A P P L IC A T IO N O F CONTINUUM MECHANICS T O C O M P A C T I O N IN T I L L A B L E SO ILS by G len Edw in Vanden B e r g AN A B S T R A C T S u b m i l l e d to t h e S c h o o l f o r A d v a n c e d G r a d u a l e S t u d i e s of M i c h i g a n S t a t e U n i v e r s i t y of A g r i c u l t u r e and A p p l i e d S c i e n c e in p a r t i a l f u l f i l l m e n t of t h e r e q u i r e m e n t s f o r th e d e g r e e of DOCTOR OF PH ILO SO PH Y D e p a r t m e n t of A g r i c u l t u r a l E n g i n e e r i n g Y e a r 1958 ABSTRACT T o s t u d y t h e c o m p a c t i o n p r o b l e m , t h e r e s u l t i n g c h a n g e in s o i l c o m p a c t i o n at e a c h point c a u s e d by a n a p p l i e d l o a d n e e d s to b e d e t e r ­ m ined, T h i s c a n b e a c c o m p l i s h e d if a s u i t a b l e m a t h e m a t i c a l m o d e l of s o i l is a v a i l a b l e , s u c h a s th e T h e o r y of E l a s t i c i t y i s f o r m e l a l s . U n f o r t u n a t e l y , s u c h a m o d e ] f o r s o i l s d o e s not e x i s t at t h i s t i m e . T h e d e v e l o p m e n t of a s u i t a b l e s o i l m e c h a n i c s r e q u i r e s s p e c i f y i n g t h e f o r c e s a c l i n g on a s m a l l v o l u m e e l e m e n t of t h e s o i l an d s p e c i f y i n g t h e d e f o r m a t i o n of t h e v o l u m e e l e m e n t r e s u l t i n g f r o m t h e applied fo rc e s. T h e m e c h a n i c s of Ihe c o n l i n u u m p r e s e n t s a r i g o r o u s m a t h e m a t i c a l m e t h o d f o r s p e c i f y i n g b olh t h e f o r c e s an d th e d e f o r m a t i o n : t h e f i r s t r e s u l l s in a s t r e s s t e n s o r a n d t h e s e c o n d in a s t r a i n l e n s o r . T h e d e s i r e d m e c h a n i c s w o u l d r e s u l t if t h e r e l a t i o n s h i p b e t w e e n Ihe two t e n s o r s could be d e te rm in e d . B e c a u s e of t h e c o m p l e x i t y of t h e s o i l , t h e d e t e r m i n a t i o n of t h e a b o v e r e l a t i o n s h i p d o e s not s e e m f e a s i b l e at the present, tim e. A s i m p l i f i c a t i o n of t h e p r o b l e m i s p o s s i b l e by i g n o r i n g t h e s h e a r i n g d e f o r m a t i o n s an d r i g i d b o d y r o t a t i o n . The volum e change or v o l u m e s t r a i n c a n b e e x p r e s s e d as a c h a n g e in b u l k d e n s i t y of t h e v o l u m e e l e m e n l i n d e p e n d e n t of t h e s h e a r i n g a n d r o t a t i o n . A stress- c o m p a c t i o n r e l a t i o n s h i p f o r s o i l s c a n t h u s b e d e v e l o p e d by r e l a t i n g t h e s t r e s s t e n s o r to b u l k d e n s i t y . T h e o re tic a l c o n sid e ra tio n s suggest t h a t t h e m e a n n o r m a l s t r e s s , an i n v a r i a n l of the s t r e s s t e n s o r , c o n t r o l s vo lu m e change. An e x p e r i m e n t w a s d e s i g n e d to t e s t t h i s h y p o t h e s i s by iii m e a s u r i n g t h e s l r e s s l e n s o r a n d t h e b u l k d e n s i t y ‘a t t h e s a m e p o i n t ’ in t h e s o i l w h i l e t h e s o i l w a s s u b j e c t e d to s t a t i c l o a d s of v a r i o u s m agnitudes, A l a b o r a t o r y m e th o d fo r t e s tin g the h y p o th e s is u s in g tr i a x i a l a p p a r a tu s w as also dev elo p ed which p e r m i t s l a b o r a t o r y c o n tr o l of t h e s o i l a n d a p p l i e d l o a d s . B e c a u s e of i t s s u i t a b l e f a c i l i t i e s , , t h e N a t i o n a l T i l l a g e M a c h i n e r y L a b o r a t o r y , Auburn., A l a b a m a , w a s c h o s e n f o r t h e t e s t i n g p r o g r a m . T h r e e different soil types - - sandy loam , silty clay loam , c l a y - - w e r e tested. L i m i t e d t i m e al t h e L a b o r a t o r y p e r m i t t e d o n ly p r e l i m i n a r y s tu d i e s with the t r i a x i a l a p p a r a tu s . T h e t i m e - c o n s u m i n g c a l c u l a t i o n s of e v a l u a t i n g i n v a r i a n t s of t h e s t r e s s t e n s o r b e s id e s th e m e a n n o r m a l s t r e s s w e r e re d u c e d by u s in g M I S T I C , an e l e c t r o n i c d i g i t a l c o m p u t e r at M i c h i g a n S t a t e U n i v e r s i t y . B e c a u s e of l a r g e v a r i a t i o n s in t h e d a t a a n d s m a l l d i f f e r e n c e in s l r e s s s t a t e s o b ta in e d , a s i m p l e , r a p id , g r a p h ic a l m eth o d of e s t i m a t i n g the s t a n d a r d e r r o r w a s u s e d . T h e d a t a i n d i c a t e d t h a t of f o u r i n v a r i a n t s of t h e s t r e s s t e n s o r in v e s tig a te d , the m e a n n o r m a l s t r e s s w as the best i n v a r i a n t . t o r e l a t e to b u l k d e n s i t y . H o w e v e r , il c o u l d not b e c o n c l u d e d t h a t o t h e r i n v a r i a n t s of i h e s t r e s s t e n s o r w e r e not r e l a t e d to b u l k density. T h i s s u p p o r t s t h e h y p o t h e s i s that m e a n n o r m a l s t r e s s is r e l a t e d to b u l k d e n s i t y , but d o e s not p r o v e it. O l h e r c o n c l u s i o n s f r o m I h e d a t a w e r e a s f o ll o w s : a) t r i a x i a l a p p a r a t u s c a n b e u s e d to s t u d y s t r e s s - b u l k d e n s i t y r e l a t i o n s h i p s in t i l l a b l e s o i l s ; t r i a x i a l d a t a in th e s a n d y l o a m s o i l a g r e e d w i t h f i e l d d a t a of t h e s a m e s o i l : t h e a g r e e m e n t v e r i f i e s b o th m e t h o d s of t e s t i n g ; b) t h e d i s t r i b u t i o n of m e a n n o r m a l s t r e s s a p p e a r s to be i n d e p e n d e n t of s o i l t y p e a n d d e t e r m i n e d o n ly by t h e g e o m e t r y of l o a d i n g ; r ) t h e m e a n n o r m a l s t r e s s - b u l k d e n s i t y r e l a t i o n s h i p a p p e a r s to b e e x p o n e n t i a l f o r a ll s o i l s s t u d i e d ; d) s t r e s s d i s t r i b u t i o n s p r e d i c t e d by t h e T h e o r y of E l a s t i c i t y d id not a g r e e w i t h m e a s u r e d d i s t r i b u t i o n s . APPLICATION OF CONTINUUM MECHANICS TO COMPACTION IN TILLABLE SOILS by G len Edw in Vanden B e r g A THESIS S u b m i t t e d to t h e S c h o o l f o r A d v a n c e d G r a d u a t e S t u d i e s of M i c h i g a n S t a t e U n i v e r s i t y of A g r i c u l t u r e an d A p p l i e d S c i e n c e i n p a r t i a l f u l f i l l m e n t of t h e r e q u i r e m e n t s f o r t h e d e g r e e of DOCTOR OF PHILOSOPHY D e p a r t m e n t of A g r i c u l t u r a l E n g i n e e r i n g Y e a r 1958 ProQuest Number: 10008564 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10008564 Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 ACKNOWLEDGEMENTS T h e au th o r w i s h e s to e x p r e s s h i s s i n c e r e a p p r e c ia t io n to th e fo llo w in g : D o c t o r A. W. F a r r a ll, fo r p r o v id in g an a s s i s la n t s h ip and th e n e c e s s a r y fu nd s to c a r r y out the in v e s t i g a t i o n s ; The g u id a n c e c o m m i t t e e c o m p o s e d of D o c t o r W. F . B u c h e l e , (c h a ir m a n ),D o c t o r A. E . E r i c k s o n , D o c t o r L u E 0 M a lv e r n , D o c t o r D 0J. M o n t g o m e r y and D o c t o r E .A . N o r d h a u s, f o r t h e ir e n c o u r a g e m e n t and a ssista n c e . T h e a u th o r is p a r t ic u l a r l y in d eb ted to D o c t o r B u c h e l e f o r h i s c o n tin u o u s g u id a n c e throughout th e p r o g r a m and to D o c t o r M a lv e r n f o r h i s m a n y h o u r s in c o n s u lta tio n ; D o c t o r W. M, C a r le t o n and P r o f e s s o r H e F. M c C o lly , f o r t h e ir l e a d e r s h i p d u r in g th e o r g a n iz a t io n of th e p r o g r a m of study; D o c t o r A. W. C o o p e r and th e s ta f f of the N a tio n a l T i l l a g e M a c h in e r y L a b o r a t o r y , A uburn, A la b a m a , f o r t h e ir v i t a l c o n tr ib u tio n to th e i n v e s t ig a t io n s ; P h y l l i s , m y w i f e , f o r h e r faith and e n c o u r a g e m e n t, and a s s i s t a n c e in p r e p a r a t io n of th e m a n u s c r ip t . Glen Edw in V anden B e r g C a n d i d a t e f o r t h e d e g r e e of D o c t o r of P h i l o s o p h y F in a l exam ination: J u n e 24 , 1958; 1 : 1 5 - 3 : 1 5 R o o m 218 , A g r i c u l t u r a l E n g i n e e r i n g D e p a r t m e n t Dis sertatio n : A p p l i c a t i o n of C o n t i n u u m M e c h a n i c s to C o m p a c t i o n in T illa b le Soils O u t l i n e of S t u d i e s : M a jo r Subject M in o r Subject s A g ricultural Engineering M a t h e m a t ic s Applied M e c h a n ic s B io g ra p h ic a l Item s: B o r n A u g u s t 12, 1930, V o lg a , S o u t h D a k o t a U n d e r g r a d u a t e s t u d i e s at S o u th D a k o t a S t a t e C o l l e g e , B r o o k i n g s , 1 9 4 8 - 1 9 5 2 (B.S. in A g r i c u l t u r a l E n g i n e e r i n g ) G r a d u a t e s t u d i e s at M i c h i g a n S t a t e U n i v e r s i t y , E a s t L a n s i n g , 1 9 5 5 - 1 9 5 6 (M.S. in A g r i c u l t u r a l E n g i n e e r i n g ) E xperience: I n te r m e d ia te s p e e d r a d io o p e r a t o r (In tern atio n al M o r s e Code) a n d T r o o p I n f o r m a t i o n and E d u c a t i o n NCO in U .S . A r m y , 195 21954; G r a d u a t e R e s e a r c h A s s i s t a n t , M ic h . S t a t e U n i v . 955-1 958 M em ber: A m e r i c a n S o c i e t y of A g r i c u l t u r a l E n g i n e e r s ; P h i K a p p a P h i ; P i Mu E p s i l o n ; S i g m a P i S i g m a ; S o c i e t y of th e S i g m a Xi; A l p h a Z e t a TABLE OF CONTENTS I N T R O D U C T I O N ................................................................. 1 R E V I E W O F L I T E R A T U R E ......................................... 4 T H EO R Y - M ECHANICS O F THE CO NTINUUM 8 F I E L D T E S T S ....................................................................... 24 T h e o r y ............................................................................. 24 I n d i c a t i n g v o l u m e t e r ................................................ 29 A p p a r a t u s ....................................................................... 31 P r o c e d u r e ........................................................................ 31 R e s u l t s ............................................................................. 33 L A B O R A T O R Y T E S T S ................................................... 60 P r o c e d u r e .................................................................... 60 R e s u l t s .......................................................................... 62 C O N C L U S I O N S ................................................................. 66 S U G G E S T I O N S F O R F U R T H E R S T U D Y .............. 67 R E F E R E N C E S ..................................................................... 68 A PPEN D IX 71 LIST OF FIG U RES F igure 1 Page S t r e s s v e c t o r on a p l a n e .............................................................. 10 S t r e s s e s o n a v o l u m e e l e m e n t .......................................... 12a 3 P r i n c i p a l s t r e s s e s on a v o l u m e e l e m e n t ........................ 15 4 P l a n e of s y m m e t r y u n d e r a c i r c u l a r l o a d .................... 25 5 S t r e s s e s on v o l u m e e l e m e n t in 6 C ell orientation 7 I n d i c a t i n g v o l u m e t e r ................................. . . .................................. 32 8 M e t h o d of w e i g h i n g in w a t e r ....................................................... 32 9 M e t h o d of l o a d i n g ............................................................................. 32 10 G e n e r a l v i e w of t e s t e q u i p m e n t ............................................. 34 11 C ell o rientation . ......................................................................... 34 12 S t a n d a r d e r r o r a n d a v e r a g e of m e a n s t r e s s v e r s u s b u l k d e n s i t y ................................................................. 39 M e a n n o r m a l s t r e s s v e r s u s b u l k d e n s i t y in H i w a s s e e s o i l ............................................................................ 41 M e a n n o r m a l s t r e s s v e r s u s b u l k d e n s i t y in L l o y d s o i l ................................................................................... 42 M e a n n o r m a l s t r e s s v e r s u s b u l k d e n s i t y in D e c a t u r s o i l .............................................................................. 43 M a x im u m n o r m a l s t r e s s v e r s u s bulk d e n s ity in H i w a s s e e s o i l .................................................................... 