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Modulus reduction dynamic analysis Purssell, Tanis Jane 1985

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MODULUS REDUCTION DYNAMIC ANALYSIS by. TANIS JANE PURSSELL A.Sc., U n i v e r s i t y Of B r i t i s h Columbia, 1982 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES 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 to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1985 © Tanis Jane P u r s s e l l , 1985 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 study. I f u r t h e r agree that permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 7 October 1985 ABSTRACT A s e m i - a n a l y t i c a l method of dynamic a n a l y s i s , capable of p r e d i c t i n g both the magnitude and p a t t e r n of earthquake induced deformations, i s presented. The a n a l y s i s i s based on a modulus r e d u c t i o n approach which uses a reduced modulus to simulate the s o f t e n i n g induced i n s o i l s d u r i n g c y c l i c l o a d i n g . The e f f e c t s of the i n e r t i a f o r c e s developed d u r i n g dynamic l o a d i n g on the induced deformations are a l s o i n c l u d e d through an a p p r o p r i a t e s e l e c t i o n of the reduced modulus. The reduced modulus i s u t i l i z e d i n a s t a t i c s t r e s s - s t r a i n a n a l y s i s to p r e d i c t the magnitude and p a t t e r n of the deformations induced d u r i n g earthquake l o a d i n g . The a p p r o p r i a t e modulus r e d u c t i o n i s determined from l a b o r a t o r y t e s t s on undisturbed s o i l samples. Three methods of computing a s u i t a b l e p o s t - c y c l i c modulus w e r e . i n v e s t i g a t e d but only the c y c l i c s t r a i n approach, i n which the modulus i s determined from c y c l i c l o a d i n g t e s t s that d u p l i c a t e the f i e l d s t r e s s c o n d i t i o n s , y i e l d s r e d u c t i o n s of s u f f i c i e n t magnitude to provide r e a l i s t i c e s t i m a t e s of earthquake induced deformations. The modulus r e d u c t i o n a n a l y s i s was used to p r e d i c t the deformations o c c u r r i n g d u r i n g dynamic l o a d i n g of a model t a i l i n g s slope i n a l a b o r a t o r y shaking t a b l e t e s t and of the Upper San Fernando Dam d u r i n g the earthquake of February, 1971. These s t u d i e s showed that the modulus r e d u c t i o n a n a l y s i s i s capable of reproducing the d y n a m i c a l l y induced deformations and that r e d u c t i o n s i n the modulus of up to 1000 times may be r e q u i r e d . U n f o r t u n a t e l y , l i m i t a t i o n s of the t e s t i n g equipment and inadequacies i n the a v a i l a b l e data r e q u i r e d that the a p p r o p r i a t e modulus r e d u c t i o n s c o u l d not be determined e n t i r e l y through l a b o r a t o r y and f i e l d i n v e s t i g a t i o n s . Some assumptions were necessary i n s e l e c t i n g the reduced modulus values used i n the a n a l y s e s . Although these case s t u d i e s were, hence, unable to p r o v i d e f u l l v e r i f i c a t i o n of the proposed method, they do demonstrate the r e l i a b i l i t y and s i m p l i c i t y of the a n a l y s i s as a method of a s s e s s i n g the performance of s o i l s t r u c t u r e s during earthquake l o a d i n g . i v TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS ix Chapter 1 INTRODUCTION 1 Chapter 2 EFFECTS OF CYCLIC LOADING ON SOIL 4 2.1 L i q u e f a c t i o n 6 2.2 C y c l i c M o b i l i t y 12 Chapter 3 CURRENT METHODS FOR EVALUATING EARTHQUAKE PERFORMANCE AND ESTIMATING EARTHQUAKE INDUCED DEFORMATIONS 16 3.1 P s e u d o - s t a t i c Method 16 3.2 Newmark A n a l y s i s 19 3.3 Seed's Dynamic S t r e s s Path Approach 21 3.4 E f f e c t i v e S t r e s s Dynamic A n a l y s i s 24 Chapter 4 PROPOSED MODULUS REDUCTION DYNAMIC ANALYSIS 26 4.1 Methods f o r C a l c u l a t i n g Modulus Reduction 27 4.2 Deformation A n a l y s i s Procedure 29 Chapter 5 VERIFICATION OF MODULUS REDUCTION ANALYSIS 38 5.1 T a i l i n g s Model T e s t s 38 5.1.1 Laboratory T e s t s 42 5.1.2 Determination of Reduced Moduli 51 5.1.3 R e s u l t s of Deformation P r e d i c t i o n s 58 V 5.2 A n a l y s i s of the Upper San Fernando Dam 70 5.2.1 D e s c r i p t i o n of Dam and Earthquake Deformations 71 5.2.2 Previous I n v e s t i g a t i o n s 74 5.2.3 Modulus Reduction A n a l y s i s 79 Chapter 6 SUMMARY AND CONCLUSIONS 89 REFERENCES 93 v i LIST OF TABLES Page Table 5-1 Summary of Hyperbolic Parameters of Ottawa Sand 50 Table 5-2 S o i l Parameters for San Fernando Dam S o i l s 78 v i i LIST OF FIGURES Page F i g u r e 2-1 L i q u e f a c t i o n Due to Monotonic or C y c l i c Loading 7 F i g u r e 2-2 T y p i c a l S t r e s s Path Followed During L i q u e f a c t i o n 9 F i g u r e 2-3 T y p i c a l S t r e s s Path Followed During C y c l i c M o b i l i t y 13 F i g u r e 4-1 P o s t - C y c l i c Modulus Approach 28 F i g u r e 4-2 C y c l i c S t r a i n Approach 30 F i g u r e 4-3 Pore Pressure Approach 31 F i g u r e 5-1 Model Container 40 F i g u r e 5-2 G r a i n S i z e D i s t r i b u t i o n of Test Sand . 41 F i g u r e 5-3 Observed Displacements of Model Slope 43 F i g u r e 5-4 S o i l Response During Monotonic T r i a x i a l T e s t s Fine Ottawa Sand - C o n f i n i n g Pressure = 50 kPa 45 F i g u r e 5-5 S o i l Response During Monotonic T r i a x i a l T e s t s Fine Ottawa Sand - C o n f i n i n g Pressure = 100 kPa 46 F i g u r e 5-6 S o i l Response During Monotonic T r i a x i a l T e s t s F i n e Ottawa Sand - C o n f i n i n g Pressure = 150 kPa 47 F i g u r e 5-7 S o i l Response During Monotonic T r i a x i a l T e s t s Fine Ottawa Sand - C o n f i n i n g Pressure = 200 kPa 48 F i g u r e 5-8 S o i l Response Before and A f t e r C y c l i c Loading F i n e Ottawa Sand - C o n f i n i n g Pressure = 50 kPa 52 F i g u r e 5-9 S o i l Response Before and A f t e r C y c l i c Loading Fine Ottawa Sand - C o n f i n i n g Pressure = 100 kPa 53 F i g u r e 5-10 S o i l Response Before and A f t e r C y c l i c Loading Fine Ottawa Sand - C o n f i n i n g Pressure = 150 kPa 54 v i i i F i g u r e 5-11 F i n i t e Element G r i d of T a i l i n g s Model used in S t a t i c - S t r e s s A n a l y s i s 59 F i g u r e 5-12 Deformations P r e d i c t e d by P o s t - C y c l i c Modulus Approach 61 Deformations P r e d i c t e d by C y c l i c S t r a i n Approach 62 Deformations P r e d i c t e d by Pore Pressure Approach 63 Deformations P r e d i c t e d by C y c l i c S t r a i n Approach F i g u r e 5-13 F i g u r e 5-14 F i g u r e 5-15 F i g u r e 5-16 F i g u r e 5-17 F i g u r e 5-18 F i g u r e 5-19 F i g u r e 5-20 F i g u r e 5-21 F i g u r e 5-22 F i g u r e 5-23 F i g u r e 5-24 F i g u r e 5-25 with Volume Change C o r r e c t i o n 64 Accelerogram of N i i g a t a Earthquake 67 Deformations of Upper San Fernando Dam d u r i n g Earthquake 72 Major S o i l Types i n Upper San Fernando Dam 75 T y p i c a l Response of H y d r a u l i c F i l l i n Drained and Undrained T r i a x i a l T e s t s 77 L i q u e f i e d Areas of Dam 80 Shear S t r a i n P o t e n t i a l s 82 F i n i t e Element G r i d used i n S t a t i c - S t r e s s A n a l y s i s of Upper San Fernando Dam 84 P r e d i c t e d Deformations 85 Required S t r a i n P o t e n t i a l s 87 P r e d i c t e d Deformations using Required S t r a i n P o t e n t i a l s 88 ix ACKNOWLEDGEMENTS I would l i k e to thank Dr. Peter M. Byrne f o r h i s guidance i n i n i t i a t i n g t h i s study and f o r making c r i t i c a l c o n t r i b u t i o n s towards i t s completion. Dr. Yogi V a i d and Edwin Chung a l s o r e q u i r e thanks f o r t h e i r d i r e c t i o n and h e l p i n performing the l a b o r a t o r y s o i l t e s t s . The f i n a n c i a l a s s i s t a n c e p r o v i d e d by a N a t u r a l Sciences and En g i n e e r i n g Research C o u n c i l of Canada postgraduate s c h o l a r s h i p i s g r a t e f u l l y a p p r e c i a t e d . 1 CHAPTER 1 INTRODUCTION The v a r i e t y of methods used to assess the dynamic response of s o i l s t r u c t u r e s to earthquake l o a d i n g has i n c r e a s e d s u b s t a n t i a l l y i n recent y e a r s . However, many of these c u r r e n t methods e i t h e r make so many s i m p l i f y i n g assumptions that they do not a c c u r a t e l y or r e a l i s t i c a l l y p r e d i c t s o i l behavior under dynamic l o a d i n g or are so complex and c o s t l y that t h e i r use may be l i m i t e d to c r i t i c a l s t r u c t u r e s . There appears to be a need f o r a r e l a t i v e l y simple method f o r a s s e s s i n g earthquake performance that i s based on a c t u a l s o i l behavior and i s capable of p r e d i c t i n g earthquake induced deformations of the c o r r e c t magnitude and p a t t e r n . The proposed modulus r e d u c t i o n approach to dynamic a n a l y s i s i s presented as such a method. From t h e i r r a t h e r rudimentary beginnings i n the pseudo-s t a t i c a n a l y s i s , dynamic response analyses have i n c r e a s i n g l y t r i e d to i n c o r p o r a t e more r e a l i s t i c models of s o i l behavior. The " e q u i v a l e n t " permanent f o r c e used i n the p s e u d o - s t a t i c a n a l y s i s was r e p l a c e d by a temporary f o r c e i n Newmark's a n a l y s i s which recognized the t r a n s i e n t nature of earthquake induced l o a d i n g s . The n o n - l i n e a r s t r a i n dependent behavior of s o i l s was i n i t i a l l y accounted f o r i n t o t a l s t r e s s f i n i t e element analyses by i n c o r p o r a t i n g h y p e r b o l i c or other s i m i l a r s t r e s s - s t r a i n models of s o i l behavior. The importance of pore pressure development and e f f e c t i v e s t r e s s e s on s o i l response d u r i n g c y c l i c l o a d i n g was then recognized and d e a l t with, f i r s t by Seed 2 i n h i s dynamic s t r e s s path approach and l a t e r by many others i n v a r i o u s r i g o r o u s n o n - l i n e a r e f f e c t i v e s t r e s s dynamic a n a l y s e s . These recent methods are extremely complex, o f t e n r e q u i r i n g parameters not commonly used in g e o t e c h n i c a l p r a c t i c e , and g e n e r a l l y l a c k i n g s u f f i c i e n t v e r i f i c a t i o n to j u s t i f y t h e i r use. The modulus r e d u c t i o n method of dynamic a n a l y s i s i s presented as a simple and r e a l i s t i c model for p r e d i c t i n g earthquake induced deformations. Although the reduced modulus i s p r i m a r i l y intended to simulate the s o f t e n i n g of a s o i l d u r i n g dynamic e x c i t a t i o n , i t can a l s o account f o r the e f f e c t s of the i n e r t i a f o r c e s on the deformations that develop d u r i n g c y c l i c l o a d i n g . Because the s t i f f n e s s degradation that r e s u l t s from the p r o g r e s s i v e r i s e i n pore pressure d u r i n g c y c l i c l o a d i n g i s represented by the reduced modulus, t h i s method i s e s p e c i a l l y s u i t e d f o r the a n a l y s i s of loose to medium dense s o i l s which may develop s i g n i f i c a n t pore p r e s s u r e s i n response to dynamic e x c i t a t i o n . However, s i n c e the e f f e c t s of the i n e r t i a f o r c e s c r e a t e d d u r i n g c y c l i c l o a d i n g are a l s o i n c l u d e d by an a p p r o p r i a t e s e l e c t i o n of the reduced modulus, the proposed method may be used to p r e d i c t the deformations r e s u l t i n g from c y c l i c l o a d i n g of s o i l s that are expected to experience only l i m i t e d pore p r e s s u r e changes duri n g c y c l i c shear. The b a s i c approach of the proposed modulus r e d u c t i o n a n a l y s i s i s s i m i l a r to Seed's s t r e s s path method except that a reduced modulus i s u t i l i z e d i n c o n j u n c t i o n with a s t a t i c s t r e s s -s t r a i n a n a l y s i s to p r e d i c t earthquake induced deformations. 3 T h i s t h e s i s e x p l a i n s the theory behind the development of the modulus r e d u c t i o n method and determines i t s v a l i d i t y on the b a s i s of two case h i s t o r i e s : a l a b o r a t o r y shaking t a b l e study intended to model the response of s a t u r a t e d t a i l i n g s t o earthquake l o a d i n g , and the Upper San Fernando Dam which experienced s u b s t a n t i a l downstream movements d u r i n g the earthquake of February, 1971. 4 CHAPTER 2 EFFECTS OF CYCLIC LOADING ON SOIL An earthquake or any other dynamic l o a d i n g c r e a t i n g c y c l i c a l l y r e v e r s i n g shear s t r e s s e s i n a s o i l s t r u c t u r e b a s i c a l l y has two e f f e c t s : i t generates t r a n s i e n t i n e r t i a f o r c e s and may cause the development of excess pore p r e s s u r e s . Although both of these c o n d i t i o n s w i l l cause earthquake induced deformations, most of the severe earthquake damage has r e s u l t e d from deformations a s s o c i a t e d with pore pressure r i s e . The extent of deformations r e s u l t i n g from changes in pore p r e s s u r e s d u r i n g c y c l i c l o a d i n g depends on the magnitude of the pore p r e s s u r e r i s e and on the mechanism of pore pressure g e n e r a t i o n . When undrained c o n d i t i o n s p r e v a i l during c y c l i c l o a d i n g of a s a t u r a t e d s o i l , pore water pr e s s u r e s w i l l i n c r ease because of the tendency of the s o i l to c o n t r a c t when subjected to c y c l i c shear. As the pore p r e s s u r e s i n c r e a s e , the s o i l s o f t e n s and s t r a i n s w i l l develop. However, s i g n i f i c a n t s t r a i n s only occur i f the pore p r e s s u r e s i n c r e a s e enough to t r i g g e r the onset of e i t h e r of two d i s t i n c t phenomena: l i q u e f a c t i o n or c y c l i c m o b i l i t y . The two terms, " l i q u e f a c t i o n " and " c y c l i c m o b i l i t y " , occur e x t e n s i v e l y i n the l i t e r a t u r e and u n f o r t u n a t e l y have been used to d e s c r i b e a v a r i e t y of phenomena by d i f f e r e n t i n v e s t i g a t o r s . Confusion p a r t i c u l a r l y a r i s e s over the term " l i q u e f a c t i o n " . Although the term has t r a d i t i o n a l l y r e f e r r e d to e i t h e r a c o n d i t i o n of zero e f f e c t i v e s t r e s s w i t h i n a s o i l or to slope 5 f a i l u r e s which resemble the flow of a v i s c o u s f l u i d , i t i s now being used to d e s c r i b e s e v e r a l phenomena observed d u r i n g l a b o r a t o r y c y c l i c l o a d i n g t e s t s . I t has become customary to use the term " l i q u e f a c t i o n " f o r the development of 5 or 10 percent s t r a i n i n a c y c l i c load t e s t . " I n i t i a l l i q u e f a c t i o n " has been d e s c r i b e d by Seed and Lee (1966) as the stage i n a c y c l i c l o a d t e s t when pore p r e s s s u r e s momentarily become equal to the c o n f i n i n g s t r e s s e s . C a s t r o and Poulos (1975), on the other hand, have d e f i n e d l i q u e f a c t i o n as a phenomenon wherein a s a t u r a t e d sand l o s e s a l a r g e percentage of i t s shear r e s i s t a n c e and flows i n a manner resembling a l i q u i d u n t i l the shear s t r e s s e s a c t i n g on the s o i l mass are as low as the reduced shear r e s i s t a n c e . V a i d and Chern (1983), l i k e w i s e , r e f e r r e d to l i q u e f a c t i o n as a c o n t r a c t i v e flow f a i l u r e . For the purpose of t h i s t h e s i s , the term " l i q u e f a c t i o n " w i l l be reserved to d e s c r i b e the flow f a i l u r e s t h a t r e s u l t from a l o s s i n s t r e n g t h d u r i n g c y c l i c l o a d i n g . The term " c y c l i c m o b i l i t y " was f i r s t introduced by Casagrande i n 1965 to r e f e r to the s t r a i n s that accumulate i n undrained c y c l i c load t e s t s on sand. He d e s c r i b e d c y c l i c m o b i l i t y as a p r o g r e s s i v e r e d u c t i o n i n the s t i f f n e s s of a s a t u r a t e d s o i l when su b j e c t e d to c y c l i c l o a d i n g . U n l i k e l i q u e f a c t i o n , the s t r a i n s a s s o c i a t e d with c y c l i c m o b i l i t y develop p r o g r e s s i v e l y d u r i n g each l o a d i n g c y c l e . They may, however, become as s i g n i f i c a n t as those r e s u l t i n g from 1 i q u e f a c t i o n . 6 The d i s t i n c t i o n between l i q u e f a c t i o n and c y c l i c m o b i l i t y i s necessary because the mechanism of s t r a i n development i n v o l v e d i n each i s e n t i r e l y d i f f e r e n t . In a d d i t i o n , the development of the two phenomena are a f f e c t e d q u i t e d i f f e r e n t l y by the i n i t i a l s t a t e of the s o i l , as d e f i n e d by v o i d r a t i o , c o n f i n i n g pressure and s t a t i c shear s t r e s s . To r e a l i s t i c a l l y simulate the s o i l response dur i n g c y c l i c l o a d i n g and to p r e d i c t the dynamically induced deformations, the d i f f e r e n t mechanisms of s t r a i n development i n v o l v e d i n the two phenomena must be f u l l y understood. 2 . 