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The influence of impermeable cores on the seismic behaviour of earth dams 1969

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THE INFLUENCE OF IMPERMEABLE CORES ON i THE SEISMIC BEHAVIOUR OF EARTH DAMS by NORMAN JOHN SERFF , B.E., U n i v e r s i t y C o l l e g e , D u b l i n , 1963 THESIS SUBMITTED IN PARTIAL FULFILMENT 0 THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969 ; In present ing th is thesis in p a r t i a l f u l f i lmen t of the requirements for an advanced degree at the Un ivers i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and Study. I fur ther agree that permission for extensive copying of th is thesis for s cho l a r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t i on of th is thes.is for f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of C i v i l Engineering The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 28, 1969. ( i i ) ABSTRACT The i n f l u e n c e of an impermeable c l a y c o r e on the s t a t i c and dynamic be h a v i o u r of an e a r t h dam i s i n v e s t i g a t e d . The c o r e s used a re of two t y p e s , c e n t r a l c o r e and upstream s l o p i n g c o r e . Recommendations a r e made on the s u i t a b i l i t y of each type of core f o r dams i n areas s u b j e c t , to s e i s m i c a c t i v i t y . The f i n i t e element method of a n a l y s i s i s used and the m a t e r i a l i s assumed to behave i n a v i s c o e l a s t i c manner. The s l o p i n g c o r e dam i s found to be l e s s d e s i r a b l e than the c e n t r a l c o r e dam f o r earthquake r e g i o n s because of the u n f a v o u r a b l e s t r e s s d i s t r i b u t i o n s i n the upper p a r t of the dam. S t a t i c t e n s i l e s t r e s s e s d evelop i n t h i s r e g i o n , which do not occur i n the c e n t r a l c o r e dam, and the e x t e n t of these s t r e s s e s i s i n c r e a s e d when the dynamic s t r e s s e s due to the earthquake a re superimposed. The a c c e l e r a t i o n s , which i n c r e a s e w i t h e l e v a t i o n i n the dam, i n d i c a t i n g the n e c e s s i t y of u s i n g a v a r i a b l e s e i s m i c c o e f f i c i e n t , a r e h i g h e r i n the s l o p i n g c o r e dam than i n the c e n t r a l core dam. I t i s found t h a t the f i r s t mode, the o n l y mode t h a t approximates a shear mode, c o n t r i b u t e s the major share to the dynamic response of the dam. ( i i i ) The f i n i t e element method i s shown to be s e n s i t i v e to i r r e g u l a r i t i e s i n the s u b d i v i s i o n of the da.u ! • . ' • • . i i n t o f i n i t e e l e m e n t s . ( i v ) TABLE OF CONTENTS • Page ABSTRACT , ( i i ) LIST OF TABLES ( v i ) LIST OF FIGURES ( v i i ) ACKNOWLEDGEMENT ( i x ) CHAPTER 1 INTRODUCTION 1 CHAPTER 2 STATIC STRESS ANALYSIS 2.1 I n t r o d u c t i o n 9 2.2 S t a t i c S t r e s s e s 10 2.3 S t a t i c D i s p l a c e m e n t s 17 CHAPTER 3 DYNAMIC RESPONSE ANALYSIS 3.1 I n t r o d u c t i o n 19 3.2 Mode Shapes and F r e q u e n c i e s 20 3.3 Power S p e c t r a l D e n s i t y E s t i m a t e s 23 3.4 P r e l i m i n a r y I n v e s t i g a t i o n of Dynamic. 26 Response 3.5 A c c e l e r a t i o n s i n Dam 31 3.6 Dynamic S t r e s s Response 37 3.7 Response by Modes 47 3.8 Modal P a r t i c i p a t i o n F a c t o r 51 3.9 A p p l i c a t i o n of R e s u l t s 54 CHAPTER 4 CONCLUSIONS 56 (v) Page BIBLIOGRAPHY 53 APPENDIX I DESCRIPTION OF THE FINITE ELEMENT 63 METHOD SEISMIC ANALYSIS 65 MODAL PARTICIPATION FACTOR 68 APPENDIX I I MODE SHAPES 69 ( v i ) LIST OF TABLES , Page TABLE 1 Modal P a r t i c i p a t i o n F a c t o r s 53 ( v i i ) LIST OF FIGURES FIGURE TITLE Page 1 Dimensions and P r o p e r t i e s of the Dams 3 2 D i v i s i o n of Dam i n t o F i n i t e Elements showing 6 Nodes used i n P r e s e n t a t i o n of R e s u l t s 3 H o r i z o n t a l S t a t i c S t r e s s e s i n Dam w i t h Core 11 E x p r e s s e d as a Percentage of the S t r e s s e s i n the Dam w i t h o u t a Core 4 S t a t i c Shear S t r e s s e s i n Dam w i t h Core 12 E x p r e s s e d as a Percentage of the S t r e s s e s i n the Dam w i t h o u t a Core 5 E l C e n t r o Earthquake - C a l i f o r n i a - May 18, 21 1940 6 Comparison of P.S.D. of E l Centro w i t h 25 P.S.D. of A c c e l e r a t i o n of C r e s t of Dam 7 D i f f e r e n c e i n Dynamic S t r e s s e s due to use 27 of e i t h e r H o r i z o n t a l A c c e l e r a t i o n o n l y or E q u a l V e r t i c a l and H o r i z o n t a l A c c e l e r a t i o n 8 D i f f e r e n c e i n Response due to use of P r i n t - 30 out I n t e r v a l s of 0.05 seconds and 0.20 seconds 9 D i f f e r e n c e i n Response between use of 5 Modes 30 and 15 Modes 10 Comparison of A b s o l u t e H o r i z o n t a l A c c e l e r a t i o n 33 of C r e s t of Dam w i t h o u t a Core w i t h H o r i z o n t a l Component of A c c e l e r a t i o n of E l Centro 11 Comparison of A b s o l u t e H o r i z o n t a l A c c e l e r a t i o n 33 of C r e s t of Dam w i t h o u t a Core when S u b j e c t e d t o E l Centro and Alameda Park Earthquakes 12 V a r i a t i o n of A b s o l u t e H o r i z o n t a l A c c e l e r a t i o n 35 w i t h H e i g h t a l o n g g of Dam w i t h o u t a Core 13 V a r i a t i o n of H o r i z o n t a l , V e r t i c a l and Shear 38 S t r e s s a l o n g the 60 F t . L e v e l of Dam w i t h o u t Core 14 V a r i a t i o n of H o r i z o n t a l , V e r t i c a l and Shear S t r e s s a l o n g the 180 F t . L e v e l of Dam w i t h o u t Core 39 ( v i i i ) FIGURE TITLE Page 15 V a r i a t i o n of Dynamic Shear S t r e s s a l o n g 41 the 125 F t . L e v e l 16 V a r i a t i o n of Dynamic H o r i z o n t a l S t r e s s w i t h 43 H e i g h t a l o n g the C e n t e r - l i n e of the S l o p i n g Core 17 V a r i a t i o n of Dynamic Shear S t r e s s w i t h H e i g ht 45 alo n g C e n t e r - l i n e of C e n t r a l Core 18 V a r i a t i o n of Dynamic Shear S t r e s s w i t h Height 46 a l o n g C e n t e r - l i n e of S l o p i n g Core 19 Dynamic Shear S t r e s s i n F i r s t Ten Modes of 48 Dam w i t h o u t Core - H o r i z o n t a l Component of E l C e n t r o o n l y 20 Dynamic Shear S t r e s s i n F i r s t F i v e Modes of 50 Dam w i t h S l o p i n g Core - H o r i z o n t a l Component of E l Centro o n l y 21 Dynamic Shear S t r e s s i n F i r s t F i v e Modes of 50 Dam w i t h o u t Core - V e r t i c a l and H o r i z o n t a l A c c e l e r a t i o n used 22 Mode Shapes 1 to 4 - Dam w i t h o u t Core 70 23 Mode Shapes 5 to 8 - Dam w i t h o u t Core 71 24 . Mode Shapes 9 and 10 - Dam w i t h o u t Core 72 25 Modes 1 to 4 - Dam w i t h C e n t r a l Core 73 26 Modes 5 to 8 - Dam w i t h C e n t r a l Core 74 27 Modes 9 and 10 - Dam w i t h C e n t r a l Core 75 28 Modes 1 to 4 - Dam w i t h S l o p i n g Core 76 29 Modes 5 to 8 - Dam w i t h S l o p i n g Core 77 30 Modes 9 and 10 - Dam w i t h S l o p i n g Core 78 ( i x ) ACKNOWLEDGEMENT The h e l p and guidance of P r o f e s s o r W. D. Liam F i n n d u r i n g the r e s e a r c h program i s g r a t e f u l l y acknowledged. Thanks are a l s o due to P r o f e s s o r P e t e r M. Byrne and Mr. John J . Emery f o r a d v i c e and h e l p g i v e n . The w r i t e r i s a l s o i n d e b t e d to Dr., F i n n f o r p e r m i s s i o n to use the d e s c r i p t i o n of the f i n i t e element method p r e s e n t e d i n the appendix to t h i s t h e s i s . .1. C H A P T E R 1 INTRODUCTION .* T h i s t h e s i s p r e s e n t s the r e s u l t s of a c o m p a r a t i v e s t u d y of the b e h a v i o u r of t h r e e types of e a r t h dam when s u b j e c t e d to the ground a c c e l e r a t i o n s of the E l Centro e a r t h - quake ( C a l i f . - 1940). The dams s t u d i e d were a homogeneous dam, a c e n t r a l c o r e dam and a s l o p i n g core dam. An e a r t h dam i s a t h r e e - d i m e n s i o n a l continuum, c o n s t r u c t e d of m a t e r i a l which i s g e n e r a l l y unhomogeneous, a n i s o t r o p i c and h a v i n g n o n - l i n e a r s t r e s s - s t r a i n r e l a t i o n s . Because of the c o m p l e x i t y of the problem of a n a l y z i n g the s t r e s s e s and d e f o r m a t i o n s i n such a s t r u c t u r e , a number of s i m p l i f y i n g assumptions have to be made. The f i r s t of these i s t h a t the dam can be r e p r e s e n t e d by a c r o s s - s e c t i o n normal to the a x i s of the dam, thus r e d u c i n g the a n a l y s i s to a two- d i m e n s i o n a l p l a n e s t r a i n problem. S e c o n d l y , i t i s assumed t h a t s o i l i s l i n e a r l y e l a s t i c w i t h v i s c o u s damping. In t h i s manner the problem i s reduced to one which can be s o l v e d by u s i n g a s t a n d a r d f i n i t e element a n a l y s i s computer program. The dam s t u d i e d i n t h i s t h e s i s was 300 f e e t h i g h , s y m m e t r i c a l about the c e n t e r - l i n e , of s i d e s l o p e s 1 i n 3 and e i t h e r homogeneous or w i t h a c o r e . Two types of c o r e , c e n t r a l and s l o p i n g , were i n c l u d e d i n the s t u d y , and the c o m p a r a t i v e b e h a v i o u r , s t a t i c and dynamic, which may be of use to the e n g i n e e r i n s e l e c t i n g a -type'of c o r e , i s p r e s e n t e d i n the f o l l o w i n g c h a p t e r s . F i g u r e 1 shows the dimensions and p r o p e r t i e s of the dam and c o r e . The Young's modulus = 81,300 p . s . i . , the P o i s s o n r a t i o u 3 0.45 and the u n i t weight y = 130 p . c . f . were used so t h a t comparison w