44 M ax im u m n o r m a l s t r e s s v e r s u s bulk d e n s ity in L l o y d s o i l ............................................................. ... 45 M a x im u m n o r m a l s t r e s s v e r s u s bulk d e n s ity in D e c a t u r s o i l ............................................ ...... ....................... 46 * 2 13 14 15 16 17 18 a p l a n e of s y m m e t r y . ....................................................................... 26 28 X F igure 19 20 21 page M a x im u m s h e a r i n g s t r e s s v e r s u s bulk d e n s ity in H i w a s s e e s o i l ............................................................................ 47 M axim um s h e a rin g s t r e s s v e r s u s bulk d e n sity in L l o y d s o i l .................................................................................... 48 M ax im u m s h e a rin g s t r e s s v e r s u s bulk d e n s ity in D e c a t u r s o i l ........................................................................... 49 S e c o n d i n v a r i a n t of d e v i a t o r v e r s u s b u l k d e n s i t y in H i w a s s e e s o i l .................................................................... 50 S e c o n d i n v a r i a n t of d e v i a t o r v e r s u s b u l k d e n s i t y in L l o y d s o i l .............................................................................. 51 S e c o n d i n v a r i a n t of d e v i a t o r v e r s u s b u l k d e n s i t y in D e c a t u r s o i l .......................................................................... 52 T h r e e d i m e n s i o n a l M o h r ' s c i r c l e s f o r 1 8 " - 1 2 ’’ a n d 1 8 ” - 2 0 " s t r e s s s t a t e s ........................... .................... 54 T h r e e d i m e n s i o n a l M o h r ’s c i r c l e s f o r 5 ” - 1 2 ” and 1 8 ” - 1 8 ” s t r e s s s ta t e s ............................... 55 D i s t r i b u t i o n of m e a n s t r e s s in t h r e e s o i l s a n d t h r e e s t r e s s s t a t e s ..................................... ........................... 58 28 T r i a x i a l a p p a r a t u s ......................................................................... 61 29 F i e l d a n d t r i a x i a l d a t a in H i w a s s e e s a n d y l o a m s o il . 22 23 24 25 26 27 64 IN TRODUCTION T h e e f f e c t of t h e p h y s i c a l p r o p e r t i e s of s o i l u p o n planl g r o w t h is r e c e i v i n g i n c r e a s e d attention. F e r t i l i z e r s , p l a c e m e n t of f e r t i l i z e r s i and s im i l a r c u ltu ra l p r a c tic e s have in c r e a s e d production. A plateau, h o w e v e r , h a s b e e n r e a c h e d w h e r e i n h i g h e r r a t e s of f e r t i l i z e r a p p l i c a t i o n , f o r e x a m p l e , do not m a t e r i a l l y i n c r e a s e p r o d u c t i o n . The re a s o n for t h i s i s a p p a r e n t l y that t h e p h y s i c a l s t a t e of t h e s o i l h a s b e c o m e t h e m a j o r l i m i t i n g f a c t o r in c r o p y i e l d s . T h e s t a t e of c o n s o l i d a t i o n of s o i l , p o p u l a r l y k n o w n a s s o i l c o m p a c t i o n , is o n e of t h e s o i l ’s p h y s i c a l properties. It h a s b e e n s h o w n that m e c h a n i c a l i m p e d a n c e to r o o t s , i n f i l t r a t i o n r a t e , p e r c o l a l i o n r a t e , o x y g e n a v a i l a b i l i t y and n i t r o g e n a v a i l a b i l i t y a r e a l l i n f l u e n c e d to s o m e d e g r e e b y t h e s t a t e of c o m p a c t i o n of t h e s o i l . ( B a v e r , 1956; G i l l , 1954; W a d l e i g h , 1957; W i e r s m a an d M o r f l a n d , 1953.) In s o m e a r e a s of C a l i f o r n i a , H a w a i i an d p o r t i o n s of t h e s o u t h e r n U n i t e d S t a t e s a s w e l l a s o t h e r r a n d o m r e g i o n s , t h e d e g r e e of c o m p a c t i o n h a s r e a c h e d a l e v e l that h a s s e r i o u s l y r e d u c e d y i e l d s an d in s o m e c a s e s r e m o v e d l a n d f r o m c u l t i v a t i o n . ( E d m i n s t e r , 1956.) S i n c e t h e s e a r e a s h a v e b e e n r e m o v e d f r o m p r o d u c t i o n , it b e h o o v e s al l c i v i l i z a t i o n s to s t u d y t h e c o m p a c t i o n p r o b l e m b e f o r e o t h e r a r e a s a r e r e m o v e d from production. Even f u rth e r , th e re d u c e d yields from i n c o r r e c t c o m p a c tio n lev els during tilla g e o p e r a tio n s , planting fo r e x a m p l e , cannot e v e n be e s tim a te d . In c e r t a i n c a s e s a d d i t i o n a l 2 c o m p a c tio n m a y be d e s ir e d . At t h e p r e s e n t l i m e , 1 h e r e i s no k n o w n m e t h o d f o r p r o d u c i n g a d e s i r e d s t a t e of c o m p a c t i o n d u r i n g t i l l a g e o p e r a t i o n s o t h e r t h a n t r i a l and e r r o r . T h e s l a t e of c o m p a c t i o n r e s u l t i n g f r o m a s i n g l e o p e r a t i o n by any given tilla g e tool cannot be p r e d ic te d except th rough e x p e r ie n c e d judgm ent. In t h i s r e s p e c l , t i l l a g e h a s not a d v a n c e d f a r f r o m t h e d a y s w h e n s e e d s w e r e p l a n t e d in a h o l e p u n c h e d in t h e s o i l w i t h a p o i n t e d stick. As co n tin u ed in v e s tig a tio n s r e v e a l new know ledge c o n c e r n in g t h e d e s i r e d d e g r e e of c o m p a c t i o n f o r a g i v e n c r o p , t h e n e e d f o r a m e t h o d to p r o d u c e s u c h a d e g r e e b e c o m e s i n c r e a s i n g l y i m p o r t a n t . T h e l a r g e s t m e a s u r e d f o r c e s a p p l i e d to Ihe s o i l w e r e u n d e r t h e r e a r w h e e l of a f a r m t r a c l o r . ( V a n d e n B e r g , e t . a l . , 1957.) W hile such m e c h a n i c a l f o r c e s a r e not I h e o n l y c a u s e s of s o i l c o m p a c t i o n , t h a y a r e th e m a j o r c a u s e today. H e a v i e r t r a c t o r s an d i m p l e m e n t s c o u p l e d w ith i n c r e a s e d n u m b e r s of t r i p s o v e r t h e s o i l ( r e s u l t i n g f r o m new c u l t u r a l p r a c t i c e s s u c h a s f e r t i l i z i n g a n d s p r a y i n g ) a p p l y r e p e a t e d l o a d s to t h e soil. T o u n d e r s t a n d t h e e f fe c t of t h e s e l o a d s on t h e s o i l , t h e a m o u n t of c o m p a c t i o n c h a n g e at e v e r y point in t h e s o i l r e s u l l i n g f r o m an a p p l i e d load must be d e te rm in e d . A s o l u t i o n to t h e p r o b l e m m a y b e s e c u r e d b y c h a n g i n g t h e m a n n e r in w h i c h t h e l o a d i s a p p l i e d a n d o b s e r v i n g t h e co rresp o n d in g com paction changes. In t h i s w a y t h e b e s t m e t h o d f o r a p p l y i n g t h e l o a d c a n b e d e t e r m i n e d ; o r at l e a s t t h e k n o w l e d g e to p r e d i c t t h e a m o u n t of c o m p a c t i o n c h a n g e c a n b e o b t a i n e d . 3 T o c a r r y out t h e s t u d y i n d i c a t e d a b o v e b y m e a s u r i n g t h e c o m p a c t i o n c h a n g e u n d e r v a r i o u s l o a d s in v a r i o u s s o i l s w o u l d s e e m i n g l y r e q u i r e a p r o h i b i t i v e a m o u n t of t i m e . It is i m p e r a t i v e , t h e r e f o r e , that a m o re o rg an ized ap proach be attem pted. If a s u i t a b l e m e c h a n i c a l m o d e l f o r s o i l s , s u c h a s t h e T h e o r y of E l a s t i c i t y p r o v i d e s f o r m e t a l s , c o u l d b e f o u n d it w o u l d b e p o s s i b l e to c a l c u l a t e t h e d e s i r e d v o l u m e change. At t h e p r e s e n t t i m e , no s u i t a b l e s o i l m e c h a n i c s m o d e l i s available. T h is t h e s i s will p r e s e n t a m e th o d w h e re b y a soil m e c h a n ic s m o d e l b a s e d u p o n t h e c o n c e p t of c o n t i n u u m m e c h a n i c s c a n b e d e v e l o p e d . T h e r e p r e s e n t a t i o n of f o r c e s a c t i n g on a n d t h e r e p r e s e n t a t i o n of c o m p a c t i o n c h a n g e in a v o l u m e e l e m e n t of s o i l w i l l b e d i s c u s s e d . An h y p o t h e s i s c o n c e r n i n g t h e r e l a t i o n s h i p b e t w e e n t h e f o r c e s a c t i n g on t h e v o l u m e e l e m e n t a n d t h e c o m p a c t i o n c h a n g e in t h e e l e m e n t w i l l b e proposed. T h e a n a l y s i s of d a t a t a k e n to t e s t t h e h y p o t h e s i s i s p r e s e n t e d and analyzed. REVIEW O F L IT E R A T U R E ‘A n a g r i c u l t u r a l s o i l i n a g o o d s t a t e of t i l t h i s a p o o r r o a d s u r f a c e a n d i n m o s t c a s e s it is d a m a g e d , at l e a s t t e m p o r a r i l y , b y t h e p a s s a g e of a t r a n s p o r t w h e e l / ( M c K i b b e n an d G r e e n , 1940.) T h is load s e t s u p a s t r e s s d i s t r i b u t i o n t h r o u g h o u t t h e s o i l a n d a c c o r d i n g to M c K i b b e n a n d G r e e n , r e s u l t s in s u b s e q u e n t d a m a g e to t h e s o il . Today t h i s m e c h a n i c a l c o n s o l i d a t i o n of t h e s o i l h a s b e e n n a m e d s o i l c o m p a c ­ tion. In s p e a k i n g of t h e s l r e s s d i s t r i b u t i o n a b o v e , H o g e n t o g l e r (1937) s t a t e d , ‘T h e v o l u m e of c o m p r e s s e d s o i l i n c l u d e d b e t w e e n t h e p o i n t s of z e r o p r e s s u r e a n d t h e b e a r i n g b l o c k (load) i s k n o w n a s t h e p r e s s u r e b u l b . . . T h e s u p p o r t i n g p o w e r of Ihe b u l b is d e v e l o p e d by t h e r e s i s t a n c e t o d e f o r m a t i o n o f f e r e d b y t h e c o h e s i o n a n d i n t e r n a l f r i c t i o n of t h e s o i l . ’ O b v i o u s l y , if t h i s s t r e s s d i s t r i b u t i o n an d t h e r e l a t i o n s h i p b e t w e e n s t r e s s and t h e r e s i s t a n c e o f f e r e d b y t h e s u p p o r t i n g b u l b d e s c r i b e d a b o v e w e r e k n o w n , 1he c h a n g e i n c o m p a c t i o n r e s u l t i n g f r o m l o a d s could be calcu lated . T h e r e h a v e b e e n s e v e r a l a t t e m p t s to e v a l u a t e s t r e s s d i s t r i b u t i o n in the soil. In 1885 B o u s s i n e s q d e v e l o p e d f o r m u l a e that g i v e s t r e s s e s in an i s o t r o p i c h o m o g e n e o u s m e d i u m thal o b ey s H o o k e ’s law w h e n s u b j e c t e d to a point lo a d . In 1934, O, K. F r o e h l i c k i n t r o d u c e d a c o n c e n t r a t i o n f a c t o r into the f o r m u l a e w hich p e r m i t s s o m e g e n e r a l i z a t i o n of t h e s t r e s s d i s t r i b u t i o n . S o e h n e (1 953) e s t i m a t e d t h e c o n t a c t a r e a u n d e r v a r i o u s t r a c t o r t i r e s a n d d i v i d e d t h e a r e a in to 25 s m a l l e r a r e a s . 5 C o n s i d e r i n g f he l o a d a c t i n g on e a c h l i t t l e a r e a a s a p oint l o a d l o c a t e d at t h e c e n t r o i d of t h e a r e a a n d u s i n g F r o e l i c k ’s f o r m u l a e h e t h e n s u m m e d a l l t h e c o n t r i b u t i o n s of e a c h a r e a at s e v e r a l p o i n t s b e l o w t h e l o a d a n d a r r i v e d al a s t r e s s d i s t r i b u t i o n . An a t t e m p t to d e t e r m i n e t h e s t r e s s d i s t r i b u t i o n u n d e r c i r c u l a r l o a d s w a s d e v e l o p e d b y L o v e and a p p l i e d b y the C o r p s of E n g i n e e r s (1 953,) T h i s a p p r o a c h w a s s o m e w h a t b e t t e r t h a n t h e o n e p r e s e n t e d by S o e h n e s i n c e it w a s not r e s t r i c t e d to po int l o a d s . L o v e 's fo rm u la e a l s o a p p l y o n l y to a n i s o t r o p i c h o m o g e n e o u s m e d i u m w h i c h o b e y s H o o k e ’s law . T h e r e i s c o n s i d e r a b l e e v i d e n c e that s o i l d o e s not o b e y H o o k e ’s l a w . T h e m o d u l u s of e l a s t i c i t y c h a n g e s not o n l y w i t h s o i l t y p e s a n d c o n d i t i o n s but a l s o a s a l o a d i s a p p l i ^ d 0 ( C o r p s of E n g i n e e r s , 1954.) F u r t h e r m o r e t h e e l a s t i c r e c o v e r y a f t e r c o m p a c t i o n is v e r y s m a ll, indicating little ela stic behavior. (Gill a n d R e a v e s , 1956.) S in ce s o ils exhibit little e la s tic b e h a v io r , the s t r e s s d is tr ib u tio n s b a s e d on e l a s t i c i t y s h o u l d b e q u e s t i o n e d . S e v e r a l a t t e m p t s h a v e b e e n m a d e to c h e c k t h e t h e o r i e s g i v i n g s t r e s s d i s t r i b u t i o n s and s o m e h av e m e t with r e a s o n a b l e s u c c e s s as r e p o r t e d b y t h e C o r p s of E n g i n e e r s (1954.) s u c c e s s f u l on s a n d . T h e i r 1es1s w e r e m o s t P l u m m e r a n d D o r e (1940) h a d t h e f o l l o w i n g to s a y about o t h e r e a r l i e r t e s t s : ‘U n f o r t u n a t e l y , a l m o s t a l