l - p ' u . - n ' p ' ^ KAo ^ i t n 1 ( l - 2 p ' hv) Aov

( 2 +n ' - 4 p 1 hv- 2 p ’hh)

r 1 ( AOn+AOj ) +a ov 1 - r ' / a

l + 2 r ' l + 2 r 'a o „ ( 1 7)

wh e r e a = - Ao v / ( A0h+ A01 ) .

Un d e r p l a n e s t r a i n c o n d i t i o n s :

a u = [ ( 4- n' ) r ' Aoh+( 2 +n , r ' )a o v] / 2 ( l + 2r ' ) ( 1 8 )

Th e p o r e p r e s s u r e c o e f f i c i e n t i s :

Aa = ( r ' s i n Ja + c o s 2a ) / ( l + 2 r ' ) ( 1 9 )

wh e r e a i s t h e a n g l e b e t we e n t h e a x i s o f a n i s o

t r o p y a n d t h e ma j o r p r i n c i p a l s t r e s s i n c r e me n t .

By u s i n g t h e p r i n c i p l e o f e f f e c t i v e s t r e s s a n d i n t r o d u c i n g ( 1 7 ) i n t o t h e s t r e s s - s t r a i n e q u a

t i o n s , t h e e l a s t i c p a r a me t e r s f o r s a t u r a t e d s o i l a r e e x p r e s s e d i n t e r ms o f t h o s e f o r t h e s o i l s k e l e t o n a s f o l l o ws ( Ur i e l , 1 9 7 0 ) :

2 ( n' - n ' p ' hh- 2 p ' hv2) ______

( 2 +n ' - 4p ' ^ - 2 }! ' ^ ) - ( 1- p ' „, - p ’hv) J

_ 2 [ ( l +2 r ' ) p ' lB, + ( l - P ,Ml - P , hv)]

( l +2 r ' J - r Ml - p ' ^ - p ' h v )

Eh ____________2 + n , - 4p ' hy - 2 p , hh__________

E ' h " ( 2 +n ' - 4 p ' hv- 2p ' hh) - ( l - p ' hh- p ' h v ) 2

( 2 0 )

P h h = l - n / 2 P h v =n /2 Ghv=G ’ t

( 2 1 )

( 2 2 )

wh e r e n = Eh / Ev i s a u s e f u l r a t i o . An o t h e r u s e f u l

r a t i o f o r u n d r a i n e d c o n d i t i o n s i s g=Ghv/Ev .

No t e t h a t t h e i n d e p e n d e n t e l a s t i c p a r a me t e r s r e d u c e t o t h r e e , a n u mb e r o f c o n s t a n t s wh i c h i s e a s i e r t o h a n d l e . Ne v e r t h e l e s s , t h e d i f f i c u l t y o f d e t e r mi n i n g t h e s h e a r mo d u l u s s t i l l r e ma i n s .

S i n c e l i n e AB i n F i g u r e 1 i s t h e l o c u s f o r s a t u r a t e d s o i l s u n d e r u n d r a i n e d c o n d i t i o n s , i t f o l l o ws t h a t t h e l i mi t i n g v a l u e s f o r t h e i r e l a s

t i c p a r a me t e r s a r e : 0 < n < 4 ; 0 < p hv<2 ; - l i p ^ i l .

Un d e r s a t u r a t e d u n d r a i n e d p l a n e s t r a i n c o n d i

t i o n s t h e s t r e s s - s t r a i n e q u a t i o n s b e c o me s i mp l y :

252 3

eh = ( a o „ - a o v ) ( 23a )

4- ne v = - TZ— ( - Aa h+AOv ) ( 23b)

’ "v

Y h v = A T h v / ® h v ( 23c )

Al t h o u g h h o mo g e n e o u s g r o u n d c o n d i t i o n s wi l l be a s s ume d , a c e r t a i n k i n d of h e t e r o g e n e i t y a r i s e s f r om t he f a c t t ha t s o i l b e h a v i o u r i s d i f f e r e n t i n l o a d i n g t h a n i n un l oa d i n g . Th i s i s s p e c i a l l y r e l e v a n t t o g r o u n d d e f o r ma t i o n s i n d u c e d b y t he e x c a v a t i o n a n d wi l l be d i s c u s s e d l a t e r on.

4 HEADI NG AND FACE STABI LI TY

Be f o r e g o i n g f ur t he r , i t i s i n t e r e s t i n g t o e s t a b l i s h t he mo s t a p p r o p r i a t e g e o me t r i c a l p a r a me t e r s . I n wh i c h f o l l o ws t he s i mp l e s t c a s e of a c i r c u l a r t u n n e l i n a h o mo g e n e o u s g r o u n d wi t h h o r i z o n t a l s u r f a c e wi l l be c o n s i d e r e d ( Fi g. 2) . Ad o p t e d a xe s n o t a t i o n i s : x, t r a n s v e r s e h o r i z o n t a l d i r e c t i o n , p o s i t i v e t o t he r i gh t ; y, l o n g i t u d i n a l h o r i z o n t a l d i r e c t i o n , p o s i t i v e o p p o s i t e t o t he t unne l a dva nc e ; z, v e r t i c a l d i r e c t i o n , p o s i t i v e downwa r d s . The o r i g i n of c o o r d i n a t e s i s t he p o i n t of i n t e r s e c t i o n of t he t u n n e l c e n t r e l i ne a t t he t u n n e l f a c e wi t h t he g r o u n d s ur f a c e .

Re g a r d i n g t he g e o me t r y of t he p r o b l e m u n d e r c o n s i d e r a t i o n , o n l y t h r e e v a r i a b l e s a r e r e qu i r e d . Fo l l o wi n g n o r ma l p r a c t i c e , t he d e p t h t o t he t u n ne l a xi s , H, a n d t he t u n n e l d i a me t e r , D, wi l l be ke p t f or t he d e f i n i t i o n of t he c r o s s - s e c t i o n . Si n c e i n mo s t c a s e s D i s c o n s t a n t wh i l s t H v a r i e s a l o n g t he t unne l , t he a u t h o r ' s r e c o mme n d a t i o n i n o r d e r t o us e d i me n s i o n l e s s c o o r d i n a t e s c o n s i s t s of d i v i d i n g t h e m b y D. Th e t h i r d v a r i a b l e i s t he l e n g t h of t he u n l i n e d h e a d i ng , P.

Co n s t r u c t i o n of s h a l l o w t u n n e l s i n s of t s o i l s u s u a l l y r e q u i r e s i mme d i a t e s u p p o r t by me a n s of a s hi e l d . Fu r t h e r mo r e , t he us e of a c e r t a i n i n t e r n a l s u p p o r t b y me a n s of c o mp r e s s e d a i r or c l a y s l u r r y u n d e r p r e s s u r e ma y a l s o be r e q u i r e d i n o r d e r t o p r e v e n t c o l l a p s e of t he h e a d i n g or f a c e of t he t unne l . Th e d e t e r mi n a t i o n of t he r e q u i r e d i n t e r n a l p r e s s u r e c a n be c o n s i d e r e d a s o n e of t he ma i n a s p e c t s of d e s i g n f or t h i s t ype of u n d e r g r o u n d c o n s t r u c t i o n .

I n p r i n c i p l e , t he i n t e r n a l p r e s s u r e c a n be g i v e n a v a l u e s o t ha t f l o w of wa t e r i n t o t he t u n n e l i s p r e v e n t e d , wh i c h me a n s t ha t t he t u nne l p r e s s u r e ha s t o be g r e a t e r t h a n t he p o r e wa t e r p r e s s u r e a t t he t u n n e l i nve r t . An o t h e r p o s s i b l e a p p r o a c h wo u l d c o n s i s t of p r o v i d i n g a n i n t e r na l p r e s s u r e e q u i v a l e n t t o t he t o t a l o v e r b u r d e n p r e s s u r e a t t he t u n n e l d e p t h b e f o r e e x c a v a t i o n . Th i s a p p r o a c h c a n c l e a r l y r e s u l t i n e x c e s s i v e o v e r a l l c os t a nd, i f e x c a v a t i o n i s d e s i g n e d u n d e r a i r p r e s s u r e , i t c a n a l s o r e s u l t i n h e a l t h r i s ks f or t he wo r ke r s . I t i s t h e r e f o r e n e c e s s a r y t o e s t a b l i s h s u i t a b l e mi n i mu m s a f e v a l u e s f or t he i n t e r na l p r e s s u r e t o be a ppl i e d .

Th e a v a i l a b l e t h e o r e t i c a l a p p r o a c h e s a r e g e n e r a l l y b a s e d on e i t h e r l i mi t e q u i l i b r i u m or l i mi t a n a l y s i s me t hods . Ap a r t f r o m t he g e o me t r i c a l v a r i a b l e s a n d t he s o i l s h e a r s t r e ng t h , a f u n d a me n t a l p a r a me t e r i n v o l v e d i n t he s e c a l c u l a t i o n s i s t he un i t we i g h t of t he s oi l .

Sz 6 c h y ( 1961) d e s c r i b e s s ome of t he e a r l y s o l u t i ons b a s e d on e s t a b l i s h i n g t he l i mi t e q u i l i b r i u m of a c e r t a i n s o i l ma s s a b o v e t he t u n n e l u n d e r i t s own we i g h t a n d t he s h e a r r e s i s t a n c e a l o n g t he p l a n e s or s u r f a c e s wh i c h f o r m t he b o u n d s of t he

Fi g u r e 2. Pr o b l e m ge ome t r y .

g r o u n d ma s s . Th e s e me t h o d s of a n a l y s i s a l l o w t he e s t i ma t e of t he s a f e t y a g a i n s t c o l l a p s e f or a n y p a r t i c u l a r t u n n e l but do not l e a d t o c l o s e d f o r m s o l u t i o n s of g e n e r a l a p p l i c a b i l i t y .

Wi t h r e s p e c t t o t he u s e of l i mi t e q u i l i b r i u m me t h o d s i n c o n n e c t i o n wi t h l a b o r a t o r y mo d e l t e s t i ng, t he f i r s t c o n t r i b u t i o n s c a n be t r a c e d b a c k t o t he wo r k f r o m Br oms a n d Be n n e r ma r k ( 1967) , wh o c o n d u c t e d e x p e r i me n t s e x t r u d i n g c l a y u n d e r v e r t i c a l p r e s s u r e t h r o u g h v e r t i c a l ope n i ngs . Th e y d e f i n e d a u s e f u l p a r a me t e r wh i c h i s c a l l e d t he1 s t a b i l i t y r a t i o 1":

s u

wh e r e o 4 d e n o t e s t u n n e l i n t e r n a l p r e s s u r e . I f no

i n t e r n a l p r e s s u r e i s a p p l i e d , t he s t a b i l i t y r a t i o c o i n c i d e s wi t h t he a l r e a d y d e f i n e d ' ove r l oa d f a c t or ' . Fr o m t h e i r o b s e r v a t i o n s , Br o ms a n d Be n n e r ma r k c o n c l u d e d t ha t N v a l u e s b e l o w 6 i n d i c a t e t ha t t he o p e n i n g i s s t a bl e .

Th e s o l u t i o n g i v e n by Br o ms a n d Be n n e r ma r k a p p l i e s t o p l a n e s t r a i n c o n d i t i o n s ( 2- D pr ob l e m) . I t i s o b v i o u s t ha t t he p r o b l e m of t he s t a b i l i t y of a t u n n e l h e a d i n g i s ma r k e d l y t h r e e - d i me n s i o n a l . Sa f e t y a g a i n s t c o l l a p s e s h o u l d d e p e n d l a r ge l y on t he r e l a t i v e l e n g t h of t he u n l i n e d h e a d i n g wh i c h v a r i e s f r o m z e r o, f or s h i e l d d r i v e n t u n ne l s , t o i n f i n i t y , f or c o mp l e t e l y u n l i n e d t u n ne l s f a r f r om t he f a c e . A s t a b i l i t y p r o b l e m r e s t r i c t e d t o t he t u n n e l f a c e c a n n o t be r e p r e s e n t e d by Br oms a nd Be n n e r ma r k ' s s o l u t i on .

Fo l l o wi n g t he l i ne of l i mi t e q u i l i b r i u m a p p r o a c he s , Ba l t h a u s ( 1989, p a p e r t o t h i s Co n f e r e nc e ) ha s p r o p o s e d a n e w me t h o d s p e c i a l l y c o n c e i v e d t o h a n d l e s l u r r y s h i e l d t unne l de s i gn , a s i t i s ma i n l y b a s e d on c o n v e n t i o n a l s t a b i l i t y c o n s i d e r a t i o n s f o r s l u r r y t r e n c h d i a p h r a g m wa l l s . Th i s a p p r o a c h a l l o ws t o t a ke i n t o a c c o u n t s oi l s t r a t i f i c a t i o n , a r c h i n g e f f e c t s a b o v e t he t unne l c r o wn a s we l l a s a n y o t h e r f e a t u r e u s u a l l y c o n s i d e r e d i n c o n v e n t i o n a l c a l c u l a t i o n s f or d i a p h r a g m wa l l s .

Wi t h r e g a r d t o t he us e of t he l i mi t t h e o r e ms of p l a s t i c i t y , i t ha s c o n s t i t u t e d t he ma i n t h e o r e t i c a l a p p r o a c h f o l l o we d a t Ca mb r i d g e Un i v e r s i t y , i n c o n j u n c t i o n wi t h e x t e n s i v e r e s e a r c h o n c e n t r i f u ga l mo d e l t e s t s .

2 5 2 4

0.2

0 . 1

O L a b o ra to r y m o d e l*

A C e n t r if u g a l m o de ls

L o w e r b o u n d

A 8 A

U p p e r b o u n d

0 .5 1.5 2 .5 3 .0

H / D

Fi g u r e 3. I n t e r n a l p r e s s u r e a t c o l l a p s e f or c ohe - s i o n l e s s s o i l s ( * ' =50 ' ) . Lo we r a n d u p p e r b o u n d s o l u t i o n s ( At k i n s o n a nd Pot t s , 1977) .

H / D

Fi gur e 5. I n t e r n a l p r e s s u r e a t c o l l a ps e . Ci r c u l a r t u n n e l s i n c l a y ( p l a i n s t r a i n) ( Ma i r , 1983) .

Fi g u r e 4. Lo we r a nd u p p e r b o u n d s o l u t i o n s f or h e a d i n g s t a b i l i t y i n c l a y ( Da vi s e t a l , 1980) .

Ak t i n s o n a n d Po t t s ( 1977) p r e s e n t e d c o mp r e h e n s i v e s o l u t i o n s f or t he s t a b i l i t y of a c i r c u l a r t unne l i n c o h e s i o n l e s s s o i l u n d e r p l a i n s t r a i n c o n d i t i ons ( i . e . , a wa y f r o m t he he a d i ng) . Fi g u r e 3 s hows a c o mp a r i s o n b e t we e n mo d e l t e s t r e s u l t s a nd t h e o r e t i c a l p r e d i c t i o n s f or t he p a r t i c u l a r c a s e of *' = 5 0 ‘ . Th e o r i g i n a l f i gu r e h a s b e e n r e e l a b o r a t e d t o be e x p r e s s e d i n t e r ms of t he p a r a me t e r s u s e d i n t h i s Re por t . I t c a n be s e e n t ha t bo t h l a b o r a t o r y a n d c e n t r i f u g e t e s t r e s u l t s l i e b e t we e n t he c a l c u l a t e d u p p e r a n d l owe r b ounds a nd, a s p o i n t e d out by t he a u t ho r s , t he b o u n d s o l u t i o n s a r e r e l a t i v e l y c l o s e t o e a c h ot he r .

Da v i s e t a l ( 1980) a d o p t e d t he s a me a p p r o a c h f or t he a n a l y s i s of h e a d i n g a n d f a c e s t a b i l i t y of s h a l l o w t u n n e l s i n c o h e s i v e s oi l u n d e r u n d r a i n e d c ond i t i ons . The a u t h o r s a n a l y z e d t h r e e p a r t i c u l a r c a s e s : t he p l a n e s t r a i n u n l i n e d c i r c u l a r t unne l , t he p l a n e s t r a i n h e a d i n g a n d t he c i r c u l a r t unne l he a d i ng , a s we l l a s o t h e r f e a t u r e s s uc h a s l oc a l c o l l a p s e a n d b l o w- o u t f a i l ur e .

Da v i s e t a l d i s c u s s t he i n f l u e n c e of t he d i f f e r e n t p a r a me t e r s i n v o l v e d i n t h e i r c a l c u l a t i o n s , s p e c i a l l y t he e f f e c t of t he r a t i o * D/ s u a n d o b

t a i n u p p e r a nd l owe r b o u n d s f or t he c a s e s p r e v i o u s l y me n t i o n e d . Fi g u r e 4 ( a l s o r e - e l a b o r a t e d ) p r e s e n t s u p p e r a n d l owe r b o u n d s f or t he c i r c u l a r t unne l h e a d i n g i n t e r ms of t he s t a b i l i t y r a t i o.I t a l s o i n c l u d e s t he s o l u t i o n f r o m Br oms a nd Be nne r ma r k , wh i c h c a n be s e e n t o c o n s t i t u t e a n a v e r a g e l owe r b o u n d f or f o r H/ D r a t i o s i n e xc e s s of 3. 5. Howe v e r , t he a u t h o r s r e c o mme n d t he i r r e s u l t s t o be u s e d f or c a l c u l a t i o n s i n wh i c h H/ D<3. 5 .

Th e c l e a r i n f l u e n c e of t he r a t i o * D/ s u on t he s t a b i l i t y of a c i r c u l a r t u nne l i n c o h e s i v e s oi l c a n be s e e n i n Fi g u r e 5, wh i c h g i v e s t he s uppor t p r e s s u r e s r e q u i r e d t o p r e v e n t c o l l a p s e a s a f u n c t i o n of t he t u n n e l de pt h. The c r i t e r i o n g i ve n i n Fi g u r e 6 ( whi c h c a n be d e d u c e d f r om Fi g u r e 5) c a n be a p p l i e d i n c o n s i d e r i n g wh e t h e r a p l a ne - s e c t i o n c i r c u l a r t u nne l i n c l a y i s l i k e l y t o be s t a b l e wi t h o u t t e mp o r a r y s u p p o r t ( Ma i r , 1983) .

Fi g u r e 6 . St a b i l i t y c r i t e r i o n f or u n s u p p o r t e d t u n n e l s i n c l a y ( p l a i n s t r a i n ) ( Ma i r , 1983) .

Ki mu r a a nd Ma i r ( 1981) a n d Ma i r ( 1983) h a v e a n a l yz e d t he e f f e c t of t he t h r e e - d i me n s i o n a l n a t u r e of a t unne l h e a d i n g on t unne l s t a b i l i t y . The r e s u l t s i n t e r ms of t he s t a b i l i t y r a t i o a r e s u mma r i z e d i n Fi g u r e 7.

At k i n s o n a n d Ma i r ( 1981) i n t r o d u c e d t he c o nc e p t of t u nne l s t a b i l i t y numbe r s , Tc , T. a n d T, , a n a

l ogous t o t he b e a r i n g c a p a c i t y f a c t o r s f or f o u n d a t i o n s Nc , Nq a nd Nr , r e s pe c t i ve l y .

Al s o , Ma i r ( 1983) i n t r o d u c e d t he ' l oa d f a c t or ' , LF, d e f i n e d as :

LF ( 25)( o ^ - o j / l o ^ - o ^ )

wh e r e o l c i s t he v a l u e of o 4 a t wh i c h c o l l a p s e

wo u l d oc c ur . By c o mp a r i n g e q u a t i o n s ( 24) a nd ( 25) i t f o l l ows t ha t t he l oa d f a c t o r i s t he r a t i o

25 2 5

Fi g u r e 7. St a b i l i t y r a t i o a t c o l l a p s e f or u n s u p p o r t e d t u n n e l h e a d i n g s i n c l a y ( Ma i r , 1983) .

b e t we e n t he s t a b i l i t y r a t i o a t wo r k i n g c o n d i t i o n s a nd t he s t a b i l i t y r a t i o a t c o l l a p s e a nd, he nc e ,LF i s t he r e c i p r o c a l of t he s a f e t y f a c t o r a g a i n s t c o l l a ps e . Sh o u l d ai be e q u a l t o t he o v e r b u r d e n

p r e s s u r e ov0, t he n LF = 0 a n d t he s a f e t y f a c t o r

i s i n f i n i t e . On t he c o n t r a r y , wh e n a i i s r e d u c e d

t o o l c, t he n LF = 1 a n d t he f a c t o r of s a f e t y i s

e q u a l t o one ( l i mi t e qu i l i b r i u m) .The r e f o r e , t he l oa d f a c t o r i s a v e r y c o n v e n i e n t

wa y t o e x p r e s s t he s a f e t y a g a i n s t c o l l a p s e of t he h e a d i n g of a t unne l a n d c a n be e a s i l y e v a l u a t e d b y u s i n g t he c u r v e s i n Fi g u r e 7. As i t s h o u l d be e xpe c t e d , t he l oa d f a c t o r ha s i mp o r t a n t i mp l i c a t i ons r e g a r d i n g t he ma g n i t u d e of g r o u n d mo v e me n t s i n d u c e d b y t he e x c a v a t i o n of s h a l l o w t unne l s .

I t c a n be c o n c l u d e d t ha t t he d e s i g n p a r a me t e r s t ha t s h o u l d be s e l e c t e d t o a n a l y z e t he h e a d i n g a n d f a c e s t a b i l i t y of a t unne l i n c l a y a r e , b a s i c a l l y , t he s t a b i l i t y r a t i o a n d t he l oa d f a c t or .

5 GROUND MOVEMENTS

Gr o u n d s u b s i d e n c e i n d u c e d b y u n d e r g r o u n d e x c a v a t i on a n d i t s e f f e c t s on s t r u c t u r e s ha s a l wa ys b e e n a ma j o r c o n c e r n f or mi n i n g e ng i n e e r s . The r a p i d l y i n c r e a s e d c o n s t r u c t i o n of t u n n e l s i n u r b a n a r e a s , s p e c i a l l y i n s of t s oi l s , ha s e x t e n d e d t h i s c o n c e r n t o t he f i e l d of t u n n e l l i n g i n c i v i l e ng i ne e r i n g .

5. 1 Pa t t e r n s of g r o u n d d i s p l a c e me n t s . Gr o u n d l os s

Fr o m a q u a l i t a t i v e v i e wp o i n t , t he p a t t e r n s of g r o u n d d i s p l a c e me n t s a s s o c i a t e d wi t h t u n n e l l i n g a r e we l l e s t a b l i s h e d a n d unde r s t o o d . I n p r a c t i c e , t he ma i n p r o b l e m l i e s on t he d i f f i c u l t y of p r e d i c t i n g s u r f a c e mo v e me n t s i n d u c e d by t he e x c a v a t i on. Fr o m t he e a r l y c o n t r i b u t i o n s of Sc hmi d t ( 1969) , ma n y a u t h o r s h a v e f a c e d t he p r o b l e m u s i n g d i f f e r e n t a ppr oa c he s .