1 L i q u e f a c t ion L i q u e f a c t i o n i s a flow f a i l u r e a s s o c i a t e d with c o n t r a c t i v e or s t r a i n s o f t e n i n g s o i l s . I t can o n l y occur i n the presence of d r i v i n g shear s t r e s s e s and i f a s u b s t a n t i a l p r o p o r t i o n of the s o i l ' s shear s t r e n g t h i s l o s t d u r i n g undrained monotonic, c y c l i c or shock l o a d i n g . The l o s s i n s t r e n g t h r e s u l t s from a c o n v e r s i o n of the s o i l mass from a p r a c t i c a l l y d r a i n e d c o n d i t i o n to a p r a c t i c a l l y undrained c o n d i t i o n of shear. For c o n t r a c t i v e s o i l s , the undrained shear s t r e n g t h may be s u b s t a n t i a l l y lower than the d r a i n e d s t r e n g t h and s t r a i n s o f t e n i n g occurs dur i n g undrained shear. Thus, although the s o i l mass i s able to support the i n - s i t u shear s t r e s s e s under d r a i n e d c o n d i t i o n s , i t f a i l s when a s u f f i c i e n t t r i g g e r i n g s t r e s s or c y c l i c l o a d i n g i s a p p l i e d undrained. F i g u r e 2-1 i l l u s t r a t e s s c h e m a t i c a l l y the c o n d i t i o n s r e q u i r e d to i n i t i a t e l i q u e f a c t i o n f a i l u r e s . During l i q u e f a c t i o n , the s o i l flows i n a manner resembling a v i s c o u s 7 f l u i d u n t i l the shear s t r e s s e s are as low as the reduced s t r e n g t h . Because flow deformations i n v o l v e l a r g e u n i d i r e c t i o n a l d isplacements, i t has been suggested (Castro and Poulos, 1975, and Poulos et a l , 1985) that the reduced s t r e n g t h reached d u r i n g l i q u e f a c t i o n i s the undrained s t e a d y - s t a t e shear s t r e n g t h . T h i s s t r e n g t h i s the s t r e n g t h that e x i s t s when the s o i l mass i s c o n t i n u o u s l y deforming at a constant volume, constant normal e f f e c t i v e s t r e s s , constant shear s t r e s s and constant v e l o c i t y . The s t e a d y - s t a t e s t r e n g t h 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 e s reached d u r i n g f a i l u r e and i s not zero. V) v> U J OC y-v> K < h i X V ) D R I V I N G S H E A R S T R E S S L I Q U E F A C T I O N D U E T O M O N O T O N I C L O A D I N G / A \ \ \ \ L I O U E F A C T I O N O C C U R S W H E N S T R E N G T H D R O P S B E L O W D R I V I N G S H E A R S T R E S S ^ L I Q U E F A C T I O N D U E T O C Y C L I C L O A D I N Q S T R A I N F i g u r e 2-1 L i q u e f a c t i o n Due to Monotonic or C y c l i c Loading (Schematic) (from Poulos et a l , 1985) In l a b o r a t o r y c y c l i c l o a d t e s t s , l i q u e f a c t i o n i s a s s o c i a t e d with a r a p i d i n c r e a s e i n pore p r e s s u r e s accompanied by a sudden development of a x i a l s t r a i n s . During c y c l i c l o a d i n g , the 8 p r o g r e s s i v e i n c r e a s e i n pore water pressure causes the e f f e c t i v e s t r e s s s t a t e of the s o i l to move s t e a d i l y towards the f a i l u r e envelope. For c o n t r a c t i v e s o i l s , l i q u e f a c t i o n i s t r i g g e r e d when the e f f e c t i v e s t r e s s s t a t e of the s o i l reaches a c r i t i c a l v a lue of the e f f e c t i v e s t r e s s r a t i o , 0 ^ / 0 3 ' , p r i o r to the f a i l u r e envelope or the s t e a d y - s t a t e l i n e . F i g u r e 2-2 i l l u s t r a t e s a t y p i c a l s t r e s s path f o l l o w e d d u r i n g l i q u e f a c t i o n . I f the s t e a d y - s t a t e of deformation i s reached d u r i n g flow, the deformations may continue u n t i l the l i m i t s of the t e s t i n g equipment are exceeded. V a i d and Chern (1983), however, found that flow deformations were o f t e n a r r e s t e d when the t r i a x i a l samples had s t r a i n e d s u f f i c i e n t l y to cause d i l a t i o n on f u r t h e r s t r a i n i n g . They r e f e r r e d to t h i s type of behavior as l i m i t e d l i q u e f a c t i o n . The s t e a d y - s t a t e i s not reached during l i m i t e d l i q u e f a c t i o n f a i l u r e s and although s t r a i n s develop r a p i d l y upon i n i t i a t i o n of f a i l u r e , they stop at the onset of d i l a t i o n . The s t r a i n t h a t develops d u r i n g such l i m i t e d flow f a i l u r e s depends on the i n i t i a l s t a t e of the sample and the p r o p e r t i e s of the s o i l . A d d i t i o n a l c y c l i c l o a d i n g , a f t e r the a r r e s t of l i q u e f a c t i o n , was found (Vaid and Chern, 1983) to cause f u r t h e r s t r a i n i n g accompanied by d i l a t i o n and a r e d u c t i o n i n pore p r e s s u r e s d u r i n g the l o a d i n g phase of each s t r e s s c y c l e . S i g n i f i c a n t accumulations of s t r a i n were only observed when the a p p l i e d c y c l i c s t r e s s was of s u f f i c i e n t magnitude to exceed the s t a t i c shear s t r e s s and cause s t r e s s r e v e r s a l s to occur. The s t r e s s path followed d u r i n g these a d d i t i o n a l c y c l e s of l o a d i n g i s shown on F i g u r e 2-2. FIGURE 2 -2 Effective Stress Path of Cyclic Loading Test on Anisotropically Consolidated Loose Sand 10 The magnitude and r a t e of pore pressure b u i l d u p d u r i n g c y c l i c l o a d i n g depends on the magnitude of the c y c l i c l o a d , the number of c y c l e s a p p l i e d , the e f f e c t i v e c o n f i n i n g p r e s s u r e s and the l e v e l of s t a t i c shear s t r e s s . The r e l a t i v e magnitudes of the c y c l i c d e v i a t o r s t r e s s and the s t a t i c shear s t r e s s are a l s o s i g n i f i c a n t . When the s t a t i c shear s t r e s s i s gr e a t e r than the c y c l i c d e v i a t o r s t r e s s , s t r e s s r e v e r s a l s during c y c l i c l o a d i n g do not occur . When s t r e s s r e v e r s a l s do occur, they s i g n i f i c a n t l y i n c r e a s e the r a t e of pore pressure g e n e r a t i o n and the magnitude of the excess pore pressures reached d u r i n g a s p e c i f i e d number of c y c l e s of l o a d i n g . The magnitude of the pore p r e s s u r e s reached d u r i n g l i q u e f a c t i o n depends on the e f f e c t i v e c o n f i n i n g s t r e s s e s and on the l e v e l of s t a t i c shear s t r e s s i n the s o i l . Chern (1981) showed that the r e s i d u a l value of the excess pore p r e s s u r e s , Ur, developed d u r i n g l i q u e f a c t i o n a f t e r the t e r m i n a t i o n of c y c l i c l o a d i n g i s given by Only i n i s o t r o p i c a l l y c o n s o l i d a t e d samples w i l l the r e s i d u a l pore p r e s s u r e become equal to the e f f e c t i v e c o n f i n i n g p r e s s u r e . In the presence of s t a t i c shear s t r e s s e s , the r e s i d u a l pore pressure i s always l e s s than the e f f e c t i v e c o n f i n i n g p r e s s u r e . However, the t r a n s i e n t value of the pore pressure generated d u r i n g c y c l i c l o a d i n g w i l l f l u c t u a t e about a mean value and w i l l exceed the r e s i d u a l value p e r i o d i c a l l y d u r i n g c y c l i c l o a d i n g as soon as l i q u e f a c t i o n has occur r e d . Thus, the maximum value of the pore p r e s s u r e d u r i n g c y c l i c l o a d i n g w i l l be higher than the Ur = o 3 c ' (1 - (Kc ~ 1 ) ( 1 - s i n 4>' ) 11 r e s i d u a l v a l u e given by Chern. The s u s c e p t i b i l i t y of a given s o i l at a given r e l a t i v e d e n s i t y to l i q u e f a c t i o n i s a f f e c t e d by the c o n s o l i d a t i o n s t r e s s , o 3 c ' , and the i n i t i a l s t a t i c shear s t r e s s , r s . Higher e f f e c t i v e c o n s o l i d a t i o n p r e s s u r e s correspond to g r e a t e r s u s c e p t i b i l i t i e s t o l i q u e f a c t i o n . For a f i x e d r e l a t i v e d e n s i t y , a s o i l w i l l e x h i b i t c o n t r a c t i v e behavior more r e a d i l y at higher c o n f i n i n g p r e s s u r e s . Thus, a s o i l that may be s u s c e p t i b l e to l i q u e f a c t i o n at high l e v e l s of c o n f i n i n g s t r e s s w i l l not be s u s c e p t i b l e at lower c o n f i n i n g s t r e s s e s . The i n f l u e n c e of s t a t i c shear on the l i q u e f a c t i o n p o t e n t i a l of l o o s e sand v a r i e s with the l e v e l of s t a t i c shear. At r e l a t i v e l y low l e v e l s of s t a t i c shear, the r e s i s t a n c e to l i q u e f a c t i o n , d e f i n e d by the l e v e l of c y c l i c shear s t r e s s r e q u i r e d t o cause flow deformation i n a f i x e d number of c y c l e s of l o a d i n g , g e n e r a l l y i n c r e a s e s as the s t a t i c shear s t r e s s i n c r e a s e s . T h i s i n c r e a s e i s a p p a r e n t l y r e l a t e d to the reduced r a t e of pore pressure gener a t i o n caused by a r e d u c t i o n i n the magnitude of the shear s t r e s s r e v e r s a l s . For higher l e v e l s of s t a t i c shear, however, i n c r e a s e s in the s t a t i c shear s t r e s s may be accompanied by s u b s t a n t i a l r e d u c t i o n s i n the r e s i s t a n c e to l i q u e f a c t i o n . The r e d u c t i o n i n the r e s i s t a n c e to l i q u e f a c t i o n at high s t a t i c shear s t r e s s l e v e l s has been a t t r i b u t e d (Vaid and Chern, 1983) to the f a c t that the i n i t i a l s t r e s s s t a t e of the sample i s so c l o s e to the c r i t i c a l e f f e c t i v e s t r e s s r a t i o l i n e t h a t only a s m a l l c y c l i c d e v i a t o r s t r e s s or only a few c y c l e s of 12 l o a d i n g are r e q u i r e d f o r the s t r e s s s t a t e to reach the l i n e and l i q u e f a c t i o n to occur. 2.2 C y c l i c M o b i l i t y . C y c l i c m o b i l i t y r e f e r s to the formation of l a r g e s t r a i n s d u r i n g undrained c y c l i c l o a d i n g t e s t s on sand. U n l i k e l i q u e f a c t i o n , i t i s c h a r a c t e r i z e d by a gradual development of pore p r e s s u r e s accompanied by a p r o g r e s s i v e accumulation of s t r a i n s d u r i n g each c y c l e of l o a d i n g . No sudden i n c r e a s e i n the pore p r e s s u r e s or a x i a l s t r a i n s o c c u r s . A l o s s i n s t r e n g t h i s not r e q u i r e d to i n i t i a t e the onset of c y c l i c m o b i l i t y . Although c y c l i c m o b i l i t y g e n e r a l l y occurs i n medium dense to dense sands which e x h i b i t d i l a t i v e r a t h e r than c o n t r a c t i v e behavior, i t may a l s o develop i n loose sands a f t e r the a r r e s t of l i q u e f a c t i o n when s u f f i c i e n t s t r a i n s have developed to cause d i l a t i o n on f u r t h e r s t r a i n i n g . L i k e l i q u e f a c t i o n , c y c l i c m o b i l i t y i s i n i t i a t e d when the e f f e c t i v e s t r e s s s t a t e of the s o i l reaches a c r i t i c a l value of the e f f e c t i v e s t r e s s r a t i o . The c r i t i c a l value of the e f f e c t i v e s t r e s s r a t i o f o r c y c l i c m o b i l i t y i s higher than that f o r l i q u e f a c t i o n . V a i d and Chern (1983) showed that the e f f e c t i v e s t r e s s r a t i o s d e f i n i n g the onset of s i g n i f i c a n t c y c l i c m o b i l i t y i n dense sands and the a r r e s t of flow f a i l u r e s i n loose sands c o i n c i d e . F i g u r e 2-3 shows a t y p i c a l e f f e c t i v e s t r e s s path f o l l o w e d d u r i n g c y c l i c m o b i l i t y . When the s t r e s s s t a t e passes the c r i t i c a l e f f e c t i v e s t r e s s r a t i o l i n e the s o i l experiences a <H- 0.4 08 1.0 1.2 1.4 (cT,'+cr3)/2 (kg/cm 2) 24 (from Chern, 1981) FIGURE 2 -3 Effective Stress Path of Cyclic Loading Test on Anisotropically Consolidated Dense Sand 00 14 s i g n i f i c a n t i n crease i n the s t r a i n developed d u r i n g a s i n g l e l o a d i n g c y c l e . Upon unloading, pore p r e s s u r e s r i s e s u b s t a n t i a l l y causing the minor e f f e c t i v e s t r e s s to momentarily go to z e r o . However, l i t t l e change in the a x i a l s t r a i n o c c u r s . Subsequent r e l o a d i n g r e s u l t s i n the development of a d d i t i o n a l s t r a i n s accompanied by d i l a t i o n and the r e d u c t i o n of pore p r e s s u r e s . R e p e t i t i o n of c y c l i c l o a d s, each c a u s i n g a d d i t i o n a l s t r a i n s , r e s u l t s i n the l a r g e accumulation of s t r a i n s a s s o c i a t e d with c y c l i c m o b i l i t y . The pore pr e s s u r e s that develop dur i n g c y c l i c m o b i l i t y f l u c t u a t e around the r e s i d u a l pore pressure value d e f i n e d by Chern. T h i s r e s i d u a l p r e s s u r e i s determined by the l e v e l of s t a t i c shear s t r e s s and i s independent of whether i t r e s u l t s from c y c l i c m o b i l i t y or l i q u e f a c t i o n (Vaid and Chern, 1983). During c y c l i c m o b i l i t y , the pore pr e s s u r e s t r a n s i e n t l y exceed the r e s i d u a l value d u r i n g each load c y c l e . The t r a n s i e n t pore p r e s s u r e s may become equal to the c o n f i n i n g p r e s s u r e s momentarily when the c y c l i c d e v i a t o r s t r e s s i s zero. When lo a d i n g c o n t i n u e s , the s o i l tends to d i l a t e and the pore p r e s s u r e s reduce. The s u s c e p t i b i l i t y of a s o i l to c y c l i c m o b i l i t y i s i n f l u e n c e d by the e f f e c t i v e c o n f i n i n g pressures and by the l e v e l of s t a t i c shear. Although i n c r e a s e d c o n f i n i n g p r e s s u r e s g e n e r a l l y i n c r e a s e the c y c l i c load necessary to cause c y c l i c m o b i l i t y , the c y c l i c m o b i l i t y r a t i o , d e f i n e d as the c y c l i c d e v i a t o r s t r e s s d i v i d e d by twice the e f f e c t i v e minor p r i n c i p a l 15 s t r e s s a t the s t a r t of c y c l i c l o a d i n g , u s u a l l y decreases with i n c r e a s i n g c o n f i n i n g s t r e s s . Thus, the r e s i s t a n c e to c y c l i c m o b i l i t y g e n e r a l l y decreases with i n c r e a s i n g c o n f i n i n g p r e s s u r e . T h i s same t r e n d with c o n f i n i n g pressure was observed f o r l i q u e f a c t i o n . L i k e the r e s i s t a n c e to l i q u e f a c t i o n , the r e s i s t a n c e of a s o i l to c y c l i c m o b i l i t y v a r i e s with the l e v e l of s t a t i c b i a s i n the s o i l but not i n the same manner. Whereas high l e v e l s of s t a t i c s t r e s s can reduce a s o i l ' s r e s i s t a n c e to l i q u e f a c t i o n , i t s r e s i s t a n c e to c y c l i c m o b i l i t y tends to i n c r e a s e with the l e v e l of s t a t i c shear. T h i s behavior i s r e l a t e d to the occ u r r e n c e and magnitude of shear s t r e s s r e v e r s a l s . C a s t r o and Poulos (1977) found that without s t r e s s r e v e r s a l , the l a r g e r s t r a i n s t h a t c o n s t i t u t e c y c l i c m o b i l i t y do not occur. 16 CHAPTER 3 CURRENT METHODS FOR EVALUATING EARTHQUAKE PERFORMANCE AND  ESTIMATING EARTHQUAKE INDUCED DEFORMATIONS 3.1 P s e u d o - s t a t i c Method The p s e u d o - s t a t i c method was the o r i g i n a l method of a n a l y s i s used f o r a s s e s s i n g the performance of e a r t h s t r u c t u r e s d u r i n g seismic e x c i t a t i o n . The method y i e l d s o n l y a f a c t o r of s a f e t y a g a i n s t f a i l u r e d u r i n g seismic l o a d i n g and does not c o n s i d e r the magnitude of deformations that may r e s u l t . For over 40 years the p s e u d o - s t a t i c method was the standard method of e v a l u a t i n g the s a f e t y of dams a g a i n s t s l i d i n g d u r i n g earthquakes. I t s p r e d i c t i v e c a p a b i l i t i e s , however, came i n t o doubt from i t s i n a b i l i t y to p r e d i c t s e v e r a l dam f a i l u r e s . Today i t s use i s g e n e r a l l y l i m i t e d to p r e l i m i n a r y design c a l c u l a t i o n s f o r s o i l s that are not expected to s u f f e r from a p p r e c i a b l e s t r e n g t h or s t i f f n e s s l o s s d u r i n g c y c l i c l o a d i n g . In the p s e u d o - s t a t i c method the e f f e c t of the earthquake i s repr e s e n t e d by an e q u i v a l e n t h o r i z o n t a l s t a t i c f o r c e c a l c u l a t e d as a f r a c t i o n of the weight of the assumed f a i l u r e mass. The f r a c t i o n of the weight, termed the seismic c o e f f i c i e n t , i s s e l e c t e d on the b a s i s of the l e v e l or s e i s m i c i t y i n the re g i o n of i n t e r e s t . The e q u i v a l e n t s t a t i c f o r c e i s asssumed to be permanent and to act i n one d i r e c t i o n o n l y , through the c e n t r o i d of the assumed f a i l u r e mass. A c o n v e n t i o n a l slope s t a b i l i t y a n a l y s i s i s then performed to determine the f a c t o r of s a f e t y 17 a g a i n s t f a i l u r e . A f a c t o r of s a f e t y i n the range of 1.0 to 1.2 i s g e n e r a l l y b e l i e v e d to be adequate (Seed, 1979). The i n i t i a l widespread acceptance of the p s e u d o - s t a t i c method f o r seismic design r e s u l t e d because i t appeared to be an extremely simple and e f f e c t i v e design approach. However, a wide v a r i a t i o n i n the ch o i c e of a n a l y t i c a l d e t a i l s i s a v a i l a b l e . Because each of these c h o i c e s can have a s i g n i f i c a n t e f f e c t on the computed f a c t o r of s a f e t y , a broad range f o r the f a c t o r of s a f e t y may e x i s t . In a d d i t i o n , although the design e f f e c t i v e n e s s of the p s e u d o - s t a t i c method seemed to be i n d i c a t e d by the f a c t that few dams designed under i t s g u i d e l i n e s had f a i l e d , t h i s apparent design adequacy c o u l d be a t t r i b u t e d to the lack of strong earthquake motions r a t h e r than to the p r e d i c t i v e c a p a b i l i t i e s of the p s e u d o - s t a t i c method. The p r e d i c t i v e c a p a b i l i t y of the p s e u d o - s t a t i c method came under q u e s t i o n f o r i t s i n a b i l i t y to p r e d i c t the 1925 S h e f f i e l d Dam f a i l u r e , the f a i l u r e of the Lower San Fernando Dam i n 1971 and the f a i l u r e s of s e v e r a l t a i l i n g s dams during the Izu Oshima Earthquake of 1978. I n v e s t i g a t i o n s i n t o the i n a b i l i t y of the p s e u d o - s t a t i c method to p r e d i c t these f a i l u r e s r e v e a l e d s e v e r a l fundamental problems with such a method of a n a l y s i s . A major problem with the p s e u d o - s t a t i c method of a n a l y s i s i s i t s use of the f a c t o r of s a f e t y . The s i g n i f i c a n c e of the computed value f o r the f a c t o r of s a f e t y i n e v a l u a t i n g s t a b i l i t y i s not c l e a r . Although a f a c t o r of s a f e t y l e s s than one i n d i c a t e s f a i l u r e i n a standard s t a t i c s t a b i l i t y a n a l y s i s , f o r 18 the p s e u d o - s t a t i c a n a l y s i s i t may i n d i c a t e only the onset of p l a s t i c deformations rather than complete c o l l a p s e . Because of the t r a n s i e n t nature of the earthquake f o r c e , the peak seismic f o r c e w i l l only act f o r b r i e f i n s t a n t s of time d u r i n g which small p l a s t i c movements may occur. The magnitude of these accumulated deformations can be s i g n i f i c a n t i n determining whether the s o i l s t r u c t u r e may continue to be used f o r i t s design purpose. The i n a b i l i t y of the p s e u d o - s t a t i c method to p r e d i c t the magnitude of such deformations r e p r e s e n t s a s e r i o u s l i m i t a t i o n of t h i s method of a n a l y s i s . A f a c t o r of s a f e t y g r e a t e r than one may f a i l to g i v e a true i n d i c a t i o n of s t a b i l i t y i n p s e u d o - s t a t i c a n a l y s e s . Slopes having f a c t o r s of s a f e t y g r e a t e r than one may f a i l i f the slope forming m a t e r i a l i s s u s c e p t i b l e to s t r e n g t h and/or s t i f f n e s s l o s s d u r i n g c y c l i c l o a d i n g . Such s o i l behavior cannot be represented i n any r a t i o n a l manner by a permanent h o r i z o n t a l s t a t i c f o r c e . The use of the p s e u d o - s t a t i c method should thus be c o n f i n e d to the a n a l y s i s of slope forming m a t e r i a l s that are not s u s c e p t i b l e to pore pressure development d u r i n g c y c l i c l o a d i n g . The v a r i a b i l i t y of the computed f a c t o r of s a f e t y and the d i f f i c u l t y i n a s s e s s i n g i t s s i g n i f i c a n c e l i m i t the u s e f u l n e s s of the p s e u d o - s t a t i c method. C l e a r l y , there are too many d e v i a t i o n s from r e a l i s t i c c o n d i t i o n s i n such an a n a l y s i s to determine the r e l i a b i l i t y of the r e s u l t . 