i t h p r e v i o u s work done i n t h i s f i e l d c o u l d be made; a l s o , the p r o p e r t i e s used are a s s o c i a t e d w i t h a shear wave v e l o c i t y of 1000 f . p . s . which i s t y p i c a l of the m a t e r i a l used i n the s h e l l of e a r t h dams. Two v a l u e s of e l a s t i c modulus f o r the i m p e r v i o u s c o r e m a t e r i a l were used, E c « 40,650 p . s . i . ( E c = 1 E d ) and E c = 8,130 p . s . i . ( E c E d) w h i c h , i t was f e l t , r e p r e s e n t e d the extreme v a l u e s l i k e l y to be e n c o u n t e r e d In p r a c t i c e . The h i g h e r v a l u e i s t y p i c a l of a s t i f f c l a y of h i g h s t r e n g t h and the lower of a s o f t c l a y w i t h good s e l f - h e a l i n g p r o p e r t i e s i n the case of c r a c k i n g due to e a r t h q u a k e s t r e s s e s . The problem of d e t e r m i n i n g the s t r e s s e s , s t r a i n s and d i s p l a c e m e n t s i n a s t r u c t u r e such as an e a r t h dam r e q u i r e s t h a t e q u i l i b r i u m and c o m p a t i b i l i t y be s a t i s f i e d w i t h i n the r e g i o n and t h a t the s t r e s s - s t r a i n r e l a t i o n s of the m a t e r i a l i • • be known. An a n a l y t i c a l s o l u t i o n , even w i t h the s i m p l i f y i n g a s s u m p t i o n s mentioned p r e v i o u s l y , would be v e r y complex and the f i n i t e element method of a n a l y s i s was used. FIG. I DIMENSIONS AND PROPERTIES OF T H E DAMS T h i s method of a n a l y s i s has been d e s c r i b e d f u l l y by a number of w r i t e r s , o'.g. Clough (1)* and a s h o r t d e s c r i p t i o n i s i n c l u d e d i n Appendix. I of t h i s t h e s i s . B r i e f l y , the method c o n s i s t s of approximating the s t r u c t u r e to be analyzed by an assemblage of elements and using an e x a c t mathematical a n a l y s i s of the approximation. Since the shape of the dam makes the use of t r i a n g u l a r elements .more convenient, t h i s shape has been adopted here. The p r o p e r t i e s of the m a t e r i a l used i n the dam are r e t a i n e d i n the i n d i v i d u a l elements and, s i n c e each element i s d e f i n e d s e p a r a t e l y , they may have d i f f e r e n t m a t e r i a l p r o p e r t i e s , thus making p o s s i b l e the a n a l y s i s of non-homogeneous systems. F a i r l y coarse networks can be used with good r e s u l t s , but, i n areas of high s t r e s s g r a d i e n t s , the network should be f i n e r . Computer storage l i m i t a t i o n s n e c e s s i t a t e the use of a coarse s u b d i v i s i o n except i n areas of expected high s t r e s s g r a d i e n t s . Once the behaviour of the dam under study has been determined, the s t a t i c behaviour of g e o m e t r i c a l l y s i m i l a r dams can be determined by the use of a p p r o p r i a t e s c a l e f a c t o r s . An i n c r e a s e i n s i z e w i l l cause a p r o p o r t i o n a l i n c r e a s e i n s t r e s s e s and s t r a i n s and w i l l cause an Increase * Numbers i n parentheses i n d i c a t e r e f e r e n c e number at the back of the t h e s i s . 5. i n d i splacements p r o p o r t i o n a l to the square of the s c a l e f a c t o r . S t r e s s e s i n homogeneous s t r u c t u r e s are u n a f f e c t e d by changes i n Young's modulus E, while s t r a i n s and d i s p l a c e m e n t s vary i n i n v e r s e p r o p o r t i o n to E. F i n a l l y , s t r a i n s and displacements a l l vary l i n e a r l y with d e n s i t y . Changes i n the Poisson r a t i o cannot be accounted f o r by a s c a l e f a c t o r , nor can a change i n the s i d e s l o p e s ; however, Clough and Woodward (2) have d e r i v e d e m p i r i c a l r e l a t i o n s which show how the s t r e s s e s , s t r a i n s and deformations v a r y with Poisson's r a t i o and the s i d e s l o p e s . T r a n s i t i o n zones and f i l t e r s have not been i n c l u d e d i n the dam, though, i n the case of earthquake d e s i g n , Sherard (3) advocates the use of a wide t r a n s i t i o n zone of well-graded sand and g r a v e l . The i n c l u s i o n of such zones i n the dam would present no d i f f i c u l t i e s i n the f i n i t e element a n a l y s i s . The dams were subdivided i n t o f i n i t e elements as shown i n F i g u r e 2. A f i n e r s u b d i v i s i o n was used i n the r e g i o n of the core i n order to examine the e f f e c t of the core i n m o d i f y i n g the s t r e s s p a t t e r n s . The asymmetrical sub- d i v i s i o n of the dam, which was necessary to i n c l u d e the c o r e s , caused some i n a c c u r a c i e s i n the r e s u l t s o btained. To examine the extent of these i n a c c u r a c i e s , a symmetrical s u b d i v i s i o n was made of the case of the dam without a core and the e r r o r s due to asymmetrical s u b d i v i s i o n were found to be s m a l l . 1600 1800 (FT.) 1600 1800 (FT) FIG. 2 DIVISION OF DAM INTO FINITE E L E M E N T S SHOWING NODES U S E D IN PRESENTATION OF RESULTS (A) DAM WITH CORE - CORE ( C E N T R A L OR SLOPING) SHOWN IN HEAVY OUTL INE (B) DAM WITHOUT CORE ( SYMMETR ICAL SUBDIVISION) The dam was d i v i d e d i n t o 129 elements with a t o t a l of 80 nodes. Nodes r e f e r r e d to i n the d i s c u s s i o n of r e s u l t s or i n diagrams are shown numbered i n Fig u r e 2. Since the base of the dam was f i x e d , nodes i n the base have no movement and the t o t a l number of degrees of freedom was 134, two f o r each node - h o r i z o n t a l and v e r t i c a l . G r a v i t y l o a d i n g was i n t r o d u c e d by lumping o n e - t h i r d of the weight of the .surrounding elements at each node. Dynamic l o a d i n g was a p p l i e d to the dam by s u b j e c t i n g the base to the a c c e l e r a t i o n s of the f i r s t 10 seconds of the North-South component of E l Centro. Chapter 2 presents the r e s u l t s of the s t a t i c s t r e s s a n a l y s i s . The normal s t r e s s e s i n the x - d i r e c t i o n ( h o r i z o n t a l ) and the y - d i r e c t i o n ( v e r t i c a l ) together with the shear s t r e s s i n the xy-plane were determined, along with the v e r t i c a l and h o r i z o n t a l displacements due to the s e l f - w e i g h t of the dam. These were determined f o r the three dams under study and m o d i f i c a t i o n s of the s t r e s s d i s t r i b u t i o n and deformations due to the two types of cores were noted. Chapter 3 deal s with the dynamic behaviour .of the dams. The modal f r e q u e n c i e s and the mode shapes were f i r s t determined and the e f f e c t s of the presence of the core was d i s c u s s e d . Next, the time h i s t o r i e s of the s t r e s s e s and a c c e l e r a t i o n s at s e l e c t e d nodes i n the dams were found with the purpose of determining how these were modified by the cor e . The power spectrum of the a c c e l e r a t i o n s of E l Centro 8. was compared with those of the a c c e l e r a t i o n s at the c r i s t of t h e dams to determine the f r e q u e n c i e s at which the energy of ithe earthquake was trans m i t t e d i n the dams. F i n a l l y , the response of the dams i n each mode was found and connections -between the mode shape and the response i n the mode were noted. The l a s t chapter presents the c o n c l u s i o n s drawn from.the study and suggestions f o r f u r t h e r r e s e a r c h on t h i s problem. C H A P T E R 2 STATIC STRESS ANALYSIS i 2.1 I n t r o d u c t i o n The d e t e r m i n a t i o n of s t a t i c s t r e s s e s and d i s p l a c e m e n t s , due to the s e l f - w e i g h t of the dam, i s the i n i t i a l s t e p i n the procedure of a n a l y z i n g an e a r t h dam s u b j e c t to earthquake l o a d i n g . The s t r e s s e s and deform- a t i o n s due to the earthquake are then superimposed on the s t a t i c s t r e s s e s to g i v e the s t r e s s d i s t r i b u t i o n throughout the dam a t any i n s t a n t w h i l e the earthquake a c t s . The g r a v i t y l o a d i n g i s a p p l i e d by assuming one- t h i r d of the w e i g h t of the s u r r o u n d i n g elements a c t s a t each node. In the a n a l y s i s h e r e i n i t i s assumed t h a t the m a t e r i a l of the dam i s l i n e a r l y e l a s t i c and i s o t r o p i c and the e f f e c t of s t o r e d water i s i g n o r e d . In t h i s a n a l y s i s , the g r a v i t a t i o n a l body f o r c e s are a p p l i e d d i r e c t l y on the completed s t r u c t u r e . In p r a c t i c e , the dam i s b u i l t up by a s u c c e s s i o n o f l i f t s . To check the v a l i d i t y of t h i s p r a c t i c e , Clough and Woodward (2) i n v e s t i g a t e d the e f f e c t s on the s t r e s s e s and d i s p l a c e m e n t of d i r e c t a p p l i c a t i o n of g r a v i t y l o a d i n g and compared them w i t h those due to s e q u e n t i a l l o a d i n g . The dam used was 100 f e e t h i g h and i t was assumed to be b u i l t up of 10 l a y e r s of 10 f e e t i n d e p t h . There was v e r y l i t t l e d i f f e r e n c e i n the 10. s t r e s s e s and the h o r i z o n t a l d i s p l a c e m e n t s between the two c a s e s , though the v e r t i c a l d i s p l a c e m e n t s o b t a i n e d from d i r e c t l o a d a p p l i c a t i o n were found to be i n a c c u r a t e , In the case of d i r e c t l o a d i n g , the maximum v e r t i c a l d e f o r m a t i o n was a t the top of the dam whereas, i n the s e q u e n t i a l l o a d i n g c a s e , the maximum v e r t i c a l d i s p l a c e m e n t o c c u r r e d a t the c e n t e r of the dam and was a p p r o x i m a t e l y h a l f the magnitude. I t was, t h e r e f o r e , d e c i d e d t h a t the method of d i r e c t l o a d a p p l i c a t i o n was adequate f o r t h i s i n v e s t i g a t i o n . I t was found t h a t the asymmetry of the i r r e g u l a r s u b d i v i s i o n of the dam i n t o f i n i t e elements n e c e s s i t a t e d by the s l o p i n g c o r e caused some e r r o r s , though s m a l l , i n the s t r e s s e s and d e f o r m a t i o n s o b t a i n e d . C o n s e q u e n t l y , a s y m m e t r i c a l s u b d i v i s i o n i n t o f i n i t e elements was made of the dam f o r the homogeneous case to check t h i s . F i g u r e 2 shows the d i v i s i o n i n t o f i n i t e elements of the two dams. 2.2 S t a t i c S t r e s s e s The s t r e s s d i s t r i b u t i o n , h o r i z o n t a l normal s t r e s s and shear s t r e s s f o r the homogeneous dam, t o g e t h e r w i t h t h a t of the s l o p i n g c o r e and c e n t r a l core dams, ex p r e s s e d as a pe r c e n t a g e of the s t r e s s i n the homogeneous dam, i s shown i n F i g u r e s 3 and 4. S t r e s s e s are i n pounds per square i n c h . (a) HORIZONTAL NORMAL STRESS IN PSI DAM W/O CORE j (b) CENTRAL CORE E c = ^ E d (C) CENTRAL CORE E c = ^ E ( (d) SLOPING CORE E c = ± E ( (e) SLOPING CORE E c = j ^ E d F I G . 