Le t t he s i mp l e s t c a s e of a c i r c u l a r t u nne l i n a h o mo g e n e o u s g r o u n d wi t h h o r i z o n t a l s u r f a c e ( Fi g. 8 ) be c ons i d e r e d . Le t a l s o be s u p p o s e d t ha t t he p r o b l e m i s t wo - d i me n s i o n a l a n d t ha t t he s t a t e of s t r e s s i n t he s oi l i s a t a n y mo me n t f a r f r om f a i l ur e . The s t r e s s e s a r o u n d t he t h e o r e t i c a l e x c a v a t i o n l i ne a r e r e mo v e d a n d t he s oi l a r o u n d d e f o r ms c o n s e q u e n t l y . The r e f o r e , o p e n i n g of a n

H o r i z o n t a l

u n d e r g r o u n d e x c a v a t i o n mo d i f i e s t he e x i s t i n g g r o u n d s t r e s s e s a nd, u n l e s s s p e c i a l me a s u r e s a r e a dop t e d , c a u s e s d e c o mp r e s s i o n a r o u n d i t . As a r e s u l t of s t r e s s r e l a x a t i o n , t he s o i l mo v e s i n wa r ds , mo r e or l e s s t o wa r d s t he c e n t r e of t he ope n i ng . The e x c a v a t i o n l i ne d e f o r ms ( e i t he r f r e e l y or r e s t r i c t e d by t he l i n i ng ) t o a n e w s h a p e wh i c h i s s ma l l e r t ha n t he t h e o r e t i c a l one . Th e a r e a e n c l o s e d b e t we e n t he t h e o r e t i c a l a nd t he d e f o r me d e x c a v a t i o n l i ne s i s r e f e r r e d t o a s t he ' gr ound l os s ' , V0.

The g r o u n d l os s d e p e n d s o n ma n y f a c t o r s s uc h a s s o i l t ype , p r e s s e n c e of wa t e r , me t h o d of c o n s t r uc t i on , r a t e of t unne l a dva nc e , t u n n e l s i z e a nd de p t h , wo r k ma n s h i p q u a l i t y , a n d f o r m of t e mp o r a r y a n d p r i ma r y s u p p o r t .

An y p o i n t not b e l o n g i n g t o t he t unne l wa l l a l s o mo v e s t o wa r d s t he ope ni ng . Th e d i s p l a c e me n t t e nds t o d e c r e a s e wi t h t he d i s t a n c e t o t he t unne l . Th i s e f f e c t c a n r e a c h t he g r o u n d s u r f a c e a n d c a n be n o t i c e a b l e or e v e n i mpor t a n t , ma i n l y d e p e n d i n g on t he t u n n e l de pt h . Th i s c o u l d be a c r i t e r i o n t o d i f f e r e n t i a t e b e t we e n d e e p a n d s h a l l o w t u nne l s f r om t he p o i n t of v i e w of g r o u n d d i s p l a c e me n t s .

I n t he s i mp l e c a s e u n d e r c o n s i d e r a t i o n , a l l t he d i s p l a c e me n t s a r e s y mme t r i c a l wi t h r e s p e c t t o t he v e r t i c a l l i ne a t t he t u n n e l a xi s . Gr o u n d s u r f a c e d i s p l a c e me n t s h a v e t wo c o mp o n e n t s : ve r t i c a l , d o wn wa r d s ( or s e t t l e me n t ) , a n d h o r i z o n t a l , t o wa r d s t he c e n t r e l i n e . Un l e s s s u f f i c i e n t i n t e r na l p r e s s u r e i s a p p l i e d o r t he l i n i n g i s e xpa nde d , s e t t l e me n t s b e c o me p o s i t i v e wh e r e a s h o r i z o n t a l d i s p l a c e me n t s a r e of o p p o s i t e s i g n a t bo t h s i de s of t he t u n n e l c e n t r e l i ne .

As a r e s u l t of t h e s e mo v e me n t s , t he g r o u n d s u r f a c e d e f o r ms a n d t e nds t o t he l i ne r e p r e s e n t e d i n Fi g u r e 8 , g i v i n g r i s e t o a t r ough. Th e d i f f e r e n c e b e t we e n t he d e f o r me d s u r f a c e a n d t he s e t t l e me n t p r o f i l e i s ne g l i g i b l e . The a r e a e n c l o s e d b e t we e n t he o r i g i n a l g r o u n d s u r f a c e a n d t he r e s u l t i n g s e t t l e me n t p r o f i l e i s k n o wn a s ' s e t t l e me n t v o l ume ' , V, , wh i c h ma i n l y d e p e n d s o n t he g r o u n d l os s .

I t i s g e n e r a l l y a d mi t t e d t ha t s ho r t t e r m s e t t l e me n t v o l u me i s s i mi l a r t o t he g r o u n d l os s . For c o h e s i v e s a t u r a t e d s o i l s u n d e r u n d r a i n e d c o n d i t i ons , t h e y a r e e q u a l s i n c e t h e r e i s no v o l u me c ha nge . For n o n - c o h e s i v e s oi l s , o r f or l ong t e r m

2 5 2 6

C A P •

Fi g u r e 9. Ga p p a r a me t e r ( Le e a n d Howe , 1988) .

c o n d i t i o n s o n c e c o n s o l i d a t i o n ha s t a k e n p l a c e a nd a ny p o s s i b l e s o i l v i s c o s i t y d e f o r ma t i o n s ha ve de ve l o p e d , t he s e t t l e me n t v o l u me c a n be e i t h e r s o me wh a t g r e a t e r or s ma l l e r t ha n t he g r o u n d l os s .

Si n c e t he l os s of g r o u n d c o n t r o l s p r i ma r i l y g r o u n d d i s p l a c e me n t s , i t s h o u l d be s e l e c t e d a s a d e s i g n p a r a me t e r by i t s e l f . Al t h o u g h t he a t t e mp t s t o e s t i ma t e g r o u n d l os s on a n a l y t i c a l ba s e s h a v e ye t f a i l e d , i t s i mp o r t a n c e c o n v e r t s i t i nt o t he o v e r r i d i n g i nput d e s i g n p a r a me t e r f or u n d e r g r o u n d c ons t r u c t i o n . Al t h o u g h v e r y d i f f i c u l t t oo, e s t i ma t e s of g r o u n d l os s c a n be a t t e mp t e d on e mp i r i c a l ba s e s , a s i t wi l l be d i s c u s s e d l a t e r .

As a n a l t e r n a t i v e p a r a me t e r t o t he g r o u n d l os s , Le e a n d Ro we ( 1988) h a v e p r o p o s e d f or l i ne d t u n ne l s e x c a v a t e d by me a n s of a s h i e l d ( a nd f or p l a ne s t r a i n c o n d i t i o n s ) t he us e of t he ' ga p pa r a me t e r ' ( Fi g. 9) , wh i c h i n c o r p o r a t e s bo t h t he l os s of g r o u n d c a u s e d by o v e r c u t t i n g a n d t he vo l u me c h a n g e i n t he r e mo u l d e d s oi l , a d e n o t e s t he g e o me t r i c l os s of g r o u n d o v e r t he s h i e l d due t o t he t a i l t h i c kne s s ; 6 d e n o t e s t he g r o u n d l os s due t o c l e a r a n c e or d i f f e r e n c e b e t we e n t he i n t e r na l s h i e l d d i a me t e r a nd t he o u t s i d e l i n i ng d i a me t e r . Fi na l l y , U i n c l ude s o v e r c u t t i n g due t o t he t h r e e - d i me n s i o n a l mo v e me n t i n t o t he f a c e , o v e r c u t t i n g due t o a l i g n me n t p r ob l e ms , a n d c o n s o l i d a t i on of t he d i s t u r b e d z one i n t he s oi l s u r r o u n d i ng t he t unne l .

Go i n g b a c k t o t he s h a p e of t he g r o u n d s u r f a c e t r ough c a u s e d by t u n n e l l i n g ( Fi g. 10) , s e t t l e me n t s s t a r t f r o m a c e r t a i n h o r i z o n t a l d i s t a n c e f r om t he t u n n e l ( t he p r a c t i c a l l i mi t of i n f l ue nc e ) a n d i n c r e a s e a s t he p o i n t c o n s i d e r e d a p p r o a c h e s t he c e n t r e l i n e . I t s e e ms t o be o b v i o u s t ha t t he ma x i mu m s e t t l e me n t c o r r e s p o n d s t o t he v e r t i c a l of t he c r own. I n ge ne r a l , t h i s i s t r ue ; but i t i s not u n d e r s o me c i r c u ms t a n c e s . Owi n g t o t he p r o b l e m s ymme t r y , t he s u r f a c e s e t t l e me n t p r o f i l e ha s t o h a v e t wo s y mme t r i c a l p o i n t s of i n f l e x i on , wh e r e i t s ma x i mu m s l o p e oc c ur s . St r u c t u r e s b u i l t i n b e t we e n t he i n f l e x i o n p o i n t s a r e s u b j e c t e d t o s a g g i n g wh e r e a s s t r u c t u r e s l y i ng o u t s i d e t h e m a r e s u b j e c t e d t o hoggi ng.

Ho r i z o n t a l d i s p l a c e me n t s a r e g e n e r a l l y s ma l l e r t ha n v e r t i c a l one s . Th e y a r e a l s o z e r o a t a c e r t a i n d i s t a n c e f r o m t he c e n t r e l i n e , wh e r e t he i n f l u e n c e of t he e x c a v a t i o n b e c o me s ne g l i g i b l e . Be c a u s e of s ymme t r y , t he h o r i z o n t a l d i s p l a c e me n t a t t he c e n t r e l i n e mus t a l s o be z e r o. The n, t he h o r i z o n t a l mo v e me n t d i s t r i b u t i o n ha s t o h a v e t wo s y mme t r i c a l p o i n t s of ma x i mu m d i s p l a c e me n t . Fi e l d e v i d e n c e s u g g e s t s t ha t p o i n t s of ma x i mu m h o r i z on t a l d i s p l a c e me n t a n d p o i n t s of i n f l e x i o n of t he s e t t l e me n t p r o f i l e a r e c l o s e t o e a c h ot he r . St r u c t u r e s l y i ng i n b e t we e n t he p o i n t s of ma x i mu m h o r i z o n t a l d i s p l a c e me n t a r e s u b j e c t e d t o h o r i z o n t a l c o mp r e s s i v e s t r a i n , wh e r e a s t h o s e one s l y i ng o u t s i d e t he m a r e s u b j e c t e d t o h o r i z o n t a l t e n s i l e s t r a i n . Th e c o mb i n a t i o n of h o g g i n g a nd ho r i z o n t a l t e n s i l e s t r a i n c o n s t i t u t e s t he mor e d a n g e r o u s s i t u a t i o n f or mos t s t r uc t u r e s .

S E T T L E M E N T S H O R I Z O N T A L

D I S P L A C E M E N T S

Fi g u r e 10. Gr o u n d d i s p l a c e me n t s a n d t he i r e f f e c t s on s t r uc t ur e s .

Al o n g t he c e n t r e l i n e t he r e a r e o n l y s e t t l e me nt s . Th e y s h o w a ma x i mu m a t t he t unne l c r o wn a nd d e c r e a s e upwa r ds . The r e f o r e , v e r t i c a l s t r a i n s a r e t e ns i l e . Soi l e l e me n t s b e t we e n t he c r o wn a nd t he g r o u n d s u r f a c e a r e s u b j e c t e d t o v e r t i c a l e x t e n s i on c o mb i n e d wi t h h o r i z o n t a l c ompr e s s i o n . The s a me h a p p e n s f or s o i l e l e me n t s l o c a t e d b e l o w t he t unne l i nve r t .

A v e r t i c a l l i ne l o c a t e d s ome d i s t a n c e a wa y f r om t he t unne l d e f o r ms a s r e p r e s e n t e d i n Fi g u r e 10. I t s ma x i mu m d i s p l a c e me n t s a r e l o c a t e d mo r e or l e s s a t t he t unne l d e p t h a nd a r e p r e d o mi n a n t l y h o r i z on t a l . The d i s p l a c e me n t s a l o n g t he s pr i ng - l i ne d e c r e a s e wi t h t he d i s t a n c e t o t he t unne l . He nc e , h o r i z o n t a l t e n s i l e s t r a i n s t a ke p l a c e a nd t he s oi l e l e me n t s i n t ha t r e g i o n d e f o r ms i n a c o mb i n a t i o n of v e r t i c a l c o mp r e s i o n a nd h o r i z o n t a l e x t e n s i o n .

Th e p r o b l e m of g r o u n d d i s p l a c e me n t s i s a c t u a l l y t h r e e - d i me n s i o n a l . Fi g u r e 11, a d a p t e d f r om At t e - we l l e t a l ( 1986) , s hows h o w s e t t l e me n t s d e v e l o p a s t he t unne l h e a d i n g a dva nc e s , a nd r e p r e s e n t s a t h r e e - d i me n s i o n a l v i e w of t he d e f o r me d g r o u n d s ur f a c e . The s o i l i n f r ont of t he f a c e t e nds t o mo v e i nwa r ds , t h i s p r o d u c i n g s e t t l e me n t s we l l a h e a d t he he a d i ng . At a c e r t a i n d i s t a n c e f r om i t , t he g r o u n d ha s not d e f o r me d ye t ; wh e r e a s a t a c e r t a i n d i s t a n c e b e h i n d i t t he g r o u n d d i s p l a c e me n t s h a v e r e a c h e d t h e i r ma x i mu m v a l u e s a n d t he d e f o r ma t i o n c o n d i t i o n s a r e t h o s e of p l a n e s t r a i n.

2 5 2 7

Fi g u r e 11. 3- D v i e w of g r o u n d s u b s i d e n c e ( a da p t e d f r om At t e we l l e t a l , 1986) .

Fi g u r e 12. Gr o u n d mo v e me n t s a t t he v e r t i c a l p l a n e a l o n g t he t u n n e l a xi s .

Su c c e s s i v e s e t t l e me n t t r o u g h s i n p l a n e s p e r p e n d i c u l a r t o t he t unne l a x i s h a v e s i mi l a r s ha pe s but g r a d u a l l y i n c r e a s i n g ma gn i t u d e s .

The v e r t i c a l p l a n e a t t he t u n n e l a xi s i s a p l a n e of s ymme t r y . The s u r f a c e s e t t l e me n t p r o f i l e a l o n g t h i s p l a n e t a ke s t he s h a p e s h o wn i n Fi g u r e 12. Th e r e i s a p o i n t of i n f l e x i on , c o r r e s p o n d i n g t o t he ma x i mu m s l o p e of t he s e t t l e me n t p r of i l e , r o u g h l y i n c o i n c i d e n c e wi t h t he v e r t i c a l a t t he t unne l f a c e . Th e c o r r e s p o n d i n g s e t t l e me n t i s c l os e t o ha l f t he ma x i mu m one . St r u c t u r e s l y i ng a he a d t he t unne l f a c e a r e s u b j e c t e d t o h o g g i ng , wh e r e a s t h o s e l o c a t e d b e h i n d i t a r e s u b j e c t e d t o s a ggi ng. Ac t u a l l y , t he s t r u c t u r e s a b o v e t he t unne l a r e f i r s t s u b j e c t e d t o h o g g i n g a n d t he n t o s a ggi ng.

At a n y mo me n t d u r i n g e x c a v a t i o n , t he a c c u mu l a t e d d i s p l a c e me n t s p a r a l l e l t o t he t unne l a xi s a r e o p p o s i t e t o t he t u n n e l a dva nc e . Th e i r d i s t r i b u t i o n p r e s e n t s a ma x i mu m a t t he v e r t i c a l of t he t unne l f a c e . Th e y d e c r e a s e wi t h t he d i s t a n c e t o t he f a c e a n d a r e z e r o a t a c e r t a i n p r a c t i c a l l i mi t . Th e s e a r e t he o n l y g r o u n d mo v e me n t s c a u s e d by t unne l e x c a v a t i o n t ha t a r e no t mo n o t o n i c a l l y i n c r e a s i n g a s t he e x c a v a t i o n p r ogr e s s e s . The y

------------ M a jo r P r in c ip a l S t re s s T r a je c t o r y ( C , )

------------ M in o r P r in c ip a l S t r e s s T r a je c to r y ( CT3 )

Fi g u r e 13. Pr i n c i p a l s t r e s s t r a j e c t o r i e s a r o u n d a n u n l i n e d t u n n e l i n c l a y ( Le e a n d Rowe , 1988) .

f i r s t i n c r e a s e a n d t h e n d e c r e a s e t o b e c o me f i n a l l y z e r o.

Th e r e i s a l a r ge b o d y of e x p e r i me n t a l e v i d e nc e , bo t h f r om r e a l t u n n e l s a n d mo d e l t u n n e l s i n t he c e n t r i f u g e , t ha t c o n f i r ms t he p r e v i o u s l y d e s c r i be d p a t t e r n s of g r o u n d mo v e me n t s a t g r o u n d s u r f a c e a n d i n s i de t he gr ound. Da t a a r e p u b l i s h e d e v e r y wh e r e i n t u n n e l l i n g l i t e r a t ur e .

5. 2 I n f l u e n c e of s t r e s s a n d s t r a i n pa t hs

I t ha s a l r e a d y b e e n s h o wn t ha t t he s t r a i n p a t h s a r o u n d a c i r c u l a r o p e n i n g a r e d i f f e r e n t f r o m one p o i n t t o a no t he r . Ho we ve r , f r om a q u a l i t a t i v e p o i n t of v i e w a l l s o i l e l e me n t s a r o u n d t he t unne l wa l l d e f o r m i n t he s a me wa y ( r a d i a l e x t e n s i o n c o mb i n e d wi t h t a n g e n t i a l c o mp r e s s i o n ) , t he o n l y d i f f e r e n c e b e i n g i n t he o r i e n t a t i o n of t he i r p r i n c i p a l s t r a i n a xe s . The r e f o r e , a c e r t a i n r a d i a l s y mme t r y e x i s t s .

Le e a n d Ro we ( 1988) a n a l y z e d t he ma j o r a nd mi n o r p r i n c i p a l s t r e s s t r a j e c t o r i e s a r o u n d a n u n l i n e d t u n n e l ( wi t h H/ D=2 . 2 ) , u s i n g f i n i t e e l e me nt s , f o r a n i s o t r o p i c e l a s t i c me d i u m u n d e r u n d r a i n e d c o n d i t i o n s . I t c a n be s e e n ( Fi g. 13) t ha t a r o u n d t he t u n n e l a n d u p t o a c o n s i d e r a b l e d i s t a n c e f r om i t ( whe r e t he b o u n d a r i e s i n f l u e n c e b e c o me s n o t i c e a b l e ) , t he ma j o r s t r e s s t r a j e c t o r i e s a r e a l mo s t c i r c u l a r a n d c o n c e n t r i c wi t h t he t unne l ; wh e r e a s t he mi n o r s t r e s s t r a j e c t o r i e s a r e a l mo s t a r a d i a t i o n of s t r a i g h t l i ne s d e p a r t i ng f r om t he c e n t r e of t he t unne l a nd mo r e or l e s s e q u a l l y s pa c e d , e v e n t h o u g h t he a s s u me d i n i t i a l K0 i s of t he o r d e r of 0. 75. I t s e e ms a g a i n t ha t t h e r e i s a r a d i a l s ymme t r y , wi t h s i mi l a r s t r e s s i n t e n s i t y a l l a r o u n d t he t unne l .

Ng a n d Lo ( 1985) p r e s e n t e d t he s t r e s s p a t h s f or t h r e e t y p i c a l s o i l e l e me n t s a r o u n d t he t u nne l i n t he s a me or s i mi l a r p r o b l e m ( Fi g. 14) . At t he s p r i ng l i n e , p o i n t C, e x c a v a t i o n i n duc e s a n i n c r e a s e i n t he t a nge n t i a l , v e r t i c a l , ma j o r p r i n c i p a l s t r e s s a n d a r e d u c t i o n i n t he r a di a l , h o r i z ont a l , mi n o r p r i n c i p a l s t r e s s . Th e s hor t ba r s on t he s t r e s s p a t h s i n d i c a t e t he d i r e c t i o n of t he ma j o r p r i n c i p a l s t r e s s . No r o t a t i o n of p r i n c i p a l s t r e s s o c c u r s a t t he s pr i ng l i ne .

2 5 2 8

Fi g u r e 14. St r e s s p a t h s a t t h r e e t y p i c a l s oi l e l e me n t s ( Ng a n d Lo, 1985) .

Ne a r t he c r own , p o i n t A, e x c a v a t i o n p r o d u c e s a r e d u c t i o n of t he r a d i a l , v e r t i c a l , i n i t i a l l y ma j o r p r i n c i p a l s t r e s s a n d a s ma l l i n c r e a s e i n t he t a n g e n t i a l , h o r i z o n t a l , i n i t i a l l y mi n o r p r i n c i pa l s t r e s s , wh i c h b e c o me s t he ma j o r one , t hus r o t a t i n g t he p r i n c i p a l s t r e s s e s by 9 0 ‘ .

At t he s h o u l d e r of t he t unne l , p o i n t B, t he r e i s a s t r e s s r o t a t i o n of 45*. Th e ma j o r p r i n c i p a l s t r e s s v a r i e s s l i ght l y .

I t i s i n t e r e s t i n g t o n o t e t ha t s h e a r s t r e n g t h mo b i l i z a t i o n i n c r e a s e s mo n o t o n i c a l l y f or p o i n t s B a n d C, wh e r e a s f or p o i n t A i t f i r s t d e c r e a s e s a nd t h e n i nc r e a s e s . Th e ne t r e s u l t i s a s u b s t a n t i a l l o a d i n g a t a ny poi nt .

Si n c e t he a s s u mp t i o n ha s be e n ma d e t ha t t he s o i l b e h a v i o u r i s e l a s t i c - p e r f e c t l y p l a s t i c , i t i s d i f f i c u l t t o u n d e r s t a n d wh y t he s t r e s s p a t h s a r e no t s t r a i g h t l i ne s . I t i s a l s o d i f f i c u l t t o u n d e r s t a n d wh y t he s t r e s s e s a t t he s p r i n g l i n e t e nd t o be s o d i f f e r e n t f r o m t h o s e a t t he c r o wn a n d t he s hou l de r , i n s t e a d of t e n d i n g a l l of t he m t o b e c o me r a t h e r e qua l . On e p o s s i b l e e x p l a n a t i o n c o u l d be t ha t p o i n t s A, B a nd C a r e s o me d i s t a n c e a wa y f r o m t he t u n n e l a n d s ome p l a s t i f i c a t i o n ha s t a k e n p l a c e f or p o i n t s c l o s e r t o t he t unne l .

Fe r n ^ n d e z - Mo y a e t a l ( 1981) a n d Ot e o a n d Sa ga - s e t a ( 1982) a n a l y z e d t he s t r e s s p a t h s a r o u n d a t unne l u s i n g f i n i t e e l e me n t s f or a n e l a s t i c i s o t r op i c me d i u m u n d e r d r a i n e d c o n d i t i o n s a nd d i f f e r e n t i a t e d t h r e e r e g i o n s ( Fi g. 15ai ) , wi t h q u i t e d i f f e r e n t s t r e s s pa t hs .

Le e a nd Ro we ( 1989) a n a l y z e d t he s t r a i n pa t hs ( Fi g. 15b) f o r t he s a me p r o b l e m i n Fi g u r e 13. The c o n t o u r s of v e r t i c a l a n d h o r i z o n t a l s t r a i n s a l l o w t o d i f f e r e n t i a t e t h r e e r e g i o n s wh i c h s h o w a v e r y g o o d a g r e e me n t wi t h t h o s e f o u n d b y Fe r n d n d e z - Mo y a

V E R T IC A L

S T R A IN

C O N T O U R S

-- - -- -E x te n s io n

- - - - --C o m p re s s io n

H O R IZ O N T A L

S T R A IN

C O N T O U R S

-- -- -- - E x te n s io n

- - - -- - -C o m p re s s io n

Fi g u r e 15. St r e s s a n d s t r a i n p a t h s c o n t o u r s ( Ot e o a n d Sa g a s e t a , 1982, a n d Le e a nd Rowe , 1989) .