19 3.2 Newmark A n a l y s i s The Newmark method of se i s m i c a n a l y s i s e v a l u a t e s the performance of a s o i l s t r u c t u r e d u r i n g earthquake l o a d i n g by computing the expected deformations r a t h e r than by e v a l u a t i n g a f a c t o r of s a f e t y . I t was f i r s t proposed by N.M. Newmark i n h i s Rankine Lectur e of 1965 (Newmark, 1965). The method i s based on the assumption that the f a i l u r e mass can be represented as a r i g i d block on an i n c l i n e d plane. Movements along the f a i l u r e plane are assumed to occur whenever the i n e r t i a f o r c e s a c t i n g on the s l i d e mass exceed the y i e l d r e s i s t a n c e along the f a i l u r e p l ane. T y p i c a l l y , s e v e r a l p e r i o d s of exceedance, r e p r e s e n t i n g only b r i e f i n s t a n t s of time, w i l l occur d u r i n g a given earthquake and w i l l r e s u l t i n the accumulation of small p l a s t i c deformations. T h i s treatment of the e f f e c t s of seismic l o a d i n g appears to be more l o g i c a l than the p s e u d o - s t a t i c method of a n a l y s i s and has the added b e n e f i t that the deformations r e s u l t i n g from se i s m i c a c t i v i t y , can be computed. Newmark showed that the earthquake induced deformations can be c a l c u l a t e d by i n t e g r a t i n g the earthquake a c c e l e r a t i o n i n excess of the y i e l d a c c e l e r a t i o n over the p e r i o d i n which the y i e l d a c c e l e r a t i o n i s exceeded. The y i e l d a c c e l e r a t i o n i s d e f i n e d as the seismic c o e f f i c i e n t , k, i n a standard pseudo-s t a t i c e q u i l i b r i u m s t a b i l i t y a n a l y s i s which y i e l d s a f a c t o r of s a f e t y a g a i n s t f a i l u r e of 1.0. To f a c i l i t a t e such c a l c u l a t i o n s and to e l i m i n a t e the r e q u i r e d i n t e g r a t i o n , Newmark pr o v i d e d c h a r t s and equations that r e l a t e d the maximum displacement of 20 the s l i d e mass to the y i e l d a c c e l e r a t i o n and to the maximum values of the s u r f a c e a c c e l e r a t i o n and the s u r f a c e v e l o c i t y caused by the earthquake. Thus, Newmark's a n a l y s i s i s capable of p r o v i d i n g a r a p i d estimate of earthquake induced deformations. The Newmark a n a l y s i s uses a s i n g l e value of the earthquake induced a c c e l e r a t i o n f o r the e n t i r e f a i l u r e mass. However, dur i n g an earthquake, the e f f e c t i v e peak a c c e l e r a t i o n w i t h i n an embankment decreases with i n c r e a s i n g depth. Seed and M a r t i n (1966) and Ambraseys and Sarma (1967) have suggested refinements to Newmark's a n a l y s i s which w i l l allow f o r v a r i a t i o n s i n the a c c e l e r a t i o n throughout the embankment and s l i d e mass. The v a r i a t i o n i n the e f f e c t i v e a c c e l e r a t i o n on a p o t e n t i a l s l i d e mass may be estimated from curves such as those presented by Makdisi and Seed (1978) which show the v a r i a t i o n i n the peak a c c e l e r a t i o n with depth. A major l i m i t a t i o n of the Newmark a n a l y s i s l i e s i n i t s assumption t h a t the s o i l behaves i n a r i g i d p l a s t i c manner dur i n g s e i s m i c l o a d i n g . Hence, i t c o n s i d e r s only the e f f e c t s of the i n e r t i a f o r c e s and ignores any displacements that may r e s u l t from s o f t e n i n g or s t r e n g t h l o s s due to pore pressure changes d u r i n g s e i s m i c l o a d i n g . For loose to medium dense c o h e s i o n l e s s s o i l s that o f t e n do not e x h i b i t a w e l l d e f i n e d y i e l d s t r e n g t h and whose behavior d u r i n g c y c l i c l o a d i n g may be complicated by the g e n e r a t i o n of l a r g e excess pore p r e s s u r e s , Newmark's a n a l y s i s w i l l f a i l to provide r e a l i s t i c estimates of 21 deformations. For t h i s reason i t s use should be r e s e r v e d f o r s o i l s where pore p r e s s u r e s do not change s i g n i f i c a n t l y d u r i n g an earthquake, that i s , f o r s i t u a t i o n s where the r i g i d p l a s t i c assumption of s o i l behavior i s reasonably v a l i d . Another problem with the r i g i d block method of a n a l y s i s i s that the p a t t e r n of deformations cannot be determined. The e n t i r e f a i l u r e mass i s assumed to move as a s i n g l e u n i t f o r the e n t i r e computed d i s t a n c e . The d i r e c t i o n of the computed movement i s a l s o not c l e a r . Such deformations can be viewed as e i t h e r h o r i z o n t a l movements or s l i d i n g movements along the s l i p s u r f a c e of the f a i l u r e mass. For a seismic model to be e n t i r e l y u s e f u l i t should not only g i v e the magnitude of deformations expected but should a l s o p r o v i d e a r e a l i s t i c p a t t e r n of deformations throughout the f a i l u r e mass. 3.3 Seed's Dynamic S t r e s s Path Approach The dynamic s t r e s s path approach, developed by H.B. Seed and h i s co-workers at the U n i v e r s i t y of C a l i f o r n i a (Seed et a l , 1969, and Seed et a l , 1973), i s a s e m i - a n a l y t i c a l method of a n a l y s i s designed f o r a n a l y z i n g s o i l s that are s u s c e p t i b l e to s i g n i f i c a n t s t r e n g t h and s t i f f n e s s r e d u c t i o n d u r i n g c y c l i c l o a d i n g . The procedure i n v o l v e s performing a simple e q u i v a l e n t l i n e a r dynamic a n a l y s i s to estimate the dynamic s t r e s s e s to be a p p l i e d to r e p r e s e n t a t i v e samples i n the l a b o r a t o r y . Earthquake induced deformations are then determined by observing the s t r a i n s or pore p r e s s u r e s developed i n the samples. 22 The steps i n v o l v e d i n t h i s dynamic s t r e s s path a n a l y s i s may be summarized as f o l l o w s : 1. s e l e c t a design earthquake motion 2. determine the i n i t i a l s t a t i c s t r e s s e s i n the s o i l s t r u c t u r e 3. determine the dynamic p r o p e r t i e s of the s o i l s 4. perform a dynamic a n a l y s i s to determine the time h i s t o r y of c y c l i c s t r e s s e s and s t r a i n s w i t h i n the s o i l 5. apply the combined s t a t i c and c y c l i c s t r e s s e s to r e p r e s e n t a t i v e samples and observe the accumulated s t r a i n s and pore p r e s s u r e s 6. estimate deformations u s i n g the observed s t r a i n s or pore p r e s s u r e s The s t r a i n s r e f e r r e d to above are the a x i a l s t r a i n s that develop i n t r i a x i a l t e s t samples when s u b j e c t e d to the s t a t i c and dynamic s t r e s s e s which reproduce as a c c u r a t e l y as p o s s i b l e the s t r e s s e s i n the f i e l d . S i n c e these s t r a i n s occur i n i s o l a t e d samples whose deformations are not r e s t r i c t e d by surrounding s o i l s , they represent p o t e n t i a l s t r a i n s r a t h e r than the s t r a i n s that may be expected to occur i n the f i e l d . These s t r a i n p o t e n t i a l s must be manipulated to produce a set of compatible s t r a i n s and deformations w i t h i n the s o i l s t r u c t u r e . Seed (1979) proposed a method i n which the shear s t r e s s corresponding to the s t r a i n p o t e n t i a l i s determined from the s t a t i c s t r e s s - s t r a i n curve. These shear s t r e s s e s are subsequently converted to e q u i v a l e n t nodal f o r c e s and a s t a t i c 23 f i n i t e element program i s used to determine p o s t - c y c l i c deformations. A l t e r n a t i v e l y , the deformations may be computed by using the pore p r e s s u r e s observed i n the c y c l i c l o a d i n g t e s t s . These excess pore p r e s s u r e s reduce the s t i f f n e s s of the s o i l and allow i t to deform under the i n - s i t u s t r e s s e s . Byrne and Janzen (1981) suggested that the observed pore p r e s s u r e s c o u l d be used i n a s t a t i c s t r e s s - s t r a i n a n a l y s i s to p r e d i c t the earthquake induced deformations. Seed r e s o r t e d to the s e m i - a n a l y t i c a l s t r e s s path technique because of the f a i l u r e of the t o t a l s t r e s s e q u i v a l e n t e l a s t i c method of dynamic a n a l y s i s to account f o r the i n f l u e n c e of pore pressure r i s e on s o i l behavior and because of i t s i n a b i l i t y to p r e d i c t permanent deformations. The e q u i v a l e n t - l i n e a r a n a l y s i s i s used only to determine the c y c l i c shear s t r e s s l e v e l that may be expected i n the f i e l d . Such shear s t r e s s e s , however, may not be an a c c u r a t e r e f l e c t i o n of the a c t u a l f i e l d s t r e s s e s because s o i l response, which i s c o n t r o l l e d by e f f e c t i v e s t r e s s e s , may not be reasonably approximated by a t o t a l s t r e s s approach. F i n n et a l (1978) found that a t o t a l s t r e s s a n a l y s i s would tend to overestimate dynamic response when pore water p r e s s u r e s exceeded about 30 percent of the e f f e c t i v e overburden p r e s s u r e . A d d i t i o n a l o v e r p r e d i c t i o n of dynamic response may a l s o r e s u l t from the development of a pseudo-resonance response, c h a r a c t e r i s t i c of e q u i v a l e n t - l i n e a r a n a l y s e s . When the fundamental p e r i o d of the earthquake motion corresponds c l o s e l y to the fundamental p e r i o d of the s o i l , the a m p l i f i c a t i o n of the response due to pseudo-resonance may be as much as 50 percent 24 (Finn et a l , 1978). Because the dynamic s t r e s s e s tend to be overestimated by a t o t a l s t r e s s e q u i v a l e n t - l i n e a r a n a l y s i s , when they are a p p l i e d to r e p r e s e n t a t i v e s o i l samples i n the l a b o r a t o r y they w i l l cause deformations i n excess of what may be expected i n the f i e l d . The r e s u l t i n g e r r o r i s , hence, on the c o n s e r v a t i v e s i d e . The degree of approximation a s s o c i a t e d with the s t r a i n p o t e n t i a l / n o d a l f o r c e c o n v e r s i o n i s not known. Although the approach appears p l a u s i b l e , there i s no t h e o r e t i c a l j u s t i f i c a t i o n f o r i t s use. The method has been v a l i d a t e d to some extent but as Seed s t a t e d i n 1979 not n e a r l y so thoroughly as one might wish f o r a procedure which c o u l d a f f e c t the s a f e t y e v a l u a t i o n of such a c r i t i c a l s t r u c t u r e as a major dam. 3•4 E f f e c t i v e S t r e s s Dynamic A n a l y s i s S e v e r a l e f f e c t i v e s t r e s s dynamic analyses are a v a i l a b l e f o r c a l c u l a t i n g the response of s a t u r a t e d c o h e s i o n l e s s s o i l s to earthquake l o a d i n g . A l l tend to be r e l a t i v e l y r i g o r o u s a n a l y t i c a l procedures i n which the s t r e s s - s t r a i n and s t r e n g t h p r o p e r t i e s of the s o i l are m o d i f i e d to account f o r pore pressure changes d u r i n g c y c l i c l o a d i n g . Siddhartan (1984) r e c e n t l y presented a two-dimensional e f f e c t i v e s t r e s s a n a l y s i s i n which the n o n - l i n e a r s t r e s s - s t r a i n behavior i s modelled by an incremental e l a s t i c approach. Less complex, one-dimensional an a l y s e s such as those used i n the computer programs DESRA (Lee and F i n n , 1978) and CHARSOIL ( S t r e e t e r et a l , 1974) are a l s o 25 a v a i l a b l e . A comparison of these l a t t e r methods i s given by Finn et a l (1978). The v a r i o u s e f f e c t i v e s t r e s s a n a l y s e s tend to d i f f e r i n the s i m p l i f y i n g assumptions made, the r e p r e s e n t a t i o n of the s t r e s s -s t r a i n r e l a t i o n s of the s o i l s , the method by which the development of pore p r e s s u r e s i s taken i n t o account and the procedures used to i n t e g r a t e the dynamic equations of motion. A l l models are capable of p r o v i d i n g a time h i s t o r y of the displacements i n the s o i l s t r u c t u r e . Since the e f f e c t s of pore pressure r i s e are i n c l u d e d i n the a n a l y s i s , the r e s u l t i n g s t r e s s e s , s t r a i n s and displacements are r e l i a b l e and hence there i s no need to r e s o r t to the s e m i - a n a l y t i c a l dynamic s t r e s s path approach. To r e a l i s t i c a l l y model the n o n l i n e a r , h y s t e r e t i c and e f f e c t i v e s t r e s s dependent s o i l behavior, these dynamic analyses have become i n c r e a s i n g l y complex. Many r e q u i r e parameters not commonly used i n g e o t e c h n i c a l p r a c t i c e and most lack s u f f i c i e n t v e r i f i c a t i o n . Although such methods are c l e a r l y more fundamental than the other forms of dynamic analyses t h e i r complexity, a s s o c i a t e d high c o s t , and lac k of v e r i f i c a t i o n g e n e r a l l y l i m i t t h e i r use. 26 CHAPTER 4 PROPOSED METHODS FOR COMPUTING MODULUS REDUCTION The proposed modulus r e d u c t i o n approach to dynamic a n a l y s i s was developed as a simple means of a n a l y s i s intended to simulate a c t u a l p h y s i c a l changes i n the s o i l under earthquake l o a d i n g . P r i m a r i l y intended f o r s o i l s prone to pore pressure r i s e d u r i n g c y c l i c l o a d i n g , i t i s very s i m i l a r to Seed's s t r e s s path approach but i s p h y s i c a l l y more r e a l i s t i c as i t e l i m i n a t e s the need to r e s o r t to an a r b i t r a r y procedure f o r c o n v e r t i n g s t r a i n p o t e n t i a l s to compatible deformations through the use of " e q u i v a l e n t " nodal f o r c e s . The proposed method i s based on the o b s e r v a t i o n that as the pore pre s s u r e s r i s e d u r i n g c y c l i c l o a d i n g , the s o i l s o f t e n s and deforms u n t i l the geometry of the s t r u c t u r e r e f l e c t s the a l t e r e d s t r e s s - s t r a i n r e l a t i o n s h i p of the s o i l . T h i s behavior can be simulated through the use of a reduced value f o r the modulus i n a s t a t i c s t r e s s - s t r a i n a n a l y s i s . A reduced s t r e n g t h may a l s o be used to r e f l e c t the l o s s i n s t r e n g t h r e s u l t i n g from l i q u e f a c t i o n f a i l u r e s should they occur. For the d i l a t i v e s o i l s s u s c e p t i b l e to c y c l i c m o b i l i t y , no r e d u c t i o n i n st r e n g t h i s used s i n c e the undrained s t r e n g t h does not decrease with s t r a i n and the steady-s t a t e undrained s t r e n g t h i s g r e a t e r than the d r a i n e d s t r e n g t h . However, because the development of the negative pore p r e s s u r e r e q u i r e d to m o b i l i z e the f u l l undrained s t r e n g t h of d i l a t i v e s o i l s should not be r e l i e d upon i n the f i e l d , the d r a i n e d s t r e n g t h i s used i n a c o n s e r v a t i v e a n a l y s i s . S i g n i f i c a n t 27 s t r a i n s may be r e q u i r e d to f u l l y m o b i l i z e t h i s s t r e n g t h due to the decreased s t i f f n e s s of the s o i l . The magnitude of modulus r e d u c t i o n r e q u i r e d to p r o v i d e r e a l i s t i c e stimates of s e i s m i c a l l y induced deformations and the method of determining such a r e d u c t i o n were not c l e a r at the onset of t h i s study. T h i s t h e s i s attempts to determine the method f o r c a l c u l a t i n g the modulus r e d u c t i o n t h a t w i l l not only provide deformations of an a p p r o p r i a t e magnitude but w i l l a l s o give a r e a l i s t i c p a t t e r n of deformations throughout the s o i l s t r u c t u r e . 