3 H O R I Z O N T A L STATIC STRESSES IN DAM WITH CORE EXPRESSED AS A PERCENTAGE OF THE STRESSES IN THE DAM WITHOUT A CORE (TOP DIAGRAM) (a) SHEAft STRESS IN PSI FOR DAM W/O CORE (d) S LOP ING CORE E c = j E FIG. 4 . STATIC SHEAR STRESSES IN DAM WITH CORE EXPRESSED AS A PERCENTAGE OF THE STRESSES IN THE DAM WITHOUT A CORE (TOP DIAGRAM ) 13. F i g u r e 3(a) shows the h o r i z o n t a l normal s t r e s s e s f o r the homogeneous dam. The maximum s t r e s s i s ap p r o x i r a a t - e l y 175 p . s . i . a t the base of tho dam, on the c e n t e r l i n e . I n the c e n t r a l c o r e dam, f o r the more r i g i d c o r e , F i g u r e 3 ( b ) , .the changes i n s t r e s s due to the core a r e c o n f i n e d m a i n l y to the upper t h i r d of the dam and to the core i t s e l f . S t r e s s e s a r e i n c r e a s e d by a maximum of 300 per cent near the c r e s t and a r e reduced to about 90 per cent a t the base of the c o r e . S t r e s s changes a r e s m a l l i n the o t h e r p a r t s of the embank- ment. For the dam w i t h the more f l e x i b l e c e n t r a l c o r e , F i g u r e 3 ( c ) , the p a t t e r n of s t r e s s change i s s i m i l a r though the v a r i a t i o n s a r e g r e a t e r , r a n g i n g from 600 per cent a t the c r e s t to 60 per cent a t the base of the c o r e . S t r e s s e s a re d e c r e a s e d s l i g h t l y , about 5 - 10 per c e n t , i n the lower two- t h i r d s of the s h e l l s . Changes i n the h o r i z o n t a l normal s t r e s s i n the s l o p i n g c o r e dams are not as g r e a t as i n the dam w i t h a c e n t r a l c o r e . For the dam w i t h a more r i g i d s l o p i n g c o r e , F i g u r e 3 ( d ) , the s t r e s s changes are a l s o g r e a t e s t i n the upper p a r t of the dam. The s t r e s s d e c r e a s e s s l i g h t l y at the base of the c o r e and the changes are g r e a t e r i n the s h e l l on the upstream s i d e of the c o r e . As the s l o p i n g core i s •made more f l e x i b l e , F i g u r e 3 ( e ) , the s t r e s s changes i n the dam a r e g r e a t e r and r e a c h a maximum of 400 per cent towards the top of the upstream s h o u l d e r of the dam, and drop to 70 per cent a t the base of the c o r e . T h i s h i g h s t r e s s i n c r e a s e a t the upstream s h o u l d e r of the s l o p i n g c o r e dam, the p a r t of the dam most l i a b l e to f a i l u r e , makes t h i s form of c o r e l e s s d e s i r a b l e than the c e n t r a l c o r e . In the upper zone of the homogeneous dam, from e l e v a t i o n 250 f e e t to the c r e s t a t 300 f e e t , t h e r e a r e some: l o c a l i z e d a r e a s of h o r i z o n t a l t e n s i l e s t r e s s e s . These a r e a s of t e n s i o n , which c o u l d cause a f i s s u r e to open and thus i n i t i a t e a f a i l u r e s u r f a c e , do not e x i s t i n the c e n t r a l c o r e dam, p r o b a b l y due to the inward and downward movement of the s h o u l d e r as the s o f t e r core i s compressed, thus r e l i e v i n g the t e n s i l e s t r e s s e s i n the embankment. The development of h o r i z o n t a l t e n s i l e s t r e s s e s i s much more s e r i o u s i n the dam w i t h a s l o p i n g c o r e . These t e n s i o n s , shown by the shaded area i n F i g u r e 3(d) and 3 ( e ) , have a l s o been noted by F i n n and Khanna ( 5 ) . As the c o r e i s made more f l e x i b l e , the t e n s i l e s t r e s s e s i n c r e a s e and extend over a s l i g h t l y l a r g e r a r e a , r e a c h i n g t h e i r maximum e x t e n t f o r a medium-soft c o r e . F u r t h e r s o f t e n i n g of the c o r e i n c r e a s e s the t e n s i l e s t r e s s e s s l i g h t l y but causes them to become more- l o c a l i z e d , D u r i n g an e a r t h q u a k e , h o r i z o n t a l t e n s i l e s t r e s s e s w i l l be superimposed p e r i o d i c a l l y on the e x i s t i n g t e n s i o n s i n the dam, g i v i n g a v e r y u n f a v o u r a b l e s t r e s s d i s t r i b u t i o n i n the top of the s l o p i n g core dam and making i t more l i a b l e to f a i l u r e than the dam w i t h a c e n t r a l core - an i m p o r t a n t p o i n t to be c o n s i d e r e d i n s e l e c t i n g a type 'of- core, f o x . an' ear t h dam. The s o f t e s t core m a t e r i a l , w i t h i t s s e l f - h e a l i n g ; I " - • " ' . ' . . • • " ' • ' ' ' • p r o p e r t i e s i n the event of c r a c k i n g , and the more l o c a l i z e d a r e a of t e n s i l e s t r e s s e s , would be the b e t t e r c h o i c e , i f o t h e r c o n s i d e r a t i o n s make i t n e c e s s a r y to use a s l o p i n g c o r e dam i n an a r e a s u b j e c t to e a r t h q u a k e s . The v e r t i c a l s t r e s s e s show l i t t l e v a r i a t i o n f o r the d i f f e r e n t dams shown h e r e , b e i n g m a i n l y dependent on the w e i g h t of the o v e r l y i n g m a t e r i a l , and are not i n v e s t i g a t e d f u r t h e r . The s t a t i c shear s t r e s s d i s t r i b u t i o n i n the xy- p l a n e f o r the homogeneous dam i s shown i n F i g u r e 4 ( a ) . The -maximum shear s t r e s s of a p p r o x i m a t e l y 30 p . s . i . o c c u r s on the base of the dam, midway between the c e n t e r - l i n e and the t o e . The e f f e c t of a c e n t r a l c o r e i s to i n c r e a s e the shear s t r e s s e s , F i g u r e 4 ( b ) , the minimum i n c r e a s e b e i n g a t the toe of the dam and becoming p r o g r e s s i v e l y l a r g e r toward the c e n t e r , where the shear s t r e s s e s a r e s i x times as g r e a t as i n the homogeneous case. However, s i n c e the s t r e s s e s i n the homogeneous dam are a t a minimum near the c e n t e r , t h i s l a r g e i n c r e a s e has l i t t l e e f f e c t on the l e v e l of the shear s t r e s s e s t h e r e . For the more f l e x i b l e c e n t r a l c o r e , F i g u r e 4 ( c ) , the shear s t r e s s e s a r e a l s o unchanged a t the t o e . About one- t h i r d of the way i n from the t o e , the s t r e s s e s d e c r e a s e to about 75 per cent and i n c r e a s e a g a i n towards the c e n t e r where the i n c r e a s e i s h i g h e r than w i t h the s t i f f e r c o r e . For the s l o p i n g core dam, the p a t t e r n of s t r e s s change i s more c o m p l i c a t e d . In the more r i g i d c o r e , F i g u r e 4 ( d ) , t h e r e i s a g e n e r a l d e c rease i n shear s t r e s s t h r o u g h o u t the c o r e , f a l l i n g to about 10 per cent of the o r i g i n a l s t r e s s towards the t o p . S t r e s s e s i n the upstream s i d e of the s h e l l a r e i n c r e a s e d s l i g h t l y ; i n the downstream s i d e they a r e g e n e r a l l y lower than i n the homogeneous dam. The s t r e s s change p a t t e r n i s s i m i l a r f o r the case of the s o f t c o r e ( F i g u r e 4 ( e ) ) , though the i n c r e a s e i n shear s t r e s s i n the upstream s h e l l i s h i g h e r , up to 400 per cent g r e a t e r than the homogeneous cas e , and the s h e a r ^ s t r e s s i n the top of the c o r e a l s o i n c r e a s e s . Thus, though the p e r c e n t a g e i n c r e a s e i n shear s t r e s s due to the presence of a core i s h i g h e r f o r the c e n t r a l c o r e dam, these l a r g e p e r c e n t a g e i n c r e a s e s occur i n area s of v e r y low shear s t r e s s and the r e s u l t i n g shear s t r e s s e s a r e not h i g h . The p e r c e n t a g e i n c r e a s e i n the s l o p i n g c o r e dam, though not as g r e a t , o c c u r s i n the upstream s h o u l d e r , a r e g i o n of h i g h shear s t r e s s i n the homogeneous dam, and a l s o the most l i k e l y r e g i o n f o r the i n i t i a t i o n of a f a i l u r e s u r f a c e . T h i s i s e s p e c i a l l y so s i n c e the h o r i z o n t a l normal s t r e s s d i s t r i b u t i o n i s a l s o most u n f a v o u r a b l e i n t h i s r e g i o n , w i t h t e n s i l e s t r e s s e s o c c u r r i n g toward the top of the s h e l l . Hence, i t would seem t h a t , from s t a t i c s t r e s s c o n s i d e r a t i o n s , the c e n t r a l c o r e dam i s the s a f e r t y p e . 