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l i * — -

Vo

i,

i -<r

Fi g u r e 16. St r a i n c o mp o n e n t s a r o u n d t unne l o p e n i ng ( Le e a nd Rowe , 1989)

e t a l ( 1981) a nd Ot e o a n d Sa g a s e t a ( 1982) i n t e r ms of s t r e s s e s .Th e s t r a i n c o mp o n e n t s a r o u n d t he o p e n i n g ha ve b e e n p l o t t e d ( Fi g. 16) by Le e a n d Ro we ( 1989) . I n t e r ms of t he ma x i mu m s h e a r s t r a i n o c c u r s a t

± 45' f r o m t he v e r t i c a l . Le e a n d Ro we c o n c l u d e d t ha t z one s of i n t e n s e s h e a r i n g d e v e l o p e d i n t he s o i l a r o u n d t he t u n n e l ' s s h o u l d e r a n d h a u n c h

2 529

Surface

Fi g u r e 17. Di s p l a c e me n t s no t a t i on .

l e ve l s . Ho we ve r , i n t e r ms of t he o c t a h e d r a l s he a r s t r a i n or d i s t o r t i o n , t he s h e a r i n g i n t e n s i t y i s mor e or l e s s t he s a me a l l a r o u n d t he t unne l .

I f h y d r o s t a t i c e x i s t i n g s t r e s s e s we r e a s s ume d, t he s t r e s s c h a n g e s i n t e r ms of s t r e s s i nva r i a n t s e x p e r i e n c e d b y s o i l e l e me n t s a t t he t unne l wa l l s h o u l d no t d i f f e r t oo mu c h f r om one po i n t t o a not he r . The c o r r e s p o n d i n g s t r e s s p a t h s s h o u l d be q u i t e s i mi l a r . The e s s e n t i a l d i f e r e n c e l i e s on t he o r i e n t a t i o n of t he p r i n c i p a l s t r e s s a xe s . The r e f o r e , t he o b s e r v e d d i f f e r e n c e s i n b e h a v i o u r a r i s e f r om i n h e r e n t a n i s o t r o p y r a t h e r t ha n f r om l o a d i ng or u n l o a d i n g ( s i nc e s h e a r s t r e n g t h mo b i l i z a t i o n t a ke s p l a c e e v e r y wh e r e ) , on o n e ha nd, a nd f r om t he f a c t t ha t s o i l b e h a v i o u r i s n e i t h e r e l a s t i c n o r l i ne a r , on t he o t h e r ha nd. I n t he a s s u mp t i o n of K0 b e i n g d i f f e r e n t f r om one a nd

e v e n t h o u g h t he s t r e s s c h a n g e s we r e i de nt i c a l , t he mo d u l u s of d e f o r ma t i o n s h o u l d be d i f f e r e n t f r om one p o i n t t o a n o t h e r a s a c o n s e q u e n c e of t he n o n - l i n e a r i t y of t he s t r e s s - s t r a i n be ha v i our . But i f t he a s s u mp t i o n i s ma d e t ha t s oi l b e h a v i o u r i s l i n e a r l y e l a s t i c , t he d i f f e r e n c e s i n mo d u l u s mus t s i mp l y be i n t e r p r e t e d i n t e r ms of a n i s o t r opy .

The d i f f e r e n c e s i n s t r e s s p a t h a r o u n d t he t u n ne l , t o g e t h e r wi t h t he d i f f i c u l t i e s i n d e t e r mi n i n g t he a p p r o p r i a t e modu l i , s p e c i a l l y i n t he c a s e of a n i s o t r o p i c be ha v i our , a r e ma j o r d r a wba c ks of t he s t r e s s me t h o d s of a na l ys i s .

5. 3 No r ma l i z a t i o n

Pr e s e n t a t i o n of r e s u l t s of g r o u n d mo v e me n t me a s u r e me n t s doe s no t a l wa y s f o l l o w t he s a me c r i t e r i on. I n o r d e r t o a n a l y z e t he v a l i d i t y of t he me t h o d s of p r e d i c t i o n , d e f i n e a d e q u a t e d e s i gn p a r a me t e r s a nd c a r r y out p a r a me t r i c s e n s i t i v i t y a na l ys e s , s ome k i n d of n o r ma l i z a t i o n i s mos t c o n v e n i e n t .

I n wh i c h f o l l ows , t he d i s p l a c e me n t s n o t a t i o n g i ve n i n Fi g u r e 17 wi l l be a dopt e d . Al l g r ound d i s p l a c e me n t s wi l l be d e n o t e d by 6 . 6 V d e n o t e s

v e r t i c a l mo v e me n t , p o s i t i v e d ownwa r ds . 6 h d e no t e s

t r a n s v e r s e h o r i z o n t a l d i s p l a c e me n t . B1 d e n o t e s

l o n g i t u d i n a l move me n t , p a r a l l e l t o t he t unne l a xi s a nd p o s i t i v e o p p o s i t e t o t he t unne l a dva nc e .

Di s p l a c e me n t s s h o u l d a l s o be e x p r e s s e d i n non- d i me n s i o n a l f or m. One p o s s i b i l i t y c o u l d c o ns i s t of u s i n g t he ma x i mu m s u r f a c e s e t t l e me n t a s a r e f e r e n c e d i s p l a c e me n t , but i t i s not a good c h o i c e b e c a u s e u n d e r s ome c i r c u ms t a n c e s t he ma x i mu m s e t t l e me n t doe s not c o r r e s p o n d t o t he a x i s of

Fi g u r e 18. Av e r a g e wa l l d i s p l a c e me n t .

s ymme t r y . The u s e of 6 „, t he a v e r a g e i nwa r ds

d i s p l a c e me n t of t he t u n n e l wa l l ( Fi g. 18) , i s p r o p o s e d i ns t e a d. Th i s p a r a me t e r i s r e l a t e d t o t he g r o u n d l os s t h r o u g h t he f o l l o wi n g e xpr e s s i o n :

S0 = V0 / t t D ( 26)

The r e f o r e , g r o u n d d i s p l a c e me n t s d i v i d e d by 6 0

wi l l be us e d. Si n c e t he l os s of g r o u n d i s a f u n d a me n t a l p a r a me t e r a nd mu s t be g u e s s e d f or a ny e s t i ma t e of g r o u n d mo v e me n t s , k n o wl e d g e of 6 0

i mme d i a t e l y f o l l ows .

5. 4 Gr o u n d l os s e s t i ma t e

At t e mp t s h a v e be e n ma d e t o c o r r e l a t e t he g r o u n d l os s wi t h e i t h e r t he s t a b i l i t y r a t i o or t he o v e r l oa d f a c t o r f or c o h e s i v e s oi l s . Ap a r t f r om o t h e r f a c t or s , i t s e e ms e v i d e n t t ha t g r o u n d l os s s h o u l d be r e l a t e d t o t he s h e a r s t r e n g t h mo b i l i z a t i o n be f o r e a n d d u r i n g e x c a va t i on .

A l a r ge a mo u n t of d a t a on g r o u n d l os s or s u r f a c e s e t t l e me n t v o l u me ha s be e n p l o t t e d a g a i ns t s t a b i l i t y r a t i o ( Fi g. 19) by c o l l e c t i n g da t a p o i n t s f r o m s e v e r a l s o u r c e s i n a s i n g l e pl ot .Mo s t of t h e m c o me f r om Cl o u g h a n d Sc h mi d t ( 1981) a n d At t e we l l e t a l ( 1986) , wh o c o l l e c t e d da t a f r om s e v e n t y t unne l s . Ma i r ' s ( 1979) r e s u l t s f r om c e n t r i f u g e t e s t s f or a r a t i o H/ D e q u a l t o 2. 2 a nd 3. 6 h a v e a l s o be e n i nc l ude d. I t i s i mp o r t a n t t o p o i n t ou t t ha t c e n t r i f u g e t e s t s we r e c a r r i e d out f or p l a n e s t r a i n c o n d i t i o n s wh e r e a s t he r e s t of t he d a t a p o i n t s c o r r e s p o n d t o t h r e e - d i me n s i o n a l t unne l he a d i ngs . St r i c t l y s pe a k i ng , b o t h k i n d s of r e s u l t s s h o u l d no t be mi x e d i n a s i n g l e pl ot .

Si n c e g r o u n d l os s e x p r e s s e d a s a p e r c e n t a g e of t he t o t a l e x c a v a t e d v o l u me r a nge s b e t we e n v e r y wi d e ma r g i n s , t he s c a l e u s e d f or i t s r e p r e s e n t a t i o n i s l o g a r i t hmi c . Fr o m e q u a t i o n ( 26) i t i s v e r y e a s y t o e x p r e s s t he s a me da t a on g r o u n d l os s i n t e r ms of t he r a t i o 6 0 / D:

6 0/ D = V. ( %) / 400 ( 27 )

Th e s c a t t e r of r e s u l t s i s v e r y l a r ge e v e n t h o u g h a l o g a r i t h mi c v e r t i c a l s c a l e ha s be e n us e d. I t c a n be s a i d t ha t , i n ge ne r a l , t he g r o u n d l os s t e nds t o i n c r e a s e wi t h t he o v e r l o a d f a c t or . But a l mo s t n o t h i n g mo r e c a n be s a i d f or r e a l t unne l s .

2 5 3 0

O v e r l o a d f a c t o r

Fi g u r e 19. Da t a on r e l a t i v e g r o u n d l os s p l o t t e d a g a i ns t o v e r l o a d f a c t or .

The r e a s o n f or t he s c a t t e r of t he d a t a f r om r e a l t u nne l s c o u l d be t ha t t h e y h a v e b e e n p l o t t e d a g a i n s t t he o v e r l o a d f a c t o r wh i c h doe s not a c c ount f o r t he t h r e e - d i me n s i o n a l n a t u r e of t he t unne l he a d i ng . Ma i r ( 1983) ha s s h o wn t ha t t he r e l a t i v e g r o u n d l os s i s b e t t e r r e l a t e d t o t he l oa d f a c t or .

Re s u l t s of c a r e f u l l y c o n d u c t e d c e n t r i f u g a l mo d e l t e s t s s u g g e s t t ha t t he l o g a r i t h m of t he v o l u me l os s i n c r e a s e s l i n e a r l y ( mor e or l e s s ) wi t h t he o v e r l o a d f a c t or . Th e s e r e s u l t s c l e a r l y s how t he ma r k e d i n f l u e n c e of t he t u nne l d e p t h ( or t he t u n n e l s i z e ) . Th e d e e p e r or s ma l l e r t he t u n ne l t he s ma l l e r t he g r o u n d l os s , a n d t he l a r ge r t he o v e r l o a d f a c t o r t he g r e a t e r t he g r o u n d l os s i n c r e a s e f o r a g i v e n r e d u c t i o n i n t he H/ D r a t i o. Re d u c i n g t he t u n n e l c o v e r or i n c r e a s i n g t he t u n ne l d i a me t e r b y a b o u t 2 mu l t i p l i e s t he g r o u n d l os s by a f a c t o r of 2 t o a s mu c h a s 1 0 , d e p e n d i n g on t he o v e r l o a d f a c t or .

I t s e e ms t ha t t he f i r s t a t t e mp t t o e mp i r i c a l l y c o r r e l a t e v o l u me l os s a n d o v e r l o a d f a c t o r wa s ma d e by Gl o s s o p ( 1977) , wh o p r o p o s e d a s i mpl e l i ne a r e x p r e s s i o n . Cl o u g h a n d Sc h mi d t ( 1981) ga ve a n e x p o n e n t i a l r e l a t i o n s h i p . Fo r d e s i g n pu r pos e s , At t e we l l a n d Ye a t e s ( 1984) s u g g e s t e d a n u p p e r l i mi t t o t he g r o u n d l os s or t he s u r f a c e s e t t l e me nt vo l ume , wh i c h e n c l o s e d 754 of t he da t a . Fi na l l y , At t e we l l e t a l ( 1986) p r o p o s e d t he us e of a b a n d wh i c h c o n t a i n e d 50% of t he a v a i l a b l e da t a po i n t s . I t f o l l o ws f r om Fi g u r e 20 t ha t no one of t he p r o p o s e d e mp i r i c a l a p p r o a c h e s r e s u l t s e n t i r e l y s a t i s f a c t o r y p r o b a b l y due t o t he c o n s i d e r a b l e s p r e a d of t he g r o u n d l os s da t a . On e of t he r e a s o n s f o r t h i s l a c k of a g r e e me n t ma y be t ha t n o n e of t h e m i n c l u d e s t he e f f e c t s of t unne l d e p t h or s i z e , wh i c h s e e ms t o be v e r y i mp o r t a n t a c c o r d i n g t o Ma i r ' s f i nd i ngs . Th e e f f e c t s of s oi l s t i f f n e s s i s a l s o i g n o r e d i n t he s e a ppr oa c he s .

Ba s e d on c a s e s t u d i e s p r e s e n t e d by Pe c k ( 1969) a nd Sc h mi d t ( 1974) , Mi t c h e l l ( 1983) p r o p o s e d a n o t h e r e mp i r i c a l e x p r e s s i o n wh i c h a c c o u n t s f or s oi l s t i f f n e s s a n d s h e a r s t r e n g t h a s we l l a s f or

-------- G l o s s o p , 1 9 7 7

-------- C l o u g h a n d S c h m i d t , 1 9 8 1

-------- A t t e w e l l a n d Y e a t e s , 1 9 8 4 , 1 9 8 6

6 oo


Fi g u r e 20. So me e mp i r i c a l a p p r o a c h e s t o g r o u n d l os s vs o v e r l o a d f a c t or .

6 o

o


Fi g u r e 21. Mi t c h e l l ' s ( 1983) e mp i r i c a l a p p r o a c h t o g r o u n d l os s v s o v e r l o a d f a c t or .

t he o v e r l o a d f a c t or :

( 28)

2531

Mi t c h e l l ' s e x p r e s s i o n ha s be e n p l o t t e d i n Fi g u r e 2 1 f or t wo c o mmo n e x t r e me v a l u e s of t he r a t i o Eu/ s u ( r a ng i ng n o r ma l l y b e t we e n 200 a n d 700) .

Mi t c h e l l s u g g e s t e d t ha t f or v e r y s e n s i t i v e c l a ys o r f or l ow c o n s t r u c t i o n qua l i t y , t he e s t i ma t e d v o l u me l os s s h o u l d be mu l t i p l i e d by a f a c t o r of t hr e e . Aga i n , e x p r e s s i o n ( 28) doe s no t a c c oun t f or t unne l s i z e .

The e l a s t i c - p l a s t i c a n a l y s i s of t he p r o b l e m ( a na l o g o u s t o t he c a v i t y e x p a n s i o n c a s e ) l e a ds t o ( Cl ough & Sc hmi dt , 1981) :

V. =s„

ee xp (-

n - O .

-1) ( 29)

I t i s i n t e r e s t i n g t o n o t e t ha t e x p r e s s i o n ( 29) i s qu i t e s i mi l a r t o t ha t p r o p o s e d b y Mi t c h e l l wi t h t he e x c e p t i o n s of a f a c t o r of 2 i n t he e x p o n e n t a nd t he g e n e r a l mu l t i p l y i n g f a c t o r 3/ e ( » 1. 10) .

Wh e n t r y i n g t o e s t i ma t e t he g r o u n d l os s i n c l a ys , t he f o l l o wi n g r e l e v a n t f a c t o r s s h o u l d be t a ke n i nt o a c c ount : s oi l p r o p e r t i e s ( u n d r a i n e d s h e a r s t r e ng t h , s t i f f n e s s , v i s c o s i t y a n d c o n s o l i da t i on) , o v e r l o a d f a c t o r ( whi c h a c c o u n t s f or de pt h) , d e p t h t o d i a me t e r r a t i o , r a t e of a dva nc e , l i n i n g ( ga p p a r a me t e r ) a nd q u a l i t y of wo r k ma n s h i p a nd c ons t r u c t i o n . Fr o m t he a n a l y s i s of t he da t a a va i l a b l e , Ur i e l p r o p o s e s a n e w e x p r e s s i o n f or t he g r o u n d l os s wh i c h e x p l i c i t l y i n c o r p o r a t e s t he e f f e c t s of c l a y s h e a r s t r e n g t h a n d s t i f f ne s s , o v e r l o a d f a c t o r a nd t u n n e l d e p t h or s i z e :

6 o

D


Fi g u r e 22. Ur i e l ' s a p p r o a c h f or g r o u n d l os s e s t i ma t e b a s e d on o v e r l o a d f a c t o r a n d t u n n e l de pt h.

— = m — e x p ( 3N— ) ( 30)D Eu H

The c o e f f i c i e n t m a c c o u n t s f or t h o s e a s p e c t s mor e d i f f i c u l t t o be a s c e r t a i n e d , s uc h a s v i s c o s i t y a n d c o n s o l i d a t i o n p r o p e r t i e s of t he c l a y, t ype of l i n i n g a n d l i n i n g e r e c t i on , r a t e of a d v a n c e a nd wo r k ma n s h i p qua l i t y . Re a s o n a b l e l i mi t s f o r m a r e0. 1 t o 0. 3. I t me a n s t ha t t he l os s of g r o u n d c a n be i n c r e a s e d by a f a c t o r of t h r e e u n d e r t he wo r s t c o n s t r u c t i o n c o nd i t i ons .

Si n c e a n o r ma l a v e r a g e v a l u e of t he r a t i o Eu/ s u

c o u l d be of t he o r d e r of 400, h e n c e t a k i n g i nt o c o n s i d e r a t i o n e x p r e s s i o n ( 27) t he f o r mu l a f or t he g r o u n d l os s a s a p e r c e n t a g e of t he t o t a l e x c a v a t e d v o l u me b e c o me s v e r y s i mpl e :

DV. ( ») = m e x p ( 3N— ) ( 31)

H

Ur i e l ' s a p p r o a c h ha s b e e n p l o t t e d i n Fi g u r e 22 f or t h r e e d i f f e r e n t v a l u e s of t he H/ D r a t i o . I t p r o d u c e s a r a d i a t i o n of ba nds , e a c h o n e c o r r e s p o n d i n g t o a d e p t h t o d i a me t e r r a t i o . The d e e p e r o r s ma l l e r t he t u n n e l t he f l a t t e r t he s l ope of t he ba nd. The v e r t i c a l wi d t h of e a c h b a n d r e f l e c t s t he i n f l u e n c e of c o n s t r u c t i o n c o n d i t i o n s t h r o u g h t he c o e f f i c i e n t m a n d i s e q u i v a l e n t t o Mi t c h e l l ' s f a c t o r of 3. As i t c a n be s e e n i n t he f i gur e , t h i s a p p r o a c h a g r e e s we l l wi t h Ma i r ' s c e n t r i f u g a l t e s t r e s u l t s f or s i mi l a r v a l u e s of t he H/ D r a t i o.

Si n c e g r o u n d l os s i n c r e a s e s l a r g e l y a s N a p p r o a c h e s 6 , h e n c e e mp i r i c a l f o r mu l a e c a n n o t h o l d f or s t a b i l i t y r a t i o s g r e a t e r t ha n a b o u t 6 . On t he o t h e r ha nd, t he s t a b i l i t y r a t i o doe s not r e f l e c t t he a c t u a l s h e a r s t r e n g t h mo b i l i z a t i o n s i nc e i t doe s not a c c o u n t f o r t he t h r e e - d i me n s i o n a l e f f e c t s a t t he h e a d i n g f or l i ne d t unne l s . Th e r e f or e , g r o u n d l os s e s t i ma t e s h o u l d be b e t t e r b a s e d on t he l oa d f a c t o r a s s u g g e s t e d by Ma i r ( 1983) .

5. 5 Pr e d i c t i o n of g r o u n d mo v e me n t s

Pr e d i c t i o n of g r o u n d mo v e me n t d i s t r i b u t i o n wi l l be c o n s i d e r e d f r o m t h r e e p o i n t s of v i e w: e mp i r i c a l a p p r o a c h e s t o f i t r e a l me a s u r e me n t s , t he us e of n u me r i c a l me t h o d s e i t h e r f or p r e d i c t i o n ( i n c o n j u n c t i o n wi t h b a c k - a n a l y s i s p r o c e d u r e s ) or t o c a r r y ou t p a r a me t r i c s e n s i t i v i t y a n a l y s e s a nd, f i na l l y , t he n e wl y a v a i l a b l e a n a l y t i c a l a ppr oa c h .

5. 5. 1 Er r o r f u n c t i o n

Fr o m a h i s t o r i c p o i n t of v i e w, t he f i r s t e mp i r i c a l a p p r o a c h t o t he u n d e r g r o u n d e x c a v a t i o n i n d u c e d s u r f a c e s e t t l e me n t p r o b l e m wa s t he Ga u s s i a n s u b s i d e n c e p r o f i l e d e r i v e d b y Li t wi n i s z y n ( 1958) , wh e n he a p p l i e d s t a t i s t i c a l me t h o d s t o s ome Ro c k Me c h a n i c s p r ob l e ms . Ma r t o s ( 1958) f o u n d a g e n e r a l a g r e e me n t b e t we e n t he me a s u r e d s e t t l e me n t t r o u g h i n mi n i n g i n d u s t r y a nd t ha t c a l c u l a t e d f r o m t he e r r o r f u n c t i o n e mp i r i c a l a ppr oa c h . Th e s a me a p p r o a c h wa s i n t r o d u c e d t o t u n n e l l i n g i n c i v i l e n g i n e e r i n g by Sc h mi d t ( 1969) .

As i t i s v e r y we l l k n o wn a n d a c c e p t e d , t he e r r o r f u n c t i o n f i t t i n g c u r v e p r o v i d e s q u i t e a c c u r a t e f i t s of r e a l or e x p e r i me n t a l s e t t l e me n t t r oughs . I t ha s b e e n r e p r e s e n t e d i n Fi g u r e 23.The s e t t l e me n t a t t he a b s c i s s a x i s :

6 V = 6 V „ e x p ( - x 2 / 2 i 2) ( 32)

The e r r o r f u n c t i o n c o n t a i n s t wo p a r a me t e r s : t he ma x i mu m s e t t l e me n t , 6 V ( a t t he t u n n e l c e n t r e

l i ne ) , a n d t he a b s c i s s a of t he i n f l e x i o n poi nt ,i . The e x p r e s s i o n f o r t he s e t t l e me n t v o l u me i s :

V. = ( 2n) *i 6VBl ul ( 33)

Th i s a l l o ws t he u s e of t he g r o u n d l os s p a r a me t e r i n s t e a d of t he ma x i mu m s e t t l e me n t . Al s o , t he

2 5 32

e r r o r f u n c t i o n c u r v e c a n e a s i l y be wr i t t e n i n n o n - d i me n s i o n a l f o r m a s p r e v i o u s l y p r o p o s e d f or n o r ma l i z a t i o n :

6 . 1 ( x / D)1— = ( tt / 2 ) * --- e x p ( - - - - - - - - - z-6 „ i / D 2( i / D)

( 34)

Fo r p r e d i c t i o n p u r p os e s , a nd a s s u mi n g t ha t t he s e t t l e me n t v o l u me ha s b e e n e s t i ma t e d , t he us e of t he e r r o r f u n c t i o n a p p r o a c h i mpl i e s t he n e e d f or a n e s t i ma t e of t he p a r a me t e r i . Th e r e i s a g e n e r a l a g r e e me n t t ha t i t s h o u l d be r e l a t e d t o H/ D i n t he s e ns e t ha t t he l a t e r a l e x t e n t of t he s e t t l e me n t t r o u g h i n c r e a s e s a s t he t u n n e l d e p t h i n c r e a s e s f or a g i v e n d i a me t e r .