4.1 Methods f o r C a l c u l a t i n g Modulus Reduction Three a l t e r n a t i v e t h e o r i e s were i n v e s t i g a t e d for-determining a s u i t a b l e modulus r e d u c t i o n . The f i r s t approach, r e f e r r e d to as the p o s t - c y c l i c modulus approach, was based on the theory that the earthquake induced deformations c o u l d be c a l c u l a t e d from the d i f f e r e n c e between the p r e - c y c l i c and post-c y c l i c s t r e s s - s t r a i n r e l a t i o n s h i p s of a s o i l . F i g u r e 4-1 d e p i c t s the earthquake induced deformations as the d i f f e r e n c e between the p r e - and p o s t - c y c l i c s t r e s s - s t r a i n curves at the s t r e s s l e v e l e x i s t i n g i n the f i e l d . The reduced modulus i s c a l c u l a t e d d i r e c t l y from the p o s t - c y c l i c s t r e s s - s t r a i n curve. The second theory f o r determining the extent of the modulus r e d u c t i o n , or c y c l i c s t r a i n approach, r e c o g n i z e s that the development of pore p r e s s u r e s d u r i n g c y c l i c l o a d i n g and the r e s u l t i n g s o i l behavior i s d i r e c t l y i n f l u e n c e d by the l e v e l of CO to Pre-Cyclic Stress-Strain Curve Post-Cyclic Stress-Strain Curve Post-Cyclic Modulus Earthquake Induced Deformations Shear Strain FIGURE 4-1 Po s t - c yc l i c modulus reduction approach for determining earthquake induced deformations M CO 29 s t a t i c shear s t r e s s i n the s o i l and the d u r a t i o n of the earthquake. Hence, the a p p r o p r i a t e modulus may be determined, as shown i n F i g u r e 4-2, from l a b o r a t o r y t e s t s which d u p l i c a t e the f i e l d s t a t i c s t r e s s s t a t e and c y c l i c l o a d i n g c o n d i t i o n s as a c c u r a t e l y as p o s s i b l e . The reduced modulus i s given by the r a t i o of the i n - s i t u shear s t r e s s and the shear s t r a i n produced d u r i n g the combined s t a t i c and c y c l i c l o a d i n g . The earthquake induced deformations are c a l c u l a t e d as the d i f f e r e n c e between the deformations computed using t h i s reduced modulus and those found from a s t a t i c a n a l y s i s using the i n - s i t u p r e - c y c l i c modulus. The advantage of t h i s method l i e s i n i t s i n c l u s i o n of both the e f f e c t s of s t r a i n s o f t e n i n g and i n e r t i a f o r c e s on the development of dynamically induced s t r a i n s . . The t h i r d a l t e r n a t i v e , or the pore pressure approach, i n v o l v e s determining the pore pressures generated during c y c l i c l o a d i n g and s u b s t i t u t i n g them i n t o a s t a t i c a n a l y s i s u t i l i z i n g d r a i n e d s t r e n g t h parameters. T h i s procedure, shown in F i g u r e 4-3, causes a r e d u c t i o n i n s t r e n g t h as w e l l as a r e d u c t i o n i n modulus. The earthquake induced deformations are again taken to be the d i f f e r e n c e between the two s t r e s s - s t r a i n curves. The pore p r e s s u r e s to be used should be determined from c y c l i c l o a d i n g t e s t s that d u p l i c a t e the f i e l d s t r e s s c o n d i t i o n s as a c c u r a t e l y as p o s s i b l e . 4.2 Deformation A n a l y s i s Procedure The a n a l y s i s procedure used i n the modulus r e d u c t i o n FIGURE 4 - 2 Cyclic strain modulus reduction approach for determining earthquake induced deformations CO o to to a> v_ 00 i_ D OJ JZ GO Shear Stress Level ^ Pre-Cyclic Stress-Strain Cu rve— Existing " \ in \ Post-Cyclic Field Initial Modulus ^X N —Stress Strain Curve Post-Cyclic Modulus ^ Earthquake Induced ^ Deformations Shear Strain FIGURE 4 - 3 Pore Pressure approach for determining earthquake induced deformations oo 32 approach to dynamic a n a l y s i s c o n s i s t s of the f o l l o w i n g b a s i c s t e p s . 1. s e l e c t a design earthquake motion and a c r o s s - s e c t i o n of the s o i l s t r u c t u r e 2. determine the i n i t i a l e f f e c t i v e s t r e s s e s i n the s o i l s t r u c t u r e by a s t a t i c s t r e s s a n a l y s i s 3. determine the time h i s t o r y of c y c l i c s t r e s s e s throughout the s o i l s t r u c t u r e by an e q u i v a l e n t - l i n e a r dynamic a n a l y s i s 4. apply the combined s t a t i c and c y c l i c s t r e s s e s to r e p r e s e n t a t i v e samples i n the l a b o r a t o r y and observe the s o i l response 5. c a l c u l a t e the a p p r o p r i a t e modulus r e d u c t i o n 6. determine the earthquake induced deformations u s i n g a s t a t i c a n a l y s i s and the reduced modulus The steps o u t l i n e d above i l l u s t r a t e that the fundamental requirements of the modulus r e d u c t i o n approach are the s t a t i c s t a b i l i t y a n a l y s i s , the dynamic a n a l y s i s and the l a b o r a t o r y s o i l t e s t s . The s t a t i c s t a b i l i t y a n a l y s i s i s used to determine the i n i t i a l e f f e c t i v e s t r e s s e s i n the s o i l s t r u c t u r e and to e v a l u a t e the magnitude of the earthquake induced deformations through the use of a reduced modulus. The most convenient method of e v a l u a t i n g the i n i t i a l s t r e s s e s and deformations i s through a f i n i t e element s t a b i l i t y a n a l y s i s such as the computer program SOILSTRESS (Byrne and Janzen, 1981). T h i s program uses a plane s t r a i n f i n i t e element fo r m u l a t i o n i n which the s o i l s k e l e t o n i s modelled as a n o n - l i n e a r e l a s t i c continuum. The n o n - l i n e a r 33 s t r e s s - s t r a i n behavior of s o i l s i s represented by e q u i v a l e n t -l i n e a r or secant moduli compatible with the l e v e l of induced s t r e s s and s t r a i n . A complete d e s c r i p t i o n of the program and i t s h y p e r b o l i c s t r e s s - s t r a i n f o r m u l a t i o n i s given by Byrne and Janzen, (1981). Experimental s t u d i e s have shown that the shear modulus of a s o i l at any s t r a i n l e v e l i s mainly a f u n c t i o n of the i n i t i a l shear modulus and the s t r e n g t h of the s o i l . I f a h y p e r b o l i c r e l a t i o n s h i p i s assumed to e x i s t between shear s t r e s s and shear s t r a i n , then the shear modulus, G, i s given by G = Gi 1^ - r__Rf where Gj = i n i t i a l shear modulus T = shear s t r e s s developed s = shear s t r e n g t h of s o i l Rf = r a t i o of s o i l s t r e n g t h to the u l t i m a t e shear s t r e s s p r e d i c t e d by the h y p e r b o l i c r e l a t i o n s h i p The i n i t i a l shear modulus, G i , i s a f u n c t i o n of the mean c o n f i n i n g s t r e s s , am', and may be expressed as Gi = kg P a f a m ' \ n [Paj where Pa = atmospheric pressure kg = shear modulus f a c t o r n = shear modulus exponent kg and n are e m p i r i c a l parameters that vary with s o i l type and with l o a d i n g c o n d i t i o n . They may be determined from c o n v e n t i o n a l d r a i n e d or undrained t r i a x i a l t e s t s or estimated 34 f r o m p u b l i s h e d p a r a m e t e r s f o r s i m i l a r s o i l s . B e c a u s e t h e y a r e n o t f u n d a m e n t a l s o i l p r o p e r t i e s , but a r e r a t h e r e m p i r i c a l c o e f f i c i e n t s whose v a l u e s r e p r e s e n t t h e b e h a v i o r o f t h e s o i l u n d er a l i m i t e d r a n g e o f c o n d i t i o n s , t h e y s h o u l d be e v a l u a t e d f r o m l a b o r a t o r y t e s t s w h i c h d u p l i c a t e f i e l d c o n d i t i o n s a s a c c u r a t e l y a s p o s s i b l e . D u r i n g t h e modulus r e d u c t i o n dynamic a n a l y s i s t h e v a l u e o f kg w i l l be r e d u c e d t o r e f l e c t t h e i n f l u e n c e o f dynamic l o a d i n g . The dynamic a n a l y s i s u t i l i z e d i n t h e modulus r e d u c t i o n a p p r o a c h f o r p r e d i c t i n g e a r t h q u a k e i n d u c e d d e f o r m a t i o n s i s r e q u i r e d t o d e t e r m i n e t h e l e v e l o f c y c l i c s h e a r s t r e s s e s t h a t i s l i k e l y t o be i n d u c e d i n t h e s o i l by t h e d e s i g n e a r t h q u a k e . B e c a u s e t h e e v a l u a t i o n of t h e d e s i g n e a r t h q u a k e i s d i f f i c u l t and s u b j e c t t o many u n c e r t a i n t i e s , t h e use of a r i g o r o u s a n a l y s i s i s g e n e r a l l y n o t j u s t i f i e d . In a d d i t i o n , t h e s o i l p a r a m e t e r s may n o t be known t o s u f f i c i e n t a c c u r a c y t o r e a s o n a b l y a p p r o x i m a t e s o i l b e h a v i o r i n t h e more r i g o r o u s a n a l y s e s . The s e l e c t i o n o f a t o t a l s t r e s s e q u i v a l e n t - l i n e a r dynamic a n a l y s i s i s c o n s i s t e n t w i t h b o t h t h e l e v e l of a c c u r a c y of t h e a v a i l a b l e d a t a and t h e d e s i r e f o r s i m p l i c i t y i n a v o i d i n g t h e a d d i t i o n a l c o m p l e x i t y of d e t e r m i n i n g an a p p r o p r i a t e p o r e p r e s s u r e g e n e r a t i o n m o d e l . N o n - l i n e a r dynamic m a t e r i a l p r o p e r t i e s a r e i n c o r p o r a t e d i n t o t h e dynamic a n a l y s i s u s i n g s t r a i n d e p e n d e n t modulus and damping v a l u e s . Seed and I d r i s s (1970) showed t h e s h e a r modulus t o be a f u n c t i o n o f t h e s q u a r e r o o t of t h e mean c o n f i n i n g s t r e s s and p r e s e n t e d modulus r e d u c t i o n c u r v e s showing t h e d e c r e a s e i n 35 modulus wi t h s t r a i n l e v e l . Such r e l a t i o n s h i p s are used to determine the r e l e v a n t shear moduli and damping r a t i o s during the dynamic a n a l y s i s . The dynamic a n a l y s i s i s performed by an i t e r a t i v e approach i n which the i n i t i a l values f o r the shear modulus and damping values are assumed and used to compute the s t r a i n s i n each element. New shear moduli and damping values are c a l c u l a t e d based on these s t r a i n s and the a n a l y s i s i s repeated u n t i l the values of modulus and damping are compatible with the s t r a i n s developed i n each element. Such a response a n a l y s i s p r o v i d e s a time h i s t o r y of s t r e s s e s and s t r a i n s w i t h i n the s o i l s t r u c t u r e . These dynamic s t r e s s e s may be converted to an e q u i v a l e n t s e r i e s of uniform s t r e s s a p p l i c a t i o n s . Lee and Chan (1972) presented a method of c o n v e r s i o n by a p p r o p r i a t e weighting of the o r d i n a t e s of the s t r e s s time h i s t o r y . The e q u i v a l e n t c y c l i c s t r e s s a p p l i c a t i o n s can then be a p p l i e d to r e p r e s e n t a t i v e samples in the l a b o r a t o r y to determine a p p r o p r i a t e modulus r e d u c t i o n s f o r computing deformations. In a d d i t i o n to c y c l i c t e s t s , standard monotonic l o a d i n g t e s t s are r e q u i r e d to evaluate s o i l behavior d u r i n g d r a i n e d and undrained l o a d i n g and to determine the a p p r o p r i a t e h y p e r b o l i c parameters. Standard t r i a x i a l and c y c l i c t r i a x i a l t e s t s are most convenient because of t h e i r r e l a t i v e s i m p l i c i t y and wide a v a i l a b i l i t y of equipment. 36 E v a l u a t i o n of the earthquake induced displacements by the p o s t - c y c l i c modulus approach r e q u i r e s the d e t e r m i n a t i o n of s t r e s s - s t r a i n curves f o r the s o i l s before and a f t e r c y c l i c l o a d i n g . Although, t h e o r e t i c a l l y , t h i s approach appears simple, i t i s i n f a c t very d i f f i c u l t to determine the p o s t - c y c l i c s t r e s s - s t r a i n r e l a t i o n s h i p e x a c t l y as represented i n F i g u r e 4-1 which shows the curve beginning at zero a x i a l s t r a i n . To determine such a curve from t r i a x i a l t e s t s , the accumulation of permanent a x i a l s t r a i n s d u r i n g c y c l i c l o a d i n g would have to be prevented by c o n s o l i d a t i n g and l o a d i n g the sample under i s o t r o p i c c o n d i t i o n s . However, s i n c e i s o t r o p i c a l l y c o n s o l i d a t e d samples g e n e r a l l y f a i l d u r i n g the e x t e n s i o n a l phase of c y c l i c l o a d i n g , any subsequent r e l o a d i n g i n compression d u r i n g p o s t - c y c l i c monotonic t e s t i n g would not r e f l e c t the • s o i l behavior expected in the f i e l d . To ensure f a i l u r e on the compression s i d e , t r i a x i a l samples g e n e r a l l y have to be t e s t e d under a n i s o t r o p i c c o n d i t i o n s with Kc values that depend on the c h a r a t e r i s t i c s of the s o i l . Under such a n i s o t r o p i c c o n d i t i o n s permanent a x i a l s t r a i n s w i l l develop d u r i n g c y c l i c l o a d i n g . I f l i q u e f a c t i o n o c c u r s , the very l a r g e a x i a l s t r a i n s that develop may exceed the l i m i t s of the t e s t i n g equipment. To ensure that l a r g e a x i a l s t r a i n s do not occur and p o s t - c y c l i c monotonic t e s t s can be performed, the development of a x i a l s t r a i n s d u r i n g l i q u e f a c t i o n must be h a l t e d at some small s t r a i n l e v e l . The sample may then be loaded m o n o t o n i c a l l y and the p o s t - c y c l i c s t r e s s - s t r a i n curve determined. The r e d u c t i o n i n the modulus f a c t o r i s determined by comparing the i n i t i a l p a r t of the p o s t -37 c y c l i c s t r e s s - s t r a i n curve to that of the p r e - c y c l i c s t r e s s -s t r a i n curve. To determine the magnitude of modulus r e d u c t i o n from the c y c l i c s t r a i n method, both standard t r i a x i a l t e s t s and c y c l i c l o a d i n g t e s t s must be performed on r e p r e s e n t a t i v e s o i l samples. The standard t r i a x i a l t e s t s are r e q u i r e d to determine the h y p e r b o l i c s t r e s s - s t r a i n parameters used to compute the i n i t i a l e f f e c t i v e s t r e s s e s w i t h i n the s o i l s t r u c t u r e . These s t r e s s e s are then a p p l i e d to r e p r e s e n t a t i v e s o i l samples i n the l a b o r a t o r y i n combination with a p p r o p r i a t e c y c l i c s t r e s s e s . The s t r a i n s r e s u l t i n g from these t e s t s are used to compute the e f f e c t i v e shear modulus duri n g c y c l i c l o a d i n g . T h i s c y c l i c l o a d i n g shear modulus i s compared to the p r e - c y c l i c value determined from the s t a t i c a n a l y s i s and an a p p r o p r i a t e r e d u c t i o n i n the modulus f a c t o r i s determined. The pore p r e s s u r e method of dynamic a n a l y s i s r e q u i r e s the same l a b o r a t o r y t e s t s as the c y c l i c s t r a i n method. However, u n l i k e the c y c l i c s t r a i n approach, the value of the modulus f a c t o r i s not reduced. The r e d u c t i o n i n the shear modulus r e s u l t s from the i n c l u s i o n of the excess pore p r e s s u r e s generated d u r i n g c y c l i c l o a d i n g i n the s t a t i c a n a l y s i s . The pore p r e s s u r e s cause a decrease i n the mean c o n f i n i n g s t r e s s and hence, a decrease i n the e f f e c t i v e shear modulus. U n l i k e the two preceding methods of modulus r e d u c t i o n , a decrease i n s t r e n g t h a l s o r e s u l t s from t h i s approach. 38 CHAPTER 5 VERIFICATION OF PROPOSED METHOD The v a l i d i t y and a p p l i c a b i l i t y of the three proposed methods f o r computing the p o s t - c y c l i c modulus can only be e v a l u a t e d by comparing p r e d i c t e d deformations to a c t u a l f i e l d deformation h i s t o r i e s . U n f o r t u n a t e l y , few e a r t h s t r u c t u r e s have been s u b j e c t e d to severe earthquake shaking and f o r the few that have, i n s u f f i c i e n t data i s a v a i l a b l e to p r o v i d e accurate s o i l and earthquake parameters for use i n such an a n a l y s i s . Because of the lack of a s u i t a b l e f i e l d case h i s t o r y , the three proposed methods f o r computing modulus r e d u c t i o n s were i n v e s t i g a t e d using the r e s u l t s of shaking t a b l e t e s t s performed at the U n i v e r s i t y of B r i t i s h .Columbia on a model that was intended to represent a sloped s a t u r a t e d t a i l i n g s d e p o s i t . The r e l i a b i l i t y of each of the proposed methods of a n a l y s i s was e v a l u a t e d by comparing the p r e d i c t e d magnitude and p a t t e r n of deformations to those observed i n the model. The method of a n a l y s i s that y i e l d e d the most a c c u r a t e deformations was then used to p r e d i c t the deformations that occurred i n the Upper San Fernando Dam d u r i n g the earthquake of February, 1971. 