2.3 S t a t i c D i s p l a c e m e n t s A s t u d y of the d i s p l a c e m e n t s i n the t h r e e dams • i shows t h a t b o t h the h o r i z o n t a l and v e r t i c a l d i s p l a c e m e n t s a r e g e n e r a l l y g r e a t e r i n the dam w i t h a c o r e . The h o r i z o n t a l d i s p l a c e m e n t s i n the s l o p i n g core dam are v e r y s i m i l a r to th o s e of the homogeneous dam i n the area between the down- s t r e a m f a c e and the edge of the c o r e . The d i f f e r e n c e i s g r e a t e s t a l o n g the c e n t e r l i n e of the core and d e c r e a s e s towards the upstream f a c e of the dam. Maximum h o r i z o n t a l d i s p l a c e m e n t f o r the homogeneous dam i s 0.97 f e e t , o c c u r r i n g a t the 150 f o o t l e v e l c l o s e to the f a c e of the s h e l l . The maximum f o r the s l o p i n g core dam i s 1.1 f e e t a t the same p o i n t i n the dam. The d i f f e r e n c e s i n the v e r t i c a l d i s p l a c e - ments f o l l o w a s i m i l a r p a t t e r n . On the downstream s i d e of the c o r e , d i f f e r e n c e s a re n e g l i g i b l e . Maximum d i f f e r e n c e s o c c u r a t the upper edge of the core and i n c r e a s e w i t h h e i g h t i n the dam. Maximum v e r t i c a l d i s p l a c e m e n t f o r the homogen- eous dam i s 2.3 f e e t , and 2.7 f e e t f o r the s l o p i n g c o r e dam. The h o r i z o n t a l d i s p l a c e m e n t s i n the core of the c e n t r a l c o r e dam a r e l e s s than a t c o r r e s p o n d i n g p o i n t s i n the homogeneous dam. At a s h o r t d i s t a n c e from the c o r e , the d i s p l a c e m e n t s a r e s i m i l a r and, towards the f a c e of the dam, a r e g r e a t e r , though the d i f f e r e n c e decreases and i s s m a l l a t the f a c e of the dam. Maximum d i s p l a c e m e n t i s 1 f o o t , o c c u r r i n g a t the same p o i n t as i n the homogeneous dam. The v e r t i c a l d i s p l a c e m e n t s i n the core and the s u r r o u n d i n g s h e l l 18. a r e g r e a t e r than at c o r r e s p o n d i n g p o i n t s i n the homogeneous dam, though i n the r e s t of the s h e l l they are s i m i l a r . Maximum v e r t i c a l d i s p l a c e m e n t i s 3 f e e t at the c r e s t . Though the c e n t r a l c o r e dam has s m a l l e r h o r i z o n t a l d i s p l a c e m e n t s than the s l o p i n g c o r e dam, i t i s s u b j e c t to g r e a t e r v e r t i c a l d i s p l a c e m e n t s . N e i t h e r type has any c l e a r advantage where s t a t i c d e f o r m a t i o n i s concerned. C H A P T E R 3 j DYNAMIC RESPONSE ANALYSIS i s i 3.1 I n t r o d u c t i o n In the f i r s t i n v e s t i g a t i o n s of the dynamic re s p o n s e of e a r t h dams by Monobe ( 6 ) , Hatanaka ( 7 ) , Ambraseys (8) and o t h e r s , the dam was r e p r e s e n t e d by a wedge- shaped v e r t i c a l shear beam. T h i s method i s r e s t r i c t e d to homogeneous, s y m m e t r i c a l c r o s s - s e c t i o n s and a l l o w s o n l y one- d i m e n s i o n a l d i s p l a c e m e n t s to o c c u r . I s h i z a k i and Hatakeyama (9) have shown t h a t the v e r t i c a l shear beam approach i s a c c u r a t e o n l y a t the c e n t e r - l i n e of the dam. E r r o r s o ccur everywhere e l s e and r e a c h a maximum a t the f a c e . S i n c e the performance of a dam d u r i n g an earthquake w i l l be determined l a r g e l y by the s t r e s s c o n d i t i o n s a t the f a c e , i t can be seen t h a t the shear beam a n a l y s i s may l e a d to s i g n i f i c a n t e r r o r s i n d e s i g n . The problem was next extended to a l l o w two- d i m e n s i o n a l d i s p l a c e m e n t s to o c c u r , by I s h i z a k i and Hatakeyama ( 9 ) , u s i n g the f i n i t e d i f f e r e n c e method. D i f f i c u l t i e s are enc o u n t e r e d i n u s i n g t h i s method when a non-homogeneous dam i s s t u d i e d , though t h i s p r e s e n t s no problem i n the f i n i t e element method used here. The f i n i t e element method was f i r s t a p p l i e d to the s e i s m i c a n a l y s i s of e a r t h dams by Clough and Chopra (10) i n 1966. Th i s method takes i n t o account non- homogeneity and s i m p l e cases of a n i s o t r o p y , but i s l i m i t e d a t p r e s e n t to l i n e a r e l a s t i c i t y f o r dynamic a n a l y s i s . I- •' ' :- V ' . -•' • - •'• The d i s t u r b i n g f o r c e a p p l i e d to the dam was the f i r s t 10 seconds of the North-South component of the E l Cent r o e a r t h q u a k e , as shown i n F i g u r e 5 ( a ) . The a c c e l e r - a t i o n s of the earthquake were s c a l e d to g i v e a maximum v a l u e of 0.28g so t h a t comparisons c o u l d be made w i t h p r e v i o u s work done a t the U n i v e r s i t y of B r i t i s h Columbia by F i n n (4) and F i n n and Khanna ( 5 ) . The v e r t i c a l component of E l C e n t r o , w hich i s not used i n t h i s a n a l y s i s , i s shown i n F i g u r e 5 ( b ) . A c o m p a r a t i v e l y h i g h v a l u e of damping of twenty per cent of c r i t i c a l i n each mode was used to take i n t o account the l a r g e a b s o r p t i o n of energy due to i n e l a s t i c d e f o r m a t i o n . 3.2 Mode Shapes and F r e q u e n c i e s The dynamic response of a s t r u c t u r e comprised of v i s c o e l a s t i c m a t e r i a l to earthquake v i b r a t i o n can be a n a l y z e d as the summation of the responses i n a number of i n d i v i d u a l modes. Thus, where a dam i s d i s t u r b e d from e q u i l i b r i u m by an e a r t h q u a k e , i t can be c o n s i d e r e d as v i b r a t i n g i n i t s d i f f e r e n t modes s i m u l t a n e o u s l y and the a c t u a l v i b r a t i o n may be o b t a i n e d by s u p e r p o s i t i o n of the responses i n the i n d i v i d u a l modes. The mode shapes and f r e q u e n c i e s depend on the geometry and e l a s t i c p r o p e r t i e s of the m a t e r i a l and not on the d i s t u r b i n g f o r c e . Thus the f i r s t s t e p i n the dynamic  a n a l y s i s i s the d e t e r m i n a t i o n of these mode shapes and f r e q u e n c i e s . The f i r s t ten mode shapes ( s c a l e e x a g g e r a t e d ) f o r ea c h dam a r e shown i n Appendix I I , F i g u r e s 22 to 30, and the n a t u r a l p e r i o d s f o r each a re g i v e n . For the dams w i t h c o r e s , the n a t u r a l p e r i o d f o r the case of the more r i g i d c o r e i s g i v e n f i r s t , f o l l o w e d by t h a t of the s o f t e r c o r e . The mode -shapes of the homogeneous dam, F i g u r e s 22 to 24, are of two b a s i c t y p e s , symmetric about the c e n t e r - l i n e , and asymmetric. The symmetric modes, which c r e a t e symmetric s t r e s s d i s t r i b u - t i o n s , w i l l n o t be e x c i t e d by h o r i z o n t a l * ground motion and t h i s can be seen when the response of the dam i s determined i n each mode i n d i v i d u a l l y . S i m i l a r l y , the asymmetric modes, whi c h cause asymmetric s t r e s s d i s t r i b u t i o n s , w i l l not be e x c i t e d by v e r t i c a l ground a c c e l e r a t i o n s . T h i s type of u n c o u p l i n g e x i s t s f o r s y m m e t r i c a l c r o s s - s e c t i o n s o n l y and i t can be seen from F i g u r e s 25 to 30 t h a t the mode shapes f o r the dams w i t h c o r e s a r e a l l asymmetric. In g e n e r a l , any mode w i l l be e x c i t e d by h o r i z o n t a l and v e r t i c a l ground a c c e l e r a t i o n s . I t i s i n t e r e s t i n g to n o t e , i n comparison w i t h the she a r wedge method which accounts o n l y f o r shear s t r e s s e s , t h a t o n l y the f i r s t mode r e p r e s e n t s * p u r e shear d i s t o r t i o n . Chopra (11) has shown t h a t the d i f f e r e n c e i n response between the c e n t e r - l i n e of the dam and the f a c e i s g r e a t e r f o r f l a t t e r s l o p e s , the r e s u l t s a p p r o a c h i n g the shear wedge s o l u t i o n as th e s l o p e s a r e made s t e e p e r . T h i s d i f f e r e n c e i s a maximum a t about o n e - t h i r d the h e i g h t of the dam. The mode shapes and modal f r e q u e n c i e s do not depend on the s i d e s l o p e s of the dam when the shear wedge method i s used, though they a r e v e r y dependent on them i n the f i n i t e element a n a l y s i s . The mode shape diagrams r e p r e s e n t the d e f l e c t e d shape of the dam i n t h a t mode and can be used as maximum d e f l e c t i o n diagrams by u s i n g the a p p r o p r i a t e s c a l e . The modal p e r i o d i n c r e a s e s as the core i s made more f l e x i b l e , the d i f f e r e n c e between the more r i g i d and the more f l e x i b l e c o r e b e i n g about ten per c e n t . The fundamental p e r i o d i s l o w e s t f o r the homogeneous dam, i n c r e a s e s f o r the s i p p i n g c o r e dam and i s h i g h e s t f o r the c e n t r a l c o r e dam. The more r i g i d the dam, the lower the fundamental p e r i o d w i l l be. 3.3 Power S p e c t r a l D e n s i t y E s t i m a t e s An earthquake causes a c c e l e r a t i o n s a t the base of a s t r u c t u r e i n d u c i n g i n e r t i a f o r c e s and consequent s t r e s s e s and s t r a i n s w i t h i n the s t r u c t u r e . These base a c c e l e r a t i o n s , though of a random n a t u r e , can be c o n s i d e r e d as the super- p o s i t i o n of a number of s i n u s o i d a l v i b r a t i o n s . A power s p e c t r a l d e n s i t y a n a l y s i s decomposes the a c c e l e r a t i o n s i n t o t h e i r b a s i s f r e q u e n c i e s and i n t h i s way a measure of the fr e q u e n c y a t which most of the energy i s b e i n g t r a n s m i t t e d can be o b t a i n e d . 