I n t he e a r l y wo r k s of Sc h mi d t ( 1969) , t he p o s i t i on of t he i n f l e x i o n p o i n t i n r e l a t i o n t o H/ D wa s s u g g e s t e d t o be s t r o n g l y d e p e n d e n t on t he s oi l t ype ( Fi g. 24) . I n a s i mi l a r wa y, Co r d i n g a nd Ha n s mi r e ( 1975) ga ve r e c o mme n d a t i o n s on t he p r a c t i c a l e x t e n t of t he s e t t l e me n t t r o u g h a s a f u n c t i o n of H/ D a n d t he s oi l t ype , u s i n g t he c o n c e p t of t he l i mi t a n g l e ( a s i t i s u s e d i n mi n i n g i ndus t r y) .

Ho we ve r , l a t e r a n a l y s e s of f i e l d me a s u r e me n t s a n d l a b o r a t o r y mo d e l t e s t d a t a l e d t o t he c o n c l u s i on t ha t t he d e p e n d e n c y of i on t he s oi l t ype i s mu c h we a k e r t h a n o r i g i n a l l y a dmi t t e d .

Ba s e d on t he wo r k f r om Li t wi n i s z y n ( 1956) , At t e we l l ( 1977) p r o p o s e d t he f o l l o wi n g s e mi - e mp i - r i c a l e qua t i on :

1 0

H / D 6

l " )

i

/

/

/

n u t " , mu»u

S o n d s A b o v e

j G ro u n dw o te r

/ L e v e l

> y s , /

/

/

/8

/

/

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o f * t o S t i

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S o n d s B<

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Fi g u r e 24. Pe c k ' s ( 1969) z o n i n g f or i a s a f u n c t i o n of t u n n e l d e p t h a n d s oi l t ype .

i / D = a ( H/ D )n/ 2 ( 35)

wh e r e a a n d n a r e p a r a me t e r s wh i c h d e p e n d on t he s t r e s s - s t r a i n b e h a v i o u r of t he s o i l a n d t he t u n n e l l i n g t e c hn i que . Ho we ve r , f or c l a y e y s oi l s , At t e we l l s u g g e s t s t o g i v e a a n d n t he v a l u e of 1 i r r e s p e c t i v e of t he me t h o d of c o n s t r u c t i o n .

Mo r e o v e r , Cl o u g h a n d Sc h mi d t ( 1981) p r o p o s e d a l s o f or c l a y e y s o i l s t he s a me u n i q u e e mp i r i c a l f o r mul a ( but g i v i n g n t he v a l u e 0 . 8 ) f or e s t i ma t i ng t he a b s c i s s a of t he i n f l e x i o n poi nt . Thi s f o r mu l a wa s s u g g e s t e d t o h o l d f or a n y s oi l t ype by Sc h mi d t ( 1988) .

A c o mp a r i s o n b e t we e n e v i d e n c e f r o m a l a r ge n u mb e r of c a s e r e c o r d s a n d l a b o r a t o r y mo d e l t e s t s a n d p r e d i c t i o n s b a s e d on e q u a t i o n ( 35) c a n be s e e n i n Fi g u r e 25 ( Sc h l o s s e r e t a l , 1985) . The l i t t l e s c a t t e r i r r e s p e c t i v e of t he s o i l t ype s u p p o r t s Sa g a s e t a ' s ( 1988) i de a s t ha t t he d i s t r i b u t i o n of d i s p l a c e me n t s i s ma i n l y c o n t r o l l e d by t he k i n e ma t i c a l c o n s t r a i n t s ; t he s oi l p r o p e r t i e s a nd c o n s t r u c t i o n p r o c e d u r e s b e i n g s e c o n d - o r d e r f a c t o r s . The i n f l u e n c e of s ome of t h e m o n t he s ha pe of t he s e t t l e me n t t r o u g h wi l l be d i s c u s s e d l a t e r on.

• At t «w*l t (1977)

O Cl ough st Schmi dt (1977)

+ Te»t « on model s ( Cl ough «t Schmi dt , 1977)

^^Cooj l der obl e con»ol ldot lon (Cl ou^i t t Schmi dt , 1977)

.- - - Sl ope 0. 8

- - - - Sl op« I

Fi g u r e 25. Ab s c i s s a of t he i n f l e x i o n p o i n t v e r s u s t unne l d e p t h f or c l a y s ( Sc h l o s s e r e t a l , 1985) .

253 3

A - -- - - -- - - -- - - -▲ C L O U G H A N D S C H M ID T ( 1 9 8 1 )

------------------PECK ( 1 9 6 9 )

o--------------- o S A G ASETA a OTEO ( 1 9 7 4 )

• C OR DIN G a H A N S M IR E ( 1 9 7 5 )

V A TT EW E L 8 FA R M E R ( 1 9 7 4 )

■ M A D R ID M E TR O

M O ST P R O B A B IL IT Y ZONE

I / R

Fi g u r e 26. Co mp a r i s o n b e t we e n Ot e o , s ( 1980) a nd Cl o u g h a n d Sc h mi d t ' s ( 1981) a p p r o a c h e s f o r t he a b s c i s s a of t he i n f l e x i o n p o i n t ( a d a p t e d f or m Ot e o a n d Sa g a s e t a , 1982) .

Ot e o ( 1980) ga ve a n o t h e r e x p r e s s i o n f or i s o l e l y a s a f u n c t i o n of H/ D. I t i n c o r p o r a t e d a mu l t i p l y i n g f a c t o r t o a c c o u n t f or t he s p r e a d of t he da t a . Fi g u r e 26, a d a p t e d f r o m Ot e o a n d Sa g a s e t a ( 1982) , s hows d a t a f r om s e v e r a l t unne l s , Pe c k ' s ( 1969) z o n i n g f or d i f f e r e n t s o i l t ype s a n d Ot e o ' s ( 1980) e mp i r i c a l a p p r o a c h wi t h t he a r e a of mos t l i k e l y v a l u e s r e s u l t i n g f r o m t he a p p l i c a t i o n of t he mu l t i p l y i n g f a c t or . The e mp i r i c a l e q u a t i o n p r o p o s e d b y Cl o u g h a n d Sc h mi d t ( 1981) ha s be e n a d d e d t o t he pl ot . For s h a l l o w t o v e r y s h a l l o w t unne l s , b o t h a p p r o a c h e s r e n d e r s i mi l a r p r e d i c t i ons , i n a z one wh e r e t he s p r e a d of t he da t a i s l a r ge . For d e e p e r t unne l s , b o t h a p p r o a c h e s d e v i a t e f r om e a c h ot he r . Ot e o ' s ( 1980) a p p r o a c h p r e d i c t s a wi d e r s e t t l e me n t t r o u g h a n d s e e ms t o a c c o mmo d a t e b e t t e r t o t he e x p e r i me n t a l da t a . Anywa y , n o n e of t h e m s e e ms s a t i s f a c t o r y e n o u g h i n s p i t e of Sc h mi d t ' s ( 1988) c l a i m t ha t t he p a r a me t e r i c a n be f o u n d wi t h good a p p r o x i ma t i o n by u s i n g Cl o u g h a nd Sc h mi d t ' s ( 1981) f or mul a .

A mo r e d e t a i l e d d i s c u s s i o n on t he i p a r a me t e r e s t i ma t e on e mp i r i c a l ba s e s ha s b e e n ma d e by At t e we l l e t a l ( 1986) . Th e y c o n c l u d e t ha t r a t he r t ha n a c c e p t i n g r e p o r t e d v a l u e s of i , i t i s p r e f e r a b l e t o p l o t t he l o g a r i t h m of t he r e c o r d e d s e t t l e me n t s a g a i n s t t he s q u a r e of t he a b s c i s s a a nd d r a w a r e g r e s s i o n l i ne . Th i s i s t he c o r r e c t p r o c e d u r e f or f i t t i n g but doe s not h e l p t o ma k e p r e d i c t i o n s f o r de s i gn.

The e r r o r f u n c t i o n a p p r o a c h t o d e s c r i b e t he t r a n s v e r s e s e t t l e me n t p r o f i l e ha s b e e n e x t e n d e d t o t he t h r e e - d i me n s i o n a l p r o b l e m ( At t e we l l e t a l , 1986) by a s s u mi n g t ha t :

X

6 . d = c o n s t.

Fi g u r e 27. Emp i r i c a l a p p r o a c h f o r s u r f a c e mo v e me n t s p r e d i c t i on .

- The l o n g i t u d i n a l s u r f a c e s e t t l e me n t p r o f i l e t a ke s t he c u mmu l a t i v e p r o b a b i l i t y f or m.

- I n t he l o n g i t u d i n a l p r o f i l e , t he s u r f a c e s e t t l e me n t a b o v e t he f a c e i s ha l f t he ma x i mu m s e t t l e me n t ( f or p o i n t s we l l b e h i n d t he t u nne l f a c e ) .

- The t r a n s v e r s e s e t t l e me n t p r o f i l e a t a ny po i n t a h e a d or b e h i n d t he t unne l f a c e i s g i v e n by t he e r r o r f u n c t i o n s c a l e d t o t he c o r r e s p o n d i n g ma x i mu m s e t t l e me nt .

On c e a ga i n , a r e a s o n a b l y g o o d a g r e e me n t b e t we e n r e c o r d e d s e t t l e me n t s a nd f i t t e d c u r v e s ha s be e n f o u n d i n ma n y c a s e s , e x c e p t f or t he f a c t t ha t , i n c l a ys , o n l y 30 t o 40 p e r c e n t of t he f i na l ma x i mu m s e t t l e me n t d e v e l o p s v e r t i c a l l y a b o v e t he t u n n e l f a c e . I n c o n n e c t i o n wi t h t ha t , At t e we l l e t a l ( 1986) p o i n t out t ha t s u c h d i s c r e p a n c y i s r e a l l y o n l y a c a d e mi c , s i nc e a l l p o i n t s a t g r o u n d s u r f a c e a h e a d of a n d wi t h i n t he b o u n d a r i e s of t he t r a n s v e r s e s e t t l e me n t t r o u g h wi l l a t s ome t i me r e a c h t h e i r u l t i ma t e ma x i mu m mo v e me n t s f or t he t r a n s l a t i n g t u n n e l s o u r c e of g r o u n d l os s .

5 . 5 . 2 Ot h e r e mp i r i c a l a p p r o a c h e s

Some o t h e r a t t e mp t s ha ve b e e n ma d e t o f i nd s u i t a b l e e mp i r i c a l e x p r e s s i o n s f or t he s e t t l e me n t t r o u g h c a u s e d by t unne l e xc a v a t i o n . No one of t h e m a p p e a r s t o ma t c h r e c o r d e d s e t t l e me n t s a s we l l a s t he e r r o r f unc t i on . As a n e xa mpl e , Ur i e l ha s p r o d u c e d a s e mi - e mp i r i c a l a p p r o a c h wh i c h c o n s i s t s of a s s u mi n g t ha t s u r f a c e d i s p l a c e me n t s a r e d i r e c t e d t o wa r d s a c e r t a i n p o i n t a t t he c e n t r e l i n e a n d t ha t t he ma g n i t u d e of t he d i s p l a c e me n t i s i n v e r s e l y p r o p o r t i o n a l t o t he d i s t a n c e b e t we e n t ha t p o i n t a n d t he one u n d e r c o n s i d e r a t i on ( Fi g. 27) . Fo l l o wi n g t he n o r ma l i z a t i o n a d o p t e d, t he e x p r e s s i o n s f or s u r f a c e mo v e me n t s a r e :

2 5 3 4

6 0 [ ( H' / D) Mx / D) J P

H ' / D

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( 36c )

( 3 6 a )

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a k i n d of v i r t u a l t unne l de p t h , a n d i t s p a r t i c u l a r s ( whi c h c a n be u s e d f or f i t t i ng ) a r e :

D

I T ”

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( 37b)

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i / 3 H' H'- = — — = 0 . 58 - ( vp = 30* ) ( 38a)D 3 D D

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No t e t ha t e q u a t i o n ( 38a ) c o i n c i d e s wi t h e q u a t i on ( 35) f o r n=l a n d a =1 . 1 6 ( a nd a l s o H' =H) . I n Fi g u r e 25, e q u a t i o n ( 38a ) i s r e p r e s e n t e d by t he d o t t e d l i ne t r a n s l a t e d u p wa r d s by a v e r y s ma l l a mmo u n t . Th e r e f o r e , t he p o s i t i o n of t he i n f l e x i o n po i n t p r o v i d e d by t h i s a p p r o a c h i s q u i t e s i mi l a r t o t ha t p r o p o s e d b y At t e we l l ( 1977) .

I n o r d e r t o a s c e r t a i n h o w we l l t h i s a p p r o a c h ma t c h e s e x p e r i me n t a l da t a , a n e x a mp l e t a ke n f r om At t e we l l e t a l ( 1986) wi l l be e xa mi ne d . Th e me a s u r e d s e t t l e me n t t r o u g h of a t u n n e l i n n o r t h - e a s t En g l a n d wh e r e t he f a c e p a s s e d t h r o u g h b o t h l a mi n a t e d c l a y a nd s t o n y c l a y ha s b e e n r e p r o d u c e d i n Fi g u r e 28. The f i t t e d e r r o r c u r v e ha s b e e n p l o t t e d .

Wh i l s t t he a g r e e me n t b e t we e n t he e r r o r f u n c t i o n a nd t he r e c o r d e d s e t t l e me n t s i s e xc e l l e n t , t he s a me c a n n o t be s a i d f o r t he a p p r o a c h u n d e r a n a l y s i s . Cu r v e I ha s b e e n f i t t e d t o t he ma x i mu m s e t t l e me n t a n d s e t t l e me n t v o l u me me a s ur e d . Se t t l e me n t p r e d i c t i o n s a r e s ma l l e r t ha n r e c o r d e d f or t he f i r s t 7 m f r o m t he c e n t r e l i n e a n d g r e a t e r a f t e r wa r d s .

For c u r v e I I , t he a b s c i s s a of t he i n f l e x i o n p o i n t ha s be e n f i t t e d t o 3. 9 m, t he s a me v a l u e of t he r e c o r d e d s e t t l e me n t p r of i l e . Th e s e t t l e me n t v o l u me ha s b e e n ke pt . Pr e d i c t e d s e t t l e me n t s a r e f a r f r om t he o b s e r v e d one s , t he p r e d i c t e d ma x i mu m s e t t l e me n t b e i n g of t he o r d e r of ha l f t he me a s u r e d o n e .

Cu r v e I I I ha s b e e n f i t t e d t o t he ma x i mu m s e t t l e me n t a n d t he a b s c i s s a of t he i n f l e x i o n poi nt . Re s u l t s a r e a l s o f a r f r om s a t i s f a c t o r y a n d t he s e t t l e me n t v o l u me i s mo r e t ha n t wi c e t he o b s e r v e d t r o u g h vol ume .

Fi na l l y , c u r v e s I V a n d V h a v e o n l y b e e n f i t t e d t o t he ma x i mu m s e t t l e me n t . The p a r a me t e r H' ha s be e n g i v e n t he a r b i t r a r y v a l u e s of 4 a n d 5, r e s p e c t i ve l y . Pr e d i c t e d s e t t l e me n t s a r e mu c h c l o s e r

D i s t a n c e t o c e n t r e l i n e ( m )

t o t he o b s e r v e d o n e s t h a n be f o r e , p a r t i c u l a r l y f or c u r v e V u p t o a d i s t a n c e of a bou t 5 m f r om t he c e n t r e l i n e . Th e c o r r e s p o n d i n g s e t t l e me n t v o l u me s a r e g r e a t e r t ha n t he me a s u r e d one .

I n e v e r y c a s e , t he c a l c u l a t e d t r o u g h e x t e n d s l a t e r a l l y mu c h f u r t h e r t ha n obs e r ve d . I n o t h e r wor ds , i f a c l o s e a g r e e me n t i s a c h i e v e d f or t he ma x i mu m s e t t l e me n t a n d f o r mo d e r a t e d i s t a n c e s f r om t he c e n t r e l i n e , t he r e s u l t i n g t r o u g h v o l u me i s mu c h g r e a t e r t h a n me a s ur e d . I t s h o u l d be c o n c l ude d , t he r e f o r e , t ha t t he e r r o r f u n c t i o n i s a mu c h b e t t e r f i t t i n g e xpr e s s i on .

5 . 5 . 3 Nu me r i c a l me t h o d s

Wi t h t he a d v e n t of p o we r f u l c o mp u t i n g f a c i l i t i e s , n u me r i c a l me t h o d s h a v e wi d e l y s p r e a d e d i n r e c e n t ye a r s . Ap p l i c a t i o n of n u me r i c a l me t h o d s t o t he p r o b l e m of g r o u n d mo v e me n t s i n d u c e d by t u n n e l l i n g i s v e r y a p p r o p r i a t e . Ex a mp l e s i n t he l i t e r a t u r e a r e s o n u me r o u s t ha t o n l y a f e w i de a s on t he s u b j e c t c a n be s i mp l y gi ve n.

Nu me r i c a l me t h o d s a r e a p p l i e d no t o n l y t o t he g r o u n d mo v e me n t p r e d i c t i o n but t o t he wh o l e t u n n e l l i n g d e s i g n pr ob l e m, i n c l u d i n g t he e x t e n t a nd d e v e l o p me n t of t he p l a s t i c z one , s i mu l a t i o n of t he e x c a v a t i o n s e q u e n c e s a n d p l a c i n g of t he l i n i ng, s o i l - l i n i n g i n t e r a c t i o n , c a l c u l a t i o n of b e n d i n g mo me n t s , e f f e c t s o r n e a r b y t unne l s , s e e p a g e a n d c o n s o l i d a t i o n , v i s c o s i t y e f f e c t s , e t c . Th e t h r e e - d i me n s i o n a l c h a r a c t e r of t he e x c a v a t i o n c a n a l s o be h a n d l e wi t h t h i s e f f i c i e n t t ool .

The mo s t r e f i n e d or s o p h i s t i c a t e d t e c h n i q u e s i n c o n t i n u u m d i s c r e t i z a t i o n , s uc h a s t he b o u n d a r y e l e me n t me t hod , t he u s e of j o i n t e l e me n t s , e t c , a r e o f t e n e mpl oye d . The a d a p t a b i l i t y t o a n y b o u n d a r y s h a p e a n d c o n d i t i o n s ma k e s t he me t h o d s p e c i a l l y s u i t a b l e f or t he a n a l y s i s o f t unne l d e s i gn p r o b l e ms , p a r t i c u l a r l y i n c a s e s of c o mp l e x g e o me t r y of t he c a v i t y t o be e xc a va t e d .

Mo d e l l i n g of s o i l b e h a v i o u r r e q u i r e s t he e s t i ma t e of c o n s t i t u t i v e e q u a t i o n p a r a me t e r s . Th e r e a r e ma n y mo d e l s a nd, t he r e f o r e , ma n y p a r a me t e r s . I n ge ne r a l , t he mo r e s o p h i s t i c a t e d t he mo d e l t he l a r ge r t he n u mb e r of p a r a me t e r s a n d t he g r e a t e r t he d i f f i c u l t y of t h e i r e s t i ma t i on .

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Fi g u r e 29. Ca r a c a s Me t r o . Me a s u r e d a n d c a l c u l a t e d s u r f a c e d i s p l a c e me n t s ( Ot e o a n d Sa ga s e t a , 1982) .

Fi g u r e 30. Ef f e c t of b o u n d a r y ba s e l o c a t i o n on s u r f a c e s e t t l e me n t s c a l c u l a t e d by n u me r i c a l me t h o d s ( Ot e o a n d Sa ga s e t a , 1982) .

Ev e n f or t he s i mp l e s t mo d e l s of s o i l be ha vi our , a s s e s s me n t of t he mo d u l u s of d e f o r ma t i o n t o be u s e d i s a r e a l l y c o mp l e x ma t t e r . I n mo s t c a s e s , e a c h l a ye r of h o mo g e n e o u s g r o u n d i s c h a r a c t e r i z e d b y a u n i q u e mo d u l u s of d e f o r ma t i o n i n s p i t e of t he a l r e a d y me n t i o n e d d i f f e r e n t s t r e s s p a t hs a r o u n d t he ope n i ng , wh i c h i n d i c a t e t ha t d i f f e r e n t mo d u l i s h o u l d be e mpl oye d .

For p r e l i mi n a r y de s i gn, t he d e f o r ma t i o n mo d u l u s c a n be d e d u c e d f r om s i t e i nve s t i g a t i o n . Owi n g t o t he u n c e r t a i n t i e s a s s o c i a t e d wi t h t he d e t e r mi n a t i on of t he d e s i g n p a r a me t e r s , g o o d a g r e e me n t b e t we e n p r e d i c t i o n s a nd r e a l i t y on t he b a s e s of s i t e i n v e s t i g a t i o n a l o n e i s , a s i n t he c a s e of f o u n d a t i o n s e t t l e me n t s ( or e v e n mor e ) , d i f f i c u l t t o a c hi e ve .

Th i s i s wh y a p r o c e d u r e of c o n t i n u o u s f i ne t u n i n g of t he d e s i g n p a r a me t e r s i s o f t e n us e d. Ac t u a l l y , d e s i g n i s no t f i n i s h e d u n t i l c o n s t r u c t i on ha s c ome t o a n e n d a n d bo t h a r e c o n t i n u o u s l y a d a p t e d t o t he c i r c u ms t a n c e s . Pe c k ' s ( 1969) s t a t e me n t t ha t ' de s i gn a n d c o n s t r u c t i o n a r e i n e x t r i c a b l y i n t e r wove n ' f u l l y a ppl i e s .

The p r o c e d u r e i s s p e c i a l l y u s e f u l wh e n a p i l o t g a l l e r y i s f i r s t o p e n e d or t he e x c a v a t i o n of t he c r o s s - s e c t i o n i s c a r r i e d out by s e ve r a l pha s e s . Ot h e r wi s e , t he t unne l i t s e l f i s u s e d a s a f ul l s c a l e mo d e l t e s t a n d me a s u r e me n t of d e f o r ma t i o n s a t t he e a r l y s t a ge s of t u n n e l c o n s t r u c t i o n a l l o ws t he i d e n t i f i c a t i o n of t he d e s i g n p a r a me t e r s . Mo n i t o r i n g a nd a n a l y s i s of me a s u r e me n t s p l a y t he n a f u n d a me n t a l r ol e .