5.1 T a i l i n g s Model Te s t s The r e s u l t s of shaking t a b l e t e s t s on s a t u r a t e d model t a i l i n g s s l o p e s were used to determine the c o r r e c t procedure fo r e v a l u a t i n g the a p p r o p r i a t e magnitude of modulus r e d u c t i o n due to c y c l i c l o a d i n g . The t e s t s were part of a study on the 39 p r e d i c t i o n of deformations of l i q u e f i e d t a i l i n g s d e p o s i t s by v i s c o u s flow theory. They were performed by B. Stuckert at the U n i v e r s i t y of B r i t i s h Columbia as p a r t of h i s master's degree r e s e a r c h . A complete d e s c r i p t i o n of the t e s t i n g equipment, procedure, and r e s u l t s i s given by S t u c k e r t , (1982). The model t a i l i n g s slopes were c o n s t r u c t e d i n the p l e x i g l a s s c o n t a i n e r shown in F i g u r e 5-1. The slopes were 81 cm long and 20 cm wide. Although v a r i o u s slope c o n f i g u r a t i o n s were t e s t e d , only s l o p e s having a f i x e d depth of 14 cm at the downstream boundary and r i s i n g with a 8 degree slope are c o n s i d e r e d f o r t h i s t h e s i s . The slopes were formed by a l l o w i n g dry sand to f a l l from a hopper i n t o the p l e x i g l a s s c o n t a i n e r that was p a r t i a l l y f i l l e d with d e a i r e d water. The hopper was used to d i s t r i b u t e the sand evenly a c r o s s the s l o p e . A scraper then smoothed the slope to the d e s i r e d angle. During d e p o s i t i o n s i l i c a beads were placed adjacent to the p l e x i g l a s s w a l l i n a g r i d p a t t e r n . The displacements of these beads were monitored dur i n g the t e s t to determine the magnitude and p a t t e r n of the dynamically induced deformat i o n s . The sand used in the study was a f i n e Ottawa sand. T h i s sand i s a c l e a n s i l i c a sand having rounded to subrounded g r a i n s and a s p e c i f i c g r a v i t y of 2.57. F i g u r e 5-2 shows the g r a i n s i z e d i s t r i b u t i o n f o r the sand. The maximum and minimum v o i d r a t i o s were determined as .86 and .56 r e s p e c t i v e l y . The method of d e p o s i t i o n used f o r forming the model slopes r e s u l t e d i n v o i d 40 H o p p a r E L E V A T I O N 8 1 c m 0«tl«t =tq t-Jr O v e r f l o w 4- I n l e t P L A N VIEW (from Byrne, Void and Stuckert, 1981) FIGURE 5-1 Model Container 41 O t t a w a S a n d 0 .02 0.1 0.5 1.0 P a r t i c l e D i a m e t e r in mm (from Byrne, Void and Stuckert,198l) FIGURE 5 - 2 Grain Size Distribution of Test Sand 42 r a t i o s of approximately .77 or about 30 percent r e l a t i v e d e n s i t y . The t e s t s were performed on a 1.2m by 2.7m shaking t a b l e . The t a b l e motions were c o n t r o l l e d by an MTS Earthquake Simulator console which p r o v i d e d s i n u s o i d a l motions at a frequency of 5 Hz. T h i s frequency was s e l e c t e d to ensure that p r a c t i c a l l y undrained c o n d i t i o n s e x i s t e d f o r the 20 c y c l e s of shaking a p p l i e d . Maximum a c c e l e r a t i o n s of .03 to .1g were used. During shaking the model slo p e s experienced l a r g e movements which r e s u l t e d i n s u b s t a n t i a l slope f l a t t e n i n g . For t e s t s i n which the a c c e l e r a t i o n s exceeded about .05g, f i n a l slope angles were l e s s than h a l f a degree. The displacement p a t t e r n i l l u s t r a t e d by the s i l i c a beads r e v e a l e d that the deformations r e s u l t e d from deep-seated movements ra t h e r than from merely a s u r f a c e t r a n s p o r t of m a t e r i a l . F i g u r e 5-3 shows the deformations which oc c u r r e d in the t a i l i n g s slope model duri n g a t e s t i n which the maximum a c c e l e r a t i o n was .08g. The displacements are t y p i c a l of those o c c u r r i n g when a s i g n i f i c a n t p o r t i o n of the slope appeared to l i q u e f y . The displacements are g r e a t e s t at the s u r f a c e and decrease with depth to zero along the model base. Maximum movements of approximately 6 to 7 cm occur near the center of the s l o p e . Such displacements i n d i c a t e a shear s t r a i n l e v e l of 30 to 40 percent. 5.1.1 Laboratory T e s t s Three s e t s of l a b o r a t o r y t r i a x i a l t e s t s were performed to FIGURE 5-3 Observed Dynamically Induced Deformations for Model Test for 8 degree Slope and .08g Maximum Acceleration 44 p r o v i d e t h e s o i l p a r a m e t e r s n e c e s s a r y f o r t h e v a r i o u s m o d u l u s r e d u c t i o n a n a l y s e s . D r a i n e d a n d u n d r a i n e d m o n o t o n i c t r i a x i a l t e s t s w e r e p e r f o r m e d t o d e t e r m i n e s o i l r e s p o n s e d u r i n g d r a i n e d a n d u n d r a i n e d l o a d i n g . C y c l i c l o a d i n g t e s t s f o l l o w e d b y u n d r a i n e d m o n o t o n i c l o a d i n g t e s t s w e r e p e r f o r m e d t o e v a l u a t e s o i l b e h a v i o r a f t e r c y c l i c l o a d i n g . F o r a l l o f t h e t e s t s , a r e l a t i v e d e n s i t y o f a p p r o x i m a t e l y 3 0 p e r c e n t w a s u s e d t o s i m u l a t e t h e . c o n d i t i o n s i n t h e m o d e l . B e c a u s e o f t h e v e r y l o w s t r e s s l e v e l s e x i s t i n g i n t h e m o d e l , i t w a s i m p o s s i b l e t o p e r f o r m t h e t r i a x i a l t e s t s o v e r t h e m o d e l s t r e s s r a n g e . A l l t e s t s w e r e p e r f o r m e d a t c o n f i n i n g p r e s s u r e s b e t w e e n 5 0 a n d 2 0 0 k P a a n d t h e r e s u l t s w e r e a s s u m e d t o b e a p p r o p r i a t e f o r t h e l o w e r s t r e s s l e v e l s i n t h e m o d e l . M o n o t o n i c T e s t s T h e r e s p o n s e o f t h e f i n e O t t a w a s a n d t o d r a i n e d a n d u n d r a i n e d m o n o t o n i c l o a d i n g i s s h o w n i n F i g u r e s 5 - 4 , 5 - 5 , 5 - 6 a n d 5 - 7 f o r c o n f i n i n g p r e s s u r e s o f 5 0 , 1 0 0 , 1 5 0 a n d 2 0 0 k P a , r e s p e c t i v e l y . F o r b o t h t h e d r a i n e d a n d u n d r a i n e d t e s t s , t h e s a m p l e s w e r e c o n s o l i d a t e d i s o t r o p i c a l l y a n d t h e n l o a d e d a x i a l l y t o f a i l u r e . T h e b e h a v i o r i s t y p i c a l o f m o s t g r a n u l a r s o i l s . A f t e r a n i n i t i a l p e r i o d o f c o m p r e s s i o n , t h e d r a i n e d s a m p l e s b e g a n t o d i l a t e . T h i s v o l u m e c h a n g e b e h a v i o r i s r e f l e c t e d i n t h e u n d r a i n e d s a m p l e s b y t h e i n i t i a l r i s e i n p o r e p r e s s u r e f o l l o w e d b y a r e d u c t i o n a t h i g h e r s t r a i n l e v e l s . T h e s t r e s s -s t r a i n c u r v e s f o r t h e d r a i n e d s a m p l e s a r e n e a r l y h y p e r b o l i c . T h e u n d r a i n e d c u r v e s r i s e s t e e p l y i n i t i a l l y , l e v e l o f f o r r e d u c e 45 2 0 0 1-D CO CO <D 00 15 *> CD Q c P / / / / / < Drair ied Test—^ / / / / / ^ / / /* O s s ^ U n d r a i m 3d Test i 4 — —i 0 2 4 6 8 Axia l Strain - Percent 10 c u CD Q_ C 'D CO o \_ -t— E O > 0.4 0.2 -0.0 -0.2 -0.4 — 2 4 6 8 Axia l Strain - Percent FIGURE 5 - 4 Soil Response During Monotonic Triaxial Loading Tests - Fine Ottawa Sand Confining Pressure = 50 kPa 4 6 FIGURE 5-5 Soil Response During Monotonic Triaxial Loading Tests - Fine Ottawa Sand Confining Pressure = 100 kPa 47 500 CL. 400 to (/> 300 d> v_ 00 u 200 _o | 1 0 0 o o Drained Test— -A & ft J^llf # / / / ...J. 0 • • ® J2>"^—Undr< T . >o-' 3ined Test 0 5 10 Axial Strain - Percent 15 c <D o a . I c 'o CO _o E o > -1- = 0 Undrained Test Drained Test 5 10 Axial Strain — Percent 200 100 -100 Q) -200 15 FIGURE 5 - 6 Soil Response During Monotonic Triaxial Loading Tests - Fine Ottawa Sand Confining Pressure = 150 kPa 48 600 O CL oo oo CD l_ 00 _o ~o "> CD Q 4 0 0 -2 0 0 -O ^ r 5 10 Axial Strain - Percent 15 c CD O i_ 0> Q_ I 'o 00 CD E O > 0.5--0.5 Q - 0 O G - - O Undrained Test •0~ O- --Q . -100 ~0 Drained Test 200 0 - -100 - 2 0 0 5 10 15 Axial Strain - Percent FIGURE 5 -7 Soil Response During Monoionic Triaxial Loading Tests - Fine Ottawa Sand Confining Pressure = 200 kPa 49 s l i g h t l y , and then s t a r t to r i s e again as the sand begins to s t r a i n harden. For the undrained t e s t at a c o n f i n i n g pressure of 50 kPa, no s i g n i f i c a n t amount of s t r a i n s o f t e n i n g occurs a f t e r the i n i t i a l peak i n the s t r e s s - s t r a i n curve. Although the t e s t s at higher c o n f i n i n g p r e s s u r e s d i d show a small but d i s t i n c t peak, the behavior i l l u s t r a t e d i n F i g u r e 5-4 f o r a c o n f i n i n g p ressure of 50 kPa i s b e l i e v e d to be more i n d i c a t i v e of the s o i l response at the s t r e s s l e v e l e x i s t i n g i n the model s l o p e s . Hence, no s i g n i f i c a n t l o s s i n s t r e n g t h i s a n t i c i p a t e d w i t h i n the model. The l a r g e displacements observed i n the model t e s t s were thus a r e s u l t of c y c l i c m o b i l i t y behavior rather than a t r u e l i q u e f a c t i o n f a i l u r e which r e q u i r e s a l o s s in s t r e n g t h a f t e r the peak i n the undrained s t r e s s - s t r a i n curve has been reached. The h y p e r b o l i c s t r e s s - s t r a i n parameters were evaluated f o l l o w i n g the procedure d e s c r i b e d by Duncan et a l (1980), and are summarized on Table 5-1. For the d r a i n e d t e s t s , the parameters are based on e f f e c t i v e s t r e s s e s , while f o r the undrained t e s t s , they are a f u n c t i o n of the i n i t i a l e f f e c t i v e s t r e s s e s . For the undrained t e s t s the bulk modulus f a c t o r may be assumed to be very high to simulate the lack of volume change that would occur under undrained c o n d i t i o n s f o r f u l l y s a t u r a t e d samples. The bulk modulus parameters f o r the d r a i n e d t e s t s were determined from the volume change measurements. C y c l i c T e s t s During the c y c l i c l o a d i n g t e s t s the samples were i n i t i a l l y 50 Table 5-1 Summary of H y p e r b o l i c Parameters F i n e Ottawa Sand dr a i n e d l o a d i n g undrained l o a d i n g shear modulus f a c t o r kg 1 35 320 shear modulus exponent n .62 .68 bulk modulus f a c t o r k b 1 57 bulk modulus exponent m .35. r e d u c t i o n r a t i o Rf .95 .71 angle of f r i c t i o n <t> 33.5 32.0 change in f r i c t i o n angle L\<t> 2. 1 4.0 c o n s o l i d a t e d a n i s o t r o p i c a l l y with a Kc value of 1.2 to ensure that f a i l u r e would occur i n compression. Because the e n t i r e model s l o p e s appeared to l i q u e f y w i t h i n the f i r s t few c y c l e s of l o a d i n g d u r i n g the shaking t a b l e t e s t s , the c y c l i c s t r e s s r a t i o was s e l e c t e d as the value that would cause l i q u e f a c t i o n i n l e s s than 10 c y c l e s . I f r i g i d body motion i s assumed i n the model, the c y c l i c s t r e s s r a t i o o c c u r r i n g d u r i n g the shaking t a b l e t e s t s may be e v a l u a t e d as .16. The c y c l i c s t r e s s r a t i o used i n the l a b o r a t o r y t e s t s was s l i g h t l y lower at .15. The development of l a r g e s t r a i n s was prevented by stopping the t e s t s a f t e r i n i t i a l l i q u e f a c t i o n had occurred at a s t r a i n l e v e l of approximately 2.5 p e r c e n t . Samples were then m o n o t o n i c a l l y loaded to provide p o s t - l i q u e f a c t i o n s t r e s s - s t r a i n c urves. During these t e s t s no a d d i t i o n a l pore pressures were generated. I n s t e a d the pore 51 pr e s s u r e s began to drop, slowly i n i t i a l l y and then at an i n c r e a s i n g r a t e i n the l a t e r stages of the t e s t s . Because the reduced pore p r e s s u r e s l e d to i n c r e a s e d c o n f i n i n g s t r e s s e s , the s t r e n g t h s at l a r g e s t r a i n s tended to be s i m i l a r to those determined from the p r e - c y c l i c undrained t e s t s . F i g u r e s 5-8, 5-9 and 5-10 show the s i m i l a r i t y between the pre- and p o s t - c y c l i c response of the sand at v a r i o u s c o n f i n i n g p r e s s u r e s d u r i n g undrained l o a d i n g . 5.1.2 Determination of Reduced Moduli The reduced modulus f a c t o r f o r the p o s t - c y c l i c modulus approach was determined from the d i f f e r e n c e i n s o i l response before and a f t e r c y c l i c l o a d i n g . The i n i t i a l shear modulus f o r each curve was e v a l u a t e d as the secant to the curve connecting the o r i g i n to the i n i t i a l peak i n the p r e - c y c l i c curve. For the p o s t - c y c l i c curve, the 2.5 percent a x i a l s t r a i n that developed dur i n g i n i t i a l l i q u e f a c t i o n was i n c l u d e d i n the c a l c u l a t i o n of the i n i t i a l shear modulus. A comparison of the two moduli y i e l d e d an apparent modulus r e d u c t i o n of approximately 50 times, corresponding to a h y p e r b o l i c modulus f a c t o r of 6. The modulus r e d u c t i o n f o r the c y c l i c s t r a i n approach was not determined d i r e c t l y from l a b o r a t o r y t e s t s . I d e a l l y , the r e d u c t i o n i n the modulus would be determined by s u b j e c t i n g u n d i s t u r b e d samples to the combined s t a t i c and c y c l i c s t r e s s e s b e l i e v e d to e x i s t i n the f i e l d and o b s e r v i n g the accumulated s t r a i n s . However, l a b o r a t o r y t e s t s at the very low s t r e s s 52 FIGURE 5 - 8 Stress-Strain Behavior Before and After Cyclic Loading - Fine Ottawa Sand Confining Pressure = 50 kPa 53 400 O oo oo CD O _o > CD Q 100-El • Pre-cyclic —^ Jul / s r f • , 0- ' "—Post-cyclic n 0 5 10 Axial Strain — Percent 15 o Q_ I 00 00 CD D_ CD i_ O C L 0 /—Post-cyclic ! P j j r e - c y c l i c — ^ «J 111 X 5 10 Axial Strain - Percent 15 FIGURE 5 - 9 Stress-Strain Behavior Before and After Cyclic Loading - Fine Ottawa Sand Confining Pressure = 100 kPa FIGURE 5-10 Stress-Strain Behavior Before and After Cyclic Loading - Fine Ottawa Sand Confining Pressure = 150 kPa 55 l e v e l s e x i s t i n g i n the model are extremely d i f f i c u l t to perform. Hence, any c y c l i c l o a d i n g t e s t s would have to be performed at s t r e s s l e v e l s s i g n i f i c a n t l y higher than those i n the model. Because the s u s c e p t i b i l i t y of a s o i l to l i q u e f a c t i o n i n c r e a s e s with i n c r e a s i n g c o n f i n i n g s t r e s s , such t e s t s would not reproduce the mechanism of s t r a i n development thought to occur i n the model. The monotonic l o a d i n g t e s t s performed on the t e s t sand at the a p p r o p r i a t e r e l a t i v e d e n s i t y i n d i c a t e d an i n c r e a s i n g tendency f o r d i l a t i o n at the lower c o n f i n i n g p r e s s u r e s . For the t r i a x i a l t e s t at the lowest c o n f i n i n g pressure of 50 kPa, the sample began to d i l a t e at s t r a i n s of between 1 and 2 percent. The s t r e s s - s t r a i n behavior of the sand at t h i s c o n f i n i n g p r e s s u r e was a l s o c h a r a c t e r i z e d by the absence of the s t r a i n s o f t e n i n g that had o c c u r r e d d u r i n g the t e s t s at the higher c o n f i n i n g p r e s s u r e s . Because the average c o n f i n i n g pressure i n the model slope was about 1 kPa - 50 times l e s s than the c o n f i n i n g p r e s s u r e used f o r the monotonic t r i a x i a l t e s t e x h i b i t i n g no s t r a i n s o f t e n i n g - the sand at the model s t r e s s c o n d i t i o n s would have been d i l a t i v e r a t h e r than c o n t r a c t i v e . Thus the development of s t r a i n s d u r i n g c y c l i c l o a d i n g would have r e s u l t e d from c y c l i c m o b i l i t y r a t h e r than from l i q u e f a c t i o n . The l a r g e deformations observed i n the model slopes represent the accumulation of the s t r a i n s o c c u r r i n g d u r i n g the l o a d i n g phase of each s t r e s s c y c l e . The speed at which the deformations o c c u r r e d can be a t t r i b u t e d to the high frequency of c y c l i c l o a d i n g that was used to ensure that undrained c o n d i t i o n s would p r e v a i l d u r i n g shaking. 56 Although the a p p r o p r i a t e c y c l i c l o a d i n g t e s t s c o u l d not be performed, the p o t e n t i a l of the s o i l to undergo l a r g e deformations r e s u l t i n g from the p r o g r e s s i v e development of s t r a i n s d u r i n g each c y c l e of s t r e s s a p p l i c a t i o n was i n v e s t i g a t e d . A c y c l i c l o a d i n g t e s t was performed to determine the c y c l i c s t r e s s amplitude r e q u i r e d to produce a p p r e c i a b l e s t r a i n s d u r i n g each c y c l i c load a p p l i c a t i o n . The c y c l i c s t r e s s r a t i o t h at would produce the magnitude of the s t r a i n s observed in the model slope during the 20 c y c l e s of shaking a p p l i e d was a l s o determined. The t e s t was performed at an o v e r a l l e f f e c t i v e c o n f i n i n g p r essure of 100 kPa on a sample c o n s o l i d a t e d a n i s o t r o p i c a l l y to a Kc value of 1.2. At these c o n d i t i o n s the sample would be s u s c e p t i b l e to l i q u e f a c t i o n and would develop c y c l i c m o b i l i t y only a f t e r an i n i t i a l flow f a i l u r e . During the i n i t i a l p a r t of the t e s t i n which l i q u e f a c t i o n was induced by a p p l y i n g a c y c l i c s t r e s s r a t i o of .15, the s t r a i n s were l i m i t e d to approximately 2.5 percent. The c y c l i c s t r e s s e s were then r e a p p l i e d but with a reduced amplitude. The a x i a l s t r a i n s r e s u l t i n g d u r i n g each l o a d i n g c y c l e were observed. The v a r i a t i o n i n the a x i a l s t r a i n s generated duri n g each l o a d i n g c y c l e with the c y c l i c s t r e s s r a t i o was determined by p r o g r e s s i v e l y i n c r e a s i n g the c y c l i c s t r e s s amplitude. The t e s t i n d i c a t e d that s i g n i f i c a n t s t r a i n s developed only when the c y c l i c s t r e s s amplitude exceeded the i n i t i a l d e v i a t o r s t r e s s used to c r e a t e a n i s o t r o p i c c o n d i t i o n s . Hence s t r e s s r e v e r s a l s were necessary f o r s i g n i f i c a n t development of s t r a i n . The t e s t a l s o showed that a c y c l i c s t r e s s r a t i o of about .10, r a t h e r than 57 .16 as was c h a r a c t e r i s t i c of the model t a i l i n g s t e s t s , would be r e q u i r e d to generate the same cumulative s t r a i n s as those observed i n the model t e s t s f o r the 20 c y c l e s of l o a d i n g a p p l i e d . However, s i n c e the model s l o p e s were s u b j e c t e d to a much lower c o n f i n i n g pressure and a somewhat higher l e v e l of s t a t i c b i a s than the l a b o r a t o r y t r i a x i a l sample, a higher c y c l i c s t r e s s l e v e l would be r e q u i r e d i n the model t e s t s to cause a comparable l e v e l of s t r a i n . Although the r e s u l t s of t h i s c y c l i c l o a d i n g t e s t cannot be used q u a n t i t a t i v e l y to p r e d i c t the modulus r e d u c t i o n i n the model t a i l i n g s slope t e s t s due to the d i f f e r i n g s t r e s s c o n d i t i o n s , they do c o n f i r m that s u b s t a n t i a l s t r a i n s w i l l d e v e l o p d u r i n g c y c l i c l o a d i n g from the accumulation of s t r a i n s o c c u r r i n g d u r i n g each l o a d c y c l e . The shear s t r a i n s used to compute the modulus r e d u c t i o n f o r the c y c l i c s t r a i n approach were assumed to be approximately 30 to 40 p e r c e n t , corresponding to those observed i n the model s l o p e s . These s t r a i n s may be s l i g h t l y lower than those determined from a p p r o p r i a t e l a b o r a t o r y t e s t s because the s t a t i c shear s t r e s s e s e x i s t i n g i n the model slopes reduced d u r i n g shaking as a r e s u l t of slope f l a t t e n i n g . The assumed s t r a i n l e v e l s caused r e d u c t i o n s i n the modulus f a c t o r of 1000 to 2000 times, r e s u l t i n g i n k g values in the .05 to .20 range. The r e d u c t i o n i n the modulus f o r the pore pressure approach i s determined from the magnitude of the pore pressure r i s e r e s u l t i n g d u r i n g c y c l i c l o a d i n g . The development of pore pr e s s u r e s d u r i n g dynamic l o a d i n g depends on both the p r o p e r t i e s 58 of the s o i l and the c h a r a c t e r i s t i c s of the dynamic e x c i t a t i o n . The magnitude of the r e s i d u a l pore pressures developed d u r i n g e i t h e r l i q u e f a c t i o n or c y c l i c m o b i l i t y i s a f u n c t i o n of the e f f e c t i v e c o n f i n i n g s t r e s s e s d u r i n g c o n s o l i d a t i o n and the l e v e l of s t a t i c b i a s . For much of the model a Kc value of 1.8 p r e v a i l e d . Thus the r e l a t i o n s h i p f o r the r e s i d u a l pore p r e s s u r e s suggested by Chern (1981) would p r e d i c t excess pore p r e s s u r e s of approximately 75 percent of the e f f e c t i v e c o n f i n i n g p r e s s u r e . These pore pressures are u t i l i z e d i n a s t a t i c s t r e s s a n a l y s i s to determine the modulus r e d u c t i o n and the earthquake induced deformations. The reduced modulus f a c t o r s or the pore p r e s s u r e s determined f o r each of the three modulus r e d u c t i o n approaches were s u b s t i t u t e d i n t o a f i n i t e element s t a t i c s t r e s s s t a b i l i t y a n a l y s i s to c a l c u l a t e the dynamically induced deformations. F i g u r e 5-11 shows the f i n i t e element g r i d used c o n j u n c t i o n with the computer program SOILSTRESS to perform the a n a l y s e s . During the a n a l y s e s , the bulk modulus f a c t o r was kept high to simulate the undrained behavior of the model sand. The e f f e c t of changes i n the geometry of the slope on the d r i v i n g shear f o r c e s d u r i n g deformation was i n c l u d e d by modifying the element geometry d u r i n g the a n a l y s i s . 