24. S t u d i e s of damage caused by earthquakes to e a r t h s t r u c t u r e s c a r r i e d out by Ambraseys (8) i n d i c a t e t h a t most damage o c c u r s when the fundamental f r e q u e n c y of the s t r u c t u r e c o r r e s p o n d s to the f r e q u e n c y at which most of the energy of the e a r t h q u a k e i s b e i n g t r a n s m i t t e d . Thus, the s t r u c t u r e w i l l r e s o n a t e when s u b j e c t e d to c e r t a i n ground m o t i o n s . Power s p e c t r a l d e n s i t y a n a l y s i s on a number of s t r o n g motion e a r t h q u a k e s i n d i c a t e t h a t most of the energy of the e a r t h q u a k e s i s t r a n s m i t t e d i n a f r e q u e n c y range of 0.25 to 5 c y c l e s per second. T h e r e f o r e , s t r u c t u r e s w i t h a fu n d a m e n t a l f r e q u e n c y g r e a t e r than about 10 c y c l e s per second can be e x p e c t e d to respond i n a r i g i d body manner and a dynamic a n a l y s i s i s not r e q u i r e d . S t r u c t u r e s w i t h a fu n d a m e n t a l f r e q u e n c y i n a range of a p p r o x i m a t e l y 0.25 to 5 c y c l e s per second can be expected to have a l a r g e response to s t r o n g motion e a r t h q u a k e s and a dynamic a n a l y s i s i s n e c e s s a r y . The power s p e c t r a l d e n s i t y (P.S.D.) a n a l y s i s of the h o r i z o n t a l component of E l Centro earthquake and of the a c c e l e r a t i o n s a t the c r e s t s of the t h r e e dams i s shown i n F i g u r e 6. From F i g u r e 6(a) i t can be seen t h a t E l Centro i s a h i g h f r e q u e n c y e a r t h q u a k e , the energy t r a n s m i t t e d b e i n g c o n c e n t r a t e d a t a f r e q u e n c y of 2 c y c l e s per second ( c . p . s . ) . S i n c e the fundamental f r e q u e n c y of the dam i s a p p r o x i m a t e l y 1 c.p.s. f o r a l l c a s e s , i t can be i n f e r r e d t h a t the response to E l C e n t r o w i l l be l a r g e . F i g u r e s 6 ( b ) , (c) and (d) show the P.S.D. of the c r e s t a c c e l e r a t i o n s of the homogeneous dam, 25. 0 I 2 3 4 5 0 1 2 3 4 5 N A T U R A L FREQUENCY (C . P.S.) FIG. 6 COMPARISON OF RS.D. OF E L CENTRO WITH RS.D. OF ACCELERATION OF C R E S T OF DAM. c e n t r a l c o r e dam and s l o p i n g core dam r e s p e c t i v e l y . There i s a peak a t 1 c.p.s. i n each c a s e , the fundamental f r e q u e n c y ^of the dam, showing t h a t most of the response of the dam i s i n : the f i r s t mode. This was c o n f i r m e d , as w i l l be seen l a t e r , when an a n a l y s i s of the response i n each s e p a r a t e mode was per f o r m e d . The secondary peak a t 2 c.p.s. c o r r e s p o n d s q u i t e c l o s e l y to the n a t u r a l f r e q u e n c y of the t h i r d mode f o r each dam. 3.4 P r e l i m i n a r y I n v e s t i g a t i o n of Dynamic Response • An earthquake w i l l a p p l y both h o r i z o n t a l and v e r t i c a l a c c e l e r a t i o n s at the base of a s t r u c t u r e . The dynamic a n a l y s i s computer program used i n the a n a l y s i s c o u l d a p p l y e i t h e r a h o r i z o n t a l base a c c e l e r a t i o n a l o n e , or v e r t i c a l and h o r i z o n t a l base a c c e l e r a t i o n s w i t h the v e r t i c a l a c c e l e r a t i o n e q u a l to the h o r i z o n t a l . The program c o u l d have been m o d i f i e d to i n c o r p o r a t e an independent v e r t i c a l a c c e l e r a t i o n , but t h i s was not done. A check on the e f f e c t of v e r t i c a l base a c c e l e r a t i o n was performed by comparing r e s u l t s u s i n g : 1) a h o r i z o n t a l base a c c e l e r a t i o n o n l y 2) the same h o r i z o n t a l and v e r t i c a l base a c c e l e r a t i o n s . The comparison i s shown i n F i g u r e 7 where the h o r i z o n t a l normal s t r e s s , the shear s t r e s s I n the x-y p l a n e and the v e r t i c a l normal s t r e s s were determined f o r a p o i n t near the 27. Ft6. 7 DIFFERENCE IN DYNAMIC STRESSES DUE TO USE OF E ITHER HORIZONTAL ACCELERATION ONLY OR EQUAL VERTICAL AND HORIZONTAL ACCELERATION ( node 3 2 - dam without core ) 28. c e n t e r - l i n e a t a p p r o x i m a t e l y t w o - t h i r d s the h e i g h t of the dam. With the e x c e p t i o n of the peak at 2.2 seconds, the h o r i z o n t a l normal s t r e s s was oi : l y s l i g h t l y a f f e c t e d , the she a r s t r e s s was a f f e c t e d even l e s s , but the change i n the v e r t i c a l normal s t r e s s was i n excess of 100 per cent at the peak v a l u e s . The response of the normal s t r e s s e s under v e r t i c a l and h o r i z o n t a l a c c e l e r a t i o n i s out of phase w i t h that under h o r i z o n t a l a c c e l e r a t i o n o n l y . The h o r i z o n t a l n o r m a l s t r e s s under combined a c c e l e r a t i o n components l e a d s t h a t under h o r i z o n t a l a c c e l e r a t i o n o n l y , w h i l e the v e r t i c a l n oraml s t r e s s e s l a g . T h i s method of a p p l y i n g v e r t i c a l m o t i o n to the dam can g i v e only an approximate i n d i c a t i o n of how the re s p o n s e w i l l be a f f e c t e d by the a c t u a l v e r t i c a l component of E l C e n t r o . I t can be seen from F i g u r e 5(b) t h a t the v e r t i c a l component of E l Centro has a much h i g h e r f r e q u e n c y of v i b r a t i o n than the a p p r o x i m a t i o n used, though the a c t u a l v a l u e of the maximum a c c e l e r a t i o n i s s i m i l a r . S i n c e the n a t u r a l f r e q u e n c i e s of the modes most a f f e c t e d by v e r t i c a l m o t i o n , i . e . modes 2 and A, are a p p r o x i m a t e l y e q u a l to the f r e q u e n c y a t which the h o r i z o n t a l component of E l Cen t r o t r a n s m i t s most of i t s energy, a resonance e f f e c t i s to be e x p e c t e d when a v e r t i c a l component of a c c e l e r a t i o n e q u a l to the h o r i z o n t a l component of E l Centro i s used. Hence i t was f e l t t h a t the a c t u a l h i g h f r e q u e n c y , v e r t i c a l component would have l e s s e f f e c t on the v e r t i c a l s t r e s s than shown i n F i g u r e 7. Thus, i t was f e l t t h a t the use of the h o r i z o n t a l component o n l y of E l Centro g i v e s r i s e to o n l y s l i g h t e r r o r i n the r e s u l t s . I d r i s s and Seed (15) have a l s o shown t h a t t h e v e r t i c a l component of earthquake a c c e l e r a t i o n has l i t t l e e f f e c t on the h o r i z o n t a l component of response of the dam or on the shear s t r e s s , but t h a t i t may i n f l u e n c e the v e r t i c a l r e s p o n s e c o n s i d e r a b l e . _ From F i g u r e 7, i t can be seen t h a t the peak i n the h o r i z o n t a l s t r e s s a t 2.2 seconds has been t r u n c a t e d f o r the case of no v e r t i c a l a c c e l e r a t i o n component. T h i s i s because, due to time l i m i t a t i o n s on the computer, p r i n t - o u t of r e s u l t s was a t 0.2 second i n t e r v a l s , though the response was c a l c u l a t e d a t 0.01 second i n t e r v a l s . To examine the e x t e n t of the e r r o r due to t h i s l i m i t a t i o n , the response a t a p o i n t near the c r e s t of the dam was p r i n t e d out a t the normal 0.2 second i n t e r v a l and a t a 0.05 second i n t e r v a l . The r e s u l t i s shown i n F i g u r e 8 and i t can be seen t h a t the e r r o r i n the response can be l a r g e . However, s i n c e t h i s s tudy i s a c o m p a r a t i v e one, i f the e r r o r s i n v o l v e d i n the p l o t t e d r e s p o n s e of each dam are s i m i l a r , the c o n c l u s i o n s s h o u l d be v a l i d . ^ B e f o r e the i n v e s t i g a t i o n c o u l d proceed f u r t h e r , i t was n e c e s s a r y to d e c i d e on the number of modes which s h o u l d be i n c l u d e d i n the i n t e g r a t i o n to o b t a i n the t o t a l response of the dam. F i g u r e 9 shows a comparison of the h o r i z o n t a l normal s t r e s s a t a p o i n t i n the embankment u s i n g 5. modes and 15 modes i n the i n t e g r a t i o n . From t h i s r e s u l t , i t was 30. FIG. 8 DIFFERENCE IN RESPONSE DUE TO USE OF PRINTOUT INTER VALS OF 0.05 SECONDS AND 0.20 SECONDS 2 0 w C L Ui Ui tc V- V) < I— I 15 10 5 - 5 tc 2 - i o -15 - 2 0 — 1 / A ' </\ j w . node 52 A 4 i no core (sym) \ 1 1 \V // _ 5 modes 5 " I 2 A 3 seconds FIG. 9 D I F F E R E N C E IN RESPONSE BETWEEN USE OF 5 MODES AND B MODES , d e c i d e d t h a t , f o r economy of computer ti m e , the use of the f i r s t 5 modes was s u f f i c i e n t . 