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Strain rate effects in one-dimensional consolidation of peat Baig, Khaliq Riaz 1970

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STRAIN RATE EFFECTS IN ONE-DIMENSIONAL CONSOLIDATION OF PEAT by KHALIQ RIAZ BAIG B . E . ( C i v i l ) , U n i v e r s i t y of Peshawar, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department o f C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f C A V I K ^ v y ^ ^ e ^ n ^ The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8. Canada Date ^ S ^ p l e w x b e r j \ O^^Q ABSTRACT A s e r i e s of one-dimensional s t r a i n r a t e c o n t r o l l e d c o n s o l i d a t i o n t e s t s were performed on remolded peat to e s t a b l i s h the v o i d r a t i o - e f f e c t i v e s t r e s s -s t r a i n r a t e r e l a t i o n s h i p f o r t h i s s o i l . The r e l a t i o n s h i p was found to be independent of the s i z e o f sample. In a d d i t i o n , the v o i d r a t i o - p e r m e a b i l i t y r e l a t i o n s h i p was a l s o determined and was found to be independent of both the s t r a i n r a t e and the s i z e of the sample. From these r e l a t i o n s h i p s the behaviour of any s i z e sample s u b j e c t e d to i n c r e m e n t a l l o a d i n g was p r e d i c t e d i n terms o f the time - s e t t l e m e n t and the time - pore p r e s s u r e c u r v e s . These p r e d i c t i o n s i n c l u d e d both 'primary' and 'secondary' s e t t l e m e n t . The comparison of observed and p r e d i c t e d time - s e t t l e m e n t and time - pore p r e s s u r e curves was found to be i n c l o s e agreement. i i i TABLE OF CONTENTS ABSTRACT LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS ACKNOWLEDGMENT Page i i v v i v i i x CHAPTER ONE INTRODUCTION TWO LITERATURE REVIEW 2.1 I n t r o d u c t i o n 2.2 One-Dimensional C o n s o l i d a t i o n of S o i l s 2.3 S t r a i n Rate C o n t r o l l e d C o n s o l i d a t i o n 2.3.a L i t e r a t u r e Review 2.3.b Theory of S t r a i n Rate C o n t r o l l e d C o n s o l i d a t i o n 8 11 THREE PREDICTION OF CONVENTIONAL TEST FROM CRS DATA 3.1 I n t r o d u c t i o n 16 3.2 The Void R a t i o - E f f e c t i v e S t r e s s -S t r a i n Rate R e l a t i o n s h i p 16 3.3 The V o i d R a t i o - P e r m e a b i l i t y R e l a t i o n s h i p 16 3.4 Governing Equation of C o n s o l i d a t i o n 21 i v CHAPTER Page 3.5 A D e s i r e d S o l u t i o n Procedure 25 3.6 The Adopted S o l u t i o n Procedure 28 FOUR RESULTS AND DISCUSSION 4.1 I n t r o d u c t i o n 30 4.2 General D i s c u s s i o n 32 4.3 e - p - £ R e l a t i o n s h i p from CRS T e s t s 34 4.4 V o i d R a t i o - P e r m e a b i l i t y R e l a t i o n s h i p 44 4.5 Comparision o f P r e d i c t e d and Observed Behaviour of Incremental Loading T e s t s 44 FIVE SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH 5.1 Summary 51 5.2 C o n c l u s i o n s 52 5.3 Suggestions f o r F u r t h e r Research 53 BIBLIOGRAPHY 54 APPENDIX 1 THE NUMERICAL SOLUTION OF THE GOVERNING CONSOLIDATION EQUATIONS 57 APPENDIX 2 ONE DIMENSIONAL CONSOLIDATION TESTS 64 V LIST OF TABLES Page TABLE 1 SOME PROPERTIES OF PEAT USED 31 TABLE 2 DESCRIPTION OF TWO INCREMENTAL TESTS 48 v i LIST OF FIGURES F i g u r e Page 3.1 N a t u r a l S c a l e P l o t of e = 0 and e = 0 0 f o r an Increment 18 3.2 Moving Drainage Boundary 23 3.3 F i n i t e D i f f e r e n c e Space - Time G r i d 26 4.1 - ~ - vs Time 33 rH o u, 4.2 — vs Time 35 a 4.3 V o i d R a t i o vs E f f e c t i v e S t r e s s a t i - 0.0213% per min 37 4.4 e - p - £ R e l a t i o n s h i p from CRS Tests 38 4.5 e vs i a t p = 0.8 kg/cm 2 40 4.6 General e - p - e R e l a t i o n s h i p f o r Peat Tested 41 4.7 f ( S t r a i n Rate) vs S t r a i n Rate R e l a t i o n s h i p a t p = 0.8 kg/cm 2 43 4.8 e - k - e R e l a t i o n s h i p f o r Peat T e s t e d 45 4.9 Comparison of Time - Pore P r e s s u r e Curves f o r Sample No. 1 46 4.10 Comparison of Time - Pore P r e s s u r e Curves f o r Sample No. 2 47 4.11 Comparison of Time - Settlement Curves f o r Sample No. 1 49 4.12 Comparison of Time - Settlement Curves f o r Sample No. 2 50 A . l F i n i t e D i f f e r e n c e Space - Time G r i d 62 A.2 Schematic Diagram of the T e s t i n g Set-Up 65 v i i LIST OF SYMBOLS E n g l i s h L e t t e r s a c o e f f i c i e n t of l i n e a r c o m p r e s s i b i l i t y a v T e r z a g h i ' s c o e f f i c i e n t of c o m p r e s s i b i l i t y b c o n s t a n t depending on the v a r i a t i o n i n v o i d r a t i o w i t h depth b r a d i m e n s i o n l e s s r a t i o i n d i c a t i n g the v a r i a t i o n i n v o i d r a t i o w i t h depth CRS Constant Rate of S t r a i n k(1+ei) C = 1 parameter r e l a t e d to p e r m e a b i l i t y c T e r z a g h i ' s c o e f f i c i e n t of c o n s o l i d a t i o n v ^ Dz t h i c k n e s s of an element of s o i l a t any time Dz i n i t i a l t h i c k n e s s of element of s o i l o e v o i d r a t i o e^ i n i t i a l v o i d r a t i o f i n a l v o i d r a t i o e^ v o i d r a t i o a t drainage boundary 6 = e - e 2 f ^ ( e ) e - p curve at e = 0 o r t = °° F X ( P ) f u n c t i o n of e f f e c t i v e s t r e s s ^2^^) f u n c t i o n of s t r a i n r a t e f ( e ) f u n c t i o n of s t r a i n r a t e H t h i c k n e s s of sample i time dimension v i i i j space dimension k c o e f f i c i e n t o f p e r m e a b i l i t y k^ i n i t i a l v a lue of k ^2 f i n a l v a lue of k p e f f e c t i v e s t r e s s p^ i n i t i a l v a l u e of p P 2 f i n a l v a l u e of p P a v average v a l u e of p over depth p^ bond s t r e s s a t end of primary P v v i s c o u s r e s i s t a n c e t o compression Pp = Pj 3+P v p l a s t i c r e s i s t a n c e t o compression r r a t e of change of average v o i d r a t i o T R = — d i m e n s i o n l e s s r a t i o r e l a t i n g the two time f a c t o r s a s s o c i a t e d w i t h d i f f u s i o n and s t r a i n r a t e e f f e c t s t t r u e time C t T = — d i m e n s i o n l e s s time f a c t o r a s s o c i a t e d w i t h a.H 2 d i f f u s i o n process T = d i m e n s i o n l e s s time f a c t o r a s s o c i a t e d w i t h S ± . 1 A ^ 3-1 s t r a i n r a t e e f f e c t s t • (Ag) " a. u pore p r e s s u r e excess u a v average pore p r e s s u r e excess over depth u^ base pore p r e s s u r e excess u = — d i m e n s i o n l e s s pore p r e s s u r e v a r i a b l e Aa v volume of sample v = ^ 7 r a t e o f change of volume d t H x = 77 d i m e n s i o n l e s s l e n g t h v a r i a b l e 0 i x z space v a r i a b l e Greek L e t t e r s a c o e f f i c i e n t r e l a t e d t o s t r a i n r a t e vs v o i d r a t i o r e l a t i o n s h i p a v a l u e of a averaged over an increment a v o f l o a d 3 index i n power law r e l a t i n g s t r a i n r a t e s to the f u n c t i o n o f s t r a i n r a t e Y u n i t weight of water Af, (e) component of t o t a l s t r e s s increment due to the b a s i c e - p r e l a t i o n s h i p Av v v o l u m e t r i c s t r a i n r a t e Aa increment of t o t a l s t r e s s e s t r a i n r a t e 'av c o e f f i c i e n t o f n o n - l i n e a r f u n c t i o n o f a. n 11 . .1 s t r a i n r a t e ( 1 + e i > e X = — T — d i m e n s i o n l e s s a.Aa u = e l ~ e degree of c o n s o l i d a t i o n e l " S 2 o t o t a l s t r e s s p s e t t l e m e n t X ACKNOWLEDGMENT The author wishes to express h i s g r a t i t u d e to Dr. P e t e r M. Byrne f o r s u g g e s t i n g the t o p i c o f t h i s i n v e s t i g a t i o n , f o r h i s s t i m u l a t i n g d i s c u s s i o n s , and f o r h i s continuous guidance and encouragement. S i n c e r e a p p r e c i a t i o n i s extended to Dr. R. G. Campanella f o r h i s re v i e w i n g o f the t h e s i s and h i s g e n e r a l guidance, Mr. Y. P. V a i d f o r many v a l u a b l e and h e l p f u l d i s c u s s i o n s , Mr. F e l i x Bucher f o r h i s p a t i e n c e i n re v i e w i n g the manuscript, and Mrs. Margaret Hyslop f o r t y p i n g t h i s t h e s i s . The r e s e a r c h f o r t h i s d i s s e r t a t i o n was supported by the N a t i o n a l Research C o u n c i l o f Canada. T h i s support i s g r a t e f u l l y acknowledged. S i n c e r e a p p r e c i a t i o n i s extended to a l l those i n Canada and P a k i s t a n , who helped each i n t h e i r own way. F i n a l l y , the author expresses h i s a p p r e c i a t i o n f o r the t e c h n i c a l a s s i s t a n c e s u p p l i e d by the s t a f f o f the C i v i l E n g i n e e r i n g Department Workshop. CHAPTER ONE INTRODUCTION A g e n e r a l problem of c u r r e n t i n t e r e s t i n s o i l mechanics, both from an academic and e n g i n e e r i n g p o i n t of view, concerns the t h e o r e t i c a l . p r e d i c t i o n s o f one di m e n s i o n a l c o n s o l i d a t i o n of s o i l s . The u s u a l method f o r e s t i m a t i n g both the r a t e and amount of s e t t l e m e n t i s to perform a c o n s o l i d a t i o n t e s t i n which the v e r t i c a l displacements are measured a t a p p r o p r i a t e time i n t e r v a l s . Each increment i s such t h a t the r a t i o o f the increment to l o a d ( AH. ) equals u n i t y . Each increment i s l e f t i n p l a c e f o r 24 hours. The data r e c o r d e d a l l o w s the d e t e r m i n a t i o n o f the c o e f f i c i e n t of c o n s o l i d a t i o n , c v , based on T e r z a g h i theory f o r a p a r t i c u l a r increment. The c v v a l u e s thus o b t a i n e d are used t o p r e d i c t the r a t e s of s e t t l e m e n t i n the f i e l d . The displacement c o r r e s p o n d i n g t o the one day r e a d i n g i s used to p l o t the v o i d r a t i o v s . e f f e c t i v e s t r e s s r e l a t i o n s h i p which i s then used t o e s t i m a t e the u l t i m a t e s e t t l e m e n t f o r the f i e l d c o n d i t i o n s . However, both l a b o r a t o r y and f i e l d t e s t s on s o i l s i n d i c a t e t h a t the r a t e of se t t l e m e n t may d e v i a t e c o n s i d e r a b l y from the T e r z a g h i p r e d i c t i o n . 