5.1.3 R e s u l t s of Deformation P r e d i c t i o n s The deformations of the model t a i l i n g s slope p r e d i c t e d by each of the three modulus re d u c t i o n approaches f o r the model FIGURE 5-11 Finite Element Grid of Tailings Model used in Static-Stress Analysis 60 t e s t with an 8 degree slope and a peak a c c e l e r a t i o n of .08g are shown i n F i g u r e s 5-12, 5-13 and 5-14. The deformations determined by the p o s t - c y c l i c modulus approach and the pore pressure approach are much smal l e r that those observed i n the model slope and have been m u l t i p l i e d 30 and 300 times, r e s p e c t i v e l y , to a r r i v e at comparable deformations. A l l approaches reproduce approximately the p a t t e r n of deformations observed i n the model. The deformations p r e d i c t e d by the c y c l i c s t r a i n approach using an average kg value of .10 are very s i m i l a r to the model deformations. The h o r i z o n t a l displacements agree very w e l l but the slope f l a t t e n i n g movements i n v o l v i n g settlements i n the upper slope and v e r t i c a l r i s e i n the lower slope are only about h a l f of those observed in the model. T h i s lack of agreement r e s u l t s because t o t a l l y undrained c o n d i t i o n s apparently d i d not e x i s t w i t h i n the model s l o p e s , a l l o w i n g some volume changes to take p l a c e . The modulus r e d u c t i o n a n a l y s i s was performed assuming that no volume changes occu r r e d . T h i s assumption i n i t i a l l y appeared reasonable because the volume of sand undergoing downward movements i n the upper part of the model slope was comparable to that undergoing upward movements i n the lower s l o p e . However, the displacement p a t t e r n d e p i c t e d by the s i l i c a beads showed a decrease i n volume in the upper slope of approximately 6 percent. For the e n t i r e model, the volume r e d u c t i o n was only about 1 p e r c e n t . Hence an i n c r e a s e i n the volume of sand i n the lower slope must a l s o have o c c u r r e d . F i g u r e 5-15 shows the deformations p r e d i c t e d when the bulk i n i t i a l slope p r e d i c t e d deformations m u l t i p l i e d 30 times FIGURE 5-12 Deformations Predicted by Post-Cyclic Modulus Approach FIGURE 5-13 Deformations Predicted by Cyclic Strain Approach p r e d i c t e d deformations m u l t i p l i e d 300 times FIGURE 5-14 Deformations Predicted by Pore Pressure Approach FIGURE 5-15 Deformations Predicted by Cyclic Strain Approach with Volume Change Correction CTi 65 modulus was p e r m i t t e d to reduce i n the upper slope to account f o r the observed changes i n volume. The bulk modulus used i n the lower slope was not permitted to reduce but no attempt was made to simulate the observed volume expansion. The deformations p r e d i c t e d u s i n g t h i s c o r r e c t i o n for volume change simulate the slope f l a t t e n i n g movements in the upper slope but cannot d u p l i c a t e the r i s e of sand i n the lower s l o p e . Apart from t h i s minor problem of p r e d i c t i n g volume changes when p a r t i a l l y undrained c o n d i t i o n s e x i s t , the c y c l i c s t r a i n approach appears capable of p r o v i d i n g very good p r e d i c t i o n s of the observed displacements, both i n the magnitude of the movements and i n the p a t t e r n of deformation. The f a i l u r e s of the other two approaches to p r o v i d e r e a l i s t i c deformation p r e d i c t i o n s r e s u l t from t h e i r i n a b i l i t y to account f o r fundamental aspects of the earthquake l o a d i n g or s o i l b ehavior. The f a i l u r e of the p o s t - c y c l i c modulus approach to p r e d i c t deformations of the a p p r o p r i a t e magnitude r e s u l t s because such a method ignores the i n f l u e n c e of earthquake d u r a t i o n on the gene r a t i o n and accumulation of s t r a i n s . I m p l i c i t i n the use of the p o s t - c y c l i c s t r e s s - s t r a i n r e l a t i o n s h i p i s the assumption that the observed s t r a i n s are caused by the s t a t i c shear s t r e s s e s d u r i n g a s i n g l e phase of s t r a i n development. However, the r e d u c t i o n i n the s t i f f n e s s of the s o i l r e s u l t i n g from c y c l i c l o a d i n g i s not s u f f i c i e n t to reproduce the observed s t r a i n s . Although s i g n i f i c a n t pore p r e s s u r e s had developed d u r i n g the c y c l i c t r i a x i a l t e s t s , they began to decrease immediately d u r i n g p o s t - c y c l i c monotonic l o a d i n g . Thus the i n i t i a l modulus f o r the 66 p o s t - c y c l i c t e s t d i d not reduce s i g n i f i c a n t l y . By moderate s t r a i n l e v e l s , the pore p r e s s u r e s e x i s t i n g i n the p o s t - c y c l i c t e s t were very s i m i l a r t o those e x i s t i n g i n the p r e - c y c l i c monotonic t e s t and enough s t r a i n hardening had oc c u r r e d that the p o s t - c y c l i c s t r e s s - s t r a i n r e l a t i o n s h i p d i d not d i f f e r s i g n i f i c a n t l y from the p r e - c y c l i c r e l a t i o n s h i p . The sequence of l o a d i n g used i n the c y c l i c t r i a x i a l t e s t s was intended to r e f l e c t the assumption that the t r a n s m i s s i o n of the earthquake induced shear s t r e s s e s through the s o i l would cease when the pore p r e s s u r e s has i n c r e a s e d s u f f i c i e n t l y to s u b s t a n t i a l l y reduce the s t i f f n e s s of the s o i l . The observed deformations would, hence, be caused e n t i r e l y by the s t a t i c shear s t r e s s e s and would r e s u l t from the r e d u c t i o n in the s t r e n g t h and/or s t i f f n e s s of the s o i l . However, Finn et a l (1976) found through an examination of an accelerogram of the N i i g a t a Earthquake that dynamic a c c e l e r a t i o n s , and hence shear s t r e s s e s , c ontinue to be t r a n s m i t t e d , even a f t e r l i q u e f a c t i o n , although at a much lower frequency. The N i i g a t a accelerogram, shown i n F i g u r e 5-16, i n d i c a t e s that the amplitude of the a c c e l e r a t i o n a f t e r l i q u e f a c t i o n approaches the a c c e l e r a t i o n amplitude p r i o r to l i q u e f a c t i o n . T h i s a b i l i t y of the s o i l to transmit shear s t r e s s e s i n d i c a t e s a r e g a i n i n s t r e n g t h d u r i n g shear, presumably as a r e s u l t of a r e d u c t i o n i n pore p r e s s u r e s . As the s o i l shears i n response to the reduced modulus, i t d i l a t e s , r e g a i n s s t r e n g t h and i s again capable of t r a n s m i t t i n g the earthquake induced shear s t r e s s e s . Thus the observed deformations r e s u l t from the accumulation of s t r a i n s that occur 67 d u r i n g s e v e r a l c y c l e s of l o a d i n g i n which the pore pre s s u r e s a l t e r n a t i v e l y r i s e and f a l l . The s t r a i n s o c c u r r i n g d u r i n g each l o a d i n g c y c l e w i l l depend on both the s t a t i c and c y c l i c shear s t r e s s l e v e l s . The magnitude of the cumulative s t r a i n depends on the number of c y c l e s of l o a d i n g which i s d i r e c t l y r e l a t e d to the d u r a t i o n of the earthquake. 8seconds .J L 1 1*7.1* F i g u r e 5-16 Earthquake Accelerogram of N i i g a t a Earthquake (from Finn et a l , 1976) The p o s t - c y c l i c modulus approach attempts to reproduce the accumulation of s t r a i n s by using o n l y a s i n g l e phase of l o a d i n g . The extent of the deformations that occur d u r i n g a s i n g l e phase of l o a d i n g , however, i s l i m i t e d by s t r a i n hardening r e s u l t i n g from d i l a t i o n and a r e d u c t i o n i n pore p r e s s u r e s . I t i s only when the pore p r e s s u r e s r i s e again d u r i n g unloading of the c y c l i c s t r e s s t h a t the s o i l w i l l s o f t e n s u f f i c i e n t l y to allow the g e n e r a t i o n of a d d i t i o n a l s t r a i n s d u r i n g r e a p p l i c a t i o n of the 68 c y c l i c s t r e s s . To reproduce the s t r a i n behavior d u r i n g c y c l i c l o a d i n g , an a n a l y s i s must be able to simulate the accumulation of s t r a i n s r e s u l t i n g from s e v e r a l c y c l e s of l o a d i n g . The post-c y c l i c modulus approach f a i l s to model t h i s p r o g r e s s i v e s t r a i n development because i t does not c o n s i d e r the e n t i r e c y c l i c s t r e s s h i s t o r y and hence ignores the s u c c e s s i v e development of s t r a i n s . R e a l i s t i c e stimates of deformations can only be achieved when the e n t i r e d u r a t i o n of the earthquake i s co n s i d e r e d i n the a n a l y s i s . An a d d i t i o n a l source of e r r o r i n the p o s t - c y c l i c modulus approach a r i s e s because i t f a i l s to account f o r the i n e r t i a f o r c e s . Although g e n e r a l l y c o n s i d e r e d l e s s s i g n i f i c a n t than the deformations r e s u l t i n g from s t i f f n e s s degradation, the i n e r t i a f o r c e s may become important when pore pressures r i s e enough to b r i n g the s o i l to a near f a i l u r e c o n d i t i o n . When these c o n d i t i o n s develop, any small i n c r e a s e i n the shear s t r e s s may r e s u l t i n s i g n i f i c a n t deformations. The f a i l u r e of the post-c y c l i c modulus approach to i n c l u d e the e f f e c t s of the i n e r t i a f o r c e s on the development of s t r a i n s w i l l , thus, c o n t r i b u t e to i t s i n a b i l i t y to p r e d i c t the observed deformations. The pore pressure approach a l s o f a i l s to p r e d i c t the observed deformations. The s i g n i f i c a n t e r r o r i n v o l v e d occurs because the shear modulus does not reduce s u b s t a n t i a l l y i n response to the changes i n the c o n f i n i n g pressures c r e a t e d by the i n c l u s i o n of the excess pore p r e s s u r e s . Whereas r e d u c t i o n s i n the modulus of 1000 times are r e q u i r e d to p r e d i c t the 69 observed deformations of the t a i l i n g s model, the pore pressure approach p r e d i c t s r e d u c t i o n s of only about 5 times. Even with the reduced s t r e n g t h that a l s o r e s u l t s from the r e d u c t i o n i n the e f f e c t i v e s t r e s s e s , the p r e d i c t e d deformations are 300 times l e s s than those observed. As f o r the p o s t - c y c l i c modulus approach, the problems with the pore p r e s s u r e approach appear to l i e i n i t s f a i l u r e to model the s u c c e s s i v e development of s t r a i n s d u r i n g c y c l i c l o a d i n g , and i n i t s n e g l e c t of the e f f e c t s of the i n e r t i a f o r c e s . At the reduced s t r e n g t h used in the pore pressure approach, the i n e r t i a f o r c e s become s i g n i f i c a n t . If the i n c r e a s e i n pore pressure i s enough to make the s o i l j u s t b a r e l y s t a b l e , any small i n c r e a s e i n shear s t r e s s e s can cause s u b s t a n t i a l deformations. Repeated c y c l e s of l o a d i n g d u r i n g which the s t r e n g t h of the s o i l i s reached would lead to the accumulation of the s i g n i f i c a n t s t r a i n s observed during c y c l i c l o a d i n g . The f a i l u r e of the pore p r e s s u r e approach to model t h i s p r o g r e s s i v e s t r a i n development and t o i n c l u d e the i n f l u e n c e of the i n e r t i a f o r c e s d u r i n g the s u c c e s s i v e p e r i o d s of l o a d i n g causes the l a r g e e r r o r i n the p r e d i c t e d d eformations. The e f f e c t i v e n e s s of the c y c l i c s t r a i n approach appears to be r e l a t e d to i t s a b i l i t y to model the g r a d u a l accumulation of s t r a i n s caused by s u c c e s s i v e c y c l e s of l o a d i n g and by i t s i n c l u s i o n of both the e f f e c t s of s o i l s o f t e n i n g and i n e r t i a f o r c e s on the development of the d y n a m i c a l l y induced s t r a i n s . Because the l a b o r a t o r y t e s t s are performed at the i n - s i t u s t r e s s l e v e l s , the pore pressure response and the r e s u l t i n g s o i l behavior i s i n d i c a t i v e of the s o i l response expected i n the 70 f i e l d . In a d d i t i o n , the i n f l u e n c e of the i n e r t i a f o r c e s and the earthquake d u r a t i o n are i n c l u d e d i n the a n a l y s i s by a p p l y i n g the e q u i v a l e n t number of uniform s t r e s s c y c l e s to the l a b o r a t o r y samples and using the observed s t r a i n s to compute the modulus r e d u c t i o n . The approach, thereby, accounts f o r the s i g n i f i c a n t aspects of the s o i l response expected i n the f i e l d and p r o v i d e s r e a l i s t i c p r e d i c t i o n s of the earthquake induced deformations. 5.2 A n a l y s i s of the Upper San Fernando Dam The a n a l y s i s of the t a i l i n g s slope model i n d i c a t e d the a b i l i t y of. the modulus r e d u c t i o n approach to p r e d i c t deformations of e a r t h s t r u c t u r e s r e s u l t i n g d u r i n g earthquake e x c i t a t i o n . However, such an a n a l y s i s was r e l a t i v e l y simple s i n c e i t i n v o l v e d only one type of s o i l whose p r o p e r t i e s were uniform throughout the model s l o p e . In a d d i t i o n the e n t i r e slope experienced s u b s t a n t i a l pore pressure r i s e and the softened s o i l was not c o n s t r a i n e d by s o i l s which r e t a i n e d a s i g n i f i c a n t p o r t i o n of t h e i r s t i f f n e s s during the dynamic e x c i t a t i o n . The a b i l i t y of the modulus r e d u c t i o n approach to p r e d i c t the earthquake induced deformations of e a r t h s t r u c t u r e s comprising s e v e r a l d i f f e r e n t s o i l types with v a r y i n g p r o p e r t i e s and d i f f e r i n g response to earthquake e x c i t a t i o n should be examined. The movements of the Upper San Fernando Dam d u r i n g the San Fernando Earthquake of 1971 p r o v i d e an a p p r o p r i a t e f i e l d h i s t o r y . 71 5.2.1 D e s c r i p t i o n of Dam and Earthquake Deformations The Upper San Fernando Dam i s an h y d r a u l i c f i l l dam l o c a t e d in the San Fernando V a l l e y of southern C a l i f o r n i a . I t was c o n s t r u c t e d between 1915 and 1925 d i r e c t l y on top of a l l u v i a l d e p o s i t s of s t i f f c l a y s and c l a y e y g r a v e l s . The dam i s 58 f e e t h i g h , c o n s i s t i n g of 40 f e e t of h y d r a u l i c f i l l m a t e r i a l topped with an 18 foot cap of r o l l e d f i l l . The completed s e c t i o n of the dam has a 2.5:1 concrete paved upstream sl o p e , a 20 foot wide c r e s t and a 2.5:1 downstream s l o p e . A 100 foot long berm comprised of h y d r a u l i c f i l l m a t e r i a l forms a s u b s t a n t i a l p a r t of the dam between the c r e s t and the downstream toe. During the e a r l y morning of February 6, 1971, the San Fernando V a l l e y region was shaken by an earthquake having a R i c h t e r magnitude of 6.6. The e p i c e n t e r of the earthquake was l o c a t e d 8.5 m i l e s to the northeast of the dam and had a f o c a l depth of 8 m i l e s . Although no major f a i l u r e of the Upper San Fernando Dam occurred, s u b s t a n t i a l downstream movements i n v o l v i n g the e n t i r e downstream slope and the v i s i b l e p a r t of the upstream slope were observed. The c r e s t of the dam moved 5 f e e t l a t e r a l l y downstream and s e t t l e d 3 f e e t . Measurements of deformations made from s u r f a c e monuments and from the o u t l e t conduit at the base of the dam i n d i c a t e d that the e n t i r e upper pa r t of the dam p a r t i c i p a t e d i n the downstream movement. F i g u r e 5-17, which shows two c r o s s - s e c t i o n s of the dam, i n d i c a t e s the extent of the movements which o c c u r r e d . The s u b s t a n t i a l downstream movements of the dam c r e a t e d 124 Or-| I2Z0|-o o tn IZOOr-2 • 100 • •8 * 1160 • c I 1140 • u 1120 • 1100-1080 -\ II i i " . Soillvoy EL I2 '2S\ I Z ' 3 2 ~> Crott-ltction btfori «orlKQuo»« Croii-ttction ofttr •orlhguokt 100 . El. I20O' Sink Holt yyi Stmi-hydrouiic n i l Cutoff Trtnch Foundation -777^7777777^7; .