l a c t u a l l e s t s h a v e b e e n c o n d u c t e d on s o i l s c o n s i s t i n g of d r y , g r a n u l a r m a s s e s s u c h as sands. T h e r e i s s o m e r e a s o n to b e l i e v e th at s o m e s o i l s a p p r o a c h , 6 i n t h e i r s u p p o r t i n g a c t i o n , th a t of a v e r y v i s c o u s m a t e r i a l s u c h a s a s p h a l t . . . W e c a n n o t at p r e s e n t p r e d i c t w i t h a n y g r e a t a c c u r a c y t h e m a g n i t u d e of s t r e s s at a g i v e n p o i n t in s u c h s o i l s . ’ F r o m t h e a b o v e it s e e m s q u i t e c l e a r that at t h e p r e s e n t t i m e t h e r e is no w a y to a c c u r a t e l y d e t e r m i n e s t r e s s d i s t r i b u t i o n s in t h e s o i l . A t t e m p t s h a v e b e e n m a d e to go d i r e c t l y f r o m t h e a p p l i e d l o a d t o s o i l c o m p a c t i o n but m a n y p r o b l e m s h a v e b e e n e n c o u n t e r e d . (G ill a n d R e a v e s , 1 9 5 6 u) T h i s r e f e r e n c e s l a t e d that ‘m a n y d i f f i c u l t i e s w e r e e n c o u n t e r e d w h e n a t t e m p t s w e r e m a d e to r e l a t e s o i l c o m p a c t i o n to th e c o m p a c t i v e p r e s s u r e s b y t i r e s to t h e s o i l . M e a s u r e m e n t of t h e c o n t a c t p r e s s u r e i s d i f f i c u l t a n d not yel s a t i s f a c t o r i l y m e a s u r e d ; y e t , g i v e n t h i s p r e s s u r e in t h e s o i l , o t h e r c o m p l i c a t i n g f a c l o r s r e m a i n . ’ T h u s t h e m o r e b a s i c a p p r o a c h of d e t e r m i n i n g t h e s t r e s s d i s t r i b u t i o n a n d t h e e x a c t r e l a t i o n s h i p b e t w e e n s t r e s s a n d c o m p a c t i o n s h o u l d l e a d to a b e l t e r s o l u t i o n of t h e p r o b l e m . S o e h n e (1 953) a s s u m e d a r e l a t i o n s h i p b e t w e e n s t r e s s an d c o m p a c t i o n a n d p r o c e e d e d to m e a s u r e t h e r e l a t i o n s h i p on u n d i s t u r b e d soils. H e a s s u m e d 1ha1 t h e m a x i m u m n o r m a l s t r e s s a c t i n g on Ihe s o i l e l e m e n t r e l a t e d b e s t to s o i l c o m p a c t i o n . He then m a d e the m e a s u r e m e n t s b y a p p l y i n g a s u r f a c e l o a d to t h e s o i l in a c o n f i n e d c o n t a i n e r . d o i n g , h e i g n o r e d t h e s i d e s t r e s s e s a c t i n g on t h e s o i l . In so It s e e m s that t h e s e s id e s t r e s s e s sh ould b e taken into account as w e ll as the m a x i m u m norm al slress. 7 The a p p l i c a t i o n of s t r a i n g a g e s in d e s i g n i n g c e l l s l e d to t h e f i r s t r e a l p r o g r e s s in a c c u r a t e l y m e a s u r i n g s o i l s t r e s s e s . T h e u s e of s e v e r a l t y p e s of t h e s e c e l l s i s r e p o r t e d in s e v e r a l s o u r c e s . (Cooper, e t . a l v, 1957; C o r p s of E n g i n e e r s , 1954; P e a t t i e a n d S p a r r o w , 1954; W i l l i t s , 1956.) T h e p e r f o r m a n c e s of t h e c e l l s p r e s e n t e d b y C o o p e r m a k e s it p r e f e r a b l e to t h e o t h e r t y p e s . T h e c e l l s of t h e C o r p s of E n g in e e r s , w hile p ro b a b ly as a c c u r a t e as th o se d eveloped by C o o p e r, a r e m o r e e x p e n s iv e and p e r h a p s l a r g e r than d e s i r a b l e fo r a p p lic a tio n to a g r i c u l t u r a l so ils . T h e r e i s l i t t l e do u b t f r o m t h e w o r k of C o o p e r a n d t h e C o r p s of E n g i n e e r s that, t h e c e l l s m e a s u r e s o i l s t r e s s a c c u ra te ly when p ro p e rly used. T H E O R Y - M ECHANICS O F TH E CONTINUUM A s m e n t i o n e d b e f o r e , i f a g o o d m a t h e m a t i c a l d e s c r i p t i o n of t h e s o i l e x i s t e d , t h e c o m p a c t i o n c h a n g e i n t h e s o i l f o r a n y c o m b i n a t i o n of applied loads could be calcu lated . n ot e x i s t today. S u c h a d e s c r i p t i o n of t h e s o i l d o e s To develop the need ed soil m e c h a n ic s , the f o rc e s a c t i n g on a v o l u m e e l e m e n t of s o i l m u s t b e r e l a t e d to t h e d e f o r m a t i o n of th e v o lu m e e le m e n t. a m ethod fo r specifying T h e m e c h a n i c s of a c o n t i n u o u s m e d i u m p r o v i d e s t h e f o r c e s a c t i n g on a v o l u m e e l e m e n t of t h e s o i l a n d a m e t h o d f o r s p e c i f y i n g t h e d e f o r m a t i o n of t h e v o l u m e e l e m e n t r e s u ltin g fro m the applied f o rc e s . A s is k n o w n , t h e f o r m e r r e s u l t s in a s t r e s s t e n s o r and the l a t t e r in a s t r a i n te n s o r . A relationship b e t w e e n t h e tw o t e n s o r s p r o v i d e s t h e d e s i r e d m e c h a n i c s m e n t i o n e d above. In d e a l i n g w i t h s o i l c o m p a c t i o n , it i s p o s s i b l e to s i m p l i f y t h e s t r a i n t e n s o r by i g n o r i n g t h e s h e a r i n g s t r a i n s a n d r i g i d b o d y r o t a t i o n s . T h e ch an g e in c o m p a c tio n i s m e r e l y t h e c h a n g e in t h e v o l u m e of t h e e l e m e n t a n d i s i n d e p e n d e n t of a n y s h e a r i n g o r r o t a t i n g that m a y t a k e p lace during stra in in g . T h e c o m p a c t i o n of t h e v o l u m e e l e m e n t c a n be c o m p l e t e l y s p e c i f i e d b y g i v i n g i t s b u l k d e n s i t y , i . e . t h e w e i g h t of o v e n d r y s o i l p e r unit b u l k v o l u m e ; t h e b u l k v o l u m e i n c l u d e s b o t h v o l u m e of s o il so lid s and so il p o r e s . T h e b u l k d e n s i t y of e a c h v o l u m e e l e m e n t in a b o d y of s o i l s p e c i f i e s c o m p l e t e l y t h e s t a t e of c o m p a c t n e s s of t h e b o d y of s o i l . A c o m p a c t i o n la w c a n t h u s b e d e v e l o p e d b y r e l a t i n g t h e 9 s t r e s s t e n s o r to b u l k d e n s i t y . . In s o d o i n g it f o l l o w s f r o m a b o v e t h a t all s h e a r i n g and r o ta ti n g d e f o r m a tio n s w ill b e ig n o re d . If a r e l a t i o n s h i p b e tw e e n the s t r e s s t e n s o r and the c o m p l e t e s t r a i n t e n s o r could be d e v e l o p e d , t h e e x a c t p o s i t i o n of e a c h s o i l e l e m e n t a s w e l l a s t h e v o l u m e c h a n g e of e a c h e l e m e n t c o u l d be a c c u r a t e l y c a l c u l a t e d . Because a c o m p l e t e s o l u t i o n i s not of i m m e d i a t e c o n c e r n , i g n o r i n g t h e s h e a r i n g a n d r o t a t i n g d e f o r m a t i o n s p r o v i d e s a s u i t a b l e m e a n s of s i m p l i f y i n g the p ro b le m . T o d e s c r i b e t h e f o r c e s a c t i n g on a v o l u m e e l e m e n t , t h e c o n c e p t s of t h e s t r e s s v e c t o r a n d t h e s t r e s s t e n s o r a r e r e q u i r e d . If an i m a g i n a r y p l a n e i s p a s s e d t h r o u g h a point in s o m e m e d i u m s u b j e c t e d to g e n e r a l f o r c e s , t h e m a t e r i a l on o n e s i d e of t h e p l a n e e x e r t s a f o r c e a c r o s s t h e p l a n e o n t h e o t h e r s i d e of t h e m a t e r i a l . f o r c e e x e r te d a c r o s s the a r e a A In F i g . 1, f o r e x a m p l e , t h e A b y p a r t (1) of t h e b o d y on p a r t (2) of t h e b o d y i s r e p r e s e n t e d by t h e v e c l o r T 2T? . The s lre s s vector at t h e p o i n t O i s t h e n d e f i n e d as x — CO PLANE £r CO LU Ct < CO O o L l_ 11 m easure t h e a v e r a g e s t r e s s o v e r an a r e a . In g e n e r a l , t h e s t r e s s v e c t o r h a s Iwo c o m p o n e n t s - o n e n o r m a l t o t h e p l a n e on w h i c h it a c t s and o n e t a n g e n l t o t h e p l a n e . These s t r e s s e s r e p r e s e n t d i f f e r e n t p h y s i c a l a c t i o n s on t h e p l a n e a n d a r e d i s t i n g u i s h e d a s s u c h in t h e n o t a t i o n . T h e f ir s t co m p o n e n t is th e n o r m a l s t r e s s and is d e n o te d by th e G r e e k l e t t e r ( T w i t h an a p p r o p r i a t e s u b s c r i p t; the l a t t e r com ponent is the s h e a r i n g s t r e s s and is d e n o te d by the G r e e k l e t t e r T , again with a p p r o p r i a t e s u b s c r i p t. T h u s in F i g , 1 w h e r e t h e p l a n e i s i d e n t i f i e d b y t h e d i r e c t i o n of i t s n o r m a l H { d r a w n o u t w a r d f r o m t h e b o d y on w h i c h t h e s t r e s s a c t s ) t h e n o r m a l s t r e s s c o m p o n e n t i s ( 7 n a n d th e s h e a r c o m p o n e n t i s T n In g e n e r a l , if a d i f f e r e n t p l a n e i s p a s s e d t h r o u g h t h e s a m e p o i n t , a d i f f e r e n t s t r e s s v e c t o r w i l l act a c r o s s it. H e n c e , the s t r e s s v e c t o r d e p e n d s not o n l y on t h e p o s i t i o n of t h e point in t h e b o d y but a l s o on t h e o r i e n t a t i o n of t h e p l a n e t h r o u g h t h e p o in t. An i n f i n i t e n u m b e r of o r i e n t a t i o n s m a y b e c h o s e n f o r a p l a n e I h r o u g h a n y p o int; it is n e c e s s a r y to b e a b l e to d e t e r m i n e t h e s t r e s s v e c t o r on a l l of t h e s e p l a n e s . It c a n be s h o w n t hat t h e s p e c i f i c a t i o n of t h e s t r e s s v e c t o r s on t h r e e m u tu a lly p e r p e n d ic u la r planes co m p le te ly d e te r m in e s the s t r e s s s ta te at a n y p o in t. F r o m t h e s e t h r e e s l r e s s v e c t o r s , t h e s l r e s s v e c t o r on a n y o t h e r p l a n e t h r o u g h t h e point c a n b e c a l c u l a t e d . The p ro c e d u re for doing this will be p r e s e n te d la te r, In F i g . 2, t h e t h r e e r e f e r e n c e p l a n e s a r e r e p r e s e n t e d a s t h r e e 12 f a c e s of a c u b e p e r p e n d i c u l a r r e s p e c t i v e l y to t h e X, Y, an d Z a x e s of conventional c a r te s ia n coordinate sy stem . The s h e a r s t r e s s com ponents in e a c h of t h e s e f a c e s a r e f u r t h e r r e s o l v e d i n t o tw o c o m p o n e n t s e a c h p a r a l l e l to a c o o r d i n a t e a x i s ; th e s y m b o l s r e p r e s e n t i n g t h e m a r e g i v e n two s u b s c r ip ts . T h e f i r s t s u b s c r i p t i d e n t i f i e s t h e p l a n e o n w h i c h it a c t s a n d t h e s e c o n d i d e n t i f i e s t h e d i r e c t i o n i n w h i c h it a c t s . These nine quantities a r e t h e c o m p o n e n t s of t h e s t r e s s t e n s o r at a point r e f e r r e d to a c h o s e n s e t of a x e s . N o t e that t h e c u b e in F i g . 2 i s i m a g i n e d to b e i n f i n i t e s i m a l s o t h a t t h e s t r e s s e s on t h e t h r e e f r o n t f a c e s act ‘at t h e point O ’ i n t h e l i m i t a s t h e v o l u m e of t h e e l e m e n t s h r i n k s to z e r o . F ro m equilibrium c o n d i t i o n s it c a n b e s h o w n T h i s l e a v e s s i x i n d e p e n d e n t v a l u e s to b e d e t e r m i n e d to d e f i n e t h e s t a t e o f s t r e s s at a p o in t. T h e s t r e s s v e c t o r on an a r b i t r a r y p l a n e c a n m o s t e a s i l y b e c a l c u l a t e d by u s in g m a t r i x a l g e b r a s in c e the s t r e s s t e n s o r c a n be r e p r e s e n t e d as a 3 x 3 m a trix . T h e c a l c u l a t i o n i s a c c o m p l i s h e d in th e follow ing m a n n e r: S u p p o s e a p l a n e i s o r i e n t e d s o that i t s n o r m a l l i e s in t h e YZ p l a n e a n d b i s e c t s t h e a n g l e b e t w e e n t h e p o s i t i v e Y an d Z a x e s . n o r m a l to t h e p l a n e w i l l h a v e d i r e c t i o n c o s i n e s 0 ; / 2 / 2 * The /s /2 FIG. 2 A STRESSES VOLUME ON ELEMENT r e s p e c t i v e l y w ith th e X ; Y and Z a x e s . T h e d ire c tio n c o s in e s can be w r i t t e n in a r o w m a t r i x a s follows:. L e t u s f u r t h e r s u p p o s e that i h e s t r e s s s t a t e i s a s p e c i a l o n e in w h i c h t w o of t h e s h e a r i n g s t r e s s e s a r e z e r o . special s tre s s If i, j } and T he s t r e s s t e n s o r fo r this s t a t e i s g i v e n by k a r e u nit v e c t o r s along the p o s itiv e X , Y, and Z ax e s r e s p e c t i v e l y , t h e n t h e c o m p o n e n t s of t h e s t r e s s v e c t o r a c t i n g on t h e p la n e d e s c r i b e d above a r e given by the m a t r i x m u ltip lic a tio n shown below . g | c e U s i n g t h e u s u a l m e t h o d of m u l t i p l y i n g r o w s in to c o l u m n s , t h e i 4a t' r 11 i c T h e m a g n i t u d e of t h e n o r m a l s t r e s s a c t i n g on t h e p l a n e c a n b e fo u n d b y o b t a i n i n g t h e s c a l a r p r o d u c t of a uni t v e c t o r n n o r m a l to t h e p l a n e a n d t h e s t r