2 T a y l o r (1940, 1942) thoroughly i n v e s t i g a t e d t h i s l a c k of agreement between the T e r z a g h i theory and the behaviour of samples i n the l a b o r a t o r y . The T e r z a g h i theory f o r the p r e d i c t i o n o f the r a t e o f se t t l e m e n t i s based on the r a t e of d i s s i p a t i o n o f excess pore water p r e s s u r e . However, T a y l o r found t h a t a p p r e c i a b l e s e t t l e m e n t s s t i l l o ccured when the excess pore p r e s s u r e was e s s e n t i a l l y zero and t o take account of t h i s T a y l o r proposed h i s Theory A. T a y l o r a l s o found t h a t t h e r e were a p p r e c i a b l e d e v i a t i o n s from the T e r z a g h i theory d u r i n g the d i s s i p a t i o n of the excess pore water p r e s s u r e and to account f o r t h i s he proposed h i s Theory B. Both of, these t h e o r i e s are e s s e n t i a l l y s a y i n g t h a t the v o i d r a t i o v ersus e f f e c t i v e s t r e s s r e l a t i o n s h i p assumed by T e r z a g h i should be r e p l a c e d by a more r e a l i s t i c r e l a t i o n s h i p which takes i n t o account the f a c t t h a t v o i d r a t i o i s a l s o a f f e c t e d by s t r a i n r a t e . T h i s v o i d r a t i o - e f f e c t i v e s t r e s s - s t r a i n r a t e r e l a t i o n s h i p can be o b t a i n e d by running c o n s t a n t r a t e of s t r a i n ( c a l l e d CRS, h e r e a f t e r ) c o n s o l i d a t i o n t e s t s on a s o i l over a range of s t r a i n r a t e s . I f the base pore p r e s s u r e s are measured d u r i n g the same t e s t s the v o i d r a t i o -p e r m e a b i l i t y r e l a t i o n s h i p can be obt a i n e d by u s i n g a r e c e n t l y proposed theory (Byrne and Aoki (1969), Smith and Wahls (1969)). 3 I t i s g e n e r a l l y b e l i e v e d t h a t t h e r e are two components o f s t r a i n , one due to pore p r e s s u r e d i s s i p a t i o n and the oth e r due to the time or s t r a i n r a t e e f f e c t . The v o i d r a t i o - e f f e c t i v e s t r e s s - s t r a i n r a t e and v o i d r a t i o -p e r m e a b i l i t y r e l a t i o n s h i p s o b t a i n e d from the s t r a i n r a t e c o n t r o l l e d t e s t s can be used i n the eq u a t i o n of c o n t i n u i t y of mass to p r e d i c t the s e t t l e m e n t behaviour of a sample of any s i z e under l o a d c o n t r o l l e d c o n d i t i o n s . The c o n s o l i d a t i o n due t o the d i s s i p a t i o n of excess pore p r e s s u r e and t h a t due to the s t r a i n r a t e e f f e c t occur c o n c u r r e n t l y . T h e r e f o r e , the s t r a i n r a t e e f f e c t i s c o n s i d e r e d d u r i n g the d i s s i p a t i o n o f excess pore' p r e s s u r e . However, a f t e r a time the excess pore water p r e s s u r e drops t o a ve r y s m a l l v a l u e and the a d d i t i o n a l time l a g i n the s e t t l e m e n t curve i s due wholly t o the s t r a i n r a t e e f f e c t . T h i s time l a g can then be o b t a i n e d by i n t e g r a t i o n o f the b a s i c v o i d r a t i o - e f f e c t i v e s t r e s s -s t r a i n r a t e r e l a t i o n s h i p a t c o n s t a n t e f f e c t i v e s t r e s s . 4 CHAPTER TWO LITERATURE REVIEW 2.1 I n t r o d u c t i o n T h i s c hapter i s d i v i d e d i n t o three s e c t i o n s . The f i r s t s e c t i o n d e a l s w i t h the equ a t i o n governing the flow o f water through a porous medium and the work of the d i f f e r e n t i n v e s t i g a t o r s to s o l v e t h i s flow problem. The ( • l i t e r a t u r e review of a r e c e n t l y (1969) proposed theory o f s t r a i n r a t e c o n t r o l l e d c o n s o l i d a t i o n i s presented i n the second s e c t i o n . The u n d e r l y i n g theory o f the s t r a i n r a t e c o n t r o l l e d c o n s o l i d a t i o n t e s t i s presented i n the t h i r d s e c t i o n . 2.2 One-Dimensional C o n s o l i d a t i o n of S o i l s T e r z a g h i (1923) developed a c o n s o l i d a t i o n e q u a t i o n 3u _ 9 2u - , 9 t V 8 z 2 where z = space v a r i a b l e , t = time v a r i a b l e , u = excess pore water p r e s s u r e , and c = k ( l + e) and i s c a l l e d v — 1 -' 0) v c o e f f i c i e n t of c o n s o l i d a t i o n , 5 where e = v o i d r a t i o k = c o e f f i c i e n t of p e r m e a b i l i t y , Y w = u n i t weight of water, and a = -de and i s c a l l e d c o e f f i c i e n t v d F of c o m p r e s s i b i l i t y , based on the f o l l o w i n g assumptions, i ) the s o i l i s homogenous and s a t u r a t e d , i i ) the water and s o l i d s are i n c o m p r e s s i b l e as compared to the s o i l s k e l e t o n , i i i ) Darcy's law i s v a l i d , i v ) the s t r a i n s and drainage occurs o n l y in~ one d i r e c t i o n ( u s u a l l y v e r t i c a l ) , v) c o e f f i c i e n t o f p e r m e a b i l i t y , k, i s a c o n s t a n t , v i ) the t o t a l s t r e s s , throughout the depth, d u r i n g c o n s o l i d a t i o n i s c o n s t a n t , v i i ) c o e f f i c i e n t of c o n s o l i d a t i o n d e f i n e d above i s c o n s t a n t f o r a p a r t i c u l a r l o a d increment i n v o l v i n g s m a l l changes i n v o i d r a t i o , and v i i i ) the r e l a t i o n s h i p between v o i d r a t i o and e f f e c t i v e s t r e s s i s l i n e a r and independent of time. D e v i a t i o n s from the T e r z a g h i theory were observed both i n the f i e l d and the l a b o r a t o r y . In the l a b o r a t o r y i t was observed t h a t c o n s i d e r a b l e s e t t l e m e n t occured a f t e r the excess pore water p r e s s u r e had e s s e n t i a l l y d i s s i p a t e d . T h i s a d d i t i o n a l s e t t l e m e n t was termed "secondary" s e t t l e m e n t by many r e s e a r c h e r s . D e v i a t i o n s from the T e r z a g h i t h e o r y were 6 a l s o observed d u r i n g the d i s s i p a t i o n of the excess pore water p r e s s u r e or "primary" c o n s o l i d a t i o n . T a y l o r (1942) put forward Theory B and Theory A t o account f o r the d e v i a t i o n s from T e r z a g h i ' s t h e o r y observed i n the l a b o r a t o r y t e s t i n g . In h i s Theory B T a y l o r proposed t h a t the i n t e r g r a n u l a r p r e s s u r e a t any time and a t any p o i n t w i t h i n the c o n s o l i d a t i n g sample may be expressed p = f b ( e ) + P p ... (2.2) where p = i n t e r g r a n u l a r p r e s s u r e f ^ (e) = the b a s i c e q u i l i b r i u m curve The e q u i l i b r i u m curve i s the curve r e p r e s e n t i n g p o i n t s where v o i d r a t i o has reached e q u i l i b r i u m under t h a t p a r t i c u l a r p r e s s u r e . T h i s curve c o u l d be c o n s i d e r e d the zero s t r a i n r a t e or t = » curve and Pp = p l a s t i c r e s i s t a n c e to compression T a y l o r f u r t h e r showed t h a t P i s a f u n c t i o n of s t r a i n r a t e P and hence from E q u a t i o n (2.2) the v o i d r a t i o i s a f u n c t i o n of the e f f e c t i v e s t r e s s and the s t r a i n r a t e , i . e . e = F 1 ( p ) + F 2 ( e ) ... (2.3) where e = r a t e of s t r a i n T a y l o r s i m p l i f i e d the problem by d i v i d i n g P^ i n t o two components P^, the bond a t the end of primary and P , the v i s c o u s r e s i s t a n c e t o compression a c t i n g d u r i n g primary. 