— Btdrock -y/f-j/xv//- - iikziie*urvn iilsxiis.UL.'. y/tw/WJi-oi* (from Seed et a l , 1973) (a) C r o s s - s e c t i o n showing d i s p l a c e d p r o f i l e of dam Tension Crocks Tension Crocks Compression [ d i s p l a c e m e n t s In f e e t ] (b) Displacements measured a f t e r earthquake —1 FIGURE 5-17 Deformations of Upper San Fernando Dam during Earthquake 73 s e v e r a l l a r g e v e r t i c a l l o n g i t u d i n a l c r a c k s i n the upstream slope that extended over almost the e n t i r e length of the dam. Another l a r g e v e r t i c a l crack was observed midway down the downstream slope of the upper r o l l e d f i l l s e c t i o n of the dam. These cracks are e x t e n s i o n a l f e a t u r e s r e s u l t i n g from the g r e a t e r downstream displacements w i t h i n the berm r e l a t i v e to the c r e s t region of the dam. On the downstream slope beyond the berm, deformations tended to decrease towards the toe of the s l o p e . A 2 foot high p r e s s u r e r i d g e developed at the downstream toe as a r e s u l t of the movements. A survey of the i n t e r i o r of the o u t l e t conduit i n d i c a t e d compressional f a i l u r e s near the toe of the dam. E x t e n s i o n a l f e a t u r e s c o n s i s t i n g of s e v e r a l one h a l f to three q u a r t e r inch c r a c k s i n the conduit were observed i n the c e n t r a l and upstream p o r t i o n s of the dam. Measurements of movements at the c o n d u i t l e v e l were c o n s i d e r a b l y l e s s than those observed on the s u r f a c e . T h i s r e d u c t i o n appears to i n d i c a t e that the major movements oc c u r r e d w i t h i n the f i l l above the c o n d u i t , although s l i p p a g e of the f i l l along the o u t s i d e edge of the conduit c o u l d a l s o account f o r the observed r e l a t i v e displacements. Some l a t e r a l spreading of the dam was i n d i c a t e d by a t r a n s v e r s e compression crack i n the paved roadway acr o s s the c r e s t of the dam and by damage to the concrete s p i l l w a y w a l l at the east abutment. The l a t e r a l displacements, however, were c o n s i d e r a b l y l e s s s i g n i f i c a n t than the t r a n s v e r s e displacements. Piezometer readings taken a f t e r the earthquake i n d i c a t e d 74 maximum pore pressure i n c r e a s e s of 8.5 to 17 f e e t of water. However, the f i r s t readings were not taken u n t i l 24 hours a f t e r the earthquake, a l l o w i n g some d i s s i p a t i o n of the excess p r e s s u r e s . In a d d i t i o n , i n the ce n t e r of the dam, the pore p r e s s u r e i n c r e a s e s were so l a r g e that the water s p i l l e d over the tops of the piezometer c a s i n g s . Thus the maximum pore p r e s s u r e s reached d u r i n g the earthquake c o u l d have been s i g n i f i c a n t l y h i g h e r than those recorded. In the area below the downstream toe, very high pore p r e s s u r e s a p p a r e n t l y o c c u r r e d i n the loose s i l t y sand f i l l l o c a t e d i n that area, causing the formation of s e v e r a l sand b o i l s . 5.2.2 P r e v i o u s I n v e s t i g a t i o n s A f t e r the earthquake, an ex t e n s i v e study of both the Upper and Lower San Fernando Dams was made by Seed et a l (1973). In-s i t u and l a b o r a t o r y s o i l t e s t s were performed to determine the c h a r a c t e r i s t i c s of the s o i l s forming the dams and t h e i r f o u n d a t i o n s . S o i l response to dynamic l o a d i n g was i n v e s t i g a t e d by c y c l i c t r i a x i a l t e s t s . A dynamic a n a l y s i s was then performed to i d e n t i f y p o s s i b l e areas w i t h i n the dam where l i q u e f a c t i o n or c y c l i c m o b i l i t y would occur and to estimate the earthquake induced deformations. F i g u r e 5-18 prese n t s an i d e a l i z e d s e c t i o n of the Upper San Fernando Dam d e l i n e a t i n g the four major zones of s o i l i n the dam. Because the a l l u v i u m and the h y d r a u l i c f i l l s o i l s formed the major p a r t of the dam and because t h e i r c h a r a c t e r i s t i c s lower a l l u v i u m FIGURE 5-18 Major Soils Types in Upper San Fernando Dam - J 76 would tend to c o n t r o l the deformations of the dam, the s o i l t e s t i n g program was c o n f i n e d to these s o i l s . Both the h y d r a u l i c f i l l and the a l l u v i u m c o n s i s t e d p r i m a r i l y of f i n e to coarse sand, although c l a y l a y e r s were common i n both s o i l s . The h y d r a u l i c f i l l c o n s i s t e d of l a y e r s of f a i r l y c l e a n sands and s i l t y to c l a y e y sands with o c c a s i o n a l l a y e r s of c l a y . The l a y e r i n g was most pronounced i n the outer p a r t s of the embankment, becoming l e s s d i s t i n c t near the c e n t r a l p a r t of the dam. The outer embankment m a t e r i a l s tended to be the c o a r s e s t , with the g r a i n s i z e d e c r e a s i n g towards the f a i r l y homogeneous f i n e sandy s i l t of the inner c o r e . The h y d r a u l i c f i l l was loose to medium-dense with an average r e l a t i v e d e n s i t y determined from f i e l d and l a b o r a t o r y compaction t e s t s of 54 pe r c e n t . N values were very low i n the s i l t and c l a y e y core of the c e n t r a l p o r t i o n of the h y d r a u l i c f i l l , but were somewhat higher i n outer sands and g r a v e l s of the s h e l l . The a l l u v i u m tended to be s l i g h t l y c o a r s e r than the h y d r a u l i c f i l l and was b e t t e r graded. I t had an average r e l a t i v e d e n s i t y of 67 percent, some 10 percent higher than the h y d r a u l i c f i l l . Shelby tube samples of the a l l u v i u m i d e n t i f i e d i t as a very heterogenous s o i l ranging from l a y e r s or pockets of c l a y to l a y e r s or pockets of sands and g r a v e l s . S t a t i c l o a d i n g t r i a x i a l t e s t s were performed on undisturbed samples of both the h y d r a u l i c f i l l and the a l l u v i u m . F i g u r e 5-19 shows a t y p i c a l response of the h y d r a u l i c f i l l to d r a i n e d and undrained l o a d i n g . A f t e r an i n i t i a l p e r i o d of compression 77 8 E o cr t_ <u a o» 3 4 cn o 2 o > Q E u sr a I 3 cl v. 3 ct c Dra ned < 0 i i 6 1 »—-• — -o-- « Undrc lined / / . 1 1/ \l IJ IC-U Te ° 3 c 8 st - Fine S 2.0 kg per ; I »ilty Sand >q. cm. i 8 12 16 Axial Strain -percent 20 24 I.O - -i.o <u c 5 I -r-L I \u, Undrained Test ^< y s Drained TE -st, Av/v-*-V O o s ; • • r 1 1.0 c a> o cl i > <1 o I CO o _ a> -10 E 3 o > 0 8 12 16 Axial Strain-percent 20 24 (from Seed et a l , 1973) FIGURE 5-19 Typical Response of Hydraulic Fill in Drained and Undrained Triaxial Tests 78 the f i l l tended to d i l a t e near and a f t e r f a i l u r e . No s i g n i f i c a n t s t r e n g t h l o s s a f t e r f a i l u r e was i n d i c a t e d by the t e s t s . Table 5-2 presents the s o i l parameters determined from the t e s t data obtained by the S t a t e of C a l i f o r n i a , Department of Water Resources. The parameters were determined from d r a i n e d t r i a x i a l t e s t s on undisturbed samples of the h y d r a u l i c f i l l and a l l u v i u m . Table 5-2 S o i l Parameters f o r Upper San Fernando Dam S o i l s f i l l a l l u v i u m shear modulus f a c t o r kg 105 120 shear modulus exponent n .85 .64 r e d u c t i o n r a t i o Rf .73 .71 angle of f r i c t i o n <j> 38.0 40.0 change in f r i c t i o n angle A<p 1 .4 2.0 C y c l i c l o a d i n g t e s t s were a l s o performed on both s o i l s . However, because the m a j o r i t y of the observed movements appeared to occur w i t h i n the h y d r a u l i c f i l l and because the f i l l was more s u s c e p t i b l e to pore pressure r i s e than the a l l u v i u m , the t e s t i n g program con c e n t r a t e d on determining the response of the h y d r a u l i c f i l l to c y c l i c l o a d i n g . None of the samples of the h y d r a u l i c f i l l experienced a true l i q u e f a c t i o n f a i l u r e i n v o l v i n g a r a p i d development of l a r g e s t r a i n s . Instead, a t r a n s i e n t l i q u e f a c t i o n c o n d i t i o n r e s u l t i n g i n the p r o g r e s s i v e development of l a r g e s t r a i n s o c c u r r e d . T h i s behavior i s c h a r a c t e r i s t i c of 79 c y c l i c m o b i l i t y r a t h e r than l i q u e f a c t i o n which i n v o l v e s a l o s s in s t r e n g t h . Seed presented e x t e n s i v e data on the l e v e l s of c y c l i c shear s t r e s s e s r e q u i r e d to cause s p e c i f i c s t r a i n l e v e l s w i t h i n the s o i l f o r v a r i o u s l e v e l s of c o n f i n i n g pressure and s t a t i c b i a s . On the b a s i s of the c y c l i c t r i a x i a l t e s t s and a dynamic a n a l y s i s , Seed i d e n t i f i e d s e v e r a l areas where i n i t i a l l i q u e f a c t i o n and 5 percent s t r a i n would occur w i t h i n the dam and concluded that the development of l a r g e s t r a i n s would be expected even i n the outer part of the embankment. F i g u r e 5-20 shows the areas that Seed c o n s i d e r e d to have l i q u e f i e d w i t h i n the h y d r a u l i c f i l l . The term " l i q u e f a c t i o n " as used by Seed r e f e r s to the development of 5 percent s t r a i n i n t r i a x i a l t e s t samples. The a l l u v i u m was not con s i d e r e d to have developed s i g n i f i c a n t pore pre s s u r e s because of i t s higher d e n s i t y and was not c o n s i d e r e d i n Seed's a n a l y s i s . Based on the r e s u l t s of these t e s t s , Seed determined the s t r a i n s that would occur i n the h y d r a u l i c f i l l i f samples were subjected to c y c l i c t r i a x i a l t e s t s that reproduced the f i e l d s t r e s s c o n d i t i o n s . 5.2.3 Modulus Reduction A n a l y s i s The a n a l y s i s of the movements of the Upper San Fernando Dam was made by the proposed c y c l i c s t r a i n method of modulus r e d u c t i o n . Although e x t e n s i v e i n v e s t i g a t i o n s and s o i l t e s t s were made by Seed et a l (1973) a f t e r the earthquake, t h e i r data i s not i n the a p p r o p r i a t e form f o r such an a n a l y s i s . Since no Zona* of fodtfft dw« to liquefoclioA indicotod bj onofy*<i after 6 seconds of shot ma. f///y\ Zone* of fdilw* due to hquefoction mdicotod by onoiym V///A r3 seconds of ehosmg FIGURE 5-20 Liquefied Areas of Dam 81 f u r t h e r t e s t i n g c o u l d be performed, the a n a l y s i s r e l i e d on the s t r a i n p o t e n t i a l s presented by Seed to estimate the modulus r e d u c t i o n by the c y c l i c s t r a i n approach. To determine the a p p r o p r i a t e modulus r e d u c t i o n by the c y c l i c s t r a i n approach the i n - s i t u shear s t r e s s e s determined from a s t a t i c s t r e s s a n a l y s i s were d i v i d e d by the shear s t r a i n p o t e n t i a l s recommended by Seedi These s t r a i n p o t e n t i a l s were assumed to represent the shear s t r a i n s which would occur in l a b o r a t o r y t e s t s performed at the f i e l d s t r e s s c o n d i t i o n s . Although Seed performed many c y c l i c l o a d i n g t e s t s , both under i s o t r o p i c and a n i s o t r o p i c c o n d i t i o n s , he was not p r i m a r i l y concerned with determining the d i s t r i b u t i o n and magnitudes of the s t r a i n s that would p o t e n t i a l l y occur throughout the dam. To determine such s t r a i n s the s t r e s s c o n d i t i o n s e x i s t i n g at s p e c i f i c l o c a t i o n s i n the dam would have to be d u p l i c a t e d i n the c y c l i c l o a d i n g t e s t s . The s t r a i n p o t e n t i a l s that Seed determined to occur throughout the dam are shown on F i g u r e 5-21. The c r i t e r i a t h a t Seed used i n s e l e c t i n g these s t r a i n p o t e n t i a l s were not s p e c i f i e d . In the c e n t r a l part of the dam, use of s t r a i n p o t e n t i a l s i n the 50 to 70 percent range r e s u l t e d i n modulus r e d u c t i o n s of 500 to 3000 times. Elements in the outer embankment on the downstream slope were not s a t u r a t e d and, hence, not expected to experience any modulus r e d u c t i o n as a r e s u l t of s o i l s o f t e n i n g . No modulus r e d u c t i o n was used i n the a l l u v i u m because the deformations observed i n the o u t l e t c o n d u i t i n d i c a t e d that the m a j o r i t y of the movement was c o n f i n e d to the h y d r a u l i c f i l l r e g i o n of the dam. The a l l u v i u m was a l s o 25 to FIGURE 5-21 Shear Strain Potentials 83 30 percent s t r o n g e r than the h y d r a u l i c f i l l under c y c l i c l o a d i n g . Hence, the magnitude of any modulus r e d u c t i o n i n the a l l u v i u m would be much l e s s s i g n i f i c a n t than i n the f i l l , a lthough some r e d u c t i o n c o u l d occur. No data was a v a i l a b l e f o r the upper r o l l e d f i l l and thus no modulus r e d u c t i o n was assumed in t h i s a r e a . The f i n i t e element program SOILSTRESS was again used to perform the s t a t i c - s t r e s s a n a l y s i s r e q u i r e d f o r computing the earthquake induced deformations. F i g u r e 5-22 shows the f i n i t e element g r i d used i n the a n a l y s i s . The reduced modulus f a c t o r s , computed f o r each element by d i v i d i n g the i n - s i t u s t a t i c shear s t r e s s by the recommended s t r a i n p o t e n t i a l , were used in the s t a t i c s t r e s s a n a l y s i s . Because undrained shear was assumed to occur d u r i n g the earthquake, the bulk modulus was not permitted to reduce from i t s i n i t i a l p r e - c y c l i c v a l u e . The geometry of the elements was c o r r e c t e d throughout the a n a l y s i s to account fo r any smal l r e d u c t i o n i n the d r i v i n g s t r e s s e s that might occur due to slope f l a t t e n i n g . The r e s u l t s of the deformation a n a l y s i s are shown on F i g u r e 5-23. The c r e s t of the dam was p r e d i c t e d to move downstream 2.5 fe e t and s e t t l e 2.5 f e e t . H o r i z o n t a l displacements i n c r e a s e d downstream of the c r e s t to 4.3 f e e t at the s t a r t of the berm where an area of extension was i n d i c a t e d by s t i l l g r e a t e r movements f u r t h e r down the berm. A f t e r reaching a maximum of 5.0 f e e t the p r e d i c t e d h o r i z o n t a l displacements decreased to 3.9 f e e t at the edge of the berm and continued to decrease to 2.3 FIGURE 5-22 Finite Element Grid of Upper San Fernando Dam used in Static Stress Analysis 2.5 (4.9) 2.5 (2.5) •2.7 (4.9) i n i t i a l dam pr o f i l e predicted post-earthquake dam pr o f i l e 1.8 (1.9) I ^ 4 - 3 (6.4) 3.9 (7.2) 3.7 (5.8) 2.3 (3.6) predicted deformations multiplied 3 times observed displacements in brackets displacements in feet co FIGURE 5-23 Predicted Deformations of Upper San Fernando Dam 86 fe e t at the t o e . Although these deformations are about h a l f of those observed, the p a t t e r n of deformation d u p l i c a t e s that of the dam. An area of compression i s i n d i c a t e d at the toe of the berm while e x t e n s i o n i s i n d i c a t e d in the c e n t r a l and upper p a r t s of the berm and dam. To p r e d i c t the deformations observed i n the dam, the s t r a i n p o t e n t i a l s used to determine the necessary r e d u c t i o n i n the modulus f a c t o r have been m o d i f i e d as shown in F i g u r e 5-24. Only elements i n which the s t r a i n p o t e n t i a l s were changed are shown with t h e i r new assumed v a l u e s ; The i n c r e a s e s i n s t r a i n p o t e n t i a l were only r e q u i r e d f o r a very few elements l o c a t e d near the toe of the dam. The use of higher s t r a i n p o t e n t i a l s i n t h i s area may be j u s t i f i e d by the high l e v e l of pore pressure development observed to occur j u s t beyond the downstream toe of the dam and because of the higher s t a t i c shear s t r e s s e s o c c u r r i n g w i t h i n the downstream slope of the berm. The deformations p r e d i c t e d using these minor changes are shown i n Fi g u r e 5-25. The p r e d i c t e d displacements of the dam agree very w e l l with the observed displacements except at the c r e s t . Apart from minor d i s c r e p a n c i e s , the modulus r e d u c t i o n dynamic a n a l y s i s p r e d i c t s deformations that agree with those observed, both i n the magnitude of the displacements and i n t h e i r p a t t e r n . B e t t e r agreement, even i n the c r e s t area, may have been achieved i f the a p p r o p r i a t e l a b o r a t o r y t e s t s were performed to determine the a c t u a l s t r a i n s to be used f o r c a l c u l a t i n g the modulus r e d u c t i o n . 