Amo n g t he ma n y e x a mp l e s t ha t c o u l d be s e l e c t e d t o i l l u s t r a t e t h i s b a c k a n a y l i s i s p r oc e d u r e , t h r e e of t he m h a v e be e n c hos e n. The f i r s t one ( Fi g. 29) r e f e r s t o t he d e s i g n a n d c o n s t r u c t i o n

of t he Ca r a c a s Me t r o wh i c h wa s d e s c r i b e d by Ot e o a n d Sa g a s e t a ( 1982) . By me a n s of f i n i t e e l e me n t t e c h n i q u e s a nd a s s u mi n g p l a n e s t r a i n c o n d i t i o n s a n d a l i n e a r l y e l a s t i c s o i l b e h a v i o u r , t he y o b t a i n e d t he e n t i r e d i s p l a c e me n t f i e l d f o r a n o mi na l v a l u e of t he mo d u l u s of e l a s t i c i t y . Th e n Ot e o a nd Sa g a s e t a f i t t e d t he ma x i mu m s e t t l e me n t by u s i n g t he e x p r e s s i o n g i v e n i n t he f i gur e . The s o c a l l e d o p e r a t i n g mo d u l u s of d e f o r ma t i o n wa s t he n d e d u c e d f or a n o r ma l r a n g e of Po i s s o n ' s r a t i o v a l ue s . Th i s b a c k - a n a l y s i s i s b a s e d on ma t c h i n g o n l y o n e me a s u r e d v a l u e a n d i s , t he r e f o r e , v e r y s i mp l e .

I n o r d e r t o a p p l y t he mode l , t he i n f l u e n c e of t he l i n i n g p l a c e me n t a nd s t i f f n e s s i s t a k e n i nt o a c c o u n t by i n t r o d u c i n g a n o v e r a l l f a c t o r f . For s h i e l d - d r i v e n t u n n e l s i n s t i f f c l a y , l i ne d wi t h c o n c r e t e s e gme n t s , a v a l u e of 1 b e t we e n 0. 4 a nd0. 5 i s a de qua t e , y i n c r e a s e s up t o 1 . 0 wh e n t he e x c a v a t i o n i s s t o p p e d f or one or mo r e da ys .

Re a s o n a b l y g o o d a g r e e me n t b e t we e n c a l c u l a t e d a n d me a s u r e d s u r f a c e d i s p l a c e me n t s wa s a c h i e ve d . Po i s s o n ' s r a t i o e f f e c t i s s e e n t o be r a t h e r s ma l l f or h o r i z o n t a l d i s p l a c e me n t s . Th e be s t f i t c o r r e s p o n d s t o a v a l u e of p e qua l t o 0. 4. Re g a r d i n g s e t t l e me n t s , t he be s t f i t c o r r e s p o n d s t o p e qua l t o 0. 3 a n d t he e f f e c t of i n c r e a s i n g t he Po i s s o n ' s r a t i o f r om 0. 3 t o 0 . 4 i s t o e x t e n d l a t e r a l l y t he s e t t l e me n t t r o u g h mu c h f u r t h e r t h a n obs e r ve d .

I t i s wo r t h c o mme t i n g on t he a n a l y s i s t ha t Ot e o a nd Sa g a s e t a ( 1982) ma k e on t he i n f l u e n c e of t he b o u n d a r i e s l oc a t i on . Th e y f o u n d t ha t t he i n f l u e n c e of t he l a t e r a l b o u n d a r y c a n be n e g l e c t e d i f i t i s a t a d i s t a n c e f r o m t he c e n t r e l i n e e q u a l t o o r g r e a t e r t h a n 9D. The i n f l u e n c e of t he r i g i d b o t t o m b a s e i s mu c h mo r e i mpor t a n t . Fi g u r e 30 s hows

' F D IS T AN C E T O T U N N E L A X IS ( X/ D)e

_ 0,0N

o

?- s «0 0,2

UJ2

0,4

N /D » 3 ; L / D * 1 0 ; V * 0 , 3

MAX, SETTLEMENTS

X------ K H * / D • 3 ,0

»— ■> ^ , / D »4 ,5

A------ * Md /D » 7 ,2

0— —0 Hd / 0 «1 2 ,0

E = S 0 I L D E F O R M A T IO N M O D U L U S

6 ^ S E T T L E M E N T

V 1 B U L K D E N S IT Y

2536

IterationFi g u r e 31. Ev o l u t i o n of t he p a r a me t e r s t o be i d e n t i f i e d t h r o u g h i t e r a t i o n i n a t unne l c o n s t r u c t i o n c a s e ( Le de s ma e t a l , 1987)

t he s u r f a c e s e t t l e me n t s f o r d i f f e r e n t v a l u e s of t he r a t i o Hd/ D, wh e r e Hd d e n o t e s t he d e p t h of t he

b o t t o m b o u n d a r y f r o m t he c e n t r e of t he t unne l . As Hd i nc r e a s e s , s e t t l e me n t s d e c r e a s e , r e s u l t i n g i n

s u r f a c e h e a v e f o r Hd/ D > 7. At t he l i mi t , wh e n

t h e r e i s no r i g i d b a s e ( i . e . , h o mo g e n e o u s e l a s t i c h a l f - s p a c e ) , t he mo v e me n t s a r e t h e o r e t i c a l l y i n f i n i t e upwa r ds . Th e s e r e s u l t s s h o w t ha t t he l o c a t i o n of t he b o t t o m b o u n d a r y i s of p r i ma r y i mpor t a nc e . Wh e n s e l e c t i n g i t , t he f o l l o wi n g f a c t o r s mu s t be t a k e n i n t o a c c ount :

- I n s ome c a s e s , t h e r e i s a r e a l r i g i d ba s e .- Th e mo d u l u s of d e f o r ma t i o n g e n e r a l l y i n

c r e a s e s wi t h de pt h.- St r a i n s d e c r e a s e wi t h t he d i s t a n c e t o t he

t unne l . I t i s we l l k n o wn t ha t t he d e f o r ma t i o n mo d u l u s a t s ma l l s t r a i n s i s s u b s t a n t i a l l y h i g h e r t ha n a t n o r ma l s t r a i ns .

The u s e of ' ad hoc ' c o mp u t e r p r o g r a ms ma ke s p o s s i b e t he n u me r i c a l e v a l u a t i o n of d e s i g n p a r a me t e r s . I n a t unne l c o n s t r u c t i o n c a s e , Le d e s ma e t a l ( 1967) u s e d t h i s b a c k - a n a l y s i s me t h o d f or p a r a me t e r s i d e n t i f i c a t i o n . Th e y ma t c h e d t he r e a d i ngs by ma k i n g us e of t he mo d u l u s of d e f o r ma t i o n of e a c h one of t he t h r e e e x i s t i n g g r o u n d l a ye r s a n d t he l a t e r a l e a r t h p r e s s u r e c o e f f i c i e n t K0.

Fi g u r e 31 s h o ws t he e v o l u t i o n of t he p a r a me t e r s t o be i d e n t i f i e d t h r o u g h t he i t e r a t i on , u s i n g a mi n i mu m s q u a r e c r i t e r i o n b e t we e n c a l c u l a t e d a nd me a s u r e d d i s p l a c e me n t s a t f our i n s t r u me n t e d s t a t i ons . Th e a g r e e me n t b e t we e n c a l c u l a t e d a nd o b s e r v e d g r o u n d mo v e me n t s f or o n e of t he a n a l y z e d s e c t i o n s c a n be s e e n i n Fi g u r e 32.

Fi g u r e 32. Be s t f i t of g r o u n d mo v e me n t s a t f our i n s t r u me n t e d s t a t i o n s a f t e r f i ne t u n i n g of d e s i gn p a r a me t e r s ( Le de s ma e t a l , 1987) .

I n a p a p e r t o t h i s Co n f e r e n c e , Ad a c h i a nd Ko j i ma( 1989) f o l l o we d a s i mi l a r p r o c e d u r e f or i d e n t i f i c a t i o n of d e s i g n pa r a me t e r s . Th e y d e s c r i b e t he b a c k - a n a l y s i s of t he n o n - l i n e a r c o n s t i t u t i v e p a r a me t e r s f or j o i n t e l e me n t s i n s i mu l a t i o n s of p a r t i c u l a r d i s c o n t i n u i t i e s o c u r r i n g i n s a ndy g r o u n d d u r i n g t u nne l e xc a va t i on . The s t r e s s - d i s - p l a c e me n t r e l a t i o n s h i p a d o p t e d f or t he j oi nt e l e me n t i n t he d i r e c t i o n t a n g e n t t o t he j oi nt i s hype r b o l i c . Th e c o n s t i t u t i v e mo d e l f or n o r ma l d i r e c t i o n i s s uc h t ha t t he c o n d i t i o n s of z e r o t e n s i l e s t r e n g t h a n d h i g h s t i f f n e s s i n c o mp r e s s i on a r e me t . The Po i s s o n ' s r a t i o i s a s s u me d t o be a k n o wn p a r a me t e r s i nc e i t n e i t h e r s i g n i f i c a n t l y f l u c t u a t e s n o r l a r g e l y a f f e c t s t he d i s p l a c e me nt s . The p l a n e s t r a i n c o n d i t i o n i s a s s ume d.

The b a c k a n a l y s i s p r o c e d u r e i s b a s e d on mi n i mi z i ng t he f unc t i on :

Nt NdJ = Z 1 ( u 1 " - U1 n ) 2 ( 39)

n 1

wh e r e u 4n a n d Uj " d e n o t e c a l c u l a t e d a nd me a s u r e d

d i s p l a c e me n t s a t n o d e i a t t i me s t e p n, r e s p e c t i ve l y . Nt i s t he n u mb e r of t i me s t e ps a nd Nd t he

nu mb e r of me a s u r e d v a l u e s of d i s p l a c e me n t s . Ad a c hi a n d Ko j i ma p r e s e n t t he e v o l u t i o n of t he v a l ue s of t he o b j e c t i v e f u n c t i o n a s we l l a s t ha t of t he p a r a me t e r s t o be i d e n t i f i e d wi t h t he i t e r a t i ve p r o c e s s f or s e v e r a l e xa mpl e s . Th e q u a l i t y of t he p r e d i c t i o n s i s a l s o s hown i n t he pa pe r .

Fr o m t he v i e wp o i n t of t hi s Re por t , one of t he mos t i n t e r e s t i n g a p p l i c a t i o n s of n u me r i c a l me t hods b a s e d o n t he d i s c r e t i z a t i o n of t he c o n t i n u u m a n d t he us e of f i n i t e e l e me n t s c o n s i s t s of p a r a me t r i c s e n s i t i v i t y a na l ys e s . I n t h i s r e s pe c t , t he wo r k s of Ro we a n d Le e a n d t he i r c o wo r k e r s c a r r i e d out a t t he Un i v e r s i t y of We s t e r n Ont a r i o , i n Ca na da , a r e l i k e l y t he mos t s ys t e ma t i c , c o mp r e h e n s i v e a n d r e l e va n t . Th e p a p e r b y Ro we a nd Le e ( 1989) s u b mi t t e d t o t h i s Co n f e r e n c e c o n t a i n s a t i gh t s u mma r y of t h e i r p r e v i o u s r e s e a r c h , wi t h

25 3 7

r e f e r e n c e t o t he wo r k s of o t h e r p e o p l e a t t he s a me Un i v e r s i t y on t he me a s u r e me n t a n d s i g n i f i c a n c e of d e f o r ma t i o n a nd s t r e n g t h b e h a v i o u r of s o f t c l a ys . Th e ma i n e mp h a s i s i s on t he r o l e of a n i s o t r o p y ( i n t e r ms of d e f o r ma b i l i t y , s t r e n g t h a n d e x i s t i n g s t r e s s e s ) . Re f e r e n c e t o t h e s e t op i c s wi l l be ma d e wh e r e a p pr opr i a t e .

5 . 5 . 4 An a l y t i c a l s o l u t i o n

To b e g i n wi t h t he s u b j e c t of t he t h e o r e t i c a l s o l u t i o n t o t he p r o b l e m of p r e d i c t i n g g r o u n d mo v e me n t s a s s o c i a t e d wi t h t unne l l i n g , i t i s i n t e r e s t i n g t o t r a n s c r i b e a p a i r of s e n t e n c e s f r om At t e we l l e t a l ( 1986) :

' Es t i ma t i o n of s e t t l e me n t c a u s e d b y t he wi t h d r a wa l of s u b s u r f a c e s u p p o r t i s a mu c h mo r e i mp r e c i s e o p e r a t i o n t h a n t he e s t i ma t i o n of s e t t l e me n t c a u s e d by s u r f a c e l oa di ng. The l a t t e r ha s b e n e f i t e d f r o m a n a l y t i c a l e x a mi n a t i o n a nd f i e l d o b s e r v a t i o n o v e r a p e r i o d of ma n y ye a r s , but t h e r e i s a s ye t no a n a l y t i c a l s o l u t i o n f or t he f o r me r ' .

Th i s wa s t r ue i n 1986 but i t s e e ms i t i s no l onge r s o t h a n k s t o Sa g a s e t a ' s ( 1987) a n a l y t i c a l a p p r oa c h , wh i c h ha s b e e n e x t e n d e d f or t he p r e s e n t Re p o r t a s i t wi l l be s h o wn l a t e r on.

Sa g a s e t a ( 1987) s o l v e d r i g o r o u s l y , i n a c l o s e d f or m, t he p r o b l e m of g r o u n d mo v e me n t s c a u s e d by t unne l l i n g , a s a n a p p l i c a t i o n of t he s t r a i n p a t h me t h o d ( Ba l i gh, 1985) . Th e o r i g i n a l s o l u t i o n wa s r e s t r i c t e d t o h o mo g e n e o u s , i s o t r o p i c a n d i n c o mp r e s s i b l e s o i l s u n d e r K0 = 1 c ond i t i o n s . Th e r e f o

r e , i s o t r o p y wa s a s s u me d not o n l y i n t e r ms of d e f o r ma t i o n but a l s o i n t e r ms of e x i s t i n g s t r e s s e s . I n c o mp r e s s i b i l i t y i s a n o r ma l a s s u mp t i o n f or s hor t t e r m d i s p l a c e me n t s c a u s e d by t u n n e l l i n g i n s of t s a t u r a t e d c l a ys .

Fi r s t l y , t he p r o b l e m of g r o u n d mo v e me n t s i n d u c e d b y a s i n k i n a n i n f i n i t e me d i u m i s c o n s i d e r e d ( Fi g. 33) . Bo u n d a r y c o n d i t i o n s a r e i n t r o d u c e d i n t e r ms of s t r a i n s ( or d i s p l a c e me n t s ) a l o n e a nd o n l y s t r a i n s ( or d i s p l a c e me n t s ) a r e wa nt e d . I f , f or a n y r e a s on , t he d i r e c t i o n of t he d i s p l a c e me n t v e c t o r i s k n o wn e v e r y wh e r e , o n l y i t s v a l u e b e i n g unknown, t he p o s t u l a t e of no v o l u me c h a n g e i s e n o u g h t o d e t e r mi n e t he d i s p l a c e me n t f i e l d. On c e t he s t r a i n s a r e known, a n y p a r t i c u l a r c o n s t i t u t i v e l a w c a n be u s e d t o o b t a i n t he c o r r e s p o n d i n g s t r e s s f i e l d.

Be c a u s e of t he s y mme t r y a bou t t he s i nk, d i s p l a c e me n t s mus t be r a di a l . Fu r t h e r mo r e , t he c o n d i t i o n of z e r o v o l u me c h a n g e i mp l i e s t ha t t he y c a n be e x p r e s s e d a s a n e x c l u s i v e f u n c t i o n of r , t he d i s t a n c e f r om t he s i nk t o t he p o i n t u n d e r c o n s i d e r a t i o n . Th o s e p o i n t s l o c a t e d a t a d i s t a n c e r f r o m t he s i nk mu s t h a v e t he i n wa r d d i s p l a c e me n t i n t e r ms of t he g r o u n d l os s g i v e n by t he f o l l o wi ng e x p r e s s i o n s f or e i t h e r t he t h r e e - d i me n s i o n a l o r t he p l a n e s t r a i n p r ob l e m:

3 - D : 8 r = V0 / 4Tt r 2

2 - D : 6 r = V0 / 2i r r

( 40a )

( 40b)

I n t he p l a n e s t r a i n c a s e V0 d e n o t e s v o l u me l os s p e r un i t l e ngt h.

Th e r e a l p r o b l e m i s t ha t of a h a l f - s p a c e wi t h z e r o s t r e s s e s a t t he s u r f a c e ( Fi g. 34) . I n o r d e r t o a c h i e v e t h e s e b o u n d a r y c o n d i t i o n s a n d e l i mi n a t e b o t h n o r ma l a n d s he a r s u r f a c e s t r e s s e s , Sa g a s e t a s u p e r i mp o s e s t wo i n f i n i t e me di a . I n one of t he m, a v i r t u a l i ma ge of t he p o i n t s i nk, s y mme t r i c wi t h r e s p e c t t o t he f r e e s ur f a c e , i s c on-

Fi g u r e 33. Po i n t s i nk: i n f i n i t e me di um.

V I R T U A L

H

F r e e

s u r f a c e

( 0 = 0 , T = 0 )

H

I M A G EI

( p o s i t i v e / n e g a t i v e )

6 h

A

/

G R O U N D

L O S S

Fi g u r e 34. Po i n t s i nk: h a l f - s pa c e .

s i de r e d . Be c a u s e of s ymme t r y , i f t he s a me g r o u n d l os s i s a s s u me d a t t he v i r t u a l i ma ge , t he n h o r i z on t a l d i s p l a c e me n t s a n d n o r ma l s t r e s s e s a t t he s u r f a c e d u p l i c a t e wh e r e a s v e r t i c a l d i s p l a c e me n t s a n d s h e a r s t r e s s e s b e c o me z e r o. I f , on t he c o n t r a r y, a v i r t u a l s o u r c e wh i c h i n t r o d u c e s t he s a me g r o u n d v o l u me i s a s s ume d , v e r t i c a l d i s p l a c e me n t s a n d s h e a r s t r e s s e s a t t he g r o u n d s u r f a c e d u p l i c a t e wh e r e a s h o r i z o n t a l d i s p l a c e me n t s a n d n o r ma l s t r e s s e s b e c o me z e r o.

Th e ne x t s t e p c o n s i s t s of d e t e r mi n i n g t he s u r f a c e s t r e s s e s a nd t he c o n s e q u e n t d i s p l a c e me n t s . Si nc e t he p r e v i o u s s t r a i n f i e l d i s a l r e a d y known, s t r e s s e s c a n be o b t a i n e d d i r e c t l y f r om t he a p p l i c a t i o n of t he c o n s t i t u t i v e l aw. I n t he c a s e of l i ne a r e l a s t i c i t y , i t i s v e r y s i mpl e . By a p p l y i n g Ce r r u t i ' s s o l u t i o n f or h o r i z o n t a l p o i n t l oa d a nd Bo u s s i n e s q ' s s o l u t i o n f or v e r t i c a l po i n t l oa d a t t he s u r f a c e of a h o mo g e n e o u s e l a s t i c ha l f - s p a c e , a n d i n t e g r a t i n g , t he c o mp l e t e s t r e s s - s t r a i n p r o b l e m i s s ol ve d. Th i s p r o v i d e s t he g r o u n d d i s p l a c e me n t s a t a n y p o i n t of t he h a l f - s p a c e due t o a g r o u n d l os s c o n c e n t r a t e d a t a p o i n t or u n i f o r ml y d i s t r i b u t e d a l o n g a h o r i z o n t a l s e g me n t wi t h i n t he h a l f - s p a c e .

For s u r f a c e po i n t s , t he p r o b l e m i s c o n s i d e r a b l y s i mp l e r a n d e v a l u a t i o n of s u r f a c e s t r e s s e s i s not n e c e s s a r y . Th e s i mp l i c i t y c o me s f r om t he f a c t

2 53 8

t ha t , f or a n i n c o mp r e s s i b l e ma t e r i a l , a s u p e r f i c i a l v e r t i c a l p o i n t l oa d p r o d u c e s z e r o h o r i z o n t a l mo v e me n t s a t t he s u r f a c e . An d a s u p e r f i c i a l h o r i z o n t a l p o i n t l oa d p r o d u c e s z e r o s u r f a c e s e t t l e me nt s . Th i s s i mp l y me a n s t ha t s u r f a c e d i s p l a c e me n t s c a u s e d b y g r o u n d l os s wi t h i n a h a l f - s p a c e a r e e x a c t l y e q u a l t o t wi c e t he mo v e me n t s c a u s e d a t t he s a me p l a n e b y t he s a me g r o u n d l os s i n t he i n f i n i t e me di um. Re me mb e r t ha t t h i s i s t r ue f or h o mo g e n e o u s e l a s t i c i s o t r o p i c a n d i n c o mp r e s s i b l e s oi l s , u n d e r i n i t i a l l y h y d r o s t a t i c s t r e s s e s .

Su r f a c e d i s p l a c e me n t s , e x p r e s s e d i n t e r ms of t he p r o p o s e d n o r ma l i z e d p a r a me t e r s , a r e :

^ =- 4 -X/ D

( x / D) 2 +( H/ D)>{ H

y/ D

t ( x / D) 2 +( y / D) 2 +( H/ D]

1

r

^ = * ______6 „ [ ( x / D) 2 +( y / D) 2 +( H/ D) 2]*

}<41a )

( 41b)

H/ D y/ D

( x / D) 2+ ( H/ D ) 2 [ ( x / D) 2 +( y / D) 2 +( H/ D) 2]*

The s u r f a c e d i s p l a c e me n t s c o r r e s p o n d i n g t o t he p l a n e s t r a i n c a s e or f i na l s i t u a t i o n ( y = ®) c o i n c i d e wi t h t h o s e p r o v i d e d b y t he s e mi - e mp i r i - c a l a p p r o a c h a l r e a d y me n t i o n e d ( e qua t i o n s 36b a nd 36c , wi t h H i n s t e a d of H 1 ). He nc e , t he p a r t i c u l a r s of t he a n a l y t i c a l s o l u t i o n a r e t he s a me . The r e f o r e , t he t h e o r e t i c a l s o l u t i o n e x h i b i t s t he s a me t e n d e n c y t o e x t e n d l a t e r a l l y t he s e t t l e me n t t r o u g h f u r t h e r t h a n u s u a l l y obs e r ve d .

The l owe r p a r t o f Fi g u r e 35 r e p r e s e n t s t r a n s v e r s e p r o f i l e s of t he s e t t l e me n t t r o u g h a s p r e d i c t e d b y Sa g a s e t a ' s ba s i c s o l u t i o n f o r H/ D = 2 a t s e ve r a l d i s t a n c e s y / D f r o m t he t u n n e l f a c e ,

} ( 41c )

— = 5

D

Fi g u r e 36. Su r f a c e d i s p l a c e me n t s a t p l a n e s p e r p e n d i c u l a r t o t he t unne l a x i s f or H/ D = 5.

— = 2

D

x / D D

Fi g u r e 35. Su r f a c e d i s p l a c e me n t s a t p l a n e s p e r p e n d i c u l a r t o t he t u nne l a x i s f or H/ D = 2.

Fi g u r e 37. Su r f a c e d i s p l a c e me n t s a t v e r t i c a l p l a n e s p a r a l l e l t o t he t unne l a x i s f or H/ D = 2.

2539

i l

6 0

5 v

6 0

- 0. 4

T -0 .2 -

0.0

0.2 -

0.4

0.6 -

0 . 8

-jj = 0 (centreline)

L â > , -

oüx|Q

\ Ü‘ 5■jj =0 (centreline)

-1 0 - 8 -4 - 2 0

_y_

D

6 8 10

Fi g u r e 38. Su r f a c e d i s p l a c e me n t s a t v e r t i c a l p l a n e s p a r a l l e l t o t he t u n n e l a x i s f o r H/ D = 5.