7 He i g n o r e d i n h i s Theory B. To c o n s i d e r the secondary compression T a y l o r proposed Theory A. He d e s i r e d t o s e t up a theory which c o u l d p r o p e r l y account f o r both primary and secondary compression. However, due to the complexity of the problem, he t r e a t e d them i n two d i f f e r e n t t h e o r i e s . He noted t h a t the combined use of i d e a s from two such t h e o r i e s must be the u l t i m a t e procedure f o r o b t a i n i n g the b e s t u n d e r s t a n d i n g of the c o n s o l i d a t i o n p r o c e s s . Bjerrum (1967) d i v i d e d the volume changes i n t o two components: (a) an ' i n s t a n t compression' which occured s i m u l t a n e o u s l y w i t h the i n c r e a s e i n e f f e c t i v e s t r e s s and caused a r e d u c t i o n i n v o i d r a t i o u n t i l an e q u i l i b r i u m v a l u e was reached a t which the s t r u c t u r e e f f e c t i v e l y supported the overburden p r e s s u r e ; (b) a 'delayed compression' r e p r e s e n t i n g r e d u c t i o n i n volume at c o n s t a n t e f f e c t i v e s t r e s s . H i s concept of having a system of e - l o g p curves f o r the c o m p r e s s i b i l i t y c h a r a c t e r i s t i c s of a c l a y showing delayed c o n s o l i d a t i o n i s s i m i l a r t o t h a t by T a y l o r (1942) and Crawford (1965). Gibson and Lo (1961) proposed a one-dimensional theory i n which the c o m p r e s s i b i l i t y of the s o i l s k e l e t o n i s r e p r e s e n t e d by a r h e l o g i c a l model comprising a Hookean s p r i n g i n s e r i e s w i t h a K e l v i n body. T h i s theory i s 1 8 i d e n t i c a l to T a y l o r ' s Theory A. In g e n e r a l , the r e c e n t approach used to p r e d i c t the behaviour of an element of s o i l i s to choose a r h e l o g i c a l model and compare the observed and p r e d i c t e d b e haviour. The parameters of the model are r e v i s e d as neces s a r y u n t i l r e a l i s t i c p r e d i c t i o n s can be made without i t bei n g necessary to a l t e r the r h e l o g i c a l parameters a t each change of t e s t c o n d i t i o n s (Barden 1965, 1968, 1969, Berry 1969, I s h i i 1951, and o t h e r s ) . 2.3 S t r a i n Rate C o n t r o l l e d C o n s o l i d a t i o n 2.3.a L i t e r a t u r e Review Hamilton and Crawford (19 59) made one of the e a r l i e s t mentions o f a co n s t a n t r a t e of s t r a i n c o n s o l i d a t i o n (CRS) t e s t . They used t h i s type o f t e s t as a r a p i d means of d e t e r m i n i n g both the p r e c o n s o l i d a t i o n p r e s s u r e and the v o i d r a t i o - e f f e c t i v e s t r e s s r e l a t i o n s h i p . When compared w i t h the c o n v e n t i o n a l t e s t s , the CRS t e s t s showed lower c o m p r e s s i b i l i t y , but the g e n e r a l shapes o f the curves were somewhat s i m i l a r . Hamilton and Crawford mentioned t h a t the pore p r e s s u r e s developed would have caused t h i s apparent decrease i n c o m p r e s s i b i l i t y but they d i d not measure the pore p r e s s u r e s . The range of s t r a i n r a t e s i n t h i s s e r i e s v a r i e d from 0.15% per min. to 0.005% per min., but even t h i s 9 l a r g e d i f f e r e n c e d i d not g r e a t l y a f f e c t the d a t a . The data f o r the slower s t r a i n r a t e s d i d show a tendency towards c l o s e r agreement w i t h the c o n v e n t i o n a l t e s t s . O b s e r v a t i o n of the t e s t specimen a f t e r d r y i n g i n d i c a t e d t h a t s i d e f r i c t i o n a p p a r e n t l y caused more s e r i o u s s t r e s s v a r i a t i o n s through the specimens i n the standard t e s t than i n the CRS t e s t . In a second paper Crawford (1964) r e p o r t e d more d a t a from the CRS t e s t s . He presented the e f f e c t of s t r a i n r a t e as a f a c t o r t h a t had been too long i g n o r e d i n s e t t l e m e n t a n a l y s i s . Crawford noted t h a t l a b o r a t o r y c o n s o l i d a t i o n r a t e s are f r e q u e n t l y s e v e r a l m i l l i o n times l a r g e r than those e x p e r i e n c e d i n the f i e l d . Data from the CRS t e s t s appeared q u i t e s i m i l a r to t h a t o b t a i n e d from the standard t e s t s u s i n g d i f f e r e n t l o a d d u r a t i o n s . During t h i s s e r i e s of t e s t s excess pore p r e s s u r e s were measured at the base of the sample. The maximum excess pore p r e s s u r e i n the CRS t e s t was approximately 5% of the a p p l i e d p r e s s u r e . Because of the low excess pore water p r e s s u r e s Crawford concluded t h a t the compression i n the CRS t e s t i s t o t a l l y secondary compression. The s t r a i n r a t e s i n t h i s s e r i e s were approximately 4% per hour to the p r e c o n s o l i d a t i o n l o a d , and then 7% to 14% per hour, depending on specimen t h i c k n e s s . In a d i s c u s s i o n of t h i s paper, Schmertmann (1965), however, i n d i c a t e d a p r e f e r e n c e f o r a c o n t r o l l e d r a t e of l o a d i n g t e s t i n which v a l u e s o f c v and k c o u l d be determined as w e l l as the p r e c o n s o l i d a t i o n p r e s s u r e and the e - p 10 r e l a t i o n s h i p . To Schmertmann the g r e a t advantage of t h i s t e s t i n g method was the s h o r t time r e q u i r e d to o b t a i n the d e s i r e d i n f o r m a t i o n . Crawford (1965) r e p o r t e d a f u r t h e r i n v e s t i g a t i o n i n which the r a t e s of s t r a i n v a r i e d from 0.133% per min. to 0.0027% per min., and the maximum excess pore p r e s s u r e measured at the base of the sample was equal t o 15% of the a p p l i e d p r e s s u r e . Pore p r e s s u r e s were measured a t the base of the samples throughout the d u r a t i o n of the t e s t and the average e f f e c t i v e s t r e s s on the sample was c a l c u l a t e d by s u b s t r a c t i n g one h a l f of the excess pore p r e s s u r e at the base from the t o t a l v e r t i c a l s t r e s s . Standard i n c r e m e n t a l c o n s o l i d a t i o n t e s t s were conducted f o r comparison purposes. From these t e s t s , Crawford concluded t h a t the s o i l s t r u c t u r e had an important time - dependent r e s i s t a n c e to compression. T e s t data showed t h a t the h i g h e r the r a t e of s t r a i n , the lower the c o m p r e s s i b i l i t y or the g r e a t e r the p l a s t i c r e s i s t a n c e . He a l s o suggested t h a t the f i n a l v o i d r a t i o f o r a p a r t i c u l a r l o a d i s mainly dependent on the average r a t e of compression and not the method by which the l o a d i s a p p l i e d . T h i s was i n d i c a t e d by the marked s i m i l a r i t y of the r e s u l t s of the i n c r e m e n t a l and the CRS t e s t s . In g e n e r a l agreement wi t h the work done by Crawford was the study conducted on remolded samples and r e p o r t e d by Wahls and DeGodoy (1965) . The s t r a i n r a t e s i n 11 t h i s s e r i e s of t e s t s v a r i e d from 0.23% per min. to 0.053% per min., and pore p r e s s u r e s were measured a t the base of the sample. Values of u b v a r i e d from 25% f o r the slowest to 75% f o r the f a s t e s t t e s t . These maximum v a l u e s r e s u l t e d from e x p o n e n t i a l type i n c r e a s e s w i t h i n c r e a s i n g s t r a i n , and were much l a r g e r than the v a l u e s recorded f o r the major p o r t i o n of the t e s t . As was the case i n Crawford's work, t h e r e was an i n c r e a s e i n c o m p r e s s i b i l i t y w i t h d e c r e a s i n g s t r a i n r a t e s . When compared wi t h standard t e s t r e s u l t s Wahls and DeGodoy observed t h a t f o r the range of s t r a i n r a t e s used a l l of the CRS t e s t r e s u l t s showed more c o m p r e s s i b i l i t y than the standard t e s t . T h i s i s i n c o n t r a s t to Crawford's work, where the o p p o s i t e was t r u e . I t has been suggested t h a t t h i s d i f f e r e n c e may be caused by the increment d u r a t i o n f o r the standard t e s t s or the d i f f e r e n t s o i l s used i n the t e s t s t u d i e s . 2.3.b Theory of S t r a i n Rate C o n t r o l l e d C o n s o l i d a t i o n Byrne and A o k i (1969) presented a theory of s t r a i n c o n t r o l l e d c o n s o l i d a t i o n . Based on assumptions (i) to (iv) S e c t i o n 2.2 i n c l u s i v e the b a s i c equation of c o n t i n u i t y of mass i s 1_ r k ( e ) i H i - 1 iSL 3z y ^z J 1 + e 3t ... (2.4) 'to Next i n t r o d u c i n g the assumption (v) the e q u a t i o n reduces to 12 k_ aju, = 1 3e ya 9 z 2 1 + e 8 t ••• (2.5) They f u r t h e r assumed t h a t the s t r a i n r a t e should be s u f f i c i e n t l y slow such t h a t the v o i d r a t i o can be assumed to be uniform throughout the sample a t a l l times, then the r i g h t hand s i d e of Equation (2.5) can be i n t e g r a t e d q u i t e s i mply. For the boundary c o n d i t i o n s u = o a t z = o f o r a l l t and ~ = o a t z = H f o r a l l t 9 z the f o l l o w i n g e x p r e s s i o n f o r the excess pore p r e s s u r e a t any time i s o b t a i n e d : . . 1 9e . Yw . ,„ z 2 u(z) = - ^ F { H z - 2~> ... (2.6) the excess pore p r e s s u r e a t the base (z = H) i s g i v e n by 1 3e . Yo) H 2  U b ~ 1 + e at k 2 ... (2.7) and s i n c e the excess pore p r e s s u r e i s p a r a b o l i c w i t h z, the average pore p r e s s u r e throughout the sample i s g i v e n by 2 2 . 