15 20 20 10 10 FIGURE 5-24 Required Shear Strain Potentials oo 2.9 (4.9) 4.0 (2.5) 5.1 (4.9) i n i t i a l dam profile-predicted post-earthquake dam profile 2.6 (1.9) 6.6 (6.4) .4 (.2) •6.1 (7.2) 5.9 (5.8) 3.7 (3.6) observed displacements in brackets displacements in feet FIGURE 5-25 Predicted Deformations using Required Shear Strain Potentials 89 CHAPTER 6 SUMMARY AND CONCLUSIONS The modulus r e d u c t i o n a n a l y s i s i s an e f f e c t i v e semi-a n a l y t i c a l method of dynamic a n a l y s i s capable of p r e d i c t i n g earthquake induced deformations of a r e a l i s t i c magnitude and p a t t e r n . The e f f e c t s of an earthquake on a s o i l are represented as a r e d u c t i o n i n the s t i f f n e s s p r o p e r t i e s of the s o i l . The a n a l y s i s i s p r i m a r i l y intended f o r s a t u r a t e d s o i l s which experience s i g n i f i c a n t s o f t e n i n g as a r e s u l t of pore pressure r i s e . However, because the e f f e c t s of the i n e r t i a f o r c e s are a l s o i n c o r p o r a t e d i n t o the modulus r e d u c t i o n a n a l y s i s , i t can be used e f f e c t i v e l y f o r s o i l s which are expected to experience only l i m i t e d changes i n pore water p r e s s u r e during c y c l i c l o a d i n g . The r e d u c t i o n s i n the modulus that are r e q u i r e d to p r e d i c t the deformations observed d u r i n g c y c l i c l o a d i n g may be as l a r g e as 1000 to 3000 times f o r s o i l s that develop l i q u e f a c t i o n or c y c l i c m o b i l i t y as a r e s u l t of t h e i r s u s c e p t i b i l i t y to s u b s t a n t i a l pore pressure r i s e . Of the three proposed methods of computing a reduced modulus, only the c y c l i c s t r a i n approach was capable of p r e d i c t i n g r e d u c t i o n s of such a magnitude. Both the p o s t - c y c l i c modulus approach and the pore p r e s s u r e approach f a i l e d to p r e d i c t r e d u c t i o n s of s u f f i c i e n t magnitude. The p o s t - c y c l i c modulus approach, in which the modulus r e d u c t i o n i s determined by comparing the s t r e s s - s t r a i n curves before and a f t e r c y c l i c l o a d i n g , f a i l s to p r e d i c t deformations 90 of s u f f i c i e n t magnitude. T h i s f a i l u r e i s presumably r e l a t e d to the i n a b i l i t y of such a method to model the s u c c e s s i v e development of s t r a i n s d u r i n g c y c l i c l o a d i n g . The p o s t - c y c l i c s t r e s s - s t r a i n r e l a t i o n s h i p that i s used f o r t h i s a n a l y s i s cannot account f o r the i n c r e a s e i n pore water pressure that develops duri n g the unloading phase of each c y c l i c s t r e s s a p p l i c a t i o n . The s t r a i n s that develop dur i n g a s i n g l e l o a d c y c l e w i l l be l i m i t e d by s t r a i n hardening, r e s u l t i n g from d i l a t i o n and a r e d u c t i o n i n pore p r e s s u r e s . Only by c o n s i d e r i n g s u c c e s s i v e p e r i o d s of s t r a i n development i n which the pore p r e s s u r e s a l t e r n a t i v e l y r i s e and f a l l w i l l an a n a l y s i s be able to model the s i g n i f i c a n t accumulation of s t r a i n s observed d u r i n g c y c l i c l o a d i n g . The pore pressure approach a l s o f a i l s to p r e d i c t r e a l i s t i c displacements of the t a i l i n g s model. Even though the i n c l u s i o n of the excess pore p r e s s u r e s c r e a t e d near f a i l u r e c o n d i t i o n s i n the model, the r e d u c t i o n i n the shear modulus was s u b s t a n t i a l l y lower than that r e q u i r e d to p r e d i c t the observed deformations. The f a i l u r e of the pore pressure approach to p r e d i c t r e a l i s t i c deformations r e s u l t s because i t ignores the s i g n i f i c a n c e of the i n e r t i a f o r c e s on the development of s t r a i n s when s u b s t a n t i a l i n c r e a s e s i n pore water pressure occur d u r i n g c y c l i c l o a d i n g . When only a minor i n c r e a s e i n the pore water pressure occurs, the i n e r t i a f o r c e s g e n e r a l l y do not cause s u b s t a n t i a l deformations u n l e s s they are very l a r g e or the s o i l i s very s e n s i t i v e . However, the i n e r t i a f o r c e s w i l l become s i g n i f i c a n t when the i n c r e a s e i n pore water pressure b r i n g s the s o i l so near 91 the f a i l u r e c o n d i t i o n that any small i n c r e a s e i n the d r i v i n g f o r c e s causes s u b s t a n t i a l deformations. F a i l u r e to i n c l u d e the e f f e c t s of the i n e r t i a f o r c e s d u r i n g s u c c e s s i v e p e r i o d s of l o a d i n g r e s u l t s i n the i n a b i l i t y of the pore pressure approach to p r e d i c t s u f f i c i e n t modulus r e d u c t i o n s and, hence, to reproduce the observed deformations. The success of the c y c l i c s t r a i n approach i n p r e d i c t i n g a s u f f i c i e n t modulus r e d u c t i o n l i e s i n i t s i n c o r p o r a t i o n of the e f f e c t s of earthquake d u r a t i o n on s o i l s o f t e n i n g and i n i t s i n c l u s i o n of the i n e r t i a f o r c e s i n the a n a l y s i s . The s t r a i n s observed in the l a b o r a t o r y t e s t s performed at the i n - s i t u s t r e s s l e v e l s with the e q u i v a l e n t number of shear s t r e s s c y c l e s , depend on both the magnitude and the d u r a t i o n of the c y c l i c l o a d i n g a p p l i e d . Because the e n t i r e e f f e c t s of the earthquake are represented by an e q u i v a l e n t c y c l i c l o a d i n g , the observed behavior w i l l reproduce the s o i l response expected in the f i e l d . The modulus r e d u c t i o n computed from the observed s t r a i n s w i l l i n c l u d e the e f f e c t s of s o i l s o f t e n i n g and i n e r t i a f o r c e s and w i l l be of s u f f i c i e n t magnitude to p r e d i c t r e a l i s t i c d eformations. The advantage of the modulus r e d u c t i o n a n a l y s i s i s that i t uses r e l a t i v e l y simple dynamic and s t a t i c - s t r e s s analyses to p r e d i c t earthquake induced deformations. Only common g e o t e c h n i c a l parameters that may be determined from simple t r i a x i a l and c y c l i c t r i a x i a l t e s t s are r e q u i r e d . The semi-a n a l y t i c a l technique permits the i n c l u s i o n of the e f f e c t s of 92 pore p r e s s u r e r i s e on s o i l behavior without having to r e s o r t to a r i g o r o u s and complex e f f e c t i v e s t r e s s a n a l y s i s . The proposed method, thus, appears to be an e f f e c t i v e and r e l a t i v e l y simple method f o r p r e d i c t i n g the deformations of s o i l s t r u c t u r e s during earthquake l o a d i n g . 93 REFERENCES Ambraseys, N.N. and Sarma, S.K. "The Response of E a r t h Dams to Strong Earthquakes," Geotechnique 17, No. 3, pp. 181-213. Annaki, M. and Lee, K.L. " E q u i v a l e n t Uniform Cycle Concept f o r S o i l Dynamics," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 103, No. GT6, Proc. Paper 12991, June 1977, pp. 549-564. Byrne, P.M. " S t a t i c F i n i t e Element A n a l y s i s of S o i l S t r u c t u r e Systems," S o i l Mechanics S e r i e s , No. 71, U n i v e r s i t y of B r i t i s h Columbia, A p r i l 1983. Byrne, P.M. and Janzen, W. "SOILSTRESS: A Computer Program f o r Non-linear A n a l y s i s of S t r e s s e s and Deformations i n S o i l , " S o i l Mechanics S e r i e s , No. 52, U n i v e r s i t y of B r i t i s h Columbia, December 1981. Byrne, P.M., V a i d , Y.P. and S t u c k e r t , B. "Model T e s t s on Earthquake S t a b i l i t y of T a i l i n g s S l o p e s , " S o i l Mechanics S e r i e s , No. 53, U n i v e r s i t y of B r i t i s h Columbia, December 1981. Cas t r o , G. and C h r i s t i a n , J.T. "Shear Strength of S o i l s and C y c l i c Loading," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 102, No. GT9, Proc. Paper 12387, September 1976, pp. 587-594. Cast r o , G. and Poulos, S.J. " F a c t o r s A f f e c t i n g L i q u e f a c t i o n and C y c l i c M o b i l i t y , " J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 103, No. GT6>.Prpc. Paper 12994, June 1977, pp. 501-516. Chern, J.C. " E f f e c t of S t a t i c Shear on Re s i s t a n c e to L i q u e f a c t i o n , " M.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada, A p r i l 1981. Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P. "Strength, S t r e s s - S t r a i n and Bulk Modulus Parameters f o r F i n i t e Element A n a l y s i s of S t r e s s e s and Movements i n S o i l Masses," Report No. UCB/GT/80-01, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , August 1980. Duncan, J.M. and Chang, C. "Non-linear A n a l y s i s of S t r e s s and S t r a i n i n S o i l s , " J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 96, No. SM5, September 1970, pp. 1629-1653. Fi n n , W.D.L. " S o i l Dynamics - L i q u e f a c t i o n of Sands," Proceedings of the I n t e r n a t i o n a l Conference on M i c r o z o n a t i o n f o r Safer C o n s t r u c t i o n Research and A p p l i c a t i o n , V o l . 1, S e a t t l e , 1972, pp. 87-111. 94 F i n n , W.D.L. " L i q u e f a c t i o n P o t e n t i a l : Developments Since 1976," Proceedings I n t e r n a t i o n a l Conference on Recent Advances i n Geo t e c h n i c a l Earthquake E n g i n e e r i n g and S o i l Dynamics, S t . Lo u i s , M i s s o u r i , 1981, pp. 655-681. F i n n , W.D.L., Byrne, P.M. and M a r t i n , G.R. "Seismic Response and L i q u e f a c t i o n of Sands," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 102, No. GT8, Proc. Paper 12323, August 1976, pp. 841-855. Fi n n , W.D.L., Lee, K.W. and M a r t i n , G.R. "An E f f e c t i v e S t r e s s Model f o r L i q u e f a c t i o n , " J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 103, No. GT6, Proc. Paper 13008, June 1977, pp. 517-533. F i n n , W.D.L., M a r t i n , G.R. and Lee, M.K.W. "Comparison of Dynamic Analyses f o r Satu r a t e d Sands," Proceedings ASCE G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n S p e c i a l t y Conference on Earthquake E n g i n e e r i n g , Pasadena, C a l i f o r n i a , June 1978, pp. 472-491. F i n n , W.D.L., P i c k e r i n g , D.J. and Bransby, P.L. " S o i l L i q u e f a c t i o n i n T r i a x i a l and Simple Shear T e s t s , " 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 . 97, No. SM4, A p r i l 1971, pp. 639-659. Hardin, B.O. and Drnevich, V.P. "Shear Modulus and Damping i n S o i l s : Measurement and Parameter E f f e c t s , " J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 98, No. SM6, Proc. Paper 8977, June 1972, pp. 603-624. Hardin, B.O. and Drnevich, V.P. "Shear Modulus and Damping i n S o i l s : Design Equations and Curves," J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 98, No. SM7, Proc. Paper 9006, J u l y 1972, pp. 667-691. I d r i s s , I.M., Seed, H.B. and S e r f f , N. "Seismic Response by V a r i a b l e Damping F i n i t e Elements," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 100, No. GT1, Proc. Paper 10284, January 1974, pp. 1-13. Lee, K.L. and Chan, K. "Number of E q u i v a l e n t S i g n i f i c a n t C y c l e s i n Strong Motion Earthquakes," Proceedings, I n t e r n a t i o n a l Conference of Mi c r o z o n a t i o n , S e a t t l e , Washington, V o l . I I , October, 1972, pp. 609-627. Lee, M.K.W. and F i n n , W.D.L. "DESRA-2, Dynamic E f f e c t i v e S t r e s s Response A n a l y s i s of S o i l D e p o s i t s with Energy T r a n s m i t t i n g Boundary i n c l u d i n g Assessment of L i q u e f a c t i o n P o t e n t i a l , " S o i l Mechanics S e r i e s , No. 38, U n i v e r s i t y of B r i t i s h Columbia, 1978. 95 Lee, K.L. and Seed, H.B. " C y c l i c S t r e s s C o n d i t i o n s Causing L i q u e f a c t i o n of Sand," Jo u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 93, No. SM1, Proc. Paper 5058, January 1967, pp. 41-70. Lee, K.L. and Seed, H.B. "Drained S t r e n g t h C h a r a c t e r i s t i c s of Sands," J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 93, No. SM6, Proc. Paper 5561, November 1967, pp. 117-141. M a k d i s i , F . I . and Seed, H.B. " S i m p l i f i e d Procedure f o r E s t i m a t i n g Embankment Earthquake-Induced Deformations," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 104, No. GT7, Proc. Paper 13898, J u l y 1978, pp. 849-867. M a r t i n , G.R., F i n n , W.D.L. and Seed, H.B. "Fundamentals of L i q u e f a c t i o n under C y c l i c Loading," J o u r n a l of the Ge o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 101, No. GT5, Proc. Paper 11284, May 1975, pp. 423-438. Newmark, N.M. " E f f e c t s of Earthquakes on Dams and Embankments," F i f t h Rankine L e c t u r e , Geotechnique, V o l . XV, No. 2, London, 1965. Poulos, S.J., Cast r o , G. and France, J.W. " L i q u e f a c t i o n E v a l u a t i o n Procedure," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 111, No. GT6, Proc. Paper 19290> June 1985, pp. 772792. Report of S o i l s T e s t i n g of Upper San Fernando Dam, State of C a l i f o r n i a , Department of Water Resources, November 1971. Seed, H.B. "A Method f o r Earthquake R e s i s t a n t Design of E a r t h Dams," J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 92, No. SM1, January 1966, pp. 13-42. Seed, H.B. "Slope S t a b i l i t y During Earthquakes," J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 93, No. SM4, Proc. Paper 5319, J u l y 1967, pp. 299-323. Seed, H.B. " L a n d s l i d e s during Earthquakes due to S o i l L i q u e f a c t i o n , " J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 94, No. SM5, September 1968, pp. 1053-1122. Seed, H.B. " C o n s i d e r a t i o n s i n the Earthquake-Resistant Design of E a r t h and R o c k f i l l Dams," Geotechnique 29, No. 3, 1979, pp. 215-263. Seed, H.B. " S o i l L i q u e f a c t i o n and C y c l i c M o b i l i t y f o r L e v e l Ground duri n g Earthquakes," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 105, No. GT2, February 1979, pp. 201-255. 96 Seed, H.B. "Earthquake-Resistant Design of E a r t h Dams," Seismic Design of Embankments and Caverns, ASCE Symposium, 1983, pp. 41-63. Seed, H.B. and I d r i s s , I.M. "Influence of S o i l C o n d i t i o n s on Ground Motions during Earthquakes," J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 95, No. SM1, Proc. Paper 6347, January 1969, pp. 99-137. Seed, H.B. and I d r i s s , I.M. " S o i l Moduli and Damping F a c t o r s f o r Dynamic Response Analyses," Report No. EERC 70-10, U n i v e r s i t y of C a l i f o r n i a , Earthquake En g i n e e r i n g Research Center, B e r k e l e y , C a l i f o r n i a , December 1970. Seed, H.B. and I d r i s s , I.M. " S i m p l i f i e d Procedure f o r E v a l u a t i n g S o i l L i q u e f a c t i o n P o t e n t i a l , " J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 97, No. SM9, Proc. Paper 8371, September 1971, pp. 1249-1273. Seed, H.B., I d r i s s , I.M., Lee, R.L. and Ma k d i s i , F . I . "Dynamic A n a l y s i s of the S l i d e i n the Lower San Fernando Dam d u r i n g the Earthquake of February, 1971," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 101, No. GT9, Proc. Paper 11541, September 1975, pp. 889-911. Seed, H.B. and Lee, K.L. " L i q u e f a c t i o n of Saturated Sands During C y c l i c Loading," J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 92, No. SM6, November 1966, pp. 105-134. Seed, H.B., Lee, K.L. and I d r i s s , I.M. " A n a l y s i s of the S h e f f i e l d Dam F a i l u r e , " J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 95, No. SM6, November 1969, pp. 1453-1490. Seed, H.B., Lee, K.L., I d r i s s , I.M. and Ma k d i s i , F. " A n a l y s i s of the S l i d e s i n the San Fernando Dams du r i n g the Earthquake of Feb. 9, 1971," Earthquake E n g i n e e r i n g Research Center, Report No. EERC 73-2, June 1973. Seed, H.B., M a k d i s i , F.I. and DeAlba, P. "Performance of E a r t h Dams durin g Earthquakes," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 104, No. GT7, Proc. Paper 13870, J u l y 1978, pp. 967-994. Seed, H.B. and Ma r t i n G.R. "The Seismic C o e f f i c i e n t i n E a r t h Dam Design," J o u r n a l of the S o i l Mechanics and Foundation E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 92, No. SM3, May 1966, pp. 25-58. Seed, H.B., M a r t i n , P.P. and Lysmer, J.L. "Pore-Water P r e s s u r e Changes During S o i l L i q u e f a c t i o n , " J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 102, No. GT4, Proc. Paper 12074, A p r i l 1976, pp. 323-346. 97 S i d d h a r t a n , R. "A Two-Dimensional Non-Linear S t a t i c and Dynamic Response A n a l y s i s of S o i l S t r u c t u r e s , " PhD T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada, March 1984. S t r e e t e r , V.L., Wylie, E.B. And R i c h a r t , F.E. " S o i l Motion Computations by C h a r a c t e r i s t i c s Method," J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 100, No. GT3, Proc. Paper 10410, March 1974, pp. 247-263. S t u c k e r t , B. "Model Study of Sloped T a i l i n g s D e p o s i t s , " M.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada, 1982. V a i d , Y.P. and F i n n , W.D.L. " S t a t i c Shear and L i q u e f a c t i o n P o t e n t i a l , " J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 105, No. GT10, Proc. Paper 14909, October 1979, pp. 1233-1246. V a i d , Y.P. and Chern, J.C. " E f f e c t of S t a t i c Shear on R e s i s t a n c e to L i q u e f a c t i o n , " S o i l s and Foundations, Japanese S o c i e t y of S o i l Mechanics and Foundation E n g i n e e r i n g , V o l . 23, No. 1, March 1 983. V a i d , Y.P. and Chern, J.C. "Mechanism of Deformation d u r i n g C y c l i c Undrained Loading of Saturated Sands," S o i l Dynamics and Earthquake E n g i n e e r i n g , V o l . 2, No. 3, 1983, pp. 171-177. Wong, K.S. and Duncan, J.M. "Hyperbolic S t r e s s - S t r a i n Parameters f o r Non-Linear F i n i t e Element Analyses of S t r e s s e s and Movements i n S o i l Masses," Report No. TE-74-3, U n i v e r s i t y of C a l i f o r n i a , B erkeley, 1974. Wilson, E.L. and Clough, R.W. "Dynamic Response by Step-by-Step M a t r i x A n a l y s i s , " Proceedings Symposium on the Use of Computers i n C i v i l E n g i n e e r i n g , L i s b o n , P o r t u g a l , October, 1962. 

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