e s s v e c t o r a c t i n g on t h e p l a n e . u n it n o r m a l v e c t o r n i s g i v e n by T in t h e d i r e c t i o n of t h e The 14 n = 0 i -r T h u s th e n o r m a l s t r e s s , CTn = 1 /2 ((TV {2 / 2 j + /2 /2 k . , on th e p la n e is (Tz ) + ^"Tyz , ( 1) T h e s t r e s s t e n s o r h a s s e v e r a l m a t h e m a t i c a l p r o p e r t i e s of interest. It i s a l w a y s p o s s i b l e to r o t a t e t h e c o o r d i n a t e a x e s (an d w it h t h e m t h e i m a g i n a r y p l a n e s b o u n d i n g t h e c u b i c a l v o l u m e e l e m e n t ) to s u c h a p o s i t i o n th a t a l l of t h e s h e a r s t r e s s e s w i l l b e z e r o a n d o n l y n o r m a l s t r e s s e s act on t h e p l a n e s . In t h i s p o s i t i o n t h e a x e s a r e c a l l e d t h e p r i n c i p a l a x e s o r p r i n c i p a l d i r e c t i o n s of t h e s t r e s s t e n s o r at t h e point. T h e t h r e e n o r m a l s t r e s s in t h e s e d i r e c t i o n s a r e t h e p r i n c i p a l s t r e s s at t h e p oint d e n o t e d b y (Tx > ( J Z > an d o r a s in F i g , 3. One of t h e s e p r i n c i p a l s t r e s s e s i s t h e l a r g e s t n o r m a l s t r e s s on a n y p o s s i b l e p l a n e t h r o u g h t h e point a n d a n o t h e r o n e i s t h e s m a l l e s t p o s s i b l e n o r m a l s t r e s s t h r o u g h t h e p o in t. The p r in c ip a l ax es o r i e n t a t i o n s and the p r i n c i p a l s t r e s s m a g n i t u d e s a r e not t h e s a m e at a l l p o i n t s . It is p o s s i b l e , h o w e v e r , to c a l c u l a t e f o r a n y o n e point t h e m a g n i t u d e an d o r i e n t a t i o n of t h e p r i n c i p a l s t r e s s e s w h e n t h e s t r e s s t e n s o r c o m p o n e n t s a r e k n o w n w i t h r e f e r e n c e to s o m e s y s t e m of c o o r d i n a t e a x e s . The p r o b l e m i s to d e t e r m i n e t h e d i r e c t i o n c o s i n e s n, so that ■A , n,J nZ t h e s t r e s s v e c t o r a s s o c i a t e d w it h t h e p l a n e d e t e r m i n e d b y t h e d i r e c t i o n 15 y FIG. 3 PRINCIPAL ON A V O L U M E STRESSES ELEMENT 16 c o s i n e s is n o r m a l to t h e p l a n e . u sed above, F o r e x a m p le , in the s p e c ia ls t r e s s s ta t e (T* m u s t b e a p r i n c i p a l s t r e s s s i n c e t h e p l a n e on w h i c h a c t s h a s no s h e a r c o m p o n e n t s . it T h e r e f o r e , t h e o t h e r tw o p r i n c i p a l s t r e s s e s m u s t l i e in t h e YZ p l a n e s i n c e a s c a n b e s e e n in F i g . 3, t h e p rin c ip a l s tr e s s e s form a m utually perp en d icu lar triad. The gen eral p r o c e d u r e i s to p e r f o r m t h e m a t r i x m u l t i p l i c a t i o n i l l u s t r a t e d b e l o w w h ich f o r the s p e c ia l s t r e s s state d is c u s s e d above is (*x *V w here ns ) . , n^. /( F 0 I o (Ty \ 0 Tyz an d p r i n c i p a l s t r e s s e s an d 0V Oz ) - (n x (T ny(T" n z (T) (2 ) a r e t h e d i r e c t i o n c o s i n e s to o n e of t h e cr i s t h e u n k n o w n v a l u e of t h e p r i n c i p a l stress. W h e n e q u a t i o n (2) i s e x p a n d e d t h e f o l l o w i n g s y s t e m of h o m o g e n e ­ ous li n e a r a lg e b ra ic equations r e s u l t s for the unknowns (G~x - C T ) rv -h 0 nx ■*" - O' 0 Rx -*T y z ( 0"y *y + o^y j + ny ( 0~z O n2 nz , n = 0 "t 'T 'y z n z -(F) ,n y ~ 0 - 0 . (^ ) T h e w e l l k n o w n r e l a t i o n s h i p b e t w e e n t h e d i r e c t i o n c o s i n e s is +- n * +- r | = 1 (4) z 17 T h e s o l u t i o n of e q u a t i o n s (3) g i v e s t h e v a l u e s f o r t h e u n k n o w n d ire c tio n cosines. T h e t r i v i a l s o l u t i o n in w h i c h a l l of t h e u n k n o w n s a r e z e r o i s of no i n t e r e s t . E x c lu d in g such a solution, the d e te r m in a n t of t h e c o e f f i c i e n t s of (3) m u s t v a n i s h , a c o n d i t i o n w h i c h l e a d s to a c u b i c e q u a t i o n in C T , w hose r o o ts a r e the t h r e e p r in c ip a l s t r e s s e s . In t h i s s p e c i a l c a s e , it i s a l r e a d y k n o w n t h a t a n d i t s d i r e c t i o n c o s i n e s w i l l b e (1, 0, 0), then If 0 " i s not e q u a l to CTx — 0 a n d e q u a t i o n s (3) r e d u c e to ( (TV ~ 0 " ) T w it h is a p r in c ip a l s t r e s s , rpr and y nz , " ’V + j + ^Tyz nz ( (T , - C ) not b o t h z e r o . ~ ”, 0 = 0 151 T h e c h a r a c t e r i s t i c e q u a t i o n of (5) IS (Jy 0~ T yz — 'T y z (Jz ~ 0 ^ E x p a n d in g th e d e te r m in a n t and c o lle c tin g t e r m s g iv e s ( T 2- ( O ', * CTy )C + (T y Cr. ~T yl = o "> E q u a t i o n (6) c a n b e s o l v e d b y a p p l y i n g t h e q u a d r a t i c f o r m u l a w h i c h g i v e s two v a lu e s f o r c r Q~1 - l / 2 ( ( T y + (Tz ) — 1/ 2 (T z - (Ty ) + 2 y2 4 (7) T h e o r i e n t a t i o n of t h e p r i n c i p a l s t r e s s i s c o m p u t e d a s f o l l o w s : th e eq u ation involving ny (T* of e q u a t i o n (5) f o r 0~z.) n z ^ T * yz n^r g i v e s Solving 18 S in c e e q u a ls z e r o , (4) ca n b e w r it te n ( 0 ~ - (T z)2 u S o lv in g f o r n„ ^ + ' nz2 1 g iv e s / *T y z nz - g \ f 2 + T ,y z (T s ) ^ (8 ) But nz w here — cos (X) i s th e d ir e c t io n b e tw e e n th e Z a x is and th e p r in c ip a l s t r e s s in q u e s tio n . B y th e tr ig o n o m e t r ic r e la t io n s b e tw e e n tan and c o s <£ , (8) b e c o m e s ta n (j) = O ' - (T z / 7 ^ yz # (9) T h e s t r e s s t e n s o r a ls o h a s th e p r o p e r ty th at t h r e e q u a n titie s c o m p u te d fr o m it a r e in v a r ia n t, m e a n in g th at t h e s e q u a n titie s a r e in d e p e n d e n t o f th e c h o ic e o f c o o r d in a te a x e s . *S JIs — = Gx G~y * (J^ ( ( T 0~Z j j ~ j + ■ + . T h e th r e e q u a n titie s a r e ( j ^ (jz Ox - 'Tyz ” T x z - ' T x y ( 10 ) 19 T *y p Txy i n s- (TV 'J 'y z Tyz (P (10, co nl ’d) T h e f i r s t i n v a r i a n t i s of p a r t i c u l a r i m p o r t a n c e . The m ean norm al s tr e s s a c t i n g at a p o in t i s d e f i n e d a s (fm s l / 3 Ia = l / s ( ( T t(T y x t ( T z (11) ) a n d b e c a u s e of i t s i n v a r i a n c e c a n b e e x p r e s s e d in p r i n c i p a l s t r e s s e s a s G“m = 1 /3 ( (T i + ffi ♦ GF} N o w a n y s t a t e of s t r e s s at a p o in t m a y b e c o n s i d e r e d a s a s u p e r p o s i t i o n of a s p h e r i c a l o r h y d r o s t a t i c s t r e s s s t a t e a n d a d e v i a t o r i c s t r e s s o r s t r e s s deviator. In t h e s p h e r i c a l s t r e s s te n s o r all s h e a r com ponents a r e z e r o a n d t h e t h r e e n o r m a l s t r e s s c o m p o n e n t s a r e e a c h e q u a l to the m ean n o rm a l s tr e s s tensor < T m T h e c o m p o n e n t s of t h e d e v i a t o r s t r e s s ■ t h e n d i f f e r f r o m t h e s t r e s s t e n s o r o n l y in t h a t fro m each n o r m a l s t r e s s com ponent. is su b tracted ( f m T h is is show n below w h e r e the f i r s t t e n s o r is th e s t r e s s t e n s o r , the seco n d is the s p h e r i c a l s t r e s s t e n s o r and th e t h i r d is th e d e v i a t o r s t r e s s t e n s o r . Txz P 0 Tyz 0 CM 0 P ° ° \ /C T ~ t h e r e l a t i o n b e t w e e n b u l k d e n s i t y a n d t h e s p h e r i c a l s t r e s s t e n s o r w o u l d t h e n s i m p l i f y to a s i n g l e s c a l a r e q u a t i o n i n v o l v i n g o n l y (Tm r a t h e r t h a n a d e p e n d e n c e on six s tr e s s e s . T h i s i s t h e h y p o t h e s i s p r o p o s e d in t h i s t h e s i s a s a s o i l c o m p a c t i o n law: M e a n n o r m a l s t r e s s is r e l a t e d to b u l k d e n s i t y f o r a l l soils. a s c a l a r f u n c t i o n of t h e f o r c e s a c t i n g on an e l e m e n t It r e l a t e s to a s c a l a r f u n c t i o n of t h e v o l u m e c h a n g e of t h e e l e m e n t . T o p r o v e t h e h y p o t h e s i s tw o t h i n g s m u s t b e d e m o n s t r a t e d - - 21 f i r s t , tha t m e a n n o r m a l s t r e s s d o e s c o r r e l a t e w i t h b u l k d e n s i t y , and s e c o n d , t h a t d e v i a t o r i c s t r e s s d o e s not c o r r e l a t e w i t h b u l k d e n s i t y . T h e m e a n n o r m a l s t r e s s is t h e o n l y m e a s u r e of t h e s p h e r i c a l s t r e s s tensor. T h e d e v i a t o r h a s m a n y e x p r e s s i o n s w h i c h a r e m e a s u r e s of i t s intensity. S i n c e t h e f i r s t i n v a r i a n t of t h e d e v i a t o r i s z e r o , t h e s e c o n d i n v a r i a n t a s g i v e n in e q u a t i o n (12) i s p e r h a p s t h e b e s t o v e r a l l m e a s u r e . T h e m a x i m u m s h e a r s t r e s s a c t i n g o n a n e l e m e n t is a f u n c t i o n of t h e deviator a n d is g i v e n b y '’T 'm a x - 1//Z ( 0 “ l ~ O il I ) (13) w h e r e Q " T is t h e a l g e b r a i c a l l y l a r g e s t p r i n c i p a l s t r e s s an d Q " j j 7 ^s the a lg e b r a ic a lly s m a lle s t p rin c ip a l s t r e s s . T h e m a x i m u m s h e a r i s of s p e c i a l i n t e r e s t s in c e the C o u l o m b - M o h r yield f o r m u l a f o r s o ils d e a ls w ith the s h e a r i n g s t r e s s . 3 — C + T h e f o r m u l a is N ta n & ( ' w h e r e S i s t h e s h e a r i n g s t r e s s on t h e p l a n e of f a i l u r e , C t h e c o h e s i o n o f t h e s o i l , N t h e n o r m a l s t r e s s a c t i n g on t h e p l a n e of f a i l u r e a n d t h e a n g l e of f r i c t i o n of t h e s o i l . & T h e s h e a r i n g s t r e s s p r e d i c t e d b y th e C o u l o m b - M o h r f o r m u l a is not n e c e s s a r i l y t h e m a x i m u m s h e a r i n g s t r e s s at t h e po in t ; t h e y i e l d s t r e s s p r e s u m a b l y a p p r o a c h e s t h e m a x i m u m s h e a r i n g s t r e s s and h e n c e the i n t e r e s t . T h e m a x i m u m s h e a r s t r e s s is a n i n v a r i a n t of t h e s t r e s s s t a t e , i. e . i n d e p e n d e n t of t h e c h o i c e of coordinate axes. assum ed T h e m a x i m u m n o r m a l s t r e s s w h i c h S o e h n e (1953) to b e r e l a t e d to b u l k d e n s i t y is a l s o a f u n c t i o n of t h e d e v i a t o r . 22 L i k e t h e m a x i m u m s h e a r s t r e s s , t h e m a x i m u m n o r m a l s t r e s s is an i n v a r i a n t of t h e s t r e s s s t a t e . T h u s if it c a n b e s h o w n th a t t h e m e a n n o r m a l s t r e s s i s r e l a t e d to b u l k d e n s i t y a n d t h a t t h e m a x i m u m n o r m a l , t h e m a x i m u m s h e a r a n d t h e s e c o n d i n v a r i a n t of t h e d e v i a t o r a r e not r e l a t e d to b u l k d e n s i t y , t h e n t h e h y p o th e sis will be proved. It is i n t e r e s t i n g t h a t if a m e a n n o r m a l s t r e s s s t a t e is a p p l i e d to a s o i l s a m p l e s u c h a s c a n b e d o n e w i t h h y d r o s t a t i c p r e s s u r e , e q u a t i o n s (12) a n d (13) r e d u c e to z e r o s i n c e 0~m = (Ji = O il — U i i i . H y d r o s t a t i c p r e s s u r e c o m p a c t s s o i l ; h e n c e it i s u n l i k e l y t h a t e i t h e r t h e m a x i m u m s h e a r o r t h e s e c o n d i n v a r i a n t of t h e d e v i a t o r i s r e l a t e d t o bulk density. T h e t e r m s s t r e s s a n d p r e s s u r e h a v e b o t h b e e n a p p l i e d to f o r c e s a c t i n g in s o i l . A s i m p l i e d a b o v e , p r e s s u r e is t h e h i g h l y s p e c i a l i z e d s t r e s s s t a t e f o u n d in a f l u i d at r e s t . H ydrostatic p r e s s u re produces a t r u l y s p h e r i c a l s t a t e of s t r e s s an d s u c h a s t a t e g e n e r a l l y w i l l not b e f o u n d in s o i l . In t h i s s t a t e , t h e s t r e s s v e c t o r s o n a l l p o s s i b l e p l a n e s t h r o u g h a p o i n t a r e e q u a l in m a g n i t u d e to t h e p r e s s u r e at t h e p o in t. T h e d i r e c t i o n s of t h e s t r e s s v e c t o r s w i l l b e n o r m a l to a ll p o s s i b l e p la n e s th ro u g h the point. p r e s s u r e at t h e p o i n t . T h e r e f o r e a ty p ic a l c e ll would m e a s u r e In a g e n e r a l s t r e s s s t a t e , n e e d not b e t h e s a m e in m a g n i t u d e o r d i r e c t i o n a point. the s t r e s s v e c t o r s fo r e v e ry plane through F o r s u c h a s t r e s s s ta te , the ty p ic a l c e ll m e a s u r e s the n o r m a l 23 c o m p o n e n t of t h e s t r e s s v e c t o r w h i c h is b y d e f i n i t i o n s t r e s s on t h e p l a n e . the n o r m a l T h e n o r m a l s t r e s s is not n e c e s s a r i l y e q u a l 1o t h e p r e s s u r e , m e a n n o r m a l s t r e s s , a c t i n g at t h e p o in t . FIELD TESTS Theory. S i n c