1 ge, Y M H 2 u a v = 3 * U b = " J ( 7 T ~ ft> ^ ' — ( 2-8) 1 + e u = ( 1 ) Jjo H_i av 3 \ _,_ 3t * k ' 2 u ' y j 1 + e 13 and p = a - u ... (2.10) The term ( y ^ ^ • I s the v o l u m e t r i c s t r a i n r a t e and w i l l be n e g a t i v e f o r a l l cases where the volume of the sample i s d e c r e a s i n g w i t h time and hence a p o s i t i v e excess pore p r e s s u r e w i l l r e s u l t . A l s o 1 3 e A * ... (2.11) 1 + e 3t v where Av = v o l u m e t r i c s t r a i n r a t e , v For one d i m e n s i o n a l c o n s o l i d a t i o n 1 ae Av 1 AH 1 + e * at v H At ... (2.12) where H = h e i g h t of the sample. S u b s t i t u t i o n of (2.11) and (2.12) i n (2.7) g i v e s AH 1 Y03 H 2  U b At ' H * k 2 (2.13) AH Taj H U b " At ' k ' 2 ... (2.14) When a c o n s t a n t deformation r a t e i s a p p l i e d to a sample wi t h base pore p r e s s u r e measured, Equation (2.14) allows the c o e f f i c i e n t o f p e r m e a b i l i t y to be c a l c u l a t e d : , H \a AH K " 2 u, * At ... (2.15) b 14 I t should be n o t e d . t h a t t h i s k i s f u n c t i o n of v o i d r a t i o of the sample. I t i s f u r t h e r assumed i n the i n t e g r a t i o n of E q u a t i o n (2.5) t h a t k i s o n l y a f u n c t i o n of time and not a f u n c t i o n of z. Smith and Wahls (1969) a l s o developed a s t r a i n r a t e c o n t r o l l e d c o n s o l i d a t i o n mathematical model. Instead of assuming co n s t a n t v o i d r a t i o throughout the depth of a sample they assumed a l i n e a r v a r i a t i o n / j.\ ^ . n b . z - 0 . 5 H \ i e ( z , t ) = e Q - r t [1 - - ( g )] _ ( 2 > 1 6 ) i n which r = - de i s r a t e of change of average v o i d r a t i o , d t b = a c o n s t a n t t h a t depends on the v a r i a t i o n i n v o i d r a t i o w i t h depth and time. The d i m e n s i o n l e s s r a t i o , i n d i c a t e s the v a r i a t i o n i n v o i d r a t i o w i t h depth. Smith and Wahls s u b s t i t u t e d E q u a t i o n (2.16) i n E q u a t i o n (2.5) and i n t e g r a t e d to g i v e an e x p r e s s i o n f o r the pore p r e s s u r e . I t can be shown t h a t Byrne and A o k i ' s theory i s m a t h e m a t i c a l l y a s p e c i a l case of Smith and Wahls' theory with — = 0 . Smith and Wahls, however, mentioned the working l i m i t s of — between 0 and 2. They recommended use of — = 1 , r J r because t h i s gave c l o s e s t agreement, when comparing c v v a l u e s from standard and CRS t e s t s . Byrne (1970) i n a d i s c u s s i o n of Smith and Wahls' (1969) paper p o i n t e d out t h a t t h e r e are reasons o t h e r than 15 the — r a t i o why the r e s u l t s of c o n v e n t i o n a l and s t r a i n c o n t r o l l e d t e s t s would y i e l d d i f f e r e n t c v a l u e s . He a l s o J v showed t h a t ^ v a l u e s w i l l depend upon s t r a i n r a t e and can be c a l c u l a t e d by b = JJ % ... (2.17) r t * r In t h i s t h e s i s the s t r a i n c o n t r o l l e d c o n s o l i d a t i o n t e s t s w i l l be used to determine the v o i d r a t i o - p e r m e a b i l i t y and the v o i d r a t i o - e f f e c t i v e s t r e s s - s t r a i n r a t e r e l a t i o n s h i p s . These r e l a t i o n s h i p s and E quation (2.4) are then used to p r e d i c t the behaviour of a c o n v e n t i o n a l t e s t . The s o l u t i o n technique f o r t h i s purpose i s d i s c u s s e d i n the next chapter. 0 16 CHAPTER THREE PREDICTION OF CONVENTIONAL TEST FROM CRS DATA 3.1 I n t r o d u c t i o n In a c o n v e n t i o n a l t e s t an increment of t o t a l s t r e s s i s added to a sample and the time - s e t t l e m e n t and time - pore p r e s s u r e curves are observed. The i n t e n t o f t h i s t h e s i s i s to show t h a t these curves can be p r e d i c t e d from d a t a o b t a i n e d from CRS t e s t s . The data o b t a i n e d from the CRS t e s t s w i l l be i n terms of v o i d r a t i o -e f f e c t i v e s t r e s s - s t r a i n r a t e and v o i d r a t i o - p e r m e a b i l i t y r e l a t i o n s h i p s . 3.2 The V o i d R a t i o - E f f e c t i v e S t r e s s -S t r a i n Rate R e l a t i o n s h i p I t i s d e s i r e d to r e p l a c e Assumption ( v i i i ) S e c t i o n 2.2, s t a t i n g "the r e l a t i o n s h i p between v o i d r a t i o and e f f e c t i v e s t r e s s i s l i n e a r and independent of time", by a more r e a l i s t i c r e l a t i o n s h i p c o n s i d e r i n g v o i d r a t i o as a f u n c t i o n o f both e f f e c t i v e s t r e s s and s t r a i n r a t e . To o b t a i n t h i s r e l a t i o n s h i p , the s t r a i n r a t e e f f e c t i s f i r s t e l i m i n a t e d by imposing a con s t a n t s t r a i n r a t e . I f the base pore p r e s s u r e i s measured, e f f e c t i v e 17 s t r e s s - v o i d r a t i o r e l a t i o n s h i p s can be p l o t t e d , f o r the p a r t i c u l a r s t r a i n r a t e . I f now on another sample of the same m a t e r i a l a d i f f e r e n t s t r a i n r a t e i s a p p l i e d , i t w i l l g i v e a d i f f e r e n t e - p p l o t . The d i f f e r e n c e i n these two curves w i l l depend upon the nature o f m a t e r i a l . In t h i s f a s h i o n a s e r i e s o f curves can be o b t a i n e d running t e s t s a t d i f f e r e n t s t r a i n r a t e s . T h e o r e t i c a l l i m i t s of s t r a i n r a t e s w i l l be e = 0 to °°. F i g u r e 3.1 shows t h a t f o r d i f f e r e n t s t r a i n r a t e s , the e f f e c t i v e s t r e s s w i l l be d i f f e r e n t f o r a g i v e n v o i d r a t i o . I t i s assumed t h a t b e f o r e p l a c i n g a s t r e s s increment, Aa, on a sample the system i s i n e q u i l i b r i u m w i t h a l l the s t r e s s e s t r a n s f e r r e d to the g r a i n bond. T h i s means t h a t i f the v o i d r a t i o i s e^ the e f f e c t i v e s t r e s s w i l l be p^ and the p o i n t l i e s on e = 0 curve. L e t the f i n a l v o i d r a t i o be e 2« A s soon as the loa d i s a p p l i e d the sample s t a r t s s t r a i n i n g . The s t r a i n r a t e s w i l l be h i g h e r near the drainage boundary and lower near the impermeable boundary. A f t e r a very s m a l l time i n t e r v a l v o i d r a t i o a t a p o i n t on the drainage boundary w i l l be e 3 and the e f f e c t i v e s t r e s s w i l l be p 2 . T h i s p o i n t w i l l then move down v e r t i c a l l y u n t i l i t reaches the f i n a l v o i d r a t i o e^. A t a p o i n t somewhat removed from the drainage boundary the v o i d r a t i o w i l l be e and the e f f e c t i v e s t r e s s w i l l be p depending on e a t some time t . As time, goes on t h i s p o i n t moves on d i f f e r e n t s t r a i n r a t e l i n e s u n t i l E f f e c t i v e S t r e s s , p F i g u r e 3.1 N a t u r a l S c a l e P l o t of f o r an Increment 19 f i n a l l y i t a l s o reaches the zero s t r a i n r a t e curve or the p o i n t ( p 2 , e 2 ) . From F i g u r e 3.1 e q u i l i b r i u m r e q u i r e s t h a t f o r every p o i n t w i t h i n the sample Aa = A f b ( e ) + f ( e ) + u ... (3.1) or DH = DF + FG + GH ... (3.2) Making the s i m p l i f y i n g assumption t h a t f^Ce) curve can be r e p l a c e d by the s t r a i g h t l i n e AB, and s i m i l a r l y a pproximating the curve KGL by s t r a i g h t l i n e KC^L. I t i s obvious t h a t the use of t h i s assumption w i l l i n v o l v e much l e s s approximation i n the case o f s m a l l e r l o a d increment r a t i o s . T h i s assumption i s s i m i l a r to the one made by. T a y l o r (1942) i n h i s Theory B. The E q u a t i o n (3.2) reduces t o DH = DF + FG + GH ... (3.3) S l " S 2 where DH = Aa = t o t a l s t r e s s increment = a 'a' b e i n g s l o p e of l i n e AFB. DF = Af^(e) = component of t o t a l s t r e s s due to the f u n c t i o n o f b a s i c e - p r e l a t i o n s h i p and e^ - e ~ a FG = f ( e ) = f u n c t i o n o f s t r a i n r a t e GH = u = excess pore p r e s s u r e 20 T h i s concept of three phase system i s i n e s s e n t i a l agreement wi t h T e r z a g h i (1941), T a y l o r (1942), Barden (1965) and o t h e r s . E q u a t i o n (3.1) can be r e w r i t t e n e - e - = u + f ( e ) ... (3.4) E q u a t i o n (3.4) i s s i m i l a r i n form t o Equation (2.3). T a y l o r r e f e r r e d to the term f ( e ) as the p l a s t i c r e s i s t a n c e to compression. T h i s term can be determined from CRS data and s u b s t i t u t e d i n Equation (3.4) to g i v e the d e s i r e d v o i d r a t i o as a f u n c t i o n of e f f e c t i v e s t r e s s and s t r a i n r a t e s . o 3.3 The V o i d R a t i o - P e r m e a b i l i t y R e l a t i o n s h i p In E q u a t i o n (2.4) the p e r m e a b i l i t y , k, i s a f u n c t i o n of the v o i d r a t i o , e. I f pore p r e s s u r e s are measured d u r i n g a CRS t e s t , E q u a t i o n (2.15) all o w s the d e t e r m i n a t i o n of the e - k r e l a t i o n s h i p . The c o e f f i c i e n t o f p e r m e a b i l i t y may or may not be a f u n c t i o n o f s t r a i n r a t e . Byrne and A o k i ' s (1969) da t a on u n d i s t u r b e d samples o f a s e n s i t i v e c l a y shows t h a t e - l o g k r e l a t i o n s h i p i s e s s e n t i a l l y a s t r a i g h t l i n e and t h a t no t r e n d w i t h s t r a i n r a t e was d i s c e r n i b l e . Smith and Wahls (1969) d i d not show e - k p l o t s . I t w i l l be assumed here t h a t the c o e f f i c i e n t o f p e r m e a b i l i t y i s not a f u n c t i o n of s t r a i n r a t e . I f the data o b t a i n e d f o r a p a r t i c u l a r s o i l shows t h a t the e - k r e l a t i o n s h i p i s dependent on s t r a i n r a t e s , then the equations may be changed a c c o r d i n g l y . 