3a 5 A c c e l e r a t i o n s i n Dam The dynamic f o r c e induced i n an element of a s t r u c t u r e d u r i n g an earthquake i s the p r o d u c t of the mass times the a b s o l u t e a c c e l e r a t i o n . I f the a c c e l e r a t i o n h i s t o r y of the s t r u c t u r e d u r i n g the earthquake can be o b t a i n e d , then the v a r i a t i o n of dynamic f o r c e s w i t h time can be d e t e r m i n e d . Thus, the s t r u c t u r e w i l l be s u b j e c t e d to a t r a n s i e n t f o r c e system i n a d d i t i o n to s t a t i c f o r c e s . With known dynamic f o r c e s , Seed (12) has o u t l i n e d a p rocedure f o r the d e s i g n of an e a r t h dam. Hence, the f i r s t s t e p i n t h i s type of a n a l y s i s i s the d e t e r m i n a t i o n of a b s o l u t e a c c e l e r a t i o n s . The f i n i t e element method used i n the dynamic a n a l y s i s assumes t h a t the m a t e r i a l behaves i n a v i s c o e l a s t i c manner. I t i s r e a l i z e d t h a t p l a s t i c d e f o r m a t i o n s w i l l occur d u r i n g s t r o n g motion earthquakes and to account f o r t h i s a r e l a t i v e l y h i g h v a l u e of the p e r c e n t a g e of c r i t i c a l damping was used . Seismographs p l a c e d i n dams show t h a t , d u r i n g e a r t h q u a k e s , the a c c e l e r a t i o n I n c r e a s e s w i t h h e i g h t (Ambraseys ( 8 ) ) and i s i n good agreement w i t h v i s c o e l a s t i c a n a l y s i s . Thus, i t would appear t h a t a v i s c o e l a s t i c r esponse a n a l y s i s p r o v i d e s a r e a s o n a b l e method f o r a s s e s s i n g the dynamic f o r c e s induced i n an embankment by an e a r t h q u a k e . Th i s i n c r e a s e w i t h h e i g h t i s demonstrated i n F i g u r e 10 which compares the h o r i z o n t a l a c c e l e r a t i o n , i n terms of g, of E l C e n t r o , a t the base of the dam, w i t h t h a t of the c r e s t . The a c c e l e r a t i o n , peaks of the c r e s t are n e a r l y double those of the base, though they are g e n e r a l l y out of phase w i t h them. F i e l d t e s t s on dams a p p r o x i m a t e l y 100 f e e t h i g h , u s i n g f o r c e d v i b r a t i o n by l a r g e machines (Seed (13)) showed t h a t peak a c c e l e r a t i o n s are developed a t c e r t a i n c h a r a c t e r - r i s t i c f r e q u e n c i e s , those of the f r e e v i b r a t i o n modes (resonance e f f e c t s ) , and a l s o t h a t the response was i n agreement w i t h t h a t p r e d i c t e d by v i s c o e l a s t i c a n a l y s i s . The response of the dam under a d i f f e r e n t e a r t h - quake i s shown i n F i g u r e 11 where the c r e s t a c c e l e r a t i o n s are compared f o r the E l Centro and Alameda Park (Mexico) e a r t h - quakes. (Both earthquakes were s c a l e d to g i v e a maximum a c c e l e r a t i o n of 0.28g.) Alameda Park earthquake i s a lower f r e q u e n c y earthquake than E l Centro w i t h a p e r i o d of two seconds compared to E l C e n t r o ' s h a l f second. S i n c e the fundamental p e r i o d of the dam i s one second, resonance e f f e c t s s h o u l d be s i m i l a r to those when E l C e n t r o a c t s , though peak a c c e l e r a t i o n s w i l l occur at d i f f e r e n t t i m e s . F i g u r e 11 shows t h a t the maximum a c c e l e r a t i o n s are s i m i l a r f o r both e a r t h q u a k e s , though the f r e q u e n c y of the response when the Alameda Park earthquake a c t s i s s l i g h t l y l o w e r , and the maximum a c c e l e r a t i o n s are reached l a t e r . 33. SECS. FIG. 10 COMPARISON OF ABSOLUTE HORIZONTAL ACCELERATION OF C R E S T OF DAM WITHOUT A CORE WITH HORIZONTAL COMPONENT OF ACCELERAT ION OF E L CENTRO O rr UJ _j ui o o < < r- Z o N oe o X SECS FIG. II COMPARISON OF ABSOLUTE HORIZONTAL ACCELERATION OF CREST OF DAM WITHOUT A CORE WHEN SUBJECTED TO EL C E N T R O AND ALAMEDA PARK EARTHQUAKES 34. The v a r i a t i o n of response w i t h h e i g h t w i t h i n the dam i s shown i n F i g u r e 12, where the a c c e l e r a t i o n s a t v a r i o u s h e i g h t s a l o n g the c e n t e r - l i n e of the homogeneous dam a r e p l o t t e d . The response a t node 43 at the base of the dam i s the h o r i z o n t a l component of E l C e n t r o , s i n c e the f o u n d a t i o n i s r i g i d , and t h a t a t node 50 i s the response a t the c r e s t , as shown i n F i g u r e 10. The a c c e l e r a t i o n , i n the e a r l y s t a g e s of v i b r a t i o n , d e c r e a s e s to a minimum a t a h e i g h t of approx- i m a t e l y 50 f e e t above the base and then i n c r e a s e s w i t h h e i g h t , r e a c h i n g a maximum at the c r e s t . However, a f t e r the f i r s t two s econds, t h e r e i s a g r a d u a l i n c r e a s e i n response from base to c r e s t . The o s c i l l a t i o n s a t the v a r i o u s h e i g h t s are g e n e r a l l y out of phase and, a t t i m e s , a c t i n o p p o s i t e d i r e c t i o n s . The e f f e c t of the c e n t r a l core i s to d e c r e a s e a c c e l e r a t i o n s a t p o i n t s w i t h i n the core by 10 per cent f o r the more r i g i d c o r e and by 30 per cent f o r the more f l e x i b l e c o r e . In the s l o p i n g c o r e , the approximate average decrease i n a c c e l e r a t i o n i s 5 per cent f o r the more r i g i d core and 15 per cent f o r the more f l e x i b l e c o r e . A c c e l e r a t i o n s at p o i n t s o u t s i d e the c o r e s are a f f e c t e d o n l y v e r y s l i g h t l y . The u s u a l c u r r e n t p r a c t i c e i n the s e i s m i c d e s i g n of e a r t h dams i s to compute the f a c t o r of s a f e t y a l o n g a p o t e n t i a l f a i l u r e s u r f a c e when a s t a t i c h o r i z o n t a l f o r c e , a c t i n g on the s l i d i n g b l o c k , i s i n c l u d e d i n the a n a l y s i s . The problem i s t r e a t e d as a s t a t i c a n a l y s i s and the h o r i z o n t a l  36. f o r c e i s e x p r e s s e d as the p r o d u c t of the weight of the s l i d i n g mass and a c o e f f i c i e n t c a l l e d the s e i s m i c c o e f f i c i e n t . The d e t e r m i n a t i o n of the s e i s m i c c o e f f i c i e n t i s not i n c l u d e d In t h i s s t u d y ; however, a method of d e t e r m i n i n g the r e q u i r e d h o r i z o n t a l f o r c e , from the r e s u l t s of the f i n i t e element a n a l y s i s , i s g i v e n by Chopra ( 1 1 ) . The l i m i t a t i o n s of t h i s p s e u d o - s t a t i c approach are d i s c u s s e d by Seed and M a r t i n (14) and a method i s proposed where the s e i s m i c c o e f f i c i e n t v a r i e s w i t h time and w i t h h e i g h t w i t h i n the dam. The i n c r e a s e i n response w i t h e l e v a t i o n i n the dam shows t h a t the s e i s m i c c o e f f i c i e n t s h o u l d be i n c r e a s e d w i t h e l e v a t i o n . Codes do not r e c o g n i z e the n o n - r i g i d n a t u r e of e a r t h dams s u b j e c t to s e i s m i c l o a d i n g and s p e c i f y a c o n s t a n t s e i s m i c c o e f f i c i e n t , w i t h the e x c e p t i o n of the R u s s i a n code, r e f e r r e d to by Ambraseys ( 8 ) , where the s e i s m i c c o e f f i c i e n t v a r i e s w i t h h e i g h t and depends on the damping p r o p e r t i e s of the embankment m a t e r i a l and the s e i s m i c i n t e n s i t y of the r e g i o n . The maximum a c c e l e r a t i o n i s developed i n a dam f o r o n l y a s h o r t p e r i o d of t i m e , hence the r e s u l t i n g d e f o r m a t i o n s may be s m a l l and, though o t h e r d e f o r m a t i o n s w i l l occur from o t h e r peaks d u r i n g the e a r t h q u a k e , i t does not seem r e a s o n a b l e to assume t h a t the t o t a l d e f o r m a t i o n s w i l l be as g r e a t as i f a s t a t i c i n e r t i a f o r c e c o r r e s p o n d i n g to the maximum a c c e l e r a t i o n were a p p l i e d f o r an u n l i m i t e d time. 37. 3.6 Dynamic S t r e s s Response j F i g u r e s 13 and 14 show the v a r i a t i o n i n dynamic s t r e s s e s a l o n g a h o r i z o n t a l p l a n e a t e l e v a t i o n s 60 f e e t and 180 f e e t r e s p e c t i v e l y i n the homogeneous dam. S i n c e the s t r u c t u r e i s s y m m e t r i c a l , the h o r i z o n t a l and v e r t i c a l normal dynamic s t r e s s e s a re zero a t the c e n t e r - l i n e of the dam and the embankment m a t e r i a l i s i n a s t a t e of pure s h e a r . The h o r i z o n t a l normal s t r e s s reaches a maximum a p p r o x i m a t e l y midway between the c e n t e r - l i n e and the embankment f a c e a t the 60 f o o t e l e v a t i o n and then f a l l s o f f s l i g h t l y towards the f a c e . I t a l s o i n c r e a s e s w i t h e l e v a t i o n , the peak v a l u e a t the 180 f o o t l e v e l b e i n g a p p r o x i m a t e l y 100 per cent g r e a t e r than t h a t a t the 60 f o o t l e v e l . At the 180 f o o t e l e v a t i o n , h o r i z o n t a l n ormal s t r e s s i s a maximum at the f a c e . The response i s out of phase a t the v a r i o u s p o i n t s between the c e n t e r - l i n e and the f a c e . The v e r t i c a l normal s t r e s s e s are i n c l u d e d f o r purpose of comparison o n l y as i t has a l r e a d y been shown t h a t the use of j u s t the h o r i z o n t a l component of E l Centro makes thes e v a l u e s q u e s t i o n a b l e . The v a r i a t i o n i n the v e r t i c a l n ormal s t r e s s f o l l o w s the same p a t t e r n as t h a t of the h o r i z o n t a l p l a n e , but the t r e n d i s r e v e r s e d w i t h e l e v a t i o n , the v e r t i c a l s t r e s s e s d e c r e a s i n g w i t h h e i g h t . At the 60 f o o t l e v e l s the v a r i a t i o n of shear s t r e s s a l o n g a h o r i z o n t a l p l a n e i s c o n s i d e r a b l e , d e c r e a s i n g from a maximum a t the c e n t e r - l i n e to a p p r o x i m a t e l y 30 per cent of t h a t FIG. 13 VARIATION OF HORIZONTAL,VERTICAL AND SHEAR STRESS ALONG THE 60 FT. LEVEL OF DAM WITHOUT CORE 39. FIG. 14 V A R I A T I O N O F H O R I Z O N T A L , V E R T I C A L A N D S H E A R S T R E S S A L O N G T H E 180 FT. L E V E L O F D A M W I T H O U T C O R E v a l u e a t the f a c e . In the f h e a r wedge s o l u t i o n , the shear s t r e s s i s c o n s t a n t from the c e r i t e r ^ l i n e to the f a c e . The s h e a r s t r e s s i s s l i g h t l y lower a t the 180 f o o t l e v e l and the v a r i a t i o n