e t h e p r i n c i p a l d i r e c t i o n s of t h e s t r e s s s t a t e a r e not g e n e r a l l y k n o w n , s i x q u a n t i t i e s n e e d to b e d e t e r m i n e d to d e f i n e s t r e s s at a p o i n t . O n l y f o u r q u a n t i t i e s n e e d to b e m e a s u r e d if a s p e c i a l s t r e s s s t a t e s u c h a s d i s c u s s e d in t h e p r e v i o u s s e c t i o n is c o n s i d e r e d . When s u r f a c e l o a d s a r e s y m m e t r i c a l l y d i s t r i b u t e d w i t h r e s p e c t to a l i n e of s y m m e t r y , t h e v e r t i c a l p l a n e c o n t a i n i n g that l i n e i s a p r i n c i p a l p l a n e of s t r e s s . T h a t t h i s is t r u e c a n b e s e e n b y r e a s o n i n g a s f o l l o w s : If t h e f o r c e s a r e s y m m e t r i c a l l y d i s t r i b u t e d , t h e f o r c e s a c t i n g on o n e s i d e of t h e p l a n e of s y m m e t r y m u s t b e e q u a l to t h e f o r c e s a c t i n g on t h e o t h e r s i d e of t h e p l a n e . T h e c o n c l u s i o n is t h a t o n l y n o r m a l f o r c e s c a n a c t o n t h e p l a n e s i n c e no s h e a r i n g f o r c e s a r e p r e s e n t . H en ce, the p l a n e m u s t b e a p r i n c i p a l p l a n e of s t r e s s s i n c e b y d e f i n i t i o n a p r i n c i p a l p l a n e h a s o n l y n o r m a l s t r e s s a c t i n g on it. If, in a d d i t i o n , t h e s u r f a c e lo a d s h av e c i r c u l a r s y m m e t r y , the s t r e s s s ta te h a s r o ta ti o n a l s y m m e t r y a b o u t a v e r t i c a l a x i s t h r o u g h th e c e n t e r of t h e l o a d c i r c l e a n d a n y p l a n e c o n t a i n i n g t h i s a x i s is a p r i n c i p a l p l a n e . Fig. 4 shows a loading sy m m e tric a lly d istributed over a circle. T h e Z a x i s is t h e v e r t i c a l a x i s of s y m m e t r y a n d t h e YZ p l a n e is a t y p i c a l p l a n e of s y m m e t r y and t h e r e f o r e a p r in c ip a l plane. T h e s t r e s s s t a t e at a n y p o i n t in t h e p l a n e of s y m m e t r y w i l l b e a s in F i g . 5. At t h e p r e s e n t t i m e no s e n s i n g e l e m e n t s a r e a v a i l a b l e f o r F I G . 4 P L A N E OF S Y M M E T R Y U N D E R A CI RCULAR LOAD 26 Z X FIG. 5 S T R E S S E S ON VOLUME E L E M E N T IN A P L A N E OF S Y M M E T R Y 27 m e a su rin g shearing s tre s s e s . The unknown sh earin g s tr e s s , 'T y z i n F i g . 5 c a n b e c a l c u l a t e d b y u s i n g e q u a t i o n (1). N o te th a t t h e s t r e s s s t a t e u s e d in d e r i v i n g e q u a t i o n (1) is p r e c i s e l y t h e s t r e s s s t a t e s h o w n in F i g . 5. H e n c e b y m e a s u r i n g CTx * ( h ' C , and the n o r m a l u CTn on a p l a n e w h o s e n o r m a l b i s e c t s t h e p o s i t i v e Y a n d Z a x e s stress a n d l i e s in t h e YZ p l a n e , t h e s t r e s s t e n s o r a c t i n g at a n y p o in t in a p l a n e of s y m m e t r y c a n b e d e t e r m i n e d . T h e f o u r m e a s u r e m e n t s a b o v e m u s t b e m a d e at t h e s a m e p o i n t in o r d e r to d e t e r m i n e t h e s t r e s s t e n s o r at t h e p o in t. The m e a su re m e n ts c a n b e m a d e b y a p p e a l i n g to t h e r o t a t i o n a l s y m m e t r y u n d e r a c i r c u l a r load. T h e s a m e s t r e s s s t a t e a c t s at a n y p o i n t on t h e c i r c u m f e r e n c e of a h o r i z o n t a l c i r c l e c e n t e r e d on t h e a x i s of s y m m e t r y . The uniform ‘s t r e s s s ta t e is t r u e f o r a h o m o g e n e o u s soil sin ce fro m r o ta tio n a l s y m m e t r y e a c h p o i n t on t h e c o n c e n t r i c c i r c l e is s u b j e c t to t h e s a m e I forces. F o u r s t r e s s c e l l s ( C o o p e r , e t. a l . , 1957) c a n b e o r i e n t e d on a c i r c l e to m e a s u r e t h e s t r e s s t e n s o r a s s h o w n in F i g . 6. In F i g . 6 b y i m a g i n i n g t h e c e l l s r o t a t e d a r o u n d t h e c i r c l e u n t i l t h e y a l l r e s t at o n e p o i n t P , it c a n b e s e e n that t h e c e l l n o r m a l s l a b e l e d and ( J 7 w ill be m u tu a lly p e r p e n d i c u l a r while ( T n CTx - G~y ■ will m a k e equal a n g l e s of 45 d e g r e e s w i t h t h e Y an d Z a x e s an d a 90 d e g r e e a n g l e w it h the X axis. B y a l s o m e a s u r i n g th e b u l k d e n s i t y at t h e p o i n t s l a b e l e d B, a m e t h o d i s p r o v i d e d to m e a s u r e b o t h t h e s t r e s s t e n s o r an d b u l k d e n s i t y lat t h e s a m e p o i n t . ’ 28 O ' G"n BALLOON FI G. 6 CELL ORIENTATION 29 Indicating V o lu m e te r. A r e v i e w of a c c e p t e d m e t h o d s f o r m e a s u r i n g b u l k d e n s i t y ( R u s s e l l a n d B e l c e r e k , 1944; T o r s t e n s s o n an d E r i c k s o n , 1936; V e i h m e y e r , 1929) r e v e a l e d th a t t h e y w o u l d not b e a c c e p t a b l e f o r t h e ex p e rim e n t indicated above. All m e th o d s r e q u i r e d a l a r g e s a m p l e c o m p a r e d w i t h t h e v o l u m e of t h e s o i l in t h e i m m e d i a t e v i c i n i t y of t h e s tr e s s cells. S i n c e a d i s t r i b u t i o n of b u l k d e n s i t y w o u l d b e e x p e c t e d , c o n s i d e r a b l e e r r o r would r e s u l t . F u r th e r m o r e , b e c a u se a soil sa m p le h a d to b e r e m o v e d , o n l y o n e s t r e s s - b u l k d e n s i t y m e a s u r e m e n t c o u l d b e m a d e f o r a g i v e n t e s t s e t up. sum ing. T h is would m a k e t e s tin g v e r y t im e c o n ­ T h e r e q u i r e m e n t s of t h e a b o v e o b j e c t i v e s w e r e m e t b y d e v e lo p in g an in d ic a tin g v o l u m e t e r . B r i e f l y , t h e v o l u m e t e r c o n s i s t s of a c a p i l l a r y t u b e c o n n e c t e d to a n o n - c o l l a p s i b l e p l a s t i c t u b in g . T h e p l a s t i c t u b i n g is in t u r n c o n n e c t e d to a s m a l l b a l l o o n ( a p p r o x i m a t e l y 15 c c . ) # W h e n t h e b a l l o o n , f i l l e d w i th s o i l , is s u r r o u n d e d w i t h a l a r g e v o l u m e of s o i l of t h e s a m e t y p e , t h e v o l u m e of t h e b a l l o o n c a n b e c o n s i d e r e d a p oint m a s s . As the s o i l c o m p a c t s , t h e b u l k v o l u m e of t h e s o i l d e c r e a s e s and t h e a i r in t h e b a l l o o n is f o r c e d o ut. T h e d e c r e a s e in v o l u m e of t h e b a l l o o n i s m e a s u r e d b y r e c o r d i n g t h e d i s p l a c e m e n t of a m e r c u r y b u b b l e in th e c a p i l l a r y tube. T h e v o l u m e p e r l i n e a r , u n it of the c a p i l l a r y t u b e is e a s i l y d eterm ined. T h e v o l u m e t e r p e r m i t s m e a s u r i n g th e v o l u m e c h a n g e at a p o in t in t h e s o i l o v e r a r a n g e of b u l k d e n s i t y w i th o u t d i s t u r b i n g t h e s o i l 30 s a m p l e u n til the t e s t is c o m p le te d . T h e f i n a l v o l u m e of t h e s o i l s a m p l e i s d e t e r m i n e d b y w e i g h i n g in a n d out of w a t e r . T h e p r i n c i p l e of t h e v o l u m e t e r w a s c h e c k e d u s i n g tw o s o i l t y p e s in a s o i l b o x . T h r e e of t h e b a l l o o n s w e r e p l a c e d in t h e c e n t e r of a b o x c o n t a i n i n g s i x i n c h e s of s o i l a n d a u n i f o r m l o a d w a s a p p l i e d o v e r t h e en tire surface. T h e c h a n g e of v o l u m e of t h e s o i l in t h e b o x w a s d e t e r ­ m i n e d b y m e a s u r i n g d i s p l a c e m e n t of t h e p l u n g e r . F r o m the in itia l w e i g h t a n d m o i s t u r e c o n t e n t of t h e s o i l in t h e b o x , t h e a v e r a g e b u l k d e n s i t y of t h e s o i l c o u l d b e c o m p u t e d . T h i s a v e r a g e w a s c o m p a r e d w it h the a v e r a g e given by the indicating v o lu m e te r. T h e r e s u l t s of t h i s t e s t a r e s h o w n in T a b l e A. Table A C o m p a r i s o n of I n d i c a t i n g V o l u m e t e r a n d S o i l B o x B u l k D e n s i t i e s in T w o S o il T y p e s M aum ee Sandy Loam Soil box g m / cc 1.14 1.20 1.2 6 1.33 1.39 1.45 1.52 Indicating V olum eter g m / cc 1.16 1.19 1.24 1.29 1.36 1.43 1.52 Soil Box g m / cc 1.24 1.31 1.37 1.44 1.51 1.57 Indicating V olum eter gm / c c 1.24 1.29 1.35 1.41 1.48 1.55 T h e o n l y q u e s t i o n a b o u t t h e p r i n c i p l e of t h e v o l u m e t e r is w h e t h e r t h e b a llo o n m e m b r a n e i n t e r f e r e s with the e q u ilib r iu m b e tw e e n the s o il in s id e and o u tsid e the balloon. d o e s not i n t e r f e r e . T he d a ta in d ic a te the b a llo o n m e m b r a n e H o v a n e s i a n (1958) f u r t h e r d e v e l o p e d t h e v o l u m e t e r 31 b y m a k i n g it r e c o r d i n g a s w e l l a s i n d i c a t i n g , a n d h i s s t u d i e s g i v e a n a d d i t i o n a l t e s t of t h e p r i n c i p l e . A pparatus. T h e f a c i l i t i e s of t h e N a t i o n a l T i l l a g e M a c h i n e r y L a b o r a t o r y , A u b u r n , A l a b a m a , w e r e m a d e a v a i l a b l e to c o n d u c t t h e t e s t s . E leven d i f f e r e n t s o il s ty p e s a r e a r r a n g e d in b in s so that the s o il c a n be p r e p a r e d m e c h a n i c a l l y ; in a d d i t i o n , o t h e r l a b o r a t o r y e q u i p m e n t f o r m a k i n g p h y s i c a l m e a s u r e m e n t s of s o i l p r o p e r t i e s e n a b l e d a m a x i m u m of i n f o r m a t i o n to b e o b t a i n e d w i t h a m i n i m u m of t i m e a n d e f f o r t . d a ta w e r e tak en with th e s e f a c ilitie s . All The s t r e s s cell used was s im iia r to t h e t y p e p r e s e n t e d b y C o o p e r , e t . a l . (1957.) M ulti-channel re c o rd in g o s c i l l o g r a p h s w i t h a p p r o p r i a t e s t r a i n g a g e a m p l i f i e r s w e r e u s e d to r e c o r d cell readings. T h e bulk d e n s ity w a s m e a s u r e d by r e a d i n g the i n d i c a t i n g v o l u m e t e r s h o w n in F i g . 7. The weight and the final v o lu m e of s o i l i n t h e b a l l o o n s w e r e o b t a i n e d u s i n g a s m a l l s c a l e a c c u r a t e to 0.01 g m . T h i s i s s h o w n in F i g . 8. T h e l o a d i n g p l a t e (20 in. in d i a m e t e r ) a n d t h e m e t h o d of a p p l y i n g t h e l o a d a r e s h o w n in F i g . 9. The larg e p o w e r c a r of t h e L a b o r a t o r y w a s c l a m p e d to t h e r a i l s of t h e s o i l b i n s to p r o v id e the n e c e s s a r y s u p p o rt f o r applying the load. P r o c edure. T h e s o i l w a s p r e p a r e d b y r o t a r y t i l l i n g t o t h e d e p t h of t h e s o i l b i n s ( a p p r o x i m a t e l y 24 in.) tests. It w a s le f t in t h i s l o o s e c o n d i t i o n f o r t h e The soil w as m o is te n e d e ith e r b e fo re o r afte r tilling depending Fig. 7 Indicating v o lu m eter 32 33 u p o n s o i l t y p e a n d c o n v e n i e n c e in m a n i p u l a t i n g t h e s o i l . This tre a tm e n t l e f t t h e s o i l in a n e a r l y h o m o g e n e o u s s t a t e . T o c o n d u c t a t e s t , a p l a t f o r m w a s p l a c e d on t h e l o o s e s o i l a n d an a r e a f o u r f e e t s q u a r e e x c a v a t e d to t h e d e p t h of t h e b i n s . The soil was r e p l a c e d b y h a n d a s c a r e f u l l y as p o s s i b l e to t h e d e s i r e d d e p t h a n d le v e le d with a te m p la te . The c e lls and balloons w e r e then p o s itio n e d as s h o w n in F i g . 6 an d 11. T h e r e m a in in g soil w as then r e p l a c e d and again leveled. T h e lo ad in g p la te w as p r o p e r l y p o sitio n e d and the i n s t r u m e n t s w e r e connected. 0 a n d 2 6 .5 p s i . T h e d a t a w e r e t a k e n at p r e s c r i b e d i n t e r v a l s b e t w e e n T h e g e n e r a l v i e w of t h e t e s t s e t up is s h o w n in F i g . 10. U p o n c o m p le tin g a te s t, the c e lls and the b a llo o n s w e r e r e m o v e d and t h e b a l l o o n s w e i g h e d in a n d out of w a t e r to d e t e r m i n e t h e i r f i n a l volum e. test. T h e e q u i p m e n t w a s m o v e d to a new a r e a to c o n d u c t t h e n ex t T h e m o i s t u r e c o n t e n t a n d o t h e r p h y s i c a l p r o p e r t i e s of t h e s o i l s t e s t e d a r e g i v e n in T a b l e B. R esults. T o p r o p e r l y test the m e a n n o r m a l s t r e s s - b u l k d e n s ity h y p o th e s is o r a n y o t h e r h y p o t h e s i s , d i f f e r e n t s t r e s s s t a t e s m u s t b e a p p l i e d to a v o l u m e e l e m e n t ; if d i f f e r e n t s t r e s s s t a t e s a r e not u s e d , o n l y p r o p o r ­ tional loading r e s u l t s . In p r o p o r t i o n a l l o a d i n g , a l l of t h e s t r e s s t e n s o r c o m p o n e n t s a r e l i n e a r l y r e l a t e d to the a p p l i e d lo a d . The lin earity m e a n s t h a t a l l of t h e i n v a r i a n t s of t h e s t r e s s t e n s o r w i l l b e l i n e a r l y r e l a t e d a n d c o n s e q u e n t l y a l l of th e i n v a r i a n t s w i l l a p p e a r to b e 34 F ig . 