21 In the T e r z a g h i theory, k i s assumed constant d u r i n g an increment. Hansbo's (1960) treatment of p e r m e a b i l i t y v a r i a t i o n , i n i n c r e m e n t a l l o a d i n g t e s t , i n d i c a t e d t h a t f o r .- ^1 < 3 k 2 where k^ = i n i t i a l p e r m e a b i l i t y and k 2 = f i n a l p e r m e a b i l i t y f o r a p a r t i c u l a r l o a d increment, the d i f f e r e n c e i n the time - s e t t l e m e n t and time -pore p r e s s u r e r e l a t i o n s h i p s i s not a p p r e c i a b l y d i f f e r e n t from the T e r z a g h i theory. 3.4 Governing E q u a t i o n o f C o n s o l i d a t i o n The major sources o f n o n - l i n e a r behaviour f o r c l a y and peat s o i l s a r e : . i ) F i n i t e s t r a i n - l e a d i n g to changes i n the l e n g t h o f d r a i n a g e path, i i ) , V a r y i n g p e r m e a b i l i t y w i t h i n the sample, i i i ) V a r y i n g c o m p r e s s i b i l i t y , and i v ) V o i d r a t i o - e f f e c t i v e s t r e s s r e l a t i o n s h i p s which are dependent on time or s t r a i n r a t e s . Although a l l of these n o n - l i n e a r i t i e s can be i n c l u d e d i n a s i n g l e g e n e r a l treatment.the p r e s e n t a t i o n i s not simple (Berry (1969)). The u s u a l a n a l y s i s i n c l u d e s o n l y those n o n - l i n e a r i t i e s which are r e l e v a n t to the s o i l under c o n s i d e r a t i o n . 22 A c o n s o l i d a t i o n e q u a t i o n i s presented i n the f o l l o w i n g pages to account f o r a l l the n o n - l i n e a r i t i e s mentioned above, i . e . f i n i t e s t r a i n , v a r y i n g p e r m e a b i l i t y , v a r y i n g c o m p r e s s i b i l i t y and the v o i d r a t i o - e f f e c t i v e s t r e s s r e l a t i o n s h i p s which are dependent on time or s t r a i n r a t e s . i ) F i n i t e S t r a i n The f o l l o w i n g r e p r e s e n t s a b r i e f summary of the development of the c o n t i n u i t y e q u a t i o n f o r f i n i t e s t r a i n . F i g u r e 3.2.a shows a s o i l l a y e r having an i n i t i a l t h i c k n e s s H which i s s u b j e c t e d to an instantaneous i n c r e a s e o J i n p r e s s u r e Ac. The width o f the s o i l l a y e r and the ext e n t of loaded area are c o n s i d e r e d to be i n f i n i t e , so t h a t the problem i s e s s e n t i a l l y one d i m e n s i o n a l . The s o i l r e s t s on an impermeable boundary and i s f r e e to d r a i n to i t s upper s u r f a c e . C o n s i d e r an element o f s o i l a t a d i s t a n c e z from o the impermeable boundary having i n i t i a l t h i c k n e s s d z Q ( F i g u r e 3.2.a). During c o n s o l i d a t i o n the t h i c k n e s s o f the element decreases and i t s p o s i t i o n i n space moves v e r t i c a l l y downwards. That i s , the element undergoes f i n i t e s t r a i n , and the subsequent development of the problem r e q u i r e s the mathematical treatment of a moving drainage boundary. L e t dz denote the new h e i g h t o f the element and z i t s d i s t a n c e from the boundary a t some time t a f t e r the s t a r t of c o n s o l i d a t i o n , as shown i n F i g u r e 3.2.b, then comparing the 23 Drainage Boundary Impermeable Boundary (a) (b). F i g u r e 3.2 Moving Drainage Boundary 24 t h i c k n e s s of the elements shown i n F i g u r e s 3.2(a) and 3.2(b) dz 1 + e dz 1 + e-. o -l ... (3.4.a) S u b s t i t u t i n g i n E q u a t i o n (2.4) (l+e x) 2 3 v ( e ) 3u , 1 de 1+e 3z (1+e) 3z * 1+e 3t ... (3.5) o w o E q u a t i o n (3.5) d e f i n e s the c o n s o l i d a t i o n w i t h r e s p e c t to the i n i t i a l s o i l c o n d i t i o n s and accounts f o r f i n i t e s t r a i n or change i n the l e n g t h of the drainage path d u r i n g c o n s o l i d a t i o n . i i ) V a r i a t i o n i n P e r m e a b i l i t y D i f f e r e n t i a t i n g Equation (3.5): (1+e,) a) dz ,k(e).3u k(e)3*u, _1 ll+e ; 3z„ 1+e 3 7 ^ 1+e-3e 3t (3.6) A f t e r the e - k r e l a t i o n s h i p i s e s t a b l i s h e d from CRS t e s t s and s u b s t i t u t e d above, Equation (3.6) w i l l account f o r both f i n i t e s t r a i n and v a r y i n g p e r m e a b i l i t y . i i i ) V a r y i n g C o m p r e s s i b i l i t y T h i s n o n - l i n e a r i t y can be i n t r o d u c e d i n t o the c o n s o l i d a t i o n e q u a t i o n , when r e l a t i n g e, u and E . 25 iv) V o i d R a t i o , a F u n c t i o n of E f f e c t i v e S t r e s s and S t r a i n Rates E q u a t i o n (3.4) r e p r e s e n t s t h i s r e l a t i o n s h i p , which can be found by running s t r a i n c o n t r o l l e d c o n s o l i d a t i o n t e s t s on the g i v e n s o i l , over a range of s t r a i n r a t e s . S u b s t i t u t i o n o f Equation (3.4) i n t o E q u a t i o n (3.6) w i l l g i v e the governing c o n s o l i d a t i o n e q u a t i o n . The r e s u l t i n g e q u a t i o n w i l l be h i g h l y n o n - l i n e a r and r e c o u r s e has t o be made to a numerical s o l u t i o n procedure. For t h a t purpose i t i s more convenient to work w i t h the two Equations (3.4) and (3.6) s i m u l t a n e o u s l y . 3.5 A D e s i r e d S o l u t i o n Procedure C o n s i d e r the g r i d shown i n F i g u r e 3.3. Assuming a l l o f the e and u v a l u e s along the depth a t any time i are known, then a p p l i c a t i o n o f e x p l i c i t c e n t r a l f i n i t e d i f f e r e n c e approximation g i v e s ( f z T ^ i j = D T ^ [ u i , j - l " 2 u i , j + U i , j + 1 ] ... (3.7) ,5u, _ 1 . v S z ' i j 2Dz Q l u i , j - l " u i , j + l J ... (3.8) . 3e. _ 1_ , . l 9 t ' i , i + l Dt l e i + l , j " e i , j J ... (3.9) The term 1 on the l e f t hand s i d e of Equation (3.6) may 1 + e be approximated by 1 , f o r the p a r t i c u l a r time 1 + e. . if!) 26 time dimension, i F i g u r e 3.3 F i n i t e D i f f e r e n c e Space-Time G r i d 27 i n t e r v a l , without much e r r o r , then /_L_ k< e> ) = _ i f k i , j - l " k i > j - H ] ( 3 1 Q ) (3z l + e ' i j 2Dz 1 1 + e. . J * ' * < J- ± U' O J O 1,] Use o f Equations (3.7) to (3.10) i n c l u s i v e i n Equation (3.7) enables the d e t e r m i n a t i o n of .., from a l l the p r e v i o u s l y known v a l u e s o f e and u. The boundary c o n d i t i o n u ( l , 0 ) = 0 can be imposed d i r e c t l y and the second boundary c o n d i t i o n , ( — ) . = 0 , can dz im be s a t i s f i e d by c o n s i d e r i n g a f i c t i t i o u s p o i n t o u t s i d e the boundary such t h a t u. ,, = u.' , and hence there i s no pore J 1,m+l i , m - l p r e s s u r e g r a d i e n t a c r o s s the impermeable boundary. A l l of the e v a l u e s along the depth a t the next time can, t h e r e f o r e , be p r e d i c t e d and the s t r a i n r a t e a t a p a r t i c u l a r time and f o r any g i v e n p o i n t can then be computed as f o l l o w s : e = 1 e i + l , j " e i , j ( 3 ± 1 ) e i , j + l 1 + e± ' Dt <J--LJ-' S u b s t i t u t i o n o f E q u a t i o n (3.11) i n t o E q u a t i o n (3.4) w i l l g i v e the u. .... Hence a l l the pore p r e s s u r e s a t the 1, ]+1 next time can be computed. New p e r m e a b i l i t y v a l u e s can a l s o be e v a l u a t e d f o r each p o i n t i n the g r i d c o r r e s p o n d i n g to each v o i d r a t i o . The s o l u t i o n , t h e r e f o r e , can proceed to the next time i n t e r v a l . Although the proposed method seems simple to apply, 28 a s t a b l e s o l u t i o n was not o b t a i n e d and hence the f o l l o w i n g a l t e r n a t e procedure was adopted i n t h i s t h e s i s . 3.6 The Adopted S o l u t i o n Procedure S t a r t i n g from Equation ( 2 . 4 ) r an average k i s assumed f o r an increment, the e r r o r i n v o l v e d depends on the f i n a l and i n i t i a l v a l u e s of p e r m e a b i l i t y . Hansbo (1960), has shown, however, t h a t the d i f f e r e n c e i n the time -s e t t l e m e n t and time - pore p r e s s u r e v a l u e s i s s m a l l i f the r a t i o _1 < 3. The e q u a t i o n , t h e r e f o r e , reduces to k 2 k a v 9 2 u _ 1 3e Y 3 z 2 1 + e 3t _ ... OJ L e t , — T — equal ^ — 7 1 + e 1 1 + then C ^ ~ = — ... (3.13) a Z 2 d t k (1 + e, ) where C = — v 1 W r i t i n g E q u a t i o n (3.4) as e - e 2 = a[u + f ^rr^- (-||)>] ... (3.14) the -ve s i g n shows decrease i n e w i t h time. E q u a t i o n s (3.13) and (3.14) are then s o l v e d f o r the u s u a l boundary c o n d i t i o n s . 