Pr e s e n t a t i o n of Sa g a s e t a ' s b a s i c s o l u t i o n c a n be c o mp l e t e d wi t h Fi g u r e 39 s h o wi n g g r o u n d d i s p l a c e me n t s ( not o n l y a t s ur f a c e ) c a u s e d by a t unne l wi t h H/ D e q u a l t o t wo. The ' s t r e a m l i ne s ' of g r o u n d mo v e me n t s h a v e b e e n d r a wn a n d t h o s e o b s e r v e d by Co r d i n g a n d Ha n s mi r e ( 1975) a t t he Wa s h i n g t o n Me t r o h a v e be e n i n c l u d e d f or c o mpa r i s on . Bo t h a r e q u i t e s i mi l a r . Howe v e r , t he o b s e r v e d l i ne s a t t he Wa s h i n g t o n Me t r o a p p e a r t o be mo r e v e r t i c a l wh i c h i s e q u i v a l e n t t o t he f a c t t ha t t he p r a c t i c a l l i mi t of i n f l u e n c e i s n a r r o we r t ha n p r e d i c t e d by t he a n a l y t i c a l s o l u t i on .

5 . 5 . 4 . 1 El a s t i c a nd s t r e s s a n i s o t r o p y

An a p p r o x i ma t e s o l u t i o n f or c o mp r e s s i b l e ma t e r i a l s wa s g i v e n by Sa g a s e t a ( 1987 a n d 1988) . I t wi l l be d e s c r i b e d l a t e r on. St i l l wi t h i n t he a s s u mp t i o n of i n c o mp r e s s i b i l i t y , Sa g a s e t a ha s e x t e n d e d t he r i g u r o u s a n a l y t i c a l s o l u t i o n t o:

- An i s o t r o p i c e l a s t i c s o i l s u n d e r e x i s t i n g i s o t r o p i c s t r e s s e s .

- I s o t r o p i c e l a s t i c s o i l s u n d e r K0 d i f f e r e n t f r o m one .

I t s h o u l d be s a i d, i n t he f i r s t p l a c e , t ha t t he p r e v i o u s r e a s o n i n g on t he e f f e c t s of s u r f a c e s t r e s s e s e l i mi n a t i o n on s u r f a c e d i s p l a c e me n t s i s a l s o v a l i d f or a n i s o t r o p i c s oi l s . I t i s o b v i o u s t ha t i t a l s o a p p l i e s f or n o n - h y d r o s t a t i c e x i s t i n g s t r e s s e s s i n c e i n a n e l a s t i c s o i l t he d i s p l a c e me n t s c a u s e d by a s t r e s s c h a n g e do no t d e p e n d on t he e x i s t i n g s t r e s s e s . As a c o n s e q u e n c e , i t i s o n l y n e c e s s a r y i n bo t h c a s e s t o f i n d a s o l u t i o n f or t he p o i n t s i n k i n t he i n f i n i t e me di um. As a ge n e r a l r e q u i r e me n t , t he s o l u t i o n mu s t t e n d t o t he b a s i c one f or i s o t r o p y a n d K0 = 1.

t hus i n c l u d i n g t he p l a n e s t r a i n p r o b l e m o r f i na l s i t ua t i on . Th e c o r r e s p o n d i n g d i s t r i b u t i o n s of h o r i z o n t a l d i s p l a c e me n t s c a n be s e e n i n t he u p p e r pa r t of t he f i gur e .

Fi g u r e 36 s hows r e s u l t s f or H/ D = 5. The e f f e c t of i n c r e a s i n g t he t u n n e l d e p t h i s t o r e d u c e t he ma g n i t u d e of s u r f a c e mo v e me n t s a n d t o e x t e n d t h e m l a t e r a l l y .

Ag a i n f or t he r a t i o H/ D e q u a l t o t wo, Fi g u r e 37 r e p r e s e n t s v e r t i c a l a n d l o n g i t u d i n a l s u r f a c e d i s p l a c e me n t s a t v e r t i c a l p l a n e s p a r a l l e l t o t he t unne l a x i s f o r s e v e r a l d i s t a n c e s x / D f r o m t he t u n n e l c e n t r e l i n e . Fi g u r e 38 g i v e s a n a l o g o u s r e s u l t s f or H/ D e q u a l t o f i ve .

d "2Z

V ( + )

u ( + )

Fi g u r e 40. Po i n t s i nk: i n f i n i t e me di um. Co mp o n e n t s of g r o u n d d i s p l a c e me n t .

( C o r d in g a n d H a n s m i r e , 1 9 7 5

(W a s h in g t o n M e t r o )

Fi g u r e 39. Gr o u n d mo v e me n t s a t t he Wa s h i n g t o n Me t r o : An a l y t i c a l a p p r o a c h a n d o b s e r v e d va l ue s .

The d i s p l a c e me n t v e c t o r a t a n y p o i n t c a u s e d by a po i n t s i n k h a s i n g e n e r a l t wo c ompon e n t s : r a d i a l a n d t a n g e n t i a l ( Fi g. 40) . Th e c o n d i t i o n of z e r o v o l u me c h a n g e p r o v i d e s a d i f f e r e n t i a l e q u a t i o n wi t h t wo unknowns :

à u u 1 j v— + - + - — = 0

à r r r ö 9( 42)

He nc e , t h e r e a r e i n f i n i t e s o l u t i ons . I t i s t he n n e c e s s a r y t o f i nd a s o l u t i o n t ha t f u l f i l s t he g e n e r a l r e q u i r e me n t t ha t i t mu s t t e n d t o t he b a s i c s o l u t i o n f or i s o t r o p i c ma t e r i a l s ( i n t e r ms of b o t h e l a s t i c i t y a n d e x i s t i n g s t r e s s e s ) . The h y p o t h e s i s of s e p a r a t i o n of v a r i a b l e s , a c l a s s i c a l me t h o d of r e s o l u t i o n of s u c h p r ob l e ms , ha s b e e n a dopt e d :

2 5 4 0

Fi g u r e 41. Ef f e c t of 9 ( 0 ) on f l ( o) .

u( r , o) = H( r ) . 0( 9) ( 43a )

v ( r , o) = ? ( r ) . 9 ( 0 ) ( 43b)

I t l e a ds t o t he f o l l o wi n g d i f f e r e n t i a l e qua t i on:

d u / d r +u / r d O/ d O

The l e f t h a n d s i de i s a f u n c t i o n of r o n l y wh e r e a s t he r i gh t h a n d s i de d e p e n d s on 0 a l one . Fo r t he m t o be e q u a l f o r a n y v a l u e s of r a n d 0 bo t h s i de s mu s t be c o n s t a n t a n d e qua l . Th e r e s u l t i s t hi s s i mp l e s y s t e m of d i f f e r e n t i a l e qua t i ons :

du u c v— + - + — = 0 ( 45a)d r r r

d 9- - - - cfl = 0 ( 45b)d 0

The t a n g e n t i a l d i s p l a c e me n t mus t be z e r o wh e n 0 e q u a l s 0 a n d 9 0 ’ , t hus h a v i n g t he d i s t r i b u t i o n s ha pe r e p r e s e n t e d i n Fi g u r e 41 ( a ) . I t me a n s t ha t t he d e r i v a t i v e of 9 a g a i n s t 0 i s p o s i t i v e a nd n e g a t i v e a s 0 va r i e s . I t f o l l o ws f r om e q u a t i o n ( 45b) t ha t t he f u n c t i o n Q( 0 ) s h o u l d c h a n g e of s i gn ( s e e Fi g u r e 41 ( b) ) wh i c h i s a bs ur d . I t doe s not s a t i s f y t he g e n e r a l r e q u i r e me n t of s y mme t r y f o r t he p a r t i c u l a r c a s e of i s o t r o p i c ma t e r i a l s . The r e f o r e , t he o n l y p o s s i b l e s o l u t i o n i s t ha t t he c o n s t a n t c i s z e r o. The n, t he s y s t e m s t r o n g l y s i mpl i f i e s . Eq u a t i o n ( 45a ) b e c o me s t he s a me of t he i s o t r o p i c c a s e a n d i t s s o l u t i o n i s 13 = k/ r .On t he o t h e r ha nd, e q u a t i o n ( 45b) l e a ds t o 9 e q u a l t o a c o n s t a n t wh i c h mus t be z e r o s i nc e i t mus t be z e r o f o r 0 = 0 a n d 0 = 90 ' .

As a c o n s e q u e n c e of t he p r e c e d i n g a na l ys i s , t he g e n e r a l s o l u t i o n f o r i n c o mp r e s s i b l e ma t e r i a l i s :

u = u( r , o) = k / r . Q( e ) ( 46a )

v = v ( r , 0 ) = 0 ( 46b)

I t me a n s t ha t t he d i s p l a c e me n t a t a ny p o i n t i s s t i l l r a d i a l ( i . e . , v = 0) . Si n c e t he e x p r e s s i o n f o r u c o n t a i n s t he f u n c t i o n Q ( o) , h e n c e t he d i s p l a c e me n t ha s not t o be s ymme t r i c a l . Th i s f u n c t i on c o u l d be a n y f u n c t i o n s a t i s f y i n g t he f o l l o wi ng c ond i t i o n s :

- I t s h o u l d be pos i t i ve .- I t s h o u l d d e p e n d on n.- I t s h o u l d be ma x i mu m a nd mi n i mu m f o r 0 = 0

a nd 0 = 90 ' .- I t s h o u l d t e n d t o 1 f o r n t e n d i n g t o 1.I n o r d e r t o f i nd a r e a s o n a b l e f u n c t i o n Q ( 0 ),

e x i s t i n g e l a s t i c s o l u t i o n s h a v e b e e n a ppl i e d .

E v

5 . 5 . 4 . 1 . 1 El a s t i c a n i s o t r o p y

Ka wa mo t o ( 1966) s o l v e d t he p r o b l e m of o b t a i n i n g t he d e f o r ma t i o n s a r o u n d a c i r c u l a r ho l e s u b j e c t e d t o i n t e r n a l p r e s s u r e i n a n o r t h o t r o p i c e l a s t i c me d i u m u n d e r p l a n e s t r a i n c o n d i t i o n s ( Fi g. 42) . Ka wa mo t o ' s g e n e r a l s o l u t i o n f o r t he c a v i t y wa l l i s v e r y c ompl e x . Howe ve r , wh e n a p p l i e d t o i n c o mp r e s s i b l e t h o u g h a n i s o t r o p i c ma t e r i a l s , i t b e c o me s v e r y s i mpl e :

u = p a a u ( Oj +02 ) ( 47a )

v = 0 ( 47b)

wh e r e :

a u = ( 4 - n ) / 4EW ( 48a )

fli = + { [ 2 - b +2 ( l - b ) * ] / b } ‘ ( 48b)

fl2 = + { [ 2 - b —2 ( 1 —b ) *] / b }* ( 48c )

b = g ( 4 - n ) ( 48d)

The c o e f f i c i e n t s a n , a nd 3 2 a r e f u n c t i o n s of

t he a n i s o t r o p i c e l a s t i c p a r a me t e r s Ev , n a n d g onl y.

Fo r t he i s o t r o p i c c a s e : n = 1, g = 1/ 3, b = 1, a u = 1/ 4G, Bj = fl2 = 1, a nd, t he r e f o r e :

u = p a / 2G ( 49)

wh i c h i s t he c o r r e c t s ol ut i on .Si n c e t he t a n g e n t i a l c o mp o n e n t of t he d i s p l a c e

me n t i s e q u a l t o z e r o a n d t he r a d i a l one doe s not d e p e n d on 0 , a f i r s t i mp o r t a n t c o n c l u s i o n c a n be de r i ve d : a c i r c u l a r h o l e i n a n i n c o mp r e s s i b l e a n i s o t r o p i c me d i u m d e f o r ms s y mme t r i c a l l y , a s i f t he me d i u m we r e i s ot r op i c , a l t h o u g h t he ma g n i t u d e of t he d e f o r ma t i o n d e p e n d s on t he a n i s o t r opy . I n o t h e r wo r ds , i n a n i n f i n i t e me di um, a c i r c l e d e f o r ms i n t o a c o n c e n t r i c c i r c l e i n s p i t e of a n i s o t r o p y . Th i s c o n c l u s i o n i s a l s o r e l e v a n t t o t he l i n i n g de s i gn . Be n d i n g mo me n t s a r e not t o be e x p e c t e d b e c a u s e of e l a s t i c c r o s s - a n i s o t r o p y .

On c e t h i s c o n c l u s i o n ha s be e n r e a c he d , i t i s i mp o r t a n t t o i n v e s t i g a t e wh e t h e r i t a p p l i e s o n l y t o t he t unne l wa l l o r t o a n y c o n c e n t r i c c i r c l e . The s t r e s s - s t r a i n e q u a t i o n s f o r a n i n c o mp r e s s i b l e

2541

i s o t r o p i e s oi l u n d e r p l a n e s t r a i n c o n d i t i o n s a r e a s f ol l ows :

3

4E

3

4E

U o h- a ov ) ( 50a )

( - Aa h+Aov ) ( 50b)

( 50c )

By c o mp a r i n g t h i s s e t of e q u a t i o n s wi t h t he mo r e g e n e r a l o n e s ( 23a , b a n d c) f o r a n i s o t r o p y , i t c a n be s e e n t ha t t he a n i s o t r o p i c c a s e i s a l mos t e q u a l t o t he i s o t r o p i c one , wi t h a n e q u i v a l e n t mo d u l u s EWI=3 Ev/ ( 4- n) . Th e o n l y d i f f e r e n c e l i e s on

t he s h e a r mo d u l u s wh i c h i s e q u a l t o E/ 3 f o r t he i s o t r o p i c c a s e wh e r e a s i t r e ma i n s i n d e p e n d e n t a nd c a n h a v e a n y v a l u e f o r t he a n i s o t r o p i c s oi l .

Fo r t he p a r t i c u l a r c a s e i n wh i c h t he i n d e p e n de n t s h e a r mo d u l u s of t he a n i s o t r o p i c s o i l i s e q u a l t o Ev/ ( 4- n) , t he s t r e s s - s t r a i n e q u a t i o n s

c o i n c i d e a n d t he s o l u t i o n s f o r t he a n i s o t r o p i c a n d t he i s o t r o p i c c a s e s a r e e x a c t l y t he s a me .

Ac t u a l v a l u e s f o r Lo n d o n c l a y ( t a ke n f r o m Wa r d e t a l , 1969) a n d f o r s ome Ca n a d i a n c l a y s ( t a ke n f r om Le e a n d Rowe , 1989, a f t e r h a v i n g c o r r e c t e d s ome e r r o r s i n Ta b l e 3 of t he p a pe r ) h a v e be e n r e p r e s e n t e d i n a g v e r s u s n p l o t i n Fi g u r e 43. I t i s i n t e r e s t i n g t o n o t e t ha t g v a r i e s s l i g h t l y f or a wi d e r a n g e of n va l ue s . Th e t h i n l i ne r e p r e s e n t s t h o s e a n i s o t r o p i c s o i l s t ha t b e h a v e i s o t r o- p i c a l l y u n d e r p l a n e s t r a i n c o n d i t i o n s . Wi t h i n t he r a n g e of n v a l u e s of i n t e r e s t , t he d i f f e r e n c e s wi t h t he r e a l g v a l u e s a r e s ma l l . Th e r e f o r e , t he a n a l y t i c a l s o l u t i o n s h o u l d be c l o s e t o t he i s o t r op i c one a n d t he i n f l u e n c e of t he s h e a r mo d u l u s s e e ms t o be mode s t . Sa g a s e t a ' s ( 1987) s o l u t i o n f o r t he i s o t r o p i c c a s e c a n be u s e d wi t h c o n f i d e n c e f o r t he a n a l y s i s of g r o u n d mo v e me n t s i n a n i s o t r o p i c s oi l s .

An o t h e r i n t e r e s t i n g c a s e of c r o s s - a n i s o t r o p i c s o i l s i s t ha t i n wh i c h t he s h e a r mo d u l u s G i s c o n s t a n t i n a l l d i r e c t i o n s . Th i s k i n d of s o i l s h a B b e e n a n a l y z e d by Ba r d e n ( 1963) a n d Gi b s o n ( 1974) . I t i s r e p r e s e n t e d b y a t h i c k l i ne i n Fi g u r e 43. Aga i n , t he d i f f e r e n c e s wi t h t he r e a l g v a l u e s wi t h i n t he r a n g e of i n t e r e s t ( f r om n = 0. 5 t o n = 2. 5) a r e s ma l l .

I n o r d e r t o c l a r i f y t h i s s ub j e c t , Sa g a s e t a ha s c a l c u l a t e d t he g r o u n d d i s p l a c e me n t s c o r r e s p o n d i n g t o t he r e f e r e n c e c a s e s h o wn i n t he l owe r pa r t of Fi g u r e 43 b y u s i n g f i n i t e e l e me n t s . Th r e e t ype s of ma t e r i a l h a v e b e e n c ons i d e r e d : i s o t r op i c , a n i s o t r o p i c but i s o t r o p i c i n p l a n e s t r a i n ( i . e . , s o i l s r e p r e s e n t e d by t he t h i n l i ne i n t he pl ot ) , a n d a n i s o t r o p i c wi t h t he s h e a r mo d u l u s c o n s t a n t i n a l l d i r e c t i o n s ( s o i l s r e p r e s e n t e d b y t he t h i c k l i ne ) . Fo r t he a n i s o t r o p i c c a s e s , t h r e e v a l u e s of n h a v e b e e n u s e d i n c a l c u l a t i o n s : 1, 2 a n d 3. The r e s u l t s s h o w t ha t :

- Th e i s o t r o p i c s o l u t i o n i s p e r f e c t l y r e p r o d u c e d, t he e r r o r s d u e t o d i s c r e t i z a t i o n b e i n g n e g l i g i b l e .

- Th e s o l u t i o n of t he s e c o n d c a s e e x a c t l y c o i n c i d e s wi t h t he i s o t r o p i c one .

- An i s o t r o p y i n f l u e n c e s t he s o l u t i o n of t he t h i r d c a s e , wi t h t he f o l l o wi n g p e c u l i a r i t i e s :

- Th e d i s p l a c e me n t s of t he o p e n i n g wa l l a r e s y mme t r i c ( i . e . ; r a d i a l a n d c ons t a n t ) . Th i s r e s u l t v a l i d a t e s Ka wa mo t o ' s ( 1966) s o l u t i on .

- Fo r c o n c e n t r i c c i r c l e s , d i s p l a c e me n t s a r e not

n * E h / E v

□ L o n d o n c l a y A C a n a d i a n c l a y s o C o m p u t e d c a s e s

( W a r d e t a l , 1 9 6 9 ) ( L e e ( R o w e , 1 9 8 9 )

R e f e r e n c e p r o b l e m : ( F . E . M . )

( i n c o m p r e s s i b l e )

Fi g u r e 43. An i s o t r o p i c s o i l s c o n s i d e r e d a n d r e f e r e n c e p r obl e m.

Fi g u r e 44. Gr o u n d d i s p l a c e me n t s d i s t r i b u t i o n a t t he c a v i t y wa l l a n d a t a c o n c e n t r i c c i r c l e .

s ymme t r i c . Ho we ve r , t he d i s p l a c e me n t s a t t he v e r t i c a l of t he c r o wn a n d a t t he h o r i z o n t a l of t he s p r i n g l i n e h a v e t he s a me va l ue , wh i c h i s a ma xi mum. The mi n i mu m v a l u e c o r r e s p o n d s t o t he h a u n c h ( 45* l i ne ) .

Th e c h a r a c t e r i s t i c s ha pe of t he d i s p l a c e me n t d i s t r i b u t i o n i s s h o wn i n Fi g u r e 44.

2 5 4 2

r

r - ' 1e 'u

E h / E v = 3 a ~ ®

0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

P o la r a n g le ( ° )

Fi g u r e 45. Di s p l a c e me n t s of p o i n t s l o c a t e d a t a c o n c e n t r i c c i r c l e of r a d i u s e qua l t o s i x t i me s t ha t of t he t u n n e l f o r t wo k i n d s of a n i s o t r opy .

Fi g u r e 45 r e p r o d u c e s t he r e s u l t s of r a d i a l a nd t a n g e n t i a l c o mp o n e n t s of t he d i s p l a c e me n t s of a c i r c l e wi t h a r a d i u s e qua l t o s i x t i me s t ha t of t he t unne l , f o r t he e x t r e me c a s e of a n i s o t r o p y u n d e r c o n s i d e r a t i o n ( n = 3) . The d i s p l a c e me n t s ha ve b e e n pu t i n a n o n - d i me n s i o n a l f o r m by d i v i d i n g t h e m b y t he c a v i t y wa l l mo v e me n t . The f o l l o wi ng g e n e r a l t r e n d s c a n be d e r i v e d f r o m t he f i gur e :

- Fo r a n i s o t r o p i c s o i l s wi t h a n i n d e p e n d e n t s he a r mo d u l u s G = Ev/ ( 4- n) , t a n g e n t i a l d i s p l a c e

me n t s a r e z e r o. The r a d i a l d i s p l a c e me n t i s t he s a me a t a n y poi n t . Th e s o l u t i o n i s s y mme t r i c a nd i s o t r o p i c .

- Fo r a n i s o t r o p i c s o i l s wi t h a s h e a r mo d u l u s G c o n s t a n t i n a l l d i r e c t i o n s , t he d i s p l a c e me n t s a r e

Fi g u r e 46. Ef f e c t of a n i s o t r o p i c r a t i o g on ma x i mu m s u r f a c e s e t t l e me n t ( Le e a n d Rowe , 1989) .

n e i t h e r r a d i a l n o r r a d i a l l y s ymme t r i c a l , but t he r e a r e t h r e e a x i s of s y mme t r y p a s s i n g t h r ough t he c e nt r e : t he v e r t i c a l , t he h o r i z o n t a l a n d t he 4 5 ’ l i ne s . Th e r a d i a l c o mp o n e n t s s h o w a ma x i mu m a t t he v e r t i c a l of t he c r o wn a nd a t t he h o r i z o n t a l of t he s p r i ng l i n e , of t he s a me va l ue , a n d a mi n i mu m a t 45" . Th e t a n g e n t i a l c o mp o n e n t s a r e r a t h e r s ma l l . The p o i n t s of ma x i mu m t a n g e n t i a l c o mp o n e n t c o i n c i d e wi t h t h o s e a t wh i c h t he r a d i a l c o mp o n e n t e q u a l s t he wa l l move me nt .

By s u mma r i z i n g f o r a n i s o t r o p i c i n c o mp r e s s i b l e s o i l s u n d e r p l a n e s t r a i n c o n d i t i o n s , t he f o l l o wi ng c o n c l u s i o n s c a n be d r a wn:

- The t u nne l wa l l d e f o r ms a l wa y s i s o t r o p i c a l l y or s y mme t r i c a l l y .

- Fo r t he p a r t i c u l a r c a s e of g = l / ( 4- n) , t he s o l u t i o n i s i s o t r o p i c a t a n y p o i n t i r r e s p e c t i v e of t he v a l u e of n. Fo r n < 2 ( whi c h i s us ua l ) , t h i s v a l u e of g i s no t f a r f r o m r e a l i t y .

- I n a ny c a s e , s i nc e t he d i s p l a c e me n t s a t t he v e r t i c a l of t he c r o wn a nd a t t he h o r i z o n t a l of t he s p r i n g l i n e a r e r a d i a l a n d e qua l , h e n c e t he i n f l u e n c e of t he s h e a r mo d u l u s i s s ma l l .