0 G e n e r a l v ie w of t e s t eq u ip m e n t (a b o v e ) F ig # 11 C e ll o r ie n ta tio n (below ) 35 Table B PHYSICAL P R O P E R T IE S OF SOILS U S E D IN T E S T S (B elow in % o r g a n i c f r e e m a t e r i a l ) H iw assee Sandy Loam D ecatur Silty C lay Loam Lloyd Clay 1.2 0,4 1.6 2.6 8.4 5.8 54 ,9 32,7 26 3 1.5 2.2 4.3 3.7 7.4 5.6 1 7,2 112 59.6 13 03 17,67 32 08 P l a s t i c i t y index 6.51 11.07 24.29 15 a t m . t e n s i o n 5,5 10.0 22.5 16 19 G ravel Very c o a r s e sand C o a r s e sand M edium sand F in e sand V e r y fine sand T o t a l s ilt F i n e s ilt C lay 0.1 2.0 '1 9.5 23.3 25.4 2.9 10,9 8.3 16.0 ( B e l o w in % m o i s t u r e ) L o w e r p la slic lim it A v e r a g e m o i s t u r e content of s o i l d u r i n g t e s t s 9 36 r e l a t e d to b u l k d e n s i t y . D i f f e r e n t s t r e s s s t a t e s w e r e o b t a i n e d by v a r y i n g t h e l o c a t i o n of t h e c e l l a n d b a l l o o n c i r c l e s . i n d i c a t e d tha t t h e b e s t v a r i a t i o n s o b t a i n a b l e P re lim in a ry tests u n d e r the 20 in. d i a m e t e r lo a d in g p la te w e r e as follows: 1. A 12 in. d i a m e t e r c i r c l e i n i t i a l l y 5 in. b e l o w t h e l o a d i n g p l a t e . 2. A 12 in. d i a m e t e r c i r c l e i n i t i a l l y 12 in. b e l o w t h e l o a d i n g p l a t e . 3. A n 18 in. d i a m e t e r c i r c l e i n i t i a l l y 18 in. b e l o w t h e l o a d i n g p l a t e . 4. A 20 in. d i a m e t e r c i r c l e i n i t i a l l y 18 in. b e lo w t h e l o a d i n g p l a t e . F i v e r e p l i c a t i o n s at e a c h s t r e s s s t a t e w e r e t a k e n an d t h e r e s u l t s a r e r e p o r t e d in T a b l e s I t h r o u g h XII in t h e A p p e n d i x . The bulk d en sity r e a d i n g s r e p o r t e d a r e a n a v e r a g e of t h e f o u r m e a s u r e m e n t s t a k e n a ro u n d the c ir c le . T h e c h o s e n m e a s u r e s of th e s p h e r i c a l an d d e v i a t o r t e n s o r s w e r e c o m p u te d f ro m the fo u r m e a s u r e d s t r e s s v a lu e s u s in g the a p p r o p r i a t e f o r m u l a e , n a m e l y e q u a t i o n s (7), (11), (12) an d (13). ordered p r i n c i p a l s t r e s s e s w e r e d e t e r m i n e d u s i n g e q u a t i o n (7) an d the m e a s u r e d v a l u e of gave The alg eb raically CTT CTx- T h e l a r g e s t v a l u e f o r ( T f r o m e q u a t i o n (7) T h e s m a l l e s t v a l u e f r o m e q u a t i o n (7) o r CTx g a v e d e p e n d in g upon w h ich was the s m a l l e r value. ( m i It s h o u l d b e n o t e d that in all c a lc u la tio n s a c o m p r e s s i v e s t r e s s h as b e e n c o n s id e re d p o sitiv e w h e r e a s u s u a l l y t h e p o s i t i v e s e n s e is r e s e r v e d f o r t e n s i o n f o r c e s ; h e n c e n e g a t i v e s t r e s s e s in T a b l e s I t h r o u g h XII i n d i c a t e t e n s i l e s t r e s s e s in the soil. E q u a t i o n (9) w a s u s e d to c a l c u l a t e t h e a n g l e ( b . The 37 la rg e st principal s tr e s s (n was substituted for Q ~ s o that r e p r e s e n t s t h e a n g l e b e t w e e n th e Z a x i s a n d the l a r g e s t p r i n c i p a l s t r e s s (Tc . S i n c e t h e o r i e n t a t i o n of principal s tre s s e s , c o m p l e t e l y d e t e r m i n e s t h e o r i e n t a t i o n of the t r i a d of p r i n c i p a l s t r e s s e s . M IS T IC , a n e l e c t r o n i c d i g i t a l c o m p u t e r at M i c h i g a n S t a t e U n i v e r s i t y , w a s u s e d to m a k e t h e l e n g t h y c a l c u l a t i o n s i n v o l v e d in e v a l u a t i n g t h e a b o v e e q u a t i o n s . T h e r e w e r e s e v e r a l s o u r c e s of e r r o r in t h e e x p e r i m e n t . The a s s u m p t i o n t hat e a c h c e l l w o u l d b e s u b j e c t e d to t h e s a m e s t r e s s s t a t e o n o n e of t h e c o n c e n t r i c c i r c l e s is not c o m p l e t e l y t r u e . This was r e v e a l e d b y v a r i a t i o n in b u l k d e n s i t y r e a d i n g s a r o u n d t h e c i r c l e a n d b y c o m p a r i s o n of t h e r e a d i n g s an d o r i e n t a t i o n s of c e l l s w h o s e n o r m a l s w e r e in t h e YZ p l a n e . An e x a m p l e of t h e e x t r e m e v a r i a t i o n o b s e r v e d in f o u r b u l k d e n s i t y r e a d i n g s a r o u n d a c i r c l e f o r a 26.5 l o a d i n g in H i w a s s e e s o i l is: 1. 56, 1.53, 1.58, 1.49. bulk density read in g s T h e a v e r a g e s p r e a d of f o u r f o r a ll t e s t s w a s a p p r o x i m a t e l y .04 g m / c c . A dditional e r r o r o c c u rre d d u r i n g c e r t a i n t e s t s w h e n s o m e of t h e c e l l s r o ta te d from th e ir o rig in al o rientations. In c a s e s w h e r e t h e r o t a t i o n w a s m o r e t h a n 10°, t h e r o t a t e d c e l l r e a d i n g w a s c o r r e c t e d b y a d j u s t i n g it c l o s e r to t h e a v e r a g e of c e l l s w h i c h m e a s u r e d th e s a m e s t r e s s c o m p o n e n t but d id not r o t a t e . n e c e s s a r y on o n l y 10 of t h e v a r i a b i l i t y of t h e s o i l i t s e l f . W h i l e t h i s is h i g h l y a r b i t r a r y , it w a s 480 t o t a l r e a d i n g s . F i n a l l y , t h e r e w a s th e T h e m a g n i t u d e of s o i l v a r i a t i o n d u e to th e f o r t u i t o u s s t r u c t u r e of t h e s o i l h a s n e v e r b e e n e v a l u a t e d but e x p e r i e n c e 38 d i c t a t e s th a t it is p r o b a b l y m u c h l a r g e r t h a n a n y of t h e a b o v e e r r o r s . F u r t h e r m o r e , s o m e of t h e v a r i a t i o n in c e l l r e a d i n g s an d b u l k d e n s i t y r e a d i n g s w a s p r o b a b l y d u e to s o i l v a r i a t i o n r a t h e r t h a n e r r o r s in ex p erim en tal procedure. T h e v a r i a t i o n w h i c h c a n b e s e e n b y e x a m i n i n g t h e d a t a in T a b l e s I t h r o u g h XII r e q u i r e s t h a t s o m e m e t h o d of s t a t i s t i c a l a n a l y s i s be em ployed. A plot of a n y of t h e c o m p u t e d v a l u e s s u c h a s or (T r e s u l t s in a s t r a i g h t l i n e w h e n t h e s t r e s s is p l o t t e d on a l o g a r i t h m i c a x i s a n d b u l k d e n s i t y on t h e l i n e a r a x i s . This exponential r e la tio n s h ip h a s b e e n o b s e r v e d b y o t h e r i n v e s t i g a t o r s (S oehne, 1953; H o v a n e s i a n , 1958.) T h e s t r a i g h t l i n e p lo t o f f e r e d a s i m p l e g r a p h i c a l m e t h o d f o r a v e r a g i n g t h e d a t a a n d o b t a i n i n g an e s t i m a t e of t h e s t a n d a r d e r r o r . T h e b e s t f i t t i n g s t r a i g h t l i n e w a s d r a w n an d a d j u s t e d s l i g h t l y u n t i l th e t o t a l d i s t a n c e s f r o m t h e p o i n t s on e a c h s i d e of t h e l i n e w e r e e q u a l . T h i s p r o c e d u r e l o c a t e d t h e a v e r a g e l i n e w i t h a r e a s o n a b l e d e g r e e of a c c u r a c y a n d c o u l d b e c a r r i e d out v e r y r a p i d l y . A ssum ing a n o rm a l d i s t r i b u t i o n of e r r o r , o n e s t a n d a r d e r r o r w o u ld i n c l u d e 68.3% of t h e p lotted points. T h u s th e s t a n d a r d e r r o r lines w e r e lo c a te d so that a p p r o x i m a t e l y 68.3% of t h e p l o t t e d p o i n t s f e l l b e t w e e n t h e two s t a n d a r d e r r o r lines. An e x a m p l e of t h i s p r o c e d u r e i s s h o w n in F i g . 12. It is a d m i t t e d t h a t t h e p r o c e d u r e of a v e r a g i n g and d e t e r m i n i n g s t a n d a r d e r r o r is n ot s t a t i s t i c a l l y p r e c i s e , but it is r a p i d an d d o e s g i v e an e s t i m a t e of v a r i a t i o n w h i c h is r e a s o n a b l y m e a n i n g f u l . : 39 F I G . 12 30- STANDARD AVERAGE VERSUS ~ (/) - 1 0 ' MEAN BULK AND STRESS DENSITY i ne rror 86- 4 - MEAN STRESS 3 20 OF ERROR DATA F R O M 5 " - 1 2 “ S T R E S S S T A T E IN D E C A T U R S O I L 2 - / 1 _ il-------------- I-------------- 11-------------- 11---------------L- 1.1 1.2 1-3 BULK DENSITY 1.4 (gm/cc) 1.5 D 40 T h e f o u r m e a s u r e s of t h e s p h e r i c a l s t r e s s t e n s o r an d t h e d ev iato r s tr e s s te n s o r - nam ely j j ^ f an d ~ w ere a v e r a g e d i n d i v i d u a l l y u s i n g t h e p r o c e d u r e s h o w n in F i g . 12 f o r e a c h s t r e s s s t a t e an d e a c h s o i l type. T h e ir individual s ta n d a rd e r r o r w a s a l s o d e t e r m i n e d an d t h e r e s u l t s s u m m a r i z e d in F i g . 13 t h r o u g h 24. If a q u a n t i t y is r e l a t e d to b u l k d e n s i t y in a g e n e r a l m a n n e r , t h e a v e r a g e l i n e s f r o m e a c h d i f f e r e n t s t r e s s s t a t e s h o u l d not b e s i g n i f i c a n t l y d i f f e r e n t f o r a g iv e n soil. F i g u r e s 13, 14, an d 15 i n d i c a t e no. s i g n i f i c a n t d i f f e r e n c e f o r t h e m e a n , n o r m a l s t r e s s in a l l t h r e e s o i l t y p e s . S im ilarly, if a q u a n t i t y is not r e l a t e d to b u l k d e n s i t y in a g e n e r a l m a n n e r , th e a v e r a g e lin e s fro m e a c h d ifferen t s t r e s s s ta te should be s ig n ific a n tly d i f f e r e n t f o r a g iven soil. and F i g u r e s 16 t h r o u g h 24 s ho w t h a t I I g t a r e not a l l s i g n i f i c a n t l y d i f f e r e n t . (Ti • T™* T h u s b a s e d upon the data, it c a n n o t b e d e t e r m i n e d w h e t h e r c o m p a c t i o n is i n d e p e n d e n t of th e deviator. F u r t h e r m o r e , b e c a u s e of t h e l a r g e v a r i a t i o n , s t r e s s is not c l e a r l y r e l a t e d to c o m p a c t i o n . mean norm al F i g u r e s 13 t h r o u g h 24 r e v e a l , h o w e v e r , that of t h e f o u r i n v a r i a n t s i n v e s t i g a t e d , t h e m e a n n o r m a l s t r e s s i s t h e b e s t i n v a r i a n t to r e l a t e to b u l k d e n s i t y . T h e d a ta in d ic a te s that f o r a g iv e n m e a s u r e d s t r e s s s ta te ( l o c a t i o n of a c i r c l e ) , t h e m a g n i t u d e of t h e p r i n c i p a l s t r e s s e s i s v e r y n e a r l y i n d e p e n d e n t of s o i l ty p e . T o o b t a i n a p h y s i c a l p i c t u r e of t h e a v e r a g e s t r e s s s t a t e s m e a s u r e d at t h e f o u r l o c a t i o n s of c i r c l e s , th e t h r e e p r i n c i p a l s t r e s s e s f o r t h e 26.5 p s i l o a d i n g in a l l s o i l t y p e s w e r e 41 FIG. 13 MEAN N ORMAL S T R E S S BULK DENSI T Y IN HI WASSEE SOIL STRESS (psi) 20 MEAN STANDARD ERROR BULK DENSITY (gm/ cc) VS 42 FIG. 14 MEAN NORMAL S T R E S S BUL K DENSITY IN LLOYD VS SOI L io S TANDARD ERROR MEAN STRESS (p s i ) 20 i BULK i DENSITY (gm/ cc) 43 FIG. 15 M E A N N O R M A L S T R E S S VS BULK D E N S I T Y IN DECAT UR SOI L 20 10 (ft Q. R R CO cn 5 UJ tt: H CO < UJ 1.2 1.4 1.3 BULK DENSITY (GM/CC) 44 FI G. 16 MAXIMUM NORMAL (psi) STRESS VS BULK D E N S ITY IN HI W A S S E E SOIL STANDARD ERROR , MAX NORMAL STRE S S 1 0 1 1.2 BULK 1.4 D EN SITY 1. 6 ( gm/ cc) 45 FI G. 17 MAXIMUM S T R E S S VS BULK NORMAL D EN SITY (psi) 20 STANDARD ERROR 5 MAX NORMAL STRESS 0 2 .9 0 BULK D E N S ITY (g m / cc) 46 FIG. 18 STRESS IN MAXIMUM NORMAL VS BULK DENSITY DECATUR SO IL 20 (A a. 10 CO CO UJ 0£ I- STANDARD ERROR e co 5 a: o X 2 < * 1 1.2 BULK 1.3 DENSITY 1.4 (g m / cc) 47 FI G. 19 M A X I M U M S H E A R I N G S T R E S S V S BULK D E N S I T Y IN HI W A S S E E S O I L 20 MAX SHEARI NG STRESS (p s i ) STANDARD ERROR / i B ULK DENSITY i i (g m / cc) 48 FIG. 2 0 M A X I M U M S H E A R I N G S T R E S S VS B U L K D E N S I T Y IN L L O Y D S O I L 20 STANDARD ERROR / Cft CL CD C/> UJ cc h c n o z cr < UJ X X i BULK DENSITY i 1 ( gm/ c c ) i 49 FIG. 21 STRESS MAXIMUM SHEARING VS BULK D E N S I T Y IN D E C A T U R S O I L 20 STANDARD ERROR CO CL 10 co co UJ (T h CO CD z or <£ UJ X CO X l 1 BULK i 1 D E N S IT Y ( i gm/ c c ) 50 FIG. 2 2 SECOND I N V A R I A N T OF D E V I A T O R VS BULK D E N S I T Y HI WAS SEE S O I L 100 STANDARD ERROR ' 50 20 - c c UJ 1 BULK 1 D EN SIT Y 1 1 U m/ cc) FIG. 2 3 S E C O N D I N V A R I A N T OF D E V I A T O R V S BULK D E N S I T Y , 00L IN L L O Y D S O I L 50 STANDARD ERROR CM (A CL 20 10 o c . < > 5 o o 2 UJ to 4b i .9 0 BULK 1 1.0 D E N S ITY 1.1 (ew/cc) 52 FIG. 2 4 SECOND INVARIANT OF D E V I A T O R V S BULK D EN SITY |0CL IN D E C A T U R SOIL CM CO Q. STANDARD ERROR < c r 5 n r* z o o UJ CO 1 i BULK DENSITY (o m / rm cc) 53 averaged. T h e s e v a l u e s w e r e t h e n u s e d to c o n s t r u c t t h r e e - d i m e n s i o n a l M o h r ’s c i r c l e s f o r e a c h s t r e s s s t a t e . F i g . 