between the c e n t e r - l i n e of the dam and the f a c e i s n o t as g r e a t . The v a r i a t i o n of the dynamic shear s t r e s s a l o n g a h o r i z o n t a l p l a n e a t e l e v a t i o n 125 f e e t i n the t h r e e dams i s shown i n F i g u r e 15. In a l l t h r e e c a s e s , the shear s t r e s s i n c r e a s e s f rom the f a c e of the dam to the c e n t e r - l i n e (node 6 on f a c e of dam, node 46 on c e n t e r - l i n e of dam). Of p a r t i c u l a r i n t e r e s t , however, i s t h a t the shear s t r e s s e s are o n l y a f f e c t e d t o any degree by the presence of a core a t p o i n t s w i t h i n the c o r e . At o t h e r p o i n t s i n the embankment, the shear s t r e s s e s a r e v i r t u a l l y unchanged. W i t h i n the more f l e x i b l e m a t e r i a l o f the c o r e , the shear s t r e s s e s are l o w e r , b e i n g a p p r o x i m a t e l y t w o - t h i r d s o f t h e . s t r e s s e s . a t . c o r r e s p o n d i n g . p o i n t s i n the homogeneous dam. The d a t a p r e s e n t e d i n F i g u r e 15 i s f o r the more r i g i d c o r e s w i t h e l a s t i c modulus e q u a l to o n e - h a l f t h a t of the embankment m a t e r i a l . The r e s u l t s f o r the more f l e x i b l e c o r e s , w i t h e l a s t i c modulus o n e - t e n t h t h a t of the embankment m a t e r i a l , i s s i m i l a r though the e f f e c t on the shear s t r e s s e s a t p o i n t s i n the embankment near the core w i l l be s l i g h t l y g r e a t e r and the shear s t r e s s e s i n the core w i l l be s m a l l e r . I t can be seen from F i g u r e s 13 to 15 t h a t the shear s t r e s s e s a l o n g a h o r i z o n t a l plane i n the dam are i n phase, w h i l e the v e r t i c a l and h o r i z o n t a l s t r e s s e s are n o t . T h i s i s 41. FIG. 15 VARIATION OF DYNAMIC SHEAR STRESS ALONG THE 125' LEVEL (See Figure 2(A) For Location of Nodes) because the response of the dam i s m a i n l y i n the f i r s t mode, p r e d o m i n a n t l y a shear mode, w i t h l i t t l e d i f f e r e n t i a l movement betwe en the f a c e of the embankment and the c e n t e r — l i n e . F i g u r e 16 p r e s e n t s the h o r i z o n t a l dynamic s t r e s s a t v a r i o u s h e i g h t s a l o n g the c e n t e r of the s l o p i n g c o r e . The s t r e s s i s seen to d e c r ease w i t h e l e v a t i o n a l o n g the s l o p i n g c o r e , though F i g u r e s 13 and 14 show an i n c r e a s e w i t h e l e v a t i o n i n the dam. The r e a son f o r the d e c r ease shown i n F i g u r e 16 i s t h a t p o i n t s h i g h e r i n the c o r e are a l s o c l o s e r to the c e n t e r - l i n e of the dam where h o r i z o n t a l dynamic s t r e s s e s are z e r o and t h i s d e c r e a s e w i t h p r o x i m i t y to the c e n t e r - l i n e more than compensates f o r the i n c r e a s e w i t h e l e v a t i o n i n the dam. From the base to h a l f the h e i g h t of the dam, the h o r i z o n t a l s t r e s s i n the c o r e i s g r e a t e r f o r the s o f t e r c o r e , j u s t s l i g h t l y below the s t r e s s a t c o r r e s p o n d i n g p o i n t s i n the homogeneous dam. The s t r e s s f o r the s o f t e r core i s a l s o o p p o s i t e i n s i g n to the o t h e r cases up to t h i s l e v e l . From h a l f the h e i g h t of the dam to the c r e s t , the s t r e s s i s a maximum f o r the s o f t e r c o r e , f o l l o w e d by the homogeneous dam and then the more r i g i d c o r e . Comparing the response bf the c e n t r a l and s l o p i n g c o r e dams, the h o r i z o n t a l dynamic s t r e s s i n the c o r e i s v e r y low f o r the c e n t r a l c o r e dam. The h i g h h o r i z o n t a l s t r e s s i n the s l o p i n g c o r e dam w i l l i nduce h i g h e r pore water p r e s s u r e s i n the impermeable core m a t e r i a l d u r i n g the earthquake and, 43. 0 2 4 seconds 6 8 1 0 FIG. 16 VARIATION OF DYNAMIC HORIZONTAL STRESS WITH HEIGHT ALONG THE CENTER-LINE OF THE SLOPING CORE hence ;, c r e a t e a l e s s d e s i r a b l e c o n d i t i o n . The e f f e c t of the core on the dynamic shear s t r e s s a t p o i n t s w i t h i n the core i s shown i n F i g u r e s 17 and 18 where the shear s t r e s s i n the core of the dam i s compared w i t h t h a t a t c o r r e s p o n d i n g p o i n t s i n the homogeneous dam. F i g u r e 17 shows the v a r i a t i o n of the dynamic shear s t r e s s w i t h h e i g h t .along the c e n t e r - l i n e of the c e n t r a l c o r e dam, f o r both s o f t and r i g i d c o r e s , compared w i t h t h a t of the homogeneous dam. I t can be seen, f o r the lower p a r t of the dam, t h a t the shear s t r e s s e s i n d u c e d by the earthquake i n the homogeneous dam a r e n e a r l y t w i c e as h i g h as i n the dam w i t h the r i g i d core and about f i v e times as h i g h as i n the dam w i t h the s o f t c o r e . Higher up i n the dam, the d i f f e r e n c e s are l e s s marked u n t i l , i n the upper f i f t h of the dam, the shear s t r e s s e s are s l i g h t l y g r e a t e r i n the c o r e than i n the homogeneous dam. Shear s t r e s s e s a t p o i n t s o u t s i d e the c o r e are a f f e c t e d v e r y l i t t l e by the presence of the c o r e . I t i s i n t e r e s t i n g to note t h a t , w h i l e the shear s t r e s s d e c r e a s e s w i t h h e i g h t a l o n g the c e n t e r - l i n e of the homogeneous dam, i t i s n e a r l y c o n s t a n t a l o n g the h e i g h t of the c e n t r a l c o r e , d e c r e a s i n g s l i g h t l y near the top f o r the r i g i d c o r e and a c t u a l l y i n c r e a s i n g w i t h h e i g h t a l o n g the c e n t e r - l i n e of the f l e x i b l e c o r e . The v a r i a t i o n of dynamic shear s t r e s s w i t h h e i g h t a l o n g the c e n t e r - l i n e of the core i n the s l o p i n g c o r e dam i s shown i n F i g u r e 18. The shear s t r e s s i s compared a t p o i n t s i n the dam w i t h a r i g i d c o r e , the dam w i t h a f l e x i b l e c o r e , 45. w X L v> 111 oc r- tc < LU I • 2 < z >• o -10 h - 2 0 h 0 2 FIG. 17 VARIATION OF DYNAMIC SHEAR STRESS WITH HEIGHT ALONG C E N T E R - L I N E OF C E N T R A L CORE 46. 10 I r — I 1—— -r V 0 2 4 seconds 6 8 FIG. 18 VARIATION OF DYNAMIC SHEAR STRESS WITH HEIGHT ALONG CENTER-LINE OF SLOPING CORE 47. and a t c o r r e s p o n d i n g p o i n t s i n the homogeneous dam. The shear s t r e s s i s h i g h e s t f o r the homogeneous dam, a p p r o x i m a t e l y t w i c e as h i g h as f o r the r i g i d core dam, and f i v e times as h i g h as f o r the dam w i t h the more f l e x i b l e c o r e . U n l i k e the c e n t r a l core dam, t h i s r a t i o i s m a i n t a i n e d throughout the l e n g t h of the c o r e , the shear s t r e s s d e c r e a s i n g w i t h h e i g h t . A common p o i n t to both c o r e s , node 49, o c c u r s 20 f e e t below the c r e s t of the dam. From F i g u r e s 17 and 18, i t i s seen from the response a t t h i s common p o i n t and from the response a t nodes 40 and 48, which are a t the same e l e v a t i o n , t h a t the shear s t r e s s i s c o n s i d e r a b l y h i g h e r i n the upper p a r t of the c e n t r a l core and i s g e n e r a l l y s l i g h t l y h i g h e r a t a l l l e v e l s than i n the s l o p i n g c o r e . 3.7 Responses by Modes When the response of the homogeneous dam i s computed f o r each mode i n d i v i d u a l l y , i t Is found t h a t modes 2, 4, 7, 8 and 10 c o n t r i b u t e n o t h i n g to the t o t a l r e s p o n s e . On r e f e r r i n g to F i g u r e s 22, 23 and 24 i n Appendix I I , i t i s seen t h a t these modes are s y m m e t r i c a l about the c e n t r a l . a x i s - they are v e r t i c a l motion modes and are not e x c i t e d by h o r i z - o n t a l ground m o t i o n . S i n c e o n l y the h o r i z o n t a l component of E l C entro was used, t h e r e i s no response i n these modes. F i g u r e 19 shows the shear s t r e s s i n each of the f i r s t ten modes f o r t h r e e p o i n t s a t the same e l e v a t i o n i n the homogeneous dam, F i g u r e 2 ( b ) . The response g e n e r a l l y d e c r e a s e d as mode 48. 15 -is i i , i m i , i 0 1 2 3 4 5 seconds FIG. 19 DYNAMIC SHEAR STRESS IN FIRST TEN MODES OF DAM WITH- OUT CORE - HORIZONTAL COMPONENT OF EL CENTRO ONLY 49. number i n c r e a s e d and the resyonse i n mode 9 was too s m a l l to p l o t . The f i r s t mode, which i s a shear mode, c o n t r i b u t e s the major p a r t to the shear s t r e s s . The next l a r g e s t c o n t r i b u t o r t o the shear s t r e s s i s not mode 3, as would be e x p e c t e d , but mode 5. The reason f o r t h i s may be seen from an e x a m i n a t i o n of the mode shapes i n F i g u r e s 22 and 23 i n Appendix I I . Mode 5 i s seen to resemble f a i r l y c l o s e l y a shear mode, w i t h bending motion a t a minimum, w h i l e mode 3 i s p r e d o m i n a n t l y a bending mode, hence the shear s t r e s s i n mode 5 w i l l be g r e a t e r . The f r e q u e n c y of mode 5 i s c l o s e to t h a t o f E l C e n t r o and hence a resonance e f f e c t i s p r o b a b l y i n v o l v e d . The f a c t t h a t mode 1, a shear mode, c o n t r i b u t e s t h e major p a r t to the response of the dam e x p l a i n s why the shear wedge a n a l y s i s can be used e f f e c t i v e l y to determine the s e i s m i c r e sponse of the e a r t h dam. The shear s t r e s s c o n t r i b u t e d by the i n d i v i d u a l modes to the t o t a l shear s t r e s s i n the dam w i t h a s l o p i n g c o r e I s p r e s e n t e d i n F i g u r e 20. The f i g u r e shows the shear s t r e s s e s i n the f i r s t 5 modes a t a p o i n t i n the s l o p i n g core a t h a l f the h e i g h t of the dam. As w i t h the homogeneous dam, the major p a r t of the response i s i n the f i r s t mode. Due to the a s y m m e t r i c a l n a t u r e of the dam, t h e r e are no s y m m e t r i c a l mode shapes and, c o n s e q u e n t l y , each mode c o n t r i b u t e s to the res p o n s e of the dam. As b e f o r e , the f i f t h mode c o n t r i b u t e s the. second l a r g e s t share to the r e s p o n s e , but the d i f f e r e n c e between modes 3 and 5 i s not as pronounced as i n the homogeneous dam. The f i r s t mode c o n t r i b u t e s a p r o p o r t i o n a l l y g r e a t e r 50. 