The l a s t c o n c l u s i o n s p a r t i a l l y i s i n c o n t r a d i c t i on wi t h t he f i n d i n g s of Ro we a n d Le e ( 1989, p a p e r t o t h i s Conf e r e n c e ) . By u s i n g f i n i t e e l e me nt s , Le e a nd Ro we ( 1989) h a v e a n a l y z e d t he e f f e c t of t he i n d e p e n d e n t s h e a r mo d u l u s on t he s e t t l e me n t t r ough. The r e s u l t s ( Fi g. 46) a r e p r e s e n t e d i n t e r ms of 6 V m o b t a i n e d f o r a g i ve n

G/ Ev r a t i o d i v i d e d by 6 V f or a n i s o t r o p i c s oi l

( G/ E = 1/ 3) . I t wa s f ound t ha t t he i n d e p e n d e n t s h e a r mo d u l u s G h a s a s i g n i f i c a n t e f f e c t upon t he s ha pe of t he s e t t l e me n t t r ough a nd i t wa s c o n c l u d e d t ha t wh e n a t t e mp t i n g t o p r e d i c t s e t t l e me n t i n d u c e d b y t unne l l i n g , a t t e n t i o n s h o u l d be g i ve n t o t he e f f e c t of e l a s t i c a n i s o t r o p y a nd, i n p a r t i c u l a r , t o t he g r a t i o. On t he c on t r a r y , t he s a me a u t h o r s c o n c l u d e d t ha t a n i s o t r o p y i n t e r ms of n a n d i n t e r ms of Po i s s o n ' s r a t i o doe s not g r e a t l y a f f e c t t he s u r f a c e s e t t l e me n t .

The r e a s o n s f o r t he d i s c r e p a n c y a r e not c l e a r . On one ha nd, i t c o u l d be due t o t he f a c t t ha t t he b o u n d a r i e s f o r t he f i n i t e e l e me n t a n a l y s i s c a r r i e d out by Le e a n d Ro we ( 1989) we r e p e r h a p s t oo c l o s e t o t he t unne l . On t he o t h e r ha nd, t he s oi l mo d e l c o n s i d e r e d i s not p e r f e c t l y e l a s t i c but

254 3

Fi gur e 47. Non- hy dr os t a t i c ex i s t i ng s t r es s es . Pr ob l em def i ni t i on.

el as t i c - p l as t i c . Al so, wher eas t he t heor et i c a l s t udy as s umes t hat s t r es s es ar e uni f or m, Lee and Rowe' s anal y s i s ac c ount s f or t he gr adi ent of s t r es s es wi t h dept h.

5. 5. 4. 1. 2 Ex i s t i ng s t r es s an i s ot r opy

Goi ng bac k t o Sagas et a ' s ex t ens i on of hi s bas i c t heor y , t he ef f ec t of s t r es s an i s o t r opy wi l l be now ex ami ned. I t was or i g i na l l y as s umed t hat s t r es s es wer e hy dr os t a t i c ( K0 = 1) .

As f or t he c as e of e l as t i c ani s ot r opy , i n or der t o s ol v e t he d i f f er ent i a l equat i on ( 42) r e l at i ng t o eac h ot her t he r adi al and t angent i a l d i s p l ac e ment s i nduc ed by a poi nt s i nk at any poi nt of an i nf i n i t e medi um, ex i s t i ng e l as t i c s o l u t i ons hav e been adopt ed. For an el as t i c i s ot r opi c mat er i a l wi t h ex i s t i ng t ot al s t r es s es p0 and Kt p0 ( Fi g.

47) , t he p r ob l em was s ol v ed i n t er ms of s t r es s es by Ki r s c h ( 1898) . I n t er ms of d i s p l a c e me n t s , Pou- l os and Dav i s ( 1974) gav e a s ol ut i on, t ak en f r om Ober t and Duv al l ( 1965) , whi c h i s er r oneous . The c aus e of t he er r or i s t hat t he i n t egr at i on of s t r a i ns t o get d i s p l ac ement s was made i n t er ms of s t r es s es i ns t ead of s t r es s i nc r ement s . I n t egr a t i ng pr oper l y , t he ex pr es s i ons f or t he r adi al and t angent i a l c omponent s of t he d i s p l ac ement ar e:

p0 1+K, a 2 1- K, a4 a2u = - - - - - - - - - - - 4( l - p) — ] c os 2G} ( 51a)

2 r 2 r 3 r2G

p0 1- K, 1 a4 a 2v = JZT —3 - 2 ( l - 2 p ) — ] s en2a ( 51b)

2G 2 2 r 3 r

For an i nc ompr es s i b l e mat er i a l ( p = i , G = E/ 3) :

" ' a p ” T [ 2

a2 1+Kt 1- Kt a

2 r

3 a4 1- K, 1 v = — pn —r— - - - - 1 — s en28

o r . 3 0 ~

( 52a)

( 52b)2E r 2 2

The c r own s et t l ement i s ( r = a, 9 = 0) :

u0 = 3p0a/ 2E ( 53)

Henc e, equat i ons ( 52a) and ( 52b) c an be wr i t t en:

u = u. — [a 1+Kt 1- Kt a

r 2(— r - 2) c os 2e] ( 54a)

2 r 2

a 1- K, 1 a2v = u . - - - - - - s- - - - - - - , s en2e

0 r 2 2 r 2( 54b)

Fi gur e 48. Ef f ec t of Kt on s ur f ac e d i s p l ac ement s f or H/ D = 3.

By i n t egr at i ng t he r adi a l d i s p l ac ement a l ong t he c av i t y wal l , a s i mpl e ex pr es s i on f or t he gr ound l oss i mmedi a t e l y f ol l ows :

V0 = ( 1+Kt ) nau0 ( 55)

Si nc e t he mat er i a l i s e l as t i c and i nc ompr es s i bl e, t he met hod of s uper pos i t i on of i mages t o ac c ount f or t he s ur f ac e ef f ec t i s v a l i d as bef or e. Henc e, s ur f ac e d i s p l ac ement s ar e s i mpl y ob t a i ned by mu l t i p l y i ng t hos e c or r es pond i ng t o t he i nf i ni t e medi um by a f ac t or of 2. The r es u l t i ng ex p r es s i ons f or s ur f ac e mov ement s under p l ane s t r ai n c ondi t i ons ar e:

H/ D

( H/ D ) 2+ ( x / D) 2^ 1

1- Kt 8 ( x / D) 4- 8( H/ D) 4+( H/ D) 2

x / D

(h / d )2+( x / d ;i Ci -

1+K*

1- Kt

1+Kt

6h/ 6„ =

4[ ( H / D ) 2+ ( x / D ) 2 ] 2

8( x / D) 4- 8 ( H/ D) 4+( H/ D) 2

}

4[ ( H/ D ) 2+ ( x / D ) 2 ] 2

x / H ( 5 6 a , b , c )

The ef f ec t of Kt i s s hown t hr ough t he quot i ent

( 1- Kt ) / ( 1+Kt ) . For hy dr os t a t i c s t r es s es , Kt = 1

and ( 1- Kt ) / ( 1+Kt ) = 0. The s ec ond t er m wi t h i n t he

par ent hes i s i n equat i ons ( 56a and b) d i s appear s and t he or i g i nal s o l u t i on f or pl ane s t r ai n c ond i t i ons i s obt ai ned.

Fi gur e 48 s hows t he ef f ec t of Kt upon s e t t l e

ment s and hor i z ont a l d i s p l ac ement s at s ur f ac e, f or a s ha l l ow t unnel wi t h a dept h t o d i amet er r at i o equal t o 3. Al l t he s et t l ement t r oughs hav e t he s ame ar ea. Ther ef or e, t he s mal l er Kt t he

nar r ower and deeper t he s et t l ement t r ough. Kt has

a v er y mar k ed i n f l uenc e on t he s et t l ement t r ough.

2544

H o r iz o n ta l D is ta n c e f r o m i . ( m )

t 5 10 15 2 0 2 5

— ...... r ~ - t i_ _ _ _ — i-------- i

- I s o t r o p ic K 0 * 1.5

K 0 = 1.0— ------ - I— I s o t r o p ic

— I s o t r o p ic * O

ii O ’->1

-U n lin e d T u n n e l

E la s t i c a l l y Is o t ro p ic

N « 4 .0

(/) oE

° o

oEo

Fi g u r e 49. Ef f e c t of l a t e r a l e a r t h p r e s s u r e c oe f . on s u r f a c e s e t t l e me n t ( Le e a n d Rowe , 1988) .

Two i n t e r e s t i n g r e s u l t s f o l l o w f r o m t he t h e o r e t i c a l s o l u t i on : on o n e ha nd, f o r h e a v i l y o v e r c o n s o l i d a t e d c l a ys , wi t h a Kt c o e f f i c i e n t s e n s i b l y

g r e a t e r t h a n one , t h e s u r f a c e s e t t l e me n t a t t he c e n t r e l i n e i s not a ma x i mum. On t h e o t h e r ha nd, a l l t he s e t t l e me n t p r o f i l e s h a v e a c o mmo n p o i n t a t a d i s t a n c e H f r o m t he c e n t r e l i n e .

Re g a r d i n g h o r i z o n t a l d i s p l a c e me n t s , t he e f f e c t of Kt i s a l s o v e r y i mpo r t a n t . As Kt i nc r e a s e s ,

t he h o r i z o n t a l d i s p l a c e me n t s e x t e n d l a t e r a l l y . Aga i n , a l l t he p r o f i l e s h a v e a c o mmo n p o i n t a t a d i s t a n c e H f r om t he c e n t r e l i ne .

Le e a n d Ro we ( 1988) e x a mi n e d t he e f f e c t of Kt

by u s i n g f i n i t e e l e me n t s . Th e i r r e s u l t s f o r H/ D e qua l t o 2 . 5 ( Fi g. 49) h a v e b e e n p l o t t e d i n F i g u r e 50 i n a n o n - d i me n s i o n a l f o r m a c c o r d i n g t o t he p r o p o s e d c r i t e r i a f o r n o r ma l i z a t i o n . I n t he s a me p l ot , t he a n a l y t i c a l r e s u l t s h a v e b e e n i nc l ude d. By c o mp a r i n g t h e o r e t i c a l a n d n u me r i c a l r e s u l t s , a c e r t a i n a g r e e me n t a n d s ome d i s c r e p a n c i e s c a n be f ound. Le e a n d Ro we ' s ( 1988) r e s u l t s s h o w t he s a me t r e nds i n t he s e n s e t ha t t he s ma l l e r Kt t he

d e e p e r t he s e t t l e me n t t r ough. Al s o , i t c a n be s e e n t ha t Le e a n d Ro we ' s l i ne s ha ve , mo r e or l e s s , a c o mmo n poi n t , wh i c h i s c l o s e r t o t he c e n t r e l i n e t h a n i n t he t h e o r e t i c a l s ol u t i on . Howe ve r , Le e a n d Ro we ' s c u r v e s ma g n i f y t he Kt

e f f e c t . Th i s i s c l e a r l y s e e n a t t he ma x i mu m s e t t l e me n t .

On c e mo r e , t he r e a s o n s f o r t he d i s c r e p a n c i e s a r e no t e n t i r e l y c l e a r . On e of t h e m c o u l d be t ha t t he b o u n d a r i e s i n t h e f i n i t e e l e me n t a n a l y s i s a r e t oo c l o s e t o t he t unne l . I n o r d e r t o t a ke i nt o

a c c o u n t a n d a n a l y z e t he b o u n d a r i e s i n f l u e n c e i n Le e a n d Ro we ' s c a l c u l a t i o n s , Sa g a s e t a ha s c o n s i d e r e d t h e i s o t r o p i c c a s e ( Kt =l ) . Fo r t he l a t e r a l

b o u n d a r i e s , t he me t h o d of s u p e r p o s i t i o n of i ma ge s ha s b e e n us e d. Fo r t he b o t t o m b o u n da r y , Sa g a s e t a ha s ma d e u s e of t he S t e i n b r e n n e r ' s me t h o d by s u b t r a c t i n g t he s e t t l e me n t a t t he r i g i d b a s e f r om t ha t a t t he s u r f a c e . As i t c a n be s e e n i n Fi g u r e 51, t he b o u n d a r i e s i n f l u e n c e i s i mpo r t a n t . The ma x i mu m a n a l y t i c a l s e t t l e me n t ( d o t t e d l i ne ) b e c o me s 0 . 52 i n s t e a d of 0 . 40 ( t he t h e o r e t i c a l v a l u e f o r t he h a l f - s p a c e ) . Le e a nd Ro we ' s ma x i mu m s e t t l e me n t f o r t he i s o t r o p i c c a s e i s 0. 70. Th e r e ma i n i n g d i f f e r e n c e c o u l d be d u e t o t he f a c t t ha t t he e x i s t i n g s t r e s s e s i n t he t h e o r e t i c a l a n a l y s i s a r e a s s u me d t o be u n i f o r m ( a l t h o u g h n o n - h y d r o s - t a t i c ) . I t me a n s t ha t t he v a r i a t i o n of s t r e s s i n t e n s i t y wi t h d e p t h i s not a c c o u n t e d f or . Th i s e f f e c t mu s t be i mp o r t a n t s i n c e t he r e l a x e d s t r e s s a l o n g t he t u n n e l wa l l i s b y no me a n s u n i f o r m i n t h i s p a r t i c u l a r c a s e of a s h a l l o w t u n n e l wi t h a H/ D r a t i o e q u a l t o 2. 5.

Ne v e r t h e l e s s , a p a r t f r o m t he q u a n t i t a t i v e d i f f e r e n c e s d i s c u s s e d a bove , f r o m a q u a l i t a t i v e v i e wp o i n t t he a n a l y t i c a l r e s u l t s a r e i n good a g r e e me n t wi t h Le e a n d Ro we ' s ( 1988) s t a t e me n t t ha t ' t he c o e f f i c i e n t of e a r t h p r e s s u r e a t r e s t s h o u l d be a d e q u a t e l y d e t e r mi n e d i f r e a s o n a b l e p r e d i c t i o n s of s e t t l e me n t a r e t o be ob t a i ne d , s i n c e i t ha s a s i g n i f i c a n t e f f e c t ' .

H o r iz o n ta l D is ta n c e f r o m ( m )

5 10 15 2 0 2 5

Is o t r o p ic K 0 = 1.5

Is o tro p ic K 0 = 0 . 7

Is o tr o p ic K 0 * I . O

--------- Lee and Rowe, 1988

-------- Sagaseta, 1989

TT = 2 . 5

Fi g u r e 50. Co mp a r i s o n of Le e a n d Ro we ' s ( 1988) FE r e s u l t s o n t he i n f l u e n c e of Kt f o r H/ D = 2. 5 wi t h

a n a l y t i c a l o n e s o b t a i n e d by Sa g a s e t a ( 1989) .

2 5 4 5

--------- Lee and R ow e ,1988

----------Sagaseta, 1989

------ — Sagaseta, 1989 , including the boundaries

D

Fi g u r e 51. Ef f e c t of b o u n d a r i e s f o r FE c a l c u l a t i o n s on t he t h e o r e t i c a l ma x i mu m s e t t l e me nt .

5 . 5 . 4 . 2 Co mp r e s s i b i l i t y

To f i n i s h wi t h t h e o r e t i c a l a na l y s e s , t he i n f l u e n c e of s oi l c o mp r e s s i b i l i t y u p o n g r o u n d mo v e me n t s wi l l be c ons i d e r e d . An e x t e n s i o n of Sa g a s e t a 1s ba s i c s o l u t i o n ha s be e n ma d e f o r t he i s o t r o p i c c a s e ( i n t e r ms of b o t h s oi l b e h a v i o u r a nd e x i s t i ng s t r e s s e s ).

Fo r t he po i n t s i nk i n a n i n f i n i t e me d i u m t he p r o b l e m i s s y mme t r i c a bou t t he s i nk. Be c a u s e of t he s ymme t r y , g r o u n d d i s p l a c e me n t s a r e r a di a l .Fo r a p o i n t a t a d i s t a n c e r f r o m t he s i nk, t he r a d i a l a nd t a n g e n t i a l s t r a i n s i n t e r ms of t he d i s p l a c e me n t a r e :

e r = " bu/ dr ( 57a )

€ e = u / r ( 57b)

The n, v o l u me t r i c a nd s h e a r s t r a i n s a r e :

e v o i = 3 u / i r +u / r ( 58a )

= | e r - e 0 1 = f bu / d r - u / r | ( 58b)

Fo r a s i nk, u i s p o s i t i v e ( u/ r > 0) a n d d e c r e a s e s a s r i n c r e a s e s ( &u / d r < 0) . The r e f o r e , | er - e 0| i s e q u a l t o u / r - du / d r .

Now, i n s t e a d of a s s u mi n g e TOl = 0 a d i l a t a n c y e q u a t i o n i s u s e d t o e x p r e s s s oi l c o mp r e s s i b i l i t y :

e v o i = = - s i n p . K^ , ( 59)

wh e r e p d e n o t e s t he a n g l e of d i l a t i on .I t i s n o t e wo r t h y t o p o i n t ou t t ha t s i n c e t he

s t r e s s c h a n g e i s s u c h t ha t t he a v e r a g e p r e s s u r e r e ma i n s c o n s t a n t , h e n c e a n e l a s t i c i s o t r o p i c ma t e r i a l d e f o r ms wi t h o u t v o l u me c h a n g e i r r e s p e c t i ve of t he Po i s s o n ' s r a t i o. Th e c o mp r e s s i b i l i t y c o n s i d e r e d h e r e a c c o u n t s f o r t he v o l u me c h a n g e i n d u c e d by s h e a r a n d i s , t he r e f o r e , of p l a s t i c n a t u r e f o r i s o t r o p i c s oi l s .

By i n t r o d u c i n g ( 58a a n d b) i n t o e q u a t i o n ( 59) , t he d i f f e r e n t i a l e q u a t i o n of d i s p l a c e me n t f o r t he p o i n t s i nk i n a n i n f i n i t e me d i u m f ol l ows :

( 1 - s i n p ) ( d u / d r ) + ( 1 +s i n p ) ( u / r ) = 0 ( 60)

Eq u a t i o n ( 60) c a n a l s o be wr i t t e n :

d u / d r + a p u / r = 0 ( 61)

wh e r e :

a p = ( 1 +s i n p ) / ( 1 - s i np) ( 62)

Fo r d e n s e o r d i l a t a n t s oi l s , p > 0 a n d t h e r e f o r e a p > 1 , wh e r e a s f o r l oos e s oi l s , P < 0 a n d a < 1 .

Eq u a t i o n ( 61) i s e a s i l y i n t e g r a t e d f o r t ne b o u n d a r y c o n d i t i o n u = u 0 f o r r = a ( a d e n o t e s t he s i n k r a di us ) :

u = u 0 ( a/ r ) ° p ( 63)

Fo r p = 0 ( ap = 1) , e q u a t i o n ( 63) i s e q u i v a l e n t t o e q u a t i o n ( 40b) f o r i n c o mp r e s s i b l e ma t e r i a l s .

Th e l os s of g r o u n d a t a d i s t a n c e r f r o m t he s i nk i s :

Vr = 2i r r u = 2 n u 0 r ( a/ r ) ° p ( 64)

At t he s i n k wa l l ( r = a) e q u a t i o n ( 64) gi ve s :

V0 = 2 n u 0a ( 65)

a nd, he nc e :

Vr = V0( a / r ) ° , _ 1 ( 6 6 )

Eq u a t i o n ( 6 6 ) s h o ws h o w t he g r o u n d l os s f o r d e n s e s o i l s v a n i s h e s wi t h t he i n c r e a s i n g d i s t a n c e wh e r e a s i t i n c r e a s e s f or l oos e ma t e r i a l s .

The ma i n d i f f i c u l t y wi t h c o mp r e s s i b l e s o i l s l i e s on a c c o u n t i n g f or t he s u r f a c e e f f e c t wh e n c h a n g i n g f r o m a n i n f i n i t e me d i u m t o a ha l f - s p a c e . On one ha nd, s i n c e t he a b s o l u t e v a l u e of t he d i s t o r t i o n ha s t o be c o n s i d e r e d i n t he d i f f e r e n t i a l e q u a t i o n s r e l a t i n g s t r a i n s t o d i s p l a c e me n t s , i t i s not t r u e f or c o mp r e s s i b l e ma t e r i a l s t ha t e i t h e r a s ymme t r i c , p o s i t i ve , i ma ge c a n c e l s s u r f a c e s h e a r s t r e s s e s or a n e g a t i v e v i r t u a l i ma ge c a n c e l s s u r f a c e n o r ma l s t r e s s e s . Th i s c a n be e a s i l y o b v i a t e d b y a s s u mi n g t ha t s o i l d i l a t e s f or K > 0 a n d c o n t r a c t s f or t < 0. Si n c e t he d i s t o r t i o n o n l y c h a n g e s s i gn f r om t he s u r f a c e u p wa r d s ( i . e . : i n t he v i r t u a l z one ) , t he a b o v e a s s u mp t i o n doe s no t p o s e a n y pr ob l e m.

On t he o t h e r ha nd, s e t t l e me n t s do t a ke p l a c e i n c o mp r e s s i b l e ma t e r i a l s wh e n t r y i n g t o e l i mi n a t e s u r f a c e s h e a r s t r e s s e s . The r e f o r e , s u r f a c e s e t t l e me n t s a r e not t wi c e t h o s e c o r r e s p o n d i n g t o t he i n f i n i t e me d i um, a s i t h a p p e n e d f o r t he c a s e of i n c o mp r e s s i b i l i t y . Some a d d i t i o n a l s e t t l e me n t s de ve l op , s p e c i a l l y c l o s e t o t he c e n t r e l i ne .

An a p p r o x i ma t e s o l u t i o n c o n s i s t s of n e g l e c t i n g t he m, a s s u mi n g t ha t mu l t i p l y i n g by t wo i s s t i l l v a l i d a n d t ha t t he d i f f e r e n c e s wi t h e x a c t s e t t l e me n t s a r e i n c l u d e d i n e x p o n e n t a p . Th e r e s u l t i n g e x p r e s s i o n s f or s u r f a c e d i s p l a c e me n t s a r e :

2 5 4 6

H/ D

»° '1 [ ( X/ D) 2+( H/ D) 2] 0

1 X/ D

4 “ ' 1 [ ( x / D) 2 +( H/ D) 2 ] 0

1

( D/ H) J

( 67b)

( 60)

( 6 7 a )C o r d i n g y H o n s m i r e ( 1 9 7 6 )

whe r e a = ( l +a p ) / 2. Fo r - 3 0 ‘ <p< 30" , 2 / 3 <a < 2.

Eq u a t i o n s ( 67a , b a n d c) c o i n c i d e wi t h t h o s e of t he b a s i c s o l u t i o n f o r a = 1 .

The s e t t l e me n t v o l u me i s o b t a i n e d b y i n t e g r a t i ng t he v e r t i c a l s u r f a c e d i s p l a c e me n t :

V„

2 2(1_a) T( a - i )( H/ D) 2' 1- » '

/ i t r ( a )( 69)

Eq u a t i o n ( 69) c a n be s u b s t i t u t e d wi t h g o o d a p p r o x i ma t i o n by:

— = 4 2(1' 0) ( H/ D) 2<1_a) [ l +f l ( a - 1 ) 2] ( 70)

wh e r e 0 = 1 f o r a > 1 a n d 0 = 3. 5 f o r a < 1.Eq u a t i o n ( 70) r e l a t i n g VB/ V0 t o H/ D ha s be e n

p l o t t e d i n Fi g u r e 52 f o r s e v e r a l v a l u e s of a. I t c a n be s e e n t ha t f o r a c l o s e t o o n e ( i n c o mp r e s s i - b i l i t y ) , a s ma l l i n c r e a s e i n t he a v a l u e g i ve s r i s e t o a d r a s t i c r e d u c t i o n i n t he r a t i o b e t we e n s e t t l e me n t v o l u me a nd g r o u n d l os s . The r e f o r e , c o mp r e s s i b i l i t y ha s a n i mp o r t a n t e f f e c t on h o w t he g r o u n d l os s i s t r a n s mi t t e d o u t s i d e t he t u nne l a nd h o w i t i s r e f l e c t e d on s u r f a c e s e t t l e me n t s .