25 a n d 26. T h e s e c i r c l e s a r e s h o w n in N ote that the r e p r e s e n t a t i o n h a s the s h e a r i n g s t r e s s plotted against n o rm a l s tr e s s . T h e s h a d e d a r e a of t h e c i r c l e s r e p r e s e n t s t h e p o s s i b l e c o m b i n a t i o n s of n o r m a l s t r e s s a n d s h e a r i n g s t r e s s ( s t r e s s v e c t o r s ) t hat c a n o c c u r on a n y p l a n e t h r o u g h t h e p o in t represented. F o r e x a m p l e , if o n l y a m e a n n o r m a l s t r e s s s t a t e e x i s t e d at a p o i n t ( h y d r o s t a t i c p r e s s u r e ) M o h r ' s c i r c l e s w o u l d s h r i n k to a point. If a c y l i n d r i c a l s t a t e of s t r e s s e x i s t e d (Q"j ( j i i i ) the c i r c l e s w o u l d b e c o m e o n e c i r c l e w i t h t h e s t r e s s v e c t o r s r e s t r i c t e d to l i e on t h e c i r c u m f e r e n c e of t h e c i r c l e . F i g u r e s 25 an d 26 c l e a r l y i n d i c a t e t h a t t h e r e w a s v e r y little* d i f f e r e n c e in t h e f o u r s t r e s s s t a t e s . C onsequently, t h e s t r e s s s t a t e s r e p r e s e n t a c a s e of n e a r l y p r o p o r t i o n a l l o a d i n g . T h e r e f o r e t h e l a c k of s i g n i f i c a n t d i f f e r e n c e in m e a s u r e s of t h e d e v i a t o r d o e s not s e e m u n u su al. In s p i t e of t h e s m a l l d i f f e r e n c e in s t r e s s s t a t e s , t h e m e a n n o r m a l s t r e s s s t i l l a p p e a r s to b e t h e b e s t i n v a r i a n t to r e l a t e to b u l k d e n s i t y . T h e l a c k of d i f f e r e n c e in s t r e s s s t a t e s p r e s e n t s s o m e i n t r i g u i n g p o ssib ilities. F r o m t h e T h e o r y of E l a s t i c i t y , p r e d i c t e d s t r e s s d i s t r i b u t i o n s u n d e r a c i r c u l a r l o a d i n d i c a t e that t h e two s m a l l e r p r i n c i p a l s t r e s s e s d e c r e a s e in m a g n i t u d e f a s t e r w i t h d e p t h t h a n t h e la rg e st prin cip al s tre s s . 0 “i to an a v e r a g e of O ( C o r p s of E n g i n e e r s , 1954.) ii an d 0 7 u T h e r a t i o of is a p p r o x i m a t e l y 3.5 at a l o c a t i o n CM CO =' Ui CO h - — cn cr LlI < -I o CM S T AT ES ID STRESS 11 18"-20" CO AND CL FOR < 18"-12" to CIRCLES to h- MOHR'S h- 3 DIMENSIONAL UJ FIG. 25 54 GO 55 o tr o. GO CO = o UJ lf> CO UJ _] o r < o lO CO £ X o o CD Si CO z UJ I< CM — CO Is y y y _ CM -JCT> = in CO UJ Q ro v > UJ o: CO AND — CD OJ CD u_ 5"-12" 00 FOR vo ii UJ STATES co n STRESS UJ c 18"-18" lO 56 s i m i l a r t o t h e 5 ” - 1 2 ” s t r e s s s t a t e in t h i s e x p e r i m e n t . F o r the 5 ” - 1 2 ” s t r e s s s t a t e , t h e m e a s u r e d r a t i o is 2.4 . At a l o c a t i o n s i m i l a r to t h e 1 8 ,x- 2 0 ’ s t r e s s s t a t e , t h e r a t i o p r e d i c t e d b y E l a s t i c i t y i s a p p r o x i m a t e l y 2 2 . 6 w h i l e t h e m e a s u r e d r a t i o is 2.7. At t h e 1 8 " - 1 2 ” s t r e s s s t a t e E l a s t i c i t y p r e d i c t s a r a t i o of 4 1 . 0 w h i l e t h e m e a s u r e d r a t i o is 6.6. These r a t i o s i n d i c a t e r a t h e r s t r o n g l y th a t s o i l in a l o o s e p h y s i c a l s t a t e d o e s not follow an e l a s t i c law. It c a n a l s o b e s h o w n t hat t h e m a g n i t u d e s of the m e a s u r e d p rin c ip a l s t r e s s e s differ g r e a tly from those p re d ic te d by E l a s t i c i t y , p a r t i c u l a r l y at g r e a t e r d e p t h s . It is a d m i t t e d th a t t h e a b o v e c o m p a r i s o n s a r e not c o m p l e t e l y r i g o r o u s , but t h e v e r y l a r g e d i f f e r e n c e s l e n d v a l i d i t y to t h e c o n c l u s i o n s . A nother i n d i c a t i o n of t h e s i m i l a r s t r e s s s t a t e s is t h e i m p l i c a t i o n of a y i e l d c o n d i t i o n f o r s o i l . T h e C o u l o m b - M o h r f o r m u l a ( e q u a t i o n 14) w a s d i s c u s s e d a s a p r o p o s e d y i e l d c o n d i t i o n f o r s o il . T he condition d o e s not s p e c i f y t h e p l a n e on w h i c h y i e l d o c c u r s b u t m e r e l y s t a t e s th a t on s o m e p la n e the im p lie d r e la tio n s h ip e x is ts . is c o r r e c t , If t h i s y i e l d c o n d i t i o n it i s p o s s i b l e that, w h e n t h e r a t i o of t h e l a r g e s t p r i n c i p a l s t r e s s to t h e s m a l l e r p r i n c i p a l s t r e s s e s e x c e e d s a c e r t a i n v a l u e , y i e l d s e t s in; t h e s o i l t h e n r e l i e v e s i t s e l f b y d e f o r m i n g s u c h t h a t a new e q u i l i b r i u m i s e s t a b l i s h e d w h i c h d o e s not v i o l a t e t h e y i e l d c o n d i t i o n . In t h i s m a n n e r , t h e p o s s i b l e s t r e s s s t a t e s t h e s o i l c a n s u p p o r t m a y b e restricted. A s s u m i n g t h e a b o v e a r g u m e n t is t r u e , p o s s i b l y t h e l a r g e s t p r i n c i p a l s t r e s s w i l l a l w a y s b e v e r y n e a r l y p r o p o r t i o n a l to t h e m e a n n o r m a l s t r e s s an d , c o n s e q u e n t l y , as w ell as , U i c a n b e r e l a t e d to b u l k d e n s i t y In p r a c t i c e , h o w e v e r , t h e m e a n n o r m a l s t r e s s is m u c h e a s i e r to m e a s u r e t h a n t h e m a x i m u m n o r m a l s t r e s s . M ean n o r m a l s t r e s s r e q u i r e s m e a s u r e m e n t of t h e n o r m a l s t r e s s o n t h r e e m u tu a lly p e r p e n d ic u la r planes. To m e a s u r e the m a x im u m n o r m a l s t r e s s , t h e e n t i r e s t r e s s t e n s o r g e n e r a l l y n e e d s to b e d e t e r m i n e d and the m a x im u m n o r m a l s t r e s s then c a lc u la te d u sin g m e th o d s d i s c u s s e d in the p r e c e d in g sectio n . Thus m e a n n o r m a l s t r e s s is p r e f e r a b l e . An in d ic a tio n about the s t r e s s d is tr ib u tio n u n d e r a c i r c u l a r load is s h o w n in F i g . 27. T h e m e a n s t r e s s f o r e a c h a p p l i e d l o a d in a g i v e n s o i l a n d at a g i v e n s t r e s s s t a t e w a s a v e r a g e d . p lo tte d a g a in s t the ap p lied load. T his a v e r a g e w as then As c a n b e s e e n , t h e d i s t r i b u t i o n of m e a n n o r m a l s t r e s s a p p e a r s to b e i n d e p e n d e n t of s o i l ty p e . The i n d e p e n d e n c e i m p l i e s t h a t t h e m e a n n o r m a l s t r e s s d i s t r i b u t i o n is d e t e r m i n e d b y t h e g e o m e t r y of t h e s i t u a t i o n an d is i n d e p e n d e n t of s o i l type. If t h e a b o v e i s g e n e r a l l y t r u e , it w i l l b e p o s s i b l e to d e t e r m i n e a s t r e s s d is tr ib u tio n f o r a given w h e e l o r im p le m e n t and the d is tr ib u tio n w i l l a p p l y to a l l s o i l s ; t h e d i f f e r e n c e in c o m p a c t i o n w i l l r e s u l t f r o m d i f f e r e n c e s in t h e s t r e s s - b u l k d e n s i t y r e l a t i o n s h i p f o r e a c h d i f f e r e n t soil. B e c a u s e of t h e c o m p l e x p h y s i c a l n a t u r e of s o i l , t h e r e i s s o m e q u e s t i o n a s t o t h e v a l i d i t y of a p p l y i n g c o n t i n u u m m e c h a n i c s to s o i l , p a r t i c u l a r l y in a l o o s e c o n d i t i o n . The Lloyd clay w as a s e v e r e test fo r 58 FI G. 2 7 28 - DISTRIBUTION S T R E S S IN T H R E E AND T H R E E S T R E S S 8 - 20 20 - APPLIED LOAD ( psi ) OF M E A N ( 3 HIWASSEE O LLOYD A 4 MEAN 6 STRESS DECATUR 8 (psi) SOILS STATES 59 the above q u estio n . T h e s o i l h a d b e e n s e v e r e l y c o m p a c t e d , and c o n s e q u e n t l y w a s l u m p y , g r a n u l a r a n d c o m p o s e d of h a r d p a r t i c l e s . T h a t a g o o d , s m o o t h r e l a t i o n s h i p c o u l d b e o b t a i n e d b e t w e e n s t r e s s and b u l k d e n s i t y r e a d i n g s i m p l i e s t h a t t h e c o n c e p t of t h e c o n t i n u u m w i l l a p p l y to l o o s e s o i l s . The sm o o th r e la tio n s h ip v e r ifie s the sta te m e n t t h a t t h e c o n t i n u u m c o n c e p t is not v i o l a t e d a s lo n g a s a r e a s l a r g e in c o m p a r i s o n to t h e p o r e s an d p a r t i c l e s a c t u a l l y p r e s e n t in s o i l a r e u s e d to m a k e m e a s u r e m e n t s . LABORATORY TESTS P ro ced u re. T h e b a s i c e q u i p m e n t u s e d in t h e l a b o r a t o r y t e s t s w a s t h e t r i a x i a l a p p a r a t u s m a n u f a c t u r e d b y S o i l t e s t , Inc. m o d e l T - 1 0 8 e m p l o y i n g f i t t i n g s f o r a 1.4 i n c h d i a m e t e r s p e c i m e n . T h e a p p a r a t u s i s s h o w n in F i g . 28. T h e e q u i p m e n t w a s m o d i f i e d s o th a t a i r d i s p l a c e d f r o m t h e s o i l s a m p l e w a s m e a s u r e d in a U - t u b e m a n o m e t e r w i t h o n e s i d e v a r i a b l e , w h i c h p e r m i t t e d m a i n t a i n i n g a t m o s p h e r i c p r e s s u r e on t h e c l o s e d s y s t e m . At t h e c o m p l e t i o n of a r u n , t h e f i n a l v o l u m e of t h e s o i l s a m p l e w a s d e t e r m i n e d b y w e ig h in g m e r c u r y d is p la c e d by the soil. The best p r o c e d u r e f o r p r e p a r i n g t h e s a m p l e w a s f o u n d to b e t r a n s f e r r i n g t h e s o i l b y s p o o n i n g d i r e c t l y in to t h e r u b b e r m e m b r a n e w h i l e in t h e t r i a x i a l apparatus. The m e m b ra n e itself produced a satisfactory sized sp ecim e n a n d at a l o w e r b u l k d e n s i t y t h a n w h e n a m o l d w a s u s e d . W i t h t h i s e q u i p m e n t it w a s p o s s i b l e to a p p l y v a r i o u s s t r e s s s t a t e s to t h e s o i l and o b s e r v e the effect upon the v o lu m e change. In t h e t r i a x i a l a p p a r a t u s , t h e f o r c e a p p l i e d to t h e v o l u m e e l e m e n t is c o n tro lle d . W a t e r p r e s s u r e a p p l i e s a m e a n s t r e s s s t a t e to t h e s p e c i m e n and is one e x t r e m e s t r e s s s ta te p o s s ib le w ith the a p p a r a t u s . T h e a p p l i c a t i o n of f o r c e a l o n g t h e l o n g e r a x i s of t h e s p e c i m e n c h a n g e s t h e s t r e s s s t a t e to a s o - c a l l e d c y l i n d r i c a l s t r e s s s t a t e w h e r e t h e two s m a l l e r p r i n c i p a l s t r e s s e s a r e eq u al (those applied by w a t e r p r e s s u r e . ) T h u s t h e p r i n c i p a l s t r e s s e s a r e m e a s u r e d d i r e c t l y in t h e t r i a x i a l 61 F ig . 28 T r ia x ia l a p p a r a tu s 62 apparatus. T h e o t h e r e x t r e m e s t r e s s s ta t e is d e t e r m i n e d by the m a g n i t u d e of t h e l a r g e s t p r i n c i p a l s t r e s s b e f o r e f a i l u r e of t h e s p e c i m e n by s h e a r o r o v e rtu rn in g . R esults. T h e t i m e a v a i l a b l e at t h e N a t i o n a l T i l l a g e M a c h i n e r y L a b o r a t o r y p e r m i t t e d o b t a i n i n g o n l y a l i m i t e d a m o u n t of d a t a . M o s t of t h e d a t a o b t a i n e d w e r e to t e s t if t r i a x i a l a p p a r a t u s c o u l d be u s e d w i t h s o i l s in a v e r y l o o s e c o n d i t i o n a n d to d e v e l o p t e c h n i q u e s f o r m a k i n g m e a s u r e m e n t s with such soils. It w a s d e t e r m i n e d tha t t r i a x i a l a p p a r a t u s c o u l d b e u s e d s u c c e s s f u l l y if p r o p e r p r e c a u t i o n s w e r e t a k e n . O n e s e r i e s of t e s t s in th e H i w a s s e e type so il w as r u n w h e re tech n iq u es had b een d ev elo p ed s u f f i c i e n t l y to w a r r a n t r e p o r t i n g t h e d a t a . It s h o u l d b e p o i n t e d out, h o w e v e r , t h a t s o m e of t h e m e a s u r e m e n t s w e r e not as p r e c i s e a s c a n b e obtained. F o u r i n d i v i d u a l s p e c i m e n s w e r e t e s t e d ; two w e r e r u n w i t h a m e a n n o r m a l s t r e s s s t a t e a p p l i e d ; two o t h e r s w e r e r u n w i t h a c y l i n d r i c a l s t r e s s s ta te applied. T a b l e C. T h e r e s u l t s of t h e s e f o u r t e s t s a r e g i v e n in T h e m o i s t u r e c o n t e n t of t h e s o i l in t h e l a b o r a t o r y t e s t r a n a p p r o x i m a t e l y 4% h i g h e r t h a n in t h e f i e l d t e s t . T h e s i g n i f i c a n c e of t h e t e s t s is t h e c l o s e a g r e e m e n t b e t w e e n l a b o r a t o r y d ata and field data. T h e a v e r a g e line f o r the m e a n s t r e s s - b u l k d e n s i t y p lo t on s e m i - l o g a r i t h m i c p a p e r and t h e s t a n d a r d e r r o r l i n e s w e r e d e t e r m i n e d a s d i s c u s s e d in F i g . 