29 Barden (1965) proposed two methods to s o l v e simultaneous p a r t i a l d i f f e r e n t i a l e q u a t i o n s . P o s k i t t (1967) d e v i s e d n u m e r i c a l procedures which take about one t e n t h o f the computing time r e q u i r e d by the methods o r i g i n a l l y proposed by Barden (1965). The P o s k i t t methods are based on the e x p l i c i t method (MCM, 1962) and the Crank - N i c o l s o n method (MCM, 1962). Both o f these methods are extremely s t a b l e and g i v e answers i n c l o s e agreement w i t h each o t h e r ( P o s k i t t (1967)). The e x p l i c i t method i s used i n t h i s a n a l y s i s (see Appendix 1). When u •*• 0, E q u a t i o n (3.14) can be d i r e c t l y i n t e g r a t e d to g i v e the p a r t of the compression which occurs a t c o n s t a n t e f f e c t i v e s t r e s s . 30 CHAPTER FOUR RESULTS AND DISCUSSION 4.1 I n t r o d u c t i o n The peat used i n t h i s i n v e s t i g a t i o n was brought from a peat farm on the i n t e r s e c t i o n o f 499 Highway and No. 5 Road i n Richmond, about ten m i l e s south of Vancouver, B.C. The data presented i n t h i s i n v e s t i g a t i o n concerns the p r o p e r t i e s o f remolded peat. The use o f remolded s o i l s i n b a s i c r e s e a r c h has important advantages, e s p e c i a l l y w i t h r e g a r d t o the u n i f o r m i t y o f t e s t specimens, H v o r s l e v (1960). Many r e l a t i o n s c o n c e r n i n g the p h y s i c a l p r o p e r t i e s o f c l a y s were f i r s t determined by means o f t e s t s on remolded specimens. The r e s u l t s o f t e s t s on remolded s o i l s , however, cannot be d i r e c t l y used f o r s o l u t i o n of p r a c t i c a l problems i n v o l v i n g u n d i s t u r b e d s o i l s . D i f f e r e n t batches of peat were remolded under vacuum a t 1,175% moisture content ( n a t u r a l water content = 600 to 700%) w i t h two mechanical s t i r r e r s f o r a p e r i o d o f 4 8 hours. A l l of these batches were then combined and mixed thoroughly i n a l a r g e r c o n t a i n e r . T h i s c o n t a i n e r was covered and kept i n a humid room. The peat i n the c o n t a i n e r was p e r i o d i c a l l y s t i r r e d 31 and was thoroughly mixed b e f o r e a sample was taken. I d e n t i f i c a t i o n t e s t s were performed t o determine some of the p r o p e r t i e s o f the s o i l used b e f o r e s t a r t i n g the main t e s t i n g programme and the r e s u l t s are shown i n Table 1. TABLE 1 SOME PROPERTIES OF PEAT USED S p e c i f i c G r a v i t y , G 1.4 N a t u r a l Water Content 600 - 700% L i q u i d L i m i t 1100 - 1200% pH 3.5 - 4.0 Nature Amorphous g r a n u l a r The procedure f o r the i n c r e m e n t a l t e s t s was i n . accordance w i t h t h a t o u t l i n e d i n Appendix 2. F u r t h e r r e f e r e n c e s were made to Lamb (1962) and Bishop and Henkel (1964). The t e s t i n g , procedure f o r the CRS t e s t s i s a l s o g i v e n i n Appendix 2. A computer programme was used t o reduce the data and to c a l c u l a t e e - p and e - k f o r d i f f e r e n t t e s t s u s i n g 32 E q u ations (2.10) and (2.15) r e s p e c t i v e l y . 4.2 General D i s c u s s i o n Byrne and Aoki's (1969) theory o f s t r a i n r a t e c o n t r o l l e d c o n s o l i d a t i o n i s based on the assumption t h a t the v o i d r a t i o i s uniform throughout the depth of the sample. The assumption i s a c c e p t a b l e when the s t r a i n r a t e s are s u f f i c i e n t l y slow and the r a t i o o f ^b are s m a l l . The a v a l i d i t y o f the assumption becomes s e r i o u s l y q u e s t i o n a b l e a t h i g h s t r a i n r a t e s . Smith and Wahls (1969) i n t h e i r development, which was d i s c u s s e d i n Chapter Two, assumed t h a t the v o i d r a t i o v a r i e d l i n e a r l y w i t h depth, E q u a t i o n (2.16) , and d i m e n s i o n l e s s r a t i o , — , i n d i c a t e s the v a r i a t i o n i n v o i d r a t i o w i t h depth. The l i m i t s o f t h i s r a t i o are from 0 to 2. I f , f o r example, i n a p a r t i c u l a r t e s t ^ e q u a l s 2, the use o f ^ equal t o 0 u b w i l l i n t r o d u c e an e r r o r o f 0.08 (—) i n the v a l u e of e f f e c t i v e s t r e s s , which can be n e g l e c t e d without l o s s of accuracy. To compare the — r a t i o f o r d i f f e r e n t samples, i t was necessary to use t h i s r a t i o per u n i t i n i t i a l t h i c k n e s s o f the sample, because of the d i f f e r e n t t h i c k n e s s e s o f the samples. F i g u r e 4.1 shows the — j j — vs time p l o t f o r d i f f e r e n t s t r a i n o k r a t e s . The maximum v a l u e o f ^JJ— from t h i s p l o t i s 0.03 and b 0 f o r t h i s t e s t — equals about 0.02, the e f f e c t i v e s t r e s s b . u h c a l c u l a t e d by u s i n g — equal to zero w i l l be 0.05 % b g r e a t e r than the one which w i l l r e s u l t by the use of — 34 e q u a l t o 0.02. On t h i s b a s i s i t was decided to use — equal to zero. U b F i g u r e 4.2 shows a ~ vs t p l o t which i s another i n d i c a t i o n of the n o n - u n i f o r m i t y of v o i d r a t i o throughout the depth. I t shows t h a t the h i g h e r the s t r a i n r a t e the u b h i g h e r the ~ . r a t i o and, t h e r e f o r e , the l a r g e r the v a r i a t i o n i n v o i d r a t i o through the depth of a sample. u. In t e s t s A - l to A-4 i n c l u s i v e the — r a t i o i s f a i r l y a c o n s t a n t throughout the d u r a t i o n of t e s t s . In t e s t A-5 the r a t i o i s i n i t i a l l y 7 0% and then drops to a minimum of about 40%. The i n i t i a l p a r t of t h i s t e s t on v o i d r a t i o -e f f e c t i v e s t r e s s p l o t was markedly d i f f e r e n t from the o t h e r s . T h i s d i f f e r e n c e c o u l d not be accounted f o r by c o n s i d e r i n g a l i n e a r v a r i a t i o n o f v o i d r a t i o . On t h i s b a s i s i t was U b d e c i d e d t h a t the — r a t i o of 50% w i l l be accepted as r e a s o n a b l e . I t may be mentioned t h a t Smith and Wahls u b d e c i d e d on the l i m i t i n g v a l u e o f — as 50% on the b a s i s of the comparison of c v v a l u e s w i t h the i n c r e m e n t a l l o a d i n g t e s t . T h i s comparison of c v v a l u e s i s , however, not attempted here because a more b a s i c comparison of i n c r e m e n t a l and CRS t e s t i s p r e s e n t e d . 4.3 e - p - e R e l a t i o n s h i p from CRS T e s t s Although the remolded samples are uniform enough t o c a r r y out b a s i c r e s e a r c h i t was thought necessary to f i r s t check the accuracy o f r e p r o d u c i b i l i t y of r e s u l t s . Time (minutes) 36 Three (A-4, RA-4, R-4) CRS t e s t s on samples 3 i n . d i a . and 2 i n . i n h e i g h t were performed with the same i n i t i a l c o n d i t i o n s and the same s t r a i n r a t e (0.0023% per min.). The e vs p r e l a t i o n s h i p s o b tained are p l o t t e d f o r these three t e s t s and l i e e s s e n t i a l l y on the same curve, F i g u r e 4.3. The r e s u l t i s t y p i c a l of a normally c o n s o l i d a t e d or remolded s o i l sample. I t was necessary to check the e f f e c t of sample s i z e on the e - p - e r e l a t i o n s h i p , because the data i s to be used f o r the p r e d i c t i o n of i n c r e m e n t a l l o a d i n g t e s t s on samples of any s i z e . A t e s t (C-l) on 5.57" high and 8.26" diameter sample was performed and e - l o g p p o i n t s f o r t h i s t e s t are a l s o p l o t t e d i n F i g u r e 4.3. I t may be seen t h a t a l l p o i n t s l i e on the same l i n e so t h a t f o r any one s t r a i n r a t e t h e r e i s o n l y one e vs p r e l a t i o n s h i p r e g a r d l e s s o f the s i z e of the t e s t specimen. Four more t e s t s ( A - l , A-2, A-3, and A-5) were performed on d i f f e r e n t s i z e s of samples w i t h v a r y i n g s t r a i n r a t e s and the r e s u l t s are p l o t t e d i n F i g u r e 4.4. These curves are e s s e n t i a l l y p a r a l l e l s t r a i g h t l i n e s on the semi l o g p l o t i n the r e g i o n of i n t e r e s t , except t e s t A-5, which has been mentioned e a r l i e r . These r e s u l t s are i n g e n e r a l agreement wi t h the d ata o b t a i n e d by Crawford (1964, 196 5), Byrne and Aoki (1969) and Smith and Wahls (1969) and show t h a t f o r a g i v e n v o i d 37 F i g u r e 4.3 V o i d R a t i o vs E f f e c t i v e S t r e s s a t t = 0.0213% per rain. 39 r a t i o the c o m p r e s s i b i l i t y i s lower f o r h i g h e r s t r a i n r a t e s . T h i s behaviour i s i n agreement wi t h the concept of s o i l as a v i s c o e l a s t i c m a t e r i a l . The v o i d r a t i o vs s t r a i n r a t e r e l a t i o n s h i p a t c o n s t a n t e f f e c t i v e s t r e s s can be d i r e c t l y o b t a i n e d from F i g u r e 4.4 and i s shown i n F i g u r e 4.5 f o r an e f f e c t i v e s t r e s s , p, of 0.8 kg/cm 2. Because F i g u r e 4.4 i s on a semi l o g p l o t , t h i s e vs e curve f o r any o t h e r e f f e c t i v e s t r e s s w i l l be s i m i l a r i n s l o p e but s h i f t e d above or below depending upon whether the s t r e s s l e v e l i s h i g h e r or lower than t h i s a r b i t r a r i l y chosen v a l u e of 0.8 kg/cm . T h i s curve i n d i c a t e s t h a t marked changes of v o i d r a t i o w i t h s t r a i n r a t e takes p l a c e f o r s t r a i n r a t e s i n the range 10" 1 to 10 - 1* p e r c e n t per min. On both the high and low s i d e of t h i s range the v a r i a t i o n i n the v o i d r a t i o a t c o n s t a n t e f f e c t i v e s t r e s s i s r e l a t i v e l y s m a l l f o r t h i s s o i l . F i g u r e 4.6 shows the g e n e r a l v o i d r a t i o - e f f e c t i v e s t r e s s - s t r a i n r a t e r e l a t i o n s h i p p l o t t e d by e x t r a p o l a t i n g the v a l u e s o f v o i d r a t i o f o r d i f f e r e n t s t r a i n r a t e s from F i g u r e 4.5 a t c onstant e f f e c t i v e s t r e s s . In t h i s f i g u r e the e = 0 l i n e has been assumed, but i t i s r e a l i z e d t h a t probably the a c t u a l p o s i t i o n of t h i s l i n e w i l l not be much d i f f e r e n t , as suggested by F i g u r e 4.5. F i g u r e 4.6 r e p r e s e n t s E q u a t i o n (2.3) f o r the s o i l t e s t e d , i . e . v o i d r a t i o as f u n c t i o n of e f f e c t i v e s t r e s s and s t r a i n r a t e . 41 F i g u r e 4.6 General e - p - e R e l a t i o n s h i p f o r Peat T e s t e d 42 E q u a t i o n (3.14) can be o b t a i n e d from F i g u r e 4.6 as e x p l a i n e d i n S e c t i o n 3.2. The v a l u e s of f ( e ) are o b t a i n e d at a p a r t i c u l a r v o i d r a t i o and are p l o t t e d i n F i g u r e 4.7. a g a i n s t c o r r e s p o n d i n g s t r a i n r a t e s , both on the l o g a r i t h m i c s c a l e . T h i s r e l a t i o n s h i p can be expressed by 1 f (E) = a(e) 3 ... (4.1) which i s s i m i l a r to Ostwald's e m p i r i c a l power law, r e l a t i n g shear s t r e s s and shear s t r a i n r a t e . " Barden (1965) used Ostwald's law i n a r h e l o g i c a l model to r e p r e s e n t the behaviour of an element of a s o i l . I t can be shown t h a t , because the v o i d r a t i o vs e f f e c t i v e s t r e s s curves f o r d i f f e r e n t s t r a i n r a t e s are p a r a l l e l l i n e s on semi l o g p l o t , the s t r a i n r a t e vs f ( s t r a i n r a t e ) curves w i l l be p a r a l l e l l i n e s on the l o g a r i t h m i c p l o t f o r d i f f e r e n t e f f e c t i v e s t r e s s v a l u e s . Hence, 6 , i n E q u a t i o n (4.1) w i l l remain co n s t a n t f o r a l l the e f f e c t i v e s t r e s s e s , but a w i l l vary as a 2 p 2 . . . ( 4 . 2 ) For the necessary s i m p l i f i c a t i o n , an average value of a w i l l be used f o r a p a r t i c u l a r l o a d increment. Hence 1 e - e 2 = a[u + n (||* 6 ... (4.3) 43 F i g u r e 4.7 f ( S t r a i n Rate) vs S t r a i n Rate R e l a t i o n s h i p a t p = 0.8 kg/cm 2 44 where n = (1 + e x ) 3 4.4 Void R a t i o - P e r m e a b i l i t y R e l a t i o n s h i p Using Equation (2.15) the c o e f f i c i e n t of p e r m e a b i l i t y , k, can be determined a t any time f o r t h a t v o i d r a t i o . F i g u r e 4.8 shows v o i d r a t i o vs l o g ( p e r m e a b i l i t y ) p l o t t e d from d i f f e r e n t CRS t e s t s . The p o i n t s are e s s e n t i a l l y l o c a t e d on the same curve and show t h a t the p e r m e a b i l i t y of t h i s s o i l i s a f u n c t i o n of the v o i d r a t i o and independent of s t r a i n r a t e s , as was assumed i n S e c t i o n 3.2. Hence, k = F(e) ... (4.4) 4.5 Comparison of P r e d i c t e d and Observed Behaviour of Incremental Loading T e s t s A computer programme was developed to p r e d i c t the behaviour of i n c r e m e n t a l t e s t s u s i n g the CRS t e s t s d a t a . The programme was t e s t e d u s i n g the assumptions of T a y l o r ' s Theory B and the r e s u l t s were i d e n t i c a l to the c l o s e d form s o l u t i o n o b t a i n e d by T a y l o r . Two i n c r e m e n t a l l o a d i n g t e s t s were performed to compare the observed and the p r e d i c t e d behaviour of samples. The two t e s t s are d e s c r i b e d i n Table 2. F i g u r e s 4.9 and 4.10 show the time vs pore p r e s s u r e Figure 4.9 Comparison of Time - Pore Pressure Curves for Sample No. 1 F i g u r e 4.10 Comparison of Time - P o r e P r e s s u r e Curves f o r Sample No. 2 48 comparison f o r the two t e s t s . . The p r e d i c t e d pore p r e s s u r e s are h i g h e r i n the e a r l i e r and lower i n the l a t e r p a r t of the curve when compared wit h the observed pore p r e s s u r e s . T h i s behaviour may be because of the use of an average p e r m e a b i l i t y . f o r the complete d u r a t i o n o f the t e s t s . I f the v a r i a t i o n i n p e r m e a b i l i t y i s accounted f o r , as suggested i n the d e s i r e d s o l u t i o n procedure, S e c t i o n 3.5, t h i s d i s c r e p a n c y i s l i k e l y to be s m a l l e r . However, the g e n e r a l agreement i s c l o s e . TABLE 2 DESCRIPTION OF TWO INCREMENTAL TESTS Sample No. Load Increment kg/cm 2 D u r a t i o n Days I n i t i a l Height cms Diameter cms 1 0.1 - 0.2 21 8.70 20.98 2 0.4 - 0.8 13 2.65 7.5 The time - se t t l e m e n t curves are presented i n F i g u r e s 4.11 and 4.12, the p r e d i c t e d and observed curves being almost i d e n t i c a l . F i g u r e 4.11 Comparison o f Time - S e t t l e m e n t Curves f o r Sample No.l Time (minutes) F i g u r e 4.12 Comparison of Time - S e t t l e m e n t Curves f o r Sample No. 2 51 CHAPTER FIVE SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH 5.1 Summary A number of one-dimensional c o n s t a n t r a t e of s t r a i n c o n s o l i d a t i o n t e s t s were performed on remolded peat to e s t a b l i s h the e - p - e and the e - k r e l a t i o n s h i p f o r t h i s s o i l . The v o i d r a t i o - p e r m e a b i l i t y r e l a t i o n s h i p was found to be f a i r l y l i n e a r on a semi l o g p l o t and independent of both the s t r a i n r a t e and the s i z e o f sample. The v o i d r a t i o - e f f e c t i v e s t r e s s - s t r a i n r a t e r e l a t i o n s h i p was found to be independent of the s i z e o f sample and was of the form e - e 2 = a(u + n ( | f ) 3 ) ... (5.1) n b e i n g the c o e f f i c i e n t o f n o n - l i n e a r s t r a i n r a t e e f f e c t s on the v o i d r a t i o - e f f e c t i v e s t r e s s r e l a t i o n s h i p and 3 being the index i n the power law r e l a t i n g the s t r a i n r a t e s to the v o i d r a t i o . The v a l u e of 3 found from CRS t e s t s was 4.92 and was con s t a n t f o r a l l v a l u e s o f e f f e c t i v e s t r e s s . To p r e d i c t the behaviour of an i n c r e m e n t a l l o a d i n g t e s t under an increment of 0.4 - 0.8 kg/cm 2 the CRS t e s t s gave the 52 v a l u e of n equal to 0.1284 and i t was found to vary d i r e c t l y w i t h e f f e c t i v e s t r e s s , p. The behaviour of samples s u b j e c t e d to i n c r e m e n t a l l o a d i n g was p r e d i c t e d by s u b s t i t u t i n g the e x p r e s s i o n f o r the v o i d r a t i o g i v e n i n E q u a t i o n (5.1) i n t o the e q u a t i o n c i i i ' = i§. 3 z 2 ^ . . . ( 5 . 2 ) The r e s u l t i n g e q u a t i o n was s o l v e d by a n u m e r i c a l i n t e g r a t i o n procedure, P o s k i t t (1967), and i s d e s c r i b e d i n Appendix 1. The p r e d i c t e d and observed time s e t t l e m e n t and time - pore p r e s s u r e curves were compared f o r two i n c r e m e n t a l l o a d i n g t e s t s on d i f f e r e n t s i z e of samples. The p r e d i c t e d and the observed behaviour was found to be i n c l o s e agreement. 5.2 C o n c l u s i o n s I t i s concluded t h a t the v o i d r a t i o - e f f e c t i v e s t r e s s - s t r a i n r a t e r e l a t i o n s h i p f o r the peat t e s t e d can be e s t a b l i s h e d by running one-dimensional CRS t e s t s over a range of s t r a i n r a t e s . T h i s r e l a t i o n s h i p was found to be independent of the s i z e of the sample. The v o i d r a t i o -p e r m e a b i l i t y r e l a t i o n s h i p can a l s o be o b t a i n e d and f o r t h i s s o i l was independent of both the s t r a i n r a t e s and s i z e of the sample. 