Fi g u r e 52 i n c l u d e s r e a l da t a f r om Co r d i n g a nd Ha n s mi r e ( 1975) , wh o d e s c r i b e d t he me c h a n i s ms of t r a n s mi s s i o n of g r o u n d l os s t o t he g r o u n d s u r f a c e as f ol l ows : ' For s ma l l v o l u me s l os t i n t o t he t unne l a l mo s t no v o l u me of s u r f a c e s e t t l e me n t s t a ke s pl a c e . Fo r l a r ge v o l u me s l os t i n t o t he t unne l t he s h e a r p l a n e s a p p r o a c h t he g r o u n d s u r f a c e a n d l i t t l e f u r t h e r v o l u me i n c r e a s e de ve l ops . Fu r t h e r d i s p l a c e me n t i n t o t he t unne l t he n r e s u l t s i n e q u a l i n c r e a s e s i n t he v o l u me l os t a t t he t unne l a nd a t t he s u r f a c e ' . I t me a n s t ha t d i l a t i on o r c o mp r e s s i o n due t o d i s t o r t i o n p l a y s a n e s s e n t i a l r ol e . Wh e n l a r ge d e f o r ma t i o n s oc c u r , t he r e i s no v o l u me c h a n g e a n d g r o u n d l os s i n c r e a s e s e q u a l s e t t l e me n t v o l u me i nc r e a s e s .

I t c a n be s e e n i n Fi g u r e 52 t ha t a g o o d a g r e e me n t wi t h r e a l da t a i s f o u n d f o r a v a l u e s of t he o r de r of 1 . 1 t o 1 . 2 , i n a z one wh e r e t he i n f l u e n c e of c o mp r e s s i b i l i t y i s mo s t i mpo r t a n t .

The s a me c o n c l u s i o n i s a c h i e v e d wh e n c o n s i d e r i ng s e t t l e me n t s a b o v e t he c r o wn i n r e l a t i o n t o t he ma x i mu m s u r f a c e s e t t l e me n t . Fo r a po i n t a t a d i s t a n c e d a b o v e t he t u n n e l c r o wn ( Fi gu r e 53) , t he ba s i c s o l u t i o n gi ve s :

= 1

( C/ D- d / D

( C/ D+l / 2 5

( 71)

Co n t i n u o u s l i ne s r e p r e s e n t e x p r e s s i o n ( 71) f or s ome v a l u e s of d / D a n d c o r r e s p o n d t o i n c o mp r e s s i b l e ma t e r i a l s . Th e d o t t e d l i ne g i v e s r e l a t i v e s e t t l e me n t a t t he c r o wn ( d/ D = 0) f or a = 2 a s p r o v i d e d b y t he a p p r o x i ma t e s o l u t i o n f or c o mp r e s s i b l e s oi l s . Fi e l d d a t a on c r o wn s e t t l e me n t s f r om Wa r d a n d Pe n d e r ( 1981) f o r b o t h c o h e s i v e a nd c o h e s i o n l e s s s o i l s h a v e b e e n a l s o p l o t t e d . The be s t a g r e e me n t wi t h r e a l d a t a i s o b t a i n e d f o r a c l os e t o one . No t e t ha t f o r a=2 a n d C/ D>2 , no

V o

Fi g u r e 52. Ra t i o b e t we e n s e t t l e me n t v o l u me a nd g r o u n d l os s f o r c o mp r e s s i b l e s o i l s a s a f u n c t i o n of t u n n e l de pt h.

S e t t l e m e n t

■ *- o

CD

Fi g u r e 53. Re l a t i v e s e t t l e me n t s f o r p o i n t s a b o v e t he c r o wn f o r d i f f e r e n t t u n n e l de p t hs . Ef f e c t of c o mp r e s s i b i l i t y .

s e t t l e me n t t a ke s p l a c e a t s ur f a c e .To f i n i s h wi t h t he a n a l y t i c a l s o l u t i o n i t mus t

be s a i d t ha t a t u n n e l i s no t a p o i n t s i nk wi t h c o n c e n t r a t e d g r o u n d l os s but a l os s of g r o u n d d i s t r i b u t e d a l o n g t he c a v i t y wa l l . Gr o u n d mo v e me n t s c a n be o b t a i n e d by i n t e g r a t i n g t he e f f e c t s of d i f f e r e n t i a l g r o u n d l os s e s a l o n g t he wa l l . The p r o b l e m h e r e i s t ha t t he g r o u n d l os s d i s t r i b u t i o n i s no t u n i f o r m a n d i s unknown. I n t he c a s e of s h i e l d d r i v e n t unne l s , i t c a n be g i v e n by t he c l e a r a n c e b e t we e n s h i e l d a n d l i ni g. Ne v e r t h e l e s s , t he s o l u t i o n f o r t he p o i n t s i n k i s a g o o d a p p r o x i ma t i o n t o t he r e a l t unne l e x c a v a t i o n p r obl e m.

2 5 4 7

6 LI NI NG DESI GNH o r i z o n ta l

5

( m )D i s t a n c e f r o m

10 15

Fi g u r e 54. Ef f e c t of u n d r a i n e d s t r e n g t h a n i s o t r o py on s e t t l e me n t t r o u g h ( Le e a n d Rowe , 1988) .

I s o t r o p ic

C - A n is o t r o p y

M - A n is o t r o p y

K - A n is o t r o p y

H o r i z o n t a l D is ta n c e f r o m t ( m )

5 10 15 2 0 2 5

U n lin e d T u n n e l

K0 » 1 . 0

E la s t i c a l l y I s o t r o p ic

C - A n i s o t r o p y

I s o t r o p ic

M - A n is o t r o p y

K - A n i s o t r o p y

5 . 5 . 5 Un d r a i n e d s t r e n g t h a n i s o t r o p y i n f l u e n c e

The i n f l u e n c e of u n d r a i n e d s t r e n g t h a n i s o t r o p y ha s b e e n i n v e s t i g a t e d by Le e a n d Ro we ( 1988) by u s i n g f i n i t e e l e me n t t e c hn i que s . Fi g u r e 54 s hows t he r e s u l t s of t he a n a l y s i s f o r t h r e e b a s i c c a t e g o r i e s of s t r e n g t h a n i s o t r o p y , t o g e t h e r wi t h t he s e t t l e me n t t r o u g h f o r i s o t r o p i c s t r e n g t h ( t he a s s u me d s t a b i l i t y r a t i o i s N = 4) .

C- a n i s o t r o p y i s t ha t i n wh i c h t he h o r i z o n t a l s t r e n g t h i s ma xi mum. I t i s u s u a l l y r e l a t e d t o h e a v i l y o v e r - c o n s o l i d a t e d c l a ys .

M- a n i s o t r o p y i s t ha t i n wh i c h t he h o r i z o n t a l s t r e n g t h i s mi n i mum. I t i s q u i t e o f t e n o b s e r v e d f r om n o r ma l l y c o n s o l i d a t e d c l a ys .

K- a n i s o t r o p y c o r r e s p o n d s t o t he c a s e wh e r e t he s ma l l e s t s t r e n g t h o c c u r s a t a n a n g l e of 45' . Th i s s p e c i a l t ype of a n i s o t r o p y i s mo r e l i ke l y e n c o u n t e r e d i n ma c r o - a n i s o t r o p i c s oi l s , f or i n s t a nc e l a mi na t e d , s t r a t i f i e d o r ma r i n e c l a ys .

The e f f e c t s of e i t h e r t ype C o r M- a n i s o t r o p y on t he s e t t l e me n t p r o f i l e a r e v e r y mode s t ; howe ve r , t ype K- a n i s o t r o p y ha s a s i g n i f i c a n t e f f e c t on t he ma g n i t u d e a n d s h a p e of t he s e t t l e me n t t r ough. Ob v i o u s l y , t he s ma l l e r t he s t a b i l i t y r a t i o t he l e s s t he i n f l u e n c e of s t r e n g t h a n i s o t r o p y , a s s hown by Le e a nd Ro we ' s ( 1988) c a l c u l a t i o n s .

Le e a nd Ro we h a v e a l s o e x a mi n e d t he e f f e c t of u n d r a i n e d s t r e n t g h a n i s o t r o p y wh e n c o mb i n e d wi t h e l a s t i c a n i s o t r o p y , t u n n e l s u p p o r t p r e s s u r e a nd ' ga p p a r a me t e r ' . Fo r a l i ne d t unne l wi t h a mo d e r a t e v a l u e of ' ga p' , t he d e t a i l e d c o n s i d e r a t i o n of s t r e n g t h a n i s o t r o p y ma y not be ne c e s s a r y .Si n c e e f f e c t i v e s t r e s s p a r a me t e r s t e n d t o e x h i b i t o n l y mo d e s t a n i s o t r o p y a s i mi l a r a n a l y s i s wo u l d s h o w l i t t l e e f f e c t on t unne l p e r f o r ma nc e .

Li n i n g d e s i g n wa s b a s e d i n t he p a s t on t h e o r i e s t ha t p r o v i d e d e a r t h p r e s s u r e s ma g n i t u d e a nd s ha pe i r r e s p e c t i v e of t he l i n i n g s t i f f n e s s a nd p l a c i n g t i me , a s u mma r y of wh i c h c a n be f o u n d i n Sz é c h y ( 1 9 6 7 ) . But t he d e s i g n of a t u n n e l l i n i n g i s e s s e n t i a l l y a s o i l - s t r u c t u r e i n t e r a c t i o n p r obl e m, wh e r e t i me p l a y s a f u n d a me n t a l r o l e a s we l l a s t he f l e x i b i l i t y of t he l i n i n g r e l a t i v e t o t he s u r r o u n d i n g s oi l .

A g o o d q u a l i t a t i v e d e s c r i p t i o n of t he i n t e r a c t i on d e v e l o p me n t i s g i v e n by Lo mb a r d i ( 1 9 7 4 ) , wh o s t a t e s t ha t ' unl e s s t he l i n i n g ha s b e e n p l a c e d v e r y c l o s e t o t he f a c e a nd wi t h i n t he h e a d i n g z one of i n f l ue nc e , a t e mp o r a r y s u p p o r t o r a p e r ma n e n t l i n i n g wi l l o n l y be s u b j e c t e d t o e a r t h p r e s s u r e s i f t he g r o u n d ha s a v i s c o u s b e h a v i o u r a n d i s , t he r e f o r e , c a p a b l e of u n d e r g o i n g s l o w d e f o r ma t i o n s . Fo r t he l i n i n g t o h a v e a s t r u c t u r a l f u n c t i o n i t i s n e c e s s a r y f o r t he p l a c e me n t g a p t o be l e s s t ha n t he s oi l d e f o r ma t i o n s t ha t mi g h t be e x p e c t e d a f t e r wa r d s ' .

The s i mp l e s t wa y t o c o n s i d e r t he i n t e r a c t i o n b e t we e n l i n i n g a nd g r o u n d l i e s on a s s u mi n g a Wi n k l e r mo d e l f o r s oi l be ha v i ou r . Th e f u n d a me n t a l p a r a me t e r f or t he a p p l i c a t i o n of t h i s me t h o d i s t he c o e f f i c i e n t of s u b g r a d e r e a c t i on , wh i c h i s not a n i n t r i n s i c p a r a me t e r s i nc e i t d e p e n d s on t he l o a d i n g a r e a . Some r e l a t i o n s h i p s b e t we e n t he mo d u l u s of d e f o r ma t i o n of t he s oi l a n d t he c o e f f i c i e n t of s u b g r a d e r e a c t i o n wh i c h a c c o u n t f or t he s i z e of t he l o a d i n g a r e a h a v e b e e n p r opos e d . Howe v e r , t h e y do no t t a ke i n t o a c c o u n t t he e f f e c t of l i n i n g c u r v a t u r e o r t he r e l a t i v e f l e x i b i l i t y of t he l i ni ng. An o t h e r i mp o r t a n t d r a wb a c k of t he me t h o d i s t ha t , b e f o r e t he s oi l s t r u c t u r e i n t e r a c t i o n t a ke s p l a c e , s ome e mp i r i c a l i nput l oa ds ha v e t o be a p p l i e d on t he l i ni ng.

The c o n v e r g e n c e - c o n f i n e me n t me t h o d of a n a l y s i s of t u n n e l s i s l i k e l y t he be s t t o d e s c r i b e a nd u n d e r s t a n d t he i n t e r a c t i o n b e t we e n l i n i n g a nd s u r r o u n d i n g g r ound. I t i s r e l a t i v e l y s i mp l e a nd a l l o ws e a s y p a r a me t r i c s e n s i t i v i t y a n a l y s i s t o be c a r r i e d out . Howe v e r , i t s s i mp l i f y i n g a s s u mp t i o n s ma k e t he me t h o d a p p l i c a b l e o n l y t o d e e p t u nne l s a nd, t he r e f o r e , t he u s e of t he c o n v e r g e n c e - c o n f i - n e me n t me t h o d i n s h a l l o w t u n n e l s i s o p e n t o q u e s t i on ( Ei s e n s t e i n a nd Br a nc o , 1 9 8 5 ) .

On l y one p a p e r ( Duf f a u t a n d Pa n e t , 1 9 8 9 ) on s e l e c t i o n of d e s i g n p a r a me t e r s f o r l i n i n g d e s i g n ha s be e n s u b mi t t e d t o t h i s Conf e r e n c e . I t r e f e r s t o t he a p p l i c a t i o n of t he c o n v e r g e n c e - c o n f i n e me n t me t h o d ( o r i g i n a l l y d e r i v e d f o r d e e p t unne l s ) t o t he c a s e of s h a l l o w t u n n e l s i n s oi l s . I t t a ke s i nt o a c c o u n t t he e f f e c t s of h e t e r o g e n e i t y , a n i s o t r o p y a nd c o n s o l i d a t i o n p r o p e r t i e s of t he s oi l wh e r e t he t u n n e l i s c ons t r u c t e d .

The f u n d a me n t a l p a r a me t e r s a r e t he r e l a x a t i o n t i me T a n d t he l os s of c o n f i n e me n t x . Th e f i r s t one i s d e d u c e d f r om c o n v e r g e n c e r e a d i n g s a nd t he us e of e mp i r i c a l f o r mu l a e s h o wi n g t he e v o l u t i o n of c o n v e r g e n c e wi t h r e a l t i me s uc h a s t he f o l l o wi ng one ( Sul e m e t a l , 1 9 8 7 ) :

C ( t ) = A { 1—[ T/ ( t +T) ] "} ( 71)

Fi g u r e 55 i l l u s t r a t e s t he a p p l i c a t i o n of t he me t hod . Th e c o n f i n e me n t l os s d e p e n d s on t he d i s t a n c e t o t he t unne l f a c e a n d on t he e v o l u t i o n of t he p l a s t i c z one . Fo r c o h e s i v e s oi l s , t h i s l a s t d e p e n d e n c y i s e x p r e s s e d t h r o u g h t he s t a b i l i t y r a t i o. Fo r v e r y s h a l l o w t unne l s , x ha s a n u p p e r l i mi t c o r r e s p o n d i n g t o t he c a s e wh e n t he p l a s t i c z one r e a c h e s t he g r o u n d s u r f a c e .

2 5 4 8

Fi g u r e 55. Fu n d a me n t a l s of t he a p p l i c a t i o n of t he c o n v e r g e n c e - c o n f i n e me n t me t h o d of l i n i n g d e s i g n ( Duf f a u t a n d Pa n e t , 1989) .

Fo r t he p u r p o s e of p a r a me t r i c s e n s i t i v i t y a n a l ys i s , a n d a l s o f o r p r e d i c t i o n , t he us e of n u me r i c a l me t h o d s i n c o n j u n c t i o n wi t h b a c k - a n a l y s i s p r o c e d u r e s c o n s t i t u t e s t he mos t f r u i t f u l wa y. Mo n i t o r i n g of l i n i n g s e c t i ons , t o me a s u r e a t t he e a r l y s t a g e s of t u n n e l c o n s t r u c t i o n t he ma g n i t u d e a nd e v o l u t i o n of g r o u n d p r e s s u r e s , a l l o ws t he f i ne t u n i n g of t he g o v e r n i n g p a r a me t e r s t o be d o n e .

7 CONCLUSI ONS

The s u b j e c t of d e s i g n p a r a me t e r s f o r u n d e r g r o u n d c o n s t r u c t i o n ha s b e e n e s s e n t i a l l y c o n s i d e r e d f r om t he p o i n t of v i e w of p a r a me t r i c s e ns i t i v i t y .

Fi r s t l y , g r o u n d b e h a v i o u r i d e a l i z a t i o n ha s be e n d i s c us s e d . The f u n d a me n t a l p a r a me t e r s t o d e s c r i b e t he e x i s t i n g s t r e s s e s a n d t he s h e a r s t r e n g t h mo b i l i z a t i o n p r i o r t o e x c a v a t i o n a r e t he l a t e r a l e a r t h p r e s s u r e c o e f f i c i e n t , t he i n i t i a l s he a r s t r e s s r a t i o a nd t he o v e r l o a d f a c t o r .

De f o r ma b i l i t y , a n d s p e c i a l l y e l a s t i c c r o s s a n i s o t r o p y , ha s r e c e i v e d g r e a t a t t e n t i o n . A r e c e n t l i ne of r e s e a r c h c o n s i s t s of a n a l y z i n g t he i n f l u e n c e of e l a s t i c a n i s o t r o p y on t he g r o u n d d e f o r ma t i o n s i n d u c e d by t u n n e l e x c a va t i on .

The s t a b i l i t y of a n u n l i n e d t u n n e l u n d e r p l a n e s t r a i n c o n d i t i o n s i s we l l d e f i n e d b y u s i n g t he c o n c e p t of t he s t a b i l i t y r a t i o. Ho we v e r , o wi n g t o t he t h r e e - d i me n s i o n a l c h a r a c t e r of t he he a d i ng , t he l o a d f a c t o r s h o u l d be u s e d i n s t e a d f o r t he a n a l y s i s of t he h e a d i n g a n d f a c e s t a b i l i t y .

The ma i n e mp h a s i s ha s b e e n g i v e n t o t he s u b j e c t of g r o u n d mo v e me n t s . Fr o m t h i s v i e wp o i n t , t he g r o u n d l os s i s t he o v e r r i d i n g i nput d e s i g n p a r a me t e r f o r u n d e r g r o u n d c o n s t r u c t i o n . A n e w s e mi - e mp i r i c a l f o r mu l a t o e s t i ma t e t he g r o u n d l os s ha s be e n p r opos e d . I t a c c o u n t s not o n l y f o r t he o v e r l oa d f a c t o r but a l s o f o r t he r a t i o b e t we e n u n d r a i n e d s h e a r s t r e n g t h a n d d e f o r ma t i o n mo d u l u s a s we l l a s f o r t he d e p t h t o d i a me t e r r a t i o ( whi c h s e e ms t o h a v e a n i mp o r t a n t e f f e c t ) .

Fi t t i n g of t he s e t t l e me n t t r o u g h i s be s t a c h i e v e d by t he u s e of t he e r r o r f u n c t i o n e mp i r i c a l a p p r o a c h .

Ap p l i c a t i o n of n u me r i c a l me t h o d s t o t he p r o b l e m of g r o u n d d i s p l a c e me n t s a s s o c i a t e d wi t h t u n n e l l i ng i s q u i t e a p p r o p r i a t e . Pr e d i c t i o n of g r o u n d mo v e me n t s c a n be v e r y a c c u r a t e i f a p r o c e d u r e of f i ne t u n i n g of t he d e s i g n p a r a me t e r s i s us e d. Mo n i t o r i n g a n d b a c k - a n a l y s i s of d e f o r ma t i o n me a s u r e me n t s a t t he e a r l y s t a g e s of t u n n e l c o n s t r u c t i on a l l o ws t he i d e n t i f i c a t i o n of t he p a r a me t e r s .

Wh e n u s i n g me t h o d s of d i s c r e t i z a t i o n of t he c o n t i n u u m, s p e c i a l a t t e n t i o n s h o u l d be g i v e n t o

t he i n f l u e n c e of t he b o u n d a r i e s l oc a t i on , p a r t i c u l a r l y t ha t of t he b o t t o m bounda r y .

Pa r a me t r i c s e n s i t i v i t y a n a l y s e s c o n s t i t u t e one of t he mo s t i n t e r e s t i n g a p p l i c a t i o n s of n u me r i c a l me t h o d s t o t he p r o b l e m of g r o u n d mo v e me n t s , s i nc e t h e y c a n be c a r r i e d out q u i t e e a s i l y .

Th e a n a l y t i c a l a p p r o a c h t o t he g r o u n d d i s p l a c e me n t s c a u s e d by u n d e r g r o u n d e x c a v a t i o n ha s be e n d e s c r i b e d wi t h de t a i l . I t wa s o r i g i n a l l y d e r i v e d f o r i n c o mp r e s i b l e , i s o t r op i c , e l a s t i c s o i l s a n d h a s j us t b e e n e x t e n d e d f o r n o n - h y d r o s t a t i c e x i s t i ng s t r e s s e s , e l a s t i c a n i s o t r o p y a n d c o mp r e s s i b l e s o i l s .

Th e t h e o r e t i c a l s o l u t i o n a l l o ws i mme d i a t e p a r a me t r i c s e n s i t i v i t y a na l ys i s . The i n f l u e n c e of t he d e p t h t o d i a me t e r r a t i o ha s b e e n s e e n t o be r e l e va n t wi t h r e s p e c t t o t he s h a p e of t he s e t t l e me n t t r ough. Ho we v e r , t he i n f l u e n c e of a n i s o t r o p y s e e ms t o b e mo d e s t ( wi t h i n t he n o r ma l r a n g e of a n i s o t r o p y e x h i b i t e d by s oi l s ) . Th e l a t e r a l e a r t h p r e s s u r e c o e f f i c i e n t h a s a n i mp o r t a n t e f f e c t on b o t h t he s e t t l e me n t p r o f i l e a n d t he h o r i z o n t a l s u r f a c e d i s p l a c e me n t d i s t r i b u t i o n . So i l c o mp r e s s i b i l i t y s h o ws a c o n s i d e r a b l e i n f l u e n c e on t he t r a n s mi s s i o n of g r o u n d l os s t o t he g r o u n d s u r f a ce . Fo r d e n s e o r d i l a t a n t s oi l s , t he r a t i o b e t we e n s e t t l e me n t v o l u me a n d g r o u n d l os s c a n be s e n s i b l y l e s s t ha n one .

ACKNOWLEDGEMENTS

Th e a u t h o r s g r a t e f u l l y a c k n o wl e d g e t he s u g g e s t i ons r e c e i v e d f r o m t he Di s c u s s i o n Le a d e r Dr . R. Ma i r d u r i n g t he p r e p a r a t i o n of t he p r e s e n t Ge n e r a l Re po r t . Th e y wi s h a l s o t o e x p r e s s t h e i r g r a t i t u d e t o Mr . L. Or t uf l o f o r h i s h e l p a nd c a r e f u l r e a d i n g of t he ma n u s c r i p t .

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