12. t r i a x i a l d a t a a r e s h o w n i n F i g . 29. T h e s e lin e s fo r the T h e p l o t t e d p o i n t s a r e f r o m th e 63 T a b le C TR IAXIAL D A T A IN HIW ASSEE LOAM SOIL S p h e r ic a l s t r e s s sta te M ean S tr e s s (p si) B u lk D e n s i t y (g m /c c ) C y lin d rica l s t r e s s sta te M ean S t r e s s (psi) B u lk D e n s i t y (g m /c c ) 2.5 5o0 7.5 12.5 20.0 30.0 38.7 1.37 1.53 1.58 1064 1.70 1.77 1,83 2.5 5.0 10.0 15.0 20.0 30.0 38,7 1,43 1.50 1.58 1.61 1.67 1.71 1.78 2.5 3.9 6.4 7.9 9.3 14.3 17.2 22.2 26„5 1,43 1.46 1.49 1.54 1.60 1.62 1.66 1.67 1.71 3.5 6.4 10.4 13.3 20. 8 23.7 26.5 36 .5 1,38 1.51 1.53 1.59 1.61 1.63 1,66 1.68 64 FIG. 2 9 FIELD AND T R I A X I A L D A T A IN H IW A S S E E SANDY LOAM S O I L 30 2 0 error overa ge MEAN STRESS (psi) s t andard N OT E : P o i n t s - Field D at a L i n e s - Tri a x i a l B U L K DENSI TY (gm/cc) Data 65 a v e r a g e l i n e s of F i g . 13 of t h e f i e l d d a t a . Since the l a b o r a t o r y soil w a s at a h i g h e r m o i s t u r e , m o r e c o m p a c tio n should o c c u r f o r a g iv en loading. ( S o e h n e , 1953; H o v a n e s i a n , 1958.) H e n c e the slight te n d e n c y f o r t h e f i e l d d a t a to b e to t h e left of t h e a v e r a g e l i n e in F i g . 29 is expected. T h i s g o o d a g r e e m e n t v e r i f i e s t h e tw o m e t h o d s an d l e n d s v a l i d i t y to b o t h t e c h n i q u e s f o r s t u d y i n g t h e r e l a t i o n s h i p b e t w e e n s t r e s s and soil c o m p actio n . T h e s u c c e s s of t h e l a b o r a t o r y t e c h n i q u e p r o v i d e s a n e f f e c t i v e m e a n s f o r p r o d u c i n g d i f f e r e n t s t r e s s s t a t e s not o b t a i n a b l e u n d e r field conditions. B e s i d e s t h e a b i l i t y to c o n t r o l t h e s t r e s s s t a t e a c c u r a t e l y , it s h o u l d b e p o s s i b l e to c o n t r o l t h e v a r i a b i l i t y of t h e s o i l s o m e w h a t b y m a i n t a i n i n g u n i f o r m s o i l c o n d i t i o n s f r o m t e s t to t e s t . P r e c i s e c o n t r o l w a s not p o s s i b l e in t h e f i e l d t e s t s e v e n w i t h t h e f a c i l i t i e s of t h e N a t i o n a l T i l l a g e M a c h i n e r y L a b o r a t o r y , a n d t h e i r long ti m e e x p e r i e n c e with handling soils. CONCLUSIONS In t h e l o o s e s o i l s t e s t e d , th e d a t a p r e s e n t e d i n d i c a t e t h e f o l l o w i n g ; 1. T h e c o n c e p t of t h e c o n t i n u u m w i l l a p p l y to l o o s e s o i l s . 2. Of t h e f o u r i n v a r i a n t s of s t r e s s i n v e s t i g a t e d , “ s' -T m ax , t h e m e a n n o r m a l s t r e s s r e l a t e s b e s t to bulk density, 3. B e c a u s e of s l i g h t d i f f e r e n c e s in a p p l i e d s t r e s s s t a t e s and l a r g e v a r i a t i o n in t e s t s , it c a n n o t b e c o n c l u d e d f r o m t h e t e s t s that s o i l c o m p a c t i o n is i n d e p e n d e n t of t h e d e v i a t o r . 4. T r i a x i a l d a t a on t h e H i w a s s e e s a n d y l o a m s o i l a g r e e d v e r y c l o s e l y w i t h f i e l d d a t a on th e s a m e s o i l , t h u s v e r i f y i n g both m e th o d s . 5. T r i a x i a l a p p a r a t u s c a n b e u s e d to s t u d y s t r e s s - c o m p a c t i o n r e l a t i o n s h i p s in l o o s e t i l l a b l e s o i l s a n d w i l l p r o v i d e a m e a n s of o b t a i n i n g l a b o r a t o r y c o n t r o l of a l l v a r i a b l e s . 6. T h e d i s t r i b u t i o n of m e a n n o r m a l s t r e s s a p p e a r s to b e i n d e p e n d e n t of s o i l t y p e an d d e t e r m i n e d o n l y b y t h e g e o m e t r y of l o a d i n g . 7 The m e a n n o rm a l s tr e s s - b u lk density re latio n sh ip was e x p o n e n tia l f o r all s o ils te s te d . 8. S t r e s s d i s t r i b u t i o n s p r e d i c t e d b y t h e T h e o r y of E l a s t i c i t y w e r e not v e r i f i e d in t h e l o o s e s o i l u s e d in th e e x p e r i m e n t . S U G G E S T I O N S F O R F U R T H E R S TU D Y 1. V e r i f y b y a d d i t i o n a l t r i a x i a l l e s t s t h e h y p o t h e s i s that m e a n n o r m a l s t r e s s i s r e l a t e d to s o i l c o m p a c t i o n , 2. I n v e s t i g a t e t h e e f f e c t of s o i l p h y s i c a l p r o p e r t i e s on t h e s t r e s s - b u l k density relationship. 3. A s s u m i n g m e a n n o r m a l s t r e s s is r e l a t e d to b u l k d e n s i l y , c a l c u l a t e , if p o s s i b l e , t h e m e a n n o r m a l s t r e s s d i s t r i b u t i o n u n d e r a p p l i e d l o a d s ; if not p o s s i b l e , m e a s u r e t h e d i s t r i b u t i o n e x p e r i m e n t a l l y . 4. D e t e r m i n e a y i e l d c o n d i t i o n f o r s o i l w h i c h is a f u n c t i o n of i n v a r i a n t s of t h e a p p l i e d s t r e s s s t a t e . 5. C o m b i n i n g t h e y i e l d c o n d i t i o n an d t h e c o m p a c t i o n c o n d i t i o n , d e v e l o p a m e t h o d f o r d e t e r m i n i n g t h e t o t a l s t r e s s d i s t r i b u t i o n in s o il . 6 D e t e r m i n e t h e d i f f e r e n c e in a b o v e r e l a t i o n s h i p s if a p p l i e d to an u n d i s t u r b e d s o il . 7. A p p l y t h e a b o v e i n f o r m a t i o n to t i l l a b l e s o i l s an d c a l c u l a t e m a g n i t u d e a n d t y p e of l o a d i n g to p r o d u c e a d e s i r e d s l a t e of c o m p a c t i o n . 8. D e s ig n and test tilla g e m a c h i n e s o r m e th o d s w hich will p r o d u c e the d e s i r e d soil p h y s ic a l state. REFERENCES B a v e r , L . D . (1956) S o i l P h y s i c s , 3 r d ed, J o h n W i l e y an d S o n s , I n c . , N ew Y o r k . C o o p e r , A . W . , G l e n E . V a n d e n B e r g , H . F . M e C o l l y an d A . E . E r i c k s o n (1957) S t r a i n g a g e c e l l m e a s u r e s s o i l p r e s s u r e , A g r . E n g r . C o r p s of E n g i n e e r s , U.S* A r m y (1953) T h e o r e t i c a l s t r e s s e s i n d u c e d b y u n i f o r m c i r c u l a r l o a d s , W a t e r w a y s Expt . S t a . , V i c k s b u r g , M i s s . C o r p s of E n g i n e e r s , U .S . A r m y (1954) H o m o g e n e o u s s a n d t e s t s e c t i o n , W a t e r w a y s E xpt. Sta., V ic k s b u rg , M is s. E d m i n s t e r , T . W . (1956) P r o g r e s s r e p o r t of c o m m i t t e e on s o i l c o m p a c ­ t i o n , P a p e r p r e s e n t e d at w i n t e r m e e t i n g of A S A E , C h i c a g o , I1L, D e c . 9- 12 C G i l l , W i l l i a m R. (1954) M e a s u r e m e n t of r o o t g r o w t h p r e s s u r e s of s e e d l i n g s , A g r o n o m y A b s t r a c t s . A n n u a l m e e t i n g of A m e r i c a n S o c i e t y of A g r o n o m y , G i l l , W i l l i a m R. a n d C a r l A, R e a v e s (1956) C o m p a c t i o n p a t t e r n s of sm ooth r u b b e r t i r e s , Agr, Engr. H o f f m a n , O. a n d G. S a c h s (1953) I n t r o d u c t i o n to t h e T h e o r y of P l a s i i c i t y f o r E n g i n e e r s , M c G r a w - H i l l B o o k C o . , I n c . , New Y o r k . H o g e n t o g l e r , C . A . , (1937) E n g i n e e r i n g P r o p e r t i e s of Soil , M c G r a w H i l l B o o k C o . , I n c . , New Y o r k 0 H o v a n e s i a n , J . D . (1958) D e v e l o p m e n t and u s e of a v o l u m e t r i c t r a n s ­ d u c e r f o r s t u d i e s of p a r a m e t e r s u p o n s o i l c o m p a c t i o n . T h e s i s f o r d e g r e e of P h . D . , M ic h , S t a t e U n i v . , E a s t L a n s i n g , M ic h . (U n p u b l i s h e d . ) J a e g e r , J . C . 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P e a t t i e , K, R. a n d R. W„ S p a r r o w (1954) T h e F u n d a m e n t a l a c t i o n of e a r t h p r e s s u r e c e l l s , J o u r n a l of t h e M e c h a n i c s an d P h y s i c s of S o l i d s , Vol. 2. P l u m m e r , F r e d L, a n d S t a n l e y M. D o r e (1940) S o i l M e c h a n i c s a n d F o u n d a t i o n s , P i t m a n P u b l i s h i n g C o r p . , New Y o r k . R e e d , I. F . (1940) A M e t h o d of s t u d y i n g s o i l p a c k i n g b y t r a c t o r s , Agr. E ngr. R u s s e l l , E . W. a n d W. B e l c e r e k (1944) T h e D e t e r m i n a t i o n of t h e v o l u m e a n d a i r s p a c e of s o i l c l o d s , J o u r n a l of A g r i . Sci, 34:123-132. S o e h n e , W a l t e r (1953) D r u c k v e r t e i l u n g i n B o d e n u n d B o d e n v e r f o r m u n g u n t e r S c h l e p p e r r e i f e n ( D i s t r i b u t i o n of p r e s s u r e in t h e s o i l and s o i l d e f o r m a t i o n u n d e r t r a c t o r t i r e s ) G r d l h n D. L a n d t e c h n , 5.49-63. T i m o s h e n k o , S. an d J. N. G o o d i e r (1951) T h e o r y of E l a s t i c i t y , M c G r a w H i l l B o o k C o . , I n c . , 2nd e d . , New Y o r k , T o r s t e n s s o n , G a n d S. E r i c k s o n (1936) A New m e t h o d f o r d e t e r m i n i n g t h e p o r o s i t y of t h e s o i l , S o il S c i . 42: 4 0 6 - 4 1 6 , V a n d e n B e r g , G . E . , A.W . C o o p e r , A . E . E r i c k s o n , W , M . C a r l e t o n (1957) S o i l p r e s s u r e d i s t r i b u t i o n u n d e r t r a c t o r an d i m p l e m e n t t r a f f i c , Agr. E n g r, V e i h m e y e r , F . J. (1929) An I m p r o v e d s o i l s a m p l i n g t u b e , S o il S ci. 27: 147-152. W a d l e i g h , C H. (1957) G r o w t h of p l a n t s . Soil: T h e Y e a r b o o k of A g r i c u l t u r e 1957 T h e U.S . Govt. P r i n t i n g Ofc, W a s h i n g t o n , D . C . 70 W ie r s m a * D. and M. M c M o rtla n d (1953) R e s p o n s e of s u g a r b e e t s to p e r o x i d e f e r t i l i z a t i o n and i t s r e l a t i o n to o x y g e n d iffu sio n . S o i l S c i. 75: 3 5 5 -3 60 * W illits * N ath an A. (1956) M e a s u r e m e n t of p r e s s u r e s in s o i l s p r o d u c e d b y t r a f f ic and th e r e l a t i o n s h ip b e tw e e n t h o s e p r e s s u r e s and c o m p a c t i o n in u n d is tu r b e d s o ils * U n p u b lish e d P h .D 0 t h e s i s p r e s e n t e d to M ic h ig a n S ta te U n iv . A PPEN D IX 72 CO C-. CQ t o CO • • O CO CO Ol tD y-_| oj LO £> t£> CO iO CO LG H4 i—I r l i—| tO I I I 1 co i> lo cc 05 to .CM •r»| CO £> cn LG) CO o O £> CO LO CO 0 5 t o 03 LO CO lO 'H CO rH 0 3 ^ 03 CO tO LO CO i> o—fl 03 *sf) LQ »—1 ^ 05 C-~ t> ^ O CO ^0- • • * ©' Q m o to o co to m • MO • * * iH 0 3 "3* lO ^ t ( • CJ> CO * I «s O • • • i—1 0 3 t> O « • • • to O to H • CO CO • • • • i CO 05 c - LO t>- 05 • • • i • • rH CO LO P- O ^ i—I I i 03 C- O • C.i i—i v-m i i r j i • • • i • • CQ CO CO to rH rH CO LO £» rH CO U'J tO O HO —I rH - 9' 9 0> CO 01 H I O to CO CO CO lO • » * • M• i—i to 00 C - t ; o H t :H tO r l 00 ^ 03 C3 rH CQ H 4 CO C.T-. tO i—i rH rH to t> to to O £** • • • « » O CO O CO t o <0 O CO£“*■ CD(H rH rH 03 tO -st* LO t o rH rH tO CO LO CO rH I—1(M vO LO to t> i> co 03 cn to • • • • « • O H C D H ♦ * • 9 9 • • « • O rH 03 CO LO 05 O rH 03 t o t o 05 i—t 03 03 0 3 CO lO tO rH CO 03 CO LO tO 0 5 CO iH 03 CO ’t!4 t o 0 5 COCO LQ 5> • * • ♦ » • • rH • • tO 05 O rH03 LO to cr> 03 CO LOtO tO C 03 COIQ O4CO CO rH CO LO^ E> O rH 03 CO '-H tO CJ3 HIWASS3E SANDY LOAM 5 " - 1 2 ■ • •r-t co 15 • • • rH 0 3 CO « • C - 05 CO H O O CO I> O O £> CO to 03 L> CO CO 03 uG CO tO O COLO COrH P 03 Cv2 U i O rH tO rH rH rH 03 £> ^ fc- {> 05 LO O LO H O CO 00 03 {—I rH rH 03 rH i—i 03 H rH CD 03 03 •H 0 3 O4 tO CO 03 1> rH rH rH to LO to O •’31 H1 t ' H* O LO 03 m rH o O 03 H1 t - O lO O rH rH 03 CO t> O CO CO 0 3 H H r l 03 CO LO CO (H £ - 03 rH rH 03 C3 LQ CQ rH LO i—I H 03 H4 to * • • 1• 03 COto oo C- 00 CO 05 05 0 5 i—ii—I CO03 P~ CO t> cq COlo H4o: O rH 03 CO to O i rH 03 03 H 4 tD G5 O H 02 to LO O 03 tO LO tO rH CO 0 2 CO tO CO O 03 CO LQCD CO O CQ tO LO--E CO CO rH 03 CO O 4 O CQ rH 03 CO ^ l o 0 3 CO 0 5 0 3 t o H 4 lO lO CO ^ • • • • • i—i i—1 <—1 i —1 H H H CQ LO t > * H t> rH rH rH r—I O to 5> H co TABLE 15 ■H ,r> B &■ * V H p CO -P O .H O to \ c: S © b£ Q — -— ' C O03 O 00 0M 2 C CO 0 2 aep> C CO 9 !.Li o 1—I CO p, '— fr ito © ♦ sft 00 LQ rH tO rH rH rH 03 ,C Q IS I 03 H 1 CH N I • 9 9 to 9 CD t o CO O • CO » • • • « » • • • « « • « I ■ rH CO LQ L'- r - l m rH i—t p- CO CO 03 LO 03 • • rH r-l CO fe • % • • ♦ iH 03 CO H4 p - O ir-H 0 3 t o H4 tO O ^ O ^ 03 CO H 4 **P LO LO *0 lO 03 C- O? 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