53 The r e l a t i o n s h i p s o b t a i n e d from CRS t e s t s a l l o w the p r e d i c t i o n o f the time - s e t t l e m e n t and the time -pore p r e s s u r e behaviour of the m a t e r i a l under i n c r e m e n t a l l o a d i n g c o n d i t i o n s . These p r e d i c t i o n s i n c l u d e both 'primary' and 'secondary' s e t t l e m e n t . The r e s u l t s suggest t h a t the behaviour of s o i l under i n c r e m e n t a l l o a d i n g c o n d i t i o n s can be p r e d i c t e d from CRS t e s t s d a t a . 5.3 Suggestions f o r F u r t h e r Research I t i s suggested t h a t the proposed method of p r e d i c t i o n should be checked f o r some o t h e r types of t e s t s , eg. c o n s t a n t r a t e , o f l o a d i n g and c o n t r o l l e d g r a d i e n t t e s t s . The a p p l i c a t i o n o f the method to p r e d i c t the f i e l d b ehaviour of s o i l should be s t u d i e d . The s o i l used i n the p r e s e n t i n v e s t i g a t i o n was a h i g h l y compressible o r g a n i c m a t e r i a l . I t i s suggested t h a t the method should be a p p l i e d to other s o i l s which are not as compressible as peat. 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Pub. 126. Lambe, T.W., (1951) S o i l T e s t i n g f o r E n g i n e e r s , John Wiley and Sons, New York. Modern Computing Methods (MCM), (1962) 'Notes on A p p l i e d S c i e n c e ' , N a t i o n a l P h y s i c s L a b o r a t o r y , No. 16, HMSO, London. 56 P o s k i t t , T . J . , (1967) 1A Note on the C o n s o l i d a t i o n of Clay w i t h Non-L i n e a r V i s c o s i t y ' , Geotechnique, V o l . 17, No. 2. Schmertmann, J.H., (1965) D i s c u s s i o n of ' I n t e r p r e t a t i o n of the C o n s o l i d a t i o n T e s t ' , J o u r n a l of the S o i l Mechanics and Foundations D i v i s i o n , ASCE, V o l . 91, No. SM 2. Smith, R.E., and Wahls, H.E. (1969) ' C o n s o l i d a t i o n Under Constant Rate of S t r a i n ' , J o u r n a l of S o i l Mechanics and Foundation D i v i s i o n , ASCE, V o l . 95, No. SM 2, Proc. Paper 6452. T a y l o r , D.W., (1942) 'Research on C o n s o l i d a t i o n of C l a y s ' , S e r i a l 82 MIT P u b l i c a t i o n . , and Merchant, W., (1940) 'A Theory of C l a y C o n s o l i d a t i o n A c c ounting f o r Secondary Compression', J o u r n a l Math, and P h y s i c s , V o l . 19. T e r z a g h i , K., (1923) 'Die Berechnung der D u r c h l a s s i g k e i t s z i f f e r des Tones aus dem V e r l a u f der hydrodynamischen Spannungserscheinungen', S i t z b e r . Akad. Wiss., Wien, Abt. I l a , V o l . 132. (1941) 'Undisturbed C l a y Samples and Undisturbed C l a y s ' , J . Boston Soc. C i v . Engrs., 28, No. 3. Wahls, H.E., and DeGodoy, N.S., (1965) D i s c u s s i o n of ' I n t e r p r e t a t i o n of the C o n s o l i d a t i o n T e s t 1 , J o u r n a l of the S o i l Mechanics and Foundations D i v i s i o n , ASCE, V o l . 91, No. SM 3. 57 APPENDIX 1 THE NUMERICAL SOLUTION OF THE GOVERNING CONSOLIDATION EQUATIONS where The equations to be s o l v e d are 3 2 u _ 3e C . a „ z 5 t ... (A.l) d Z a n d e - e 2 = a(u + n (§§>B> .... (A.2) C = k ( 1 + e l ) parameter r e l a t e d to p e r m e a b i l i t y u the excess pore p r e s s u r e z the space v a r i a b l e e v o i d r a t i o t time v a r i a b l e e 2 f i n a l v a l u e of v o i d r a t i o a c o e f f i c i e n t o f l i n e a r c o m p r e s s i b i l i t y - a a v c o e f f i c i e n t o f n o n - l i n e a r f u n c t i o n ^ ~~ , .1 of s t r a i n r a t e (1 +e 1) g-6 index i n power law r e l a t i n g s t r a i n r a t e s to the f u n c t i o n o f s t r a i n r a t e e^ i n i t i a l v a l u e of v o i d r a t i o 3 e 3 e P u t t i n g e" = e - e 2 and = y^-58 x = — / where H i s the l e n g t h of drainage path u U = Aa (e - e) y = 1 , the degree of c o n s o l i d a t i o n ( e x - e 2) X = 1 - y = a(Aa) E l i m i n a t i n g u from Equations (A.l) and (A.2) C. 8 2 [ | - n ( f § ) 3 I = f f ••• <A-3> 3(xH) D i v i d i n g Aa and r e a r r a n g i n g 3 2 r 1 / 3 ,. *1-3 3X i , [ X - (a.n (Aa) . ^r) 3 ] . o dt 3x ' c at 3X aH 2 3X P u t t i n g (A.4) 3T C ' 3t ' the d i m e n s i o n l e s s time f a c t o r governing the d i f f u s i o n p r o c e s s i s : T = ^ . — ... (A.5) a H z P u t t i n g (an 3(Aa) 1 _ B |£) J = (|^_) ... (A. 6) the d i m e n s i o n l e s s time f a c t o r a s s o c i a t e d w i t h the s t r a i n r a t e e f f e c t s i s T where: s 59 t ^ j A c n ^ ... ( f i. 7 ) a . n T g i s be s t expressed by means of the d i m e n s i o n l e s s r a t i o : R = h = V n V i ••• (A-8) r s H 2 (Aa ) 1 3 1 S i n c e E q u a t i o n (A.4) i s n o n - l i n e a r , Barden (1965) found i t convenient to work wi t h the f o l l o w i n g two simultaneous e q u a t i o n s : = ¥A tA 9) - ... ( .yjOX ^ = - | " u ) 3 ... (A. 10) P o s k i t t (1967), however, used the f o l l o w i n g f o r m u l a t i o n which i s more s t a b l e and e a s i e r to apply. From Equations (A.9) and (A.10), the f o l l o w i n g e q u a t i o n may be found * = u + (-R — ) ... (A.11) 8x2 S u b s t i t u t i o n o f Equation (A.11) i n Eq u a t i o n (A.9) g i v e s | - [ a + ( - R 0 ) t r ] = 0 . . . ( A .12) or d i f f e r e n t i a t i n g 60 1 j£ - \ [ ( ^ ) ] 2g ^—^ = ... (A.13) d T p 3x 2 a x 23T 3x 2 Boundary and I n i t i a l C o n d i t i o n s Boundary C o n d i t i o n s : ~ = 0 a t x = 0 f o r any T. d X T h i s i m p l i e s no flow a c r o s s the impermeable boundary, and u = 0 at x = 1 and any T t h a t i s f r e e d r ainage to the top of the sample. I n i t i a l C o n d i t i o n : The i n i t i a l c o n d i t i o n o f the sample i s t h a t the v o i d r a t i o i s everywhere equal to e, so t h a t A = 1 a t T = 0 and any x. S u b s t i t u t i n g t h i s v a l u e i n t o E q u a t i o n (A.10), the i n i t i a l excess pore water p r e s s u r e d i s t r i b u t i o n i s g i v e n by ^ = - | (1 - u ) 3 ... (A.14) 3 X 2 R In the T e r z a g h i theory the i n i t i a l excess pore water p r e s s u r e i s uniform and equal to the a p p l i e d l o a d , t h a t i s , ft(x,0) = 1. However, as p o i n t e d out by T a y l o r 61 (1942) and as shown by E q u a t i o n (A.14), t h i s i s not q u i t e so, due to the s t r a i n r a t e e f f e c t s . E q u a t i o n (A.14) must be s o l v e d s u b j e c t to the boundary c o n d i t i o n s . du dx = 0 a t x = 0; u = 0 a t x = 1 As R approaches zero, however, the i n i t i a l pore p r e s s u r e d i s t r i b u t i o n approaches the T e r z a g h i v a l u e , i . e . , u(x,0) = 1, R e f e r r i n g to E q u a t i o n (A.13) and w r i t i n g i t i n f i n i t e d i f f e r e n c e form f o r the i t h p o i n t along the x - a x i s (see F i g u r e A . l ) and the j t h p o i n t along the T - a x i s , then the s o l u t i o n may be forwarded i n time steps A T by the formula ... (A.15) where u. u. u = • u P = u N Time, T Drainage boundary F i g u r e A . l F i n i t e D i f f e r e n c e Space - Time G r i d 63 P. = 1 u. , - 2u. + u. ^ , ) l - l l i + l (Ax) A = (1 + 2Q 1) - Q. 2Q. (1 + 2Q 2) - Q 2 r Q N ( 1 + 2 Q N and 1 (1-B ) Q = lit' K P i ) 2 ] 2 &  1 B ' 3 . ( A x ) 2 When s o l v i n g the t r i d i a g o n a l equations (A.. ^  P..) use was made of the s u b r o u t i n e s a v a i l a b l e on the U.B.C. Computer l i b r a r y . 64 APPENDIX 2 ONE DIMENSIONAL CONSOLIDATION TESTS A schematic diagram of the CRS t e s t i n g set-up i s shown i n Fig u r e A.2. Two power d r i v e n constant deformation r a t e machines were a v a i l a b l e w i t h a wide range ^of speeds. To exclude the temperature e f f e c t s , a l l t e s t s were performed i n a temperature c o n t r o l l e d room which was kept constant at 20°C (± 0.5°C). To ensure accurate measurement of the base pore pressure, great care was taken to s a t u r a t e the sample. The base porous stone was b o i l e d f o r f i f t e e n minutes to remove a i r and then cooled to room temperature. To reduce f r i c t i o n between the sample and the r i n g , a t h i n f i l m of s i l i c o n l u b r i c a n t was a p p l i e d to the r i n g . The peat was poured i n the c o n s o l i d a t i o n r i n g . The base of the sample was sealed preventing drainage and the base pore pressures were measured. A l l samples were c o n s o l i d a t e d under a s t r e s s of 0.044 kg/cm 2 f o r 24 hours. A g u i d i n g rod was used to avoid t i p p i n g of the top porous stone due to the low s t r e n g t h a s s o c i a t e d w i t h the high i n i t i a l water content of the samples. This rod was replaced by a b a l l before the samples were subjected to the d e s i r e d s t r a i n r a t e . For an incremental t e s t , however, increments of load were added at d e s i r e d i n t e r v a l s of time. F i g u r e A.2 Schematic Diagram of T e s t i n g Set-Up 66 P e r i o d i c a l readings of the v e r t i c a l displacement base pore p r e s s u r e and t o t a l l o a d were recorded by v e r t i c a l d i a l gauge, pore p r e s s u r e t r a n s d u c e r and a lo a d c e l l r e s p e c t i v e l y . Barometric readings were a l s o taken to apply c o r r e c t i o n s t o the pore p r e s s u r e r e a d i n g s . 

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