15 seconds FIG. 20 DYNAMIC SHEAR STRESS IN FIRST FIVE MODES OF DAM WITH SLOPING CORE - HORIZONTAL COMPONENT OF EL CENTRO ONLY 15 -20 I 1 1 : -1 — — 1 — » 0 I 2 3 4 5 seconds FIG. 21 DYNAMIC SHEAR STRESS IN FIRST FIVE MODES OF DAM WITHOUT CORE - VERTICAL AND HORIZONTAL ACCELERATION USED 51. s h are of the response than i n the homogeneous dam and the e r r o r would be s m a l l i f t h i s mode al o n e were used to determine i • | ithe shear s t r e s s . In F i g u r e 21, the response i n the f i r s t 5 modes a t a p o i n t i n the homogeneous dam i s p r e s e n t e d assuming t h a t a v e r t i c a l component e q u a l to the h o r i z o n t a l a c c e l e r a t i o n component of the earthquake i s a c t i n g . When compared w i t h the same p o i n t i n F i g u r e 19, the response due to the h o r i z - o n t a l component of the earthquake o n l y , i t i s seen t h a t modes which a re r e p r e s e n t e d i n F i g u r e 19 are unchanged h e r e . The e f f e c t of the v e r t i c a l component of earthquake a c c e l e r a t i o n i s to e x c i t e response i n the s y m m e t r i c a l modes o n l y , l e a v i n g the o t h e r modes unchanged. - 3.8 Modal P a r t i c i p a t i o n F a c t o r The response i n each mode can a l s o be a n a l y z e d i n terms of the modal p a r t i c i p a t i o n f a c t o r (M.P.F.). The M.P.F. f o r any mode g i v e s a measure of the p r o p o r t i o n of the t o t a l r e s p o n s e o c c u r r i n g i n t h a t mode. I t depends o n l y on the mode shape and d i r e c t i o n of the earthquake base a c c e l e r a - t i o n . For h o r i z o n t a l base a c c e l e r a t i o n a l o n e , o n l y mode shapes w i t h h o r i z o n t a l d i s p l a c e m e n t s c o n t r i b u t e to the r e s p o n s e , w h i l e , f o r v e r t i c a l base a c c e l e r a t i o n a l o n e , o n l y mode shapes w i t h v e r t i c a l d i s p l a c e m e n t s c o n t r i b u t e . The a p p l i c a b l e M.P.F. f o r each case i s g i v e n by e q u a t i o n s 12 and 13 of Appendix I . For both h o r i z o n t a l and v e r t i c a l base 52. a c c e l e r a t i o n s a l l modes w i l l c o n t r i b u t e to the re s p o n s e s and the M.P.F. i s g i v e n by e q u a t i o n 14 of Appendix I . S i n c e M.P.F. fs depend o n l y on the mode shapes and d i r e c t i o n of the earthquake base a c c e l e r a t i o n , they can be determined p r i o r to i n t e g r a t i n g the e q u a t i o n s of motion and, hence, the c o n t r i - b u t i o n of each mode i s p r e d e t e r m i n e d . The M.P.F. f o r each type of dam, f o r modes 1 to 5, f o r h o r i z o n t a l earthquake a c c e l e r a t i o n o n l y and f o r combined h o r i z o n t a l and v e r t i c a l a c c e l e r a t i o n , i s shown i n Table 1. The M.P.F. f o r modes 2 and 4 of the s y m m e t r i c a l homogeneous dam, modes i n which t h e r e was no r e s p o n s e , as shown i n F i g u r e 19, i s z e r o . When a v e r t i c a l a c c e l e r a t i o n component i s i n c l u d e d i n the e a r t h q u a k e , the M.P.F. f o r t h e s e two modes i s now n o n - z e r o , the M.P.F. f o r the o t h e r modes remains unchanged. T h i s u n c o u p l i n g a p p l i e s o n l y to the s y m m e t r i c a l dam, though the c e n t r a l c o r e dam i s l e s s a f f e c t e d than the more asymmetric s l o p i n g c o r e dam. In the response of the more f l e x i b l e c o r e dams, the i n t r o d u c t i o n of a v e r t i c a l a c c e l e r a t i o n component d e c r e a s e s the response i n some modes, n o t a b l y mode 5 of the c e n t r a l c o r e dam. E x a m i n a t i o n of F i g u r e s 19, 20 and 21 c o n f i r m s t h a t the modes w i t h the h i g h e s t M.P.F. c o n t r i b u t e most to the response of the dam. Table 1 a l s o shows t h a t , f o r the dams w i t h the more f l e x i b l e c o r e s , the response c o n t r i b u t e d by each mode i s q u i t e d i f f e r e n t than f o r the more r i g i d c o r e . Mode 4 i s the second l a r g e s t c o n t r i b u t o r to the response of the f l e x i b l e core dams, a mode which c o n t r i b u t e s n o t h i n g to the response of the homogeneous dam, though mode 1 T A B L E 1 MODAL PARTICIPATION FACTOR MODE HOMOGENEOUS DAM CENTRAL C ORE DAM SLOPING CORE DAM R i g i d Core F l e x i b l e Core R i g i d Core F l e x i b l e Core H o r i z o n t a l Component o n l y H o r i z o n t a l & V e r t i c a l 11 H & V H H & V H H & V H H & V 1 1.54 1.54 -1.54 -1.54 -1.65 -1.63 1.58 1.56 1.61 1.56 2 0 0.45 -0.02 -0.63 0.06 0.91 0.05 0.58 -0.29 -0.88 3 -0.33 -0.33 0.44 0.45 0.60 1.44 -0.36 -0.40 0.42 0.10 4 0 -0.002 0.20 0. 20 0.95 0.61 -0.10 -0.06 -0.66 -1.00 5 1.00 1.00 0.8 9 0.81 0.22 -0.01 -0.97 -1.09 0.25 0.86 54. remains the l a r g e s t c o n t r i b u t o r i n every case. The e f f e c t of i n c l u d i n g a v e r t i c a l a c c e l e r a t i o n component i n the modal r e s p o n s e d i s t r i b u t i o n i s more pronounced i n the case of the - s o f t e r c o r e . Thus, i t i s shown t h a t the M.P.F. i s v e r y u s e f u l i n d e t e r m i n i n g the modes t h a t are the major c o n t r i b u t o r s to the r e s p o n s e b e f o r e the dynamic a n a l y s i s i s run and, hence, an a c c u r a t e e s t i m a t e of the number of modes needed f o r the r e q u i r e d a c c u r a c y i n the i n t e g r a t i o n can be made. 3.9 A p p l i c a t i o n of R e s u l t s The f i n i t e element method y i e l d s the s t a t e of s t r e s s and s t r a i n i n the dam f o r s t a t i c l o a d i n g cases and, the s t r e s s and a b s o l u t e a c c e l e r a t i o n h i s t o r y of the dam f o r dynamic l o a d i n g cases such as e a r t h q u a k e s . However, due to the h i g h v a l u e of damping used In each mode to take i n t o a c c o u n t the i n e l a s t i c d e f o r m a t i o n , the a c t u a l d i s p l a c e m e n t and s t r a i n h i s t o r y of the r e a l dam i s not known from such a v i s c o e l a s t i c f i n i t e element a n a l y s i s . To be of use to the d e s i g n e r , i t must be p o s s i b l e to r e l a t e the a v a i l a b l e r e s u l t s to the a c t u a l performance of the m a t e r i a l used i n the dam and p r e d i c t the u l t i m a t e d i s p l a c e m e n t s of the dam due to the e a r t h q u a k e . I t i s n e c e s s a r y to r e c o g n i z e the importance of b a s i n g d e s i g n on d i s p l a c e m e n t s produced, r a t h e r than on f a c t o r of s a f e t y , s i n c e a f a c t o r of s a f e t y of l e s s than u n i t y can e x i s t f o r a s h o r t p e r i o d of time w i t h o u t e x c e s s i v e d i s p l a c e m e n t s b e f o r e the s t r e s s c y c l e i s r e v e r s e d (Newmark ( 1 6 ) ) . S i n c e the v i s c o e l a s t i c f i n i t e element a n a l y s i s does not y i e l d an adequate d i s p l a c e m e n t h i s t o r y , assessment of the a c t u a l d e f o r m a t i o n of the dam w i l l r e q u i r e the use of e i t h e r the s t r e s s or a b s o l u t e a c c e l e r a t i o n h i s t o r y . F i n n (4) s u g g e s t s u s i n g c y c l i c l o a d i n g t e s t s on r e p r e s e n t a t i v e samples to d e t e r m i n e s t r a i n s w i t h i n the dam. F i r s t , the s t a t i c f i n i t e element a n a l y s i s i s used to determine the a n i s o t r o p i c c o n s o l i d a t i o n s t r e s s e s to be used, and then the c y c l i c s t r e s s h i s t o r y i s taken from the v i s c o e l a s t i c f i n i t e element s o l u t i o n f o r the dynamic l o a d i n g . From the s t r a i n s o b t a i n e d i n such c y c l i c t e s t s , the d e f o r m a t i o n of the dam can be o b t a i n e d . On the o t h e r hand, Seed (12) uses the a b s o l u t e a c c e l e r a t i o n h i s t o r y to d e t e r m i n e the i n e r t i a f o r c e s to i n t r o d u c e i n t o a s l i p - c i r c l e a n a l y s i s to d etermine the s t a t e of s t r e s s on the p o t e n t i a l f a i l u r e s u r f a c e . These s t r e s s e s are then used i n c y c l i c l o a d i n g t e s t s and the s t r a i n s o b t a i n e d . 56. C H A P T E R 4 CONCLUSIONS i That a case f o r the i m t r o v e < * s e i s m i c d e s i g n of e a r t h dams i s e v i d e n t i s shown by Ambraseys (17) who s t a t e s t h a t i n an i n v e s t i g a t i o n of dams, l e v e e s and embankments s u b j e c t e d to s t r o n g e a r t h q u a k e s , most dams were s e v e r e l y damaged w h i l e a l l l e v e e s and embankments were d e s t r o y e d . In r e c e n t y e a r s , t h e r e has been no f a i l u r e of a major e a r t h dam due to e a r t h q u a k e a c t i o n , but t h i s i s p r o b a b l y because no l a r g e dam has been s u b j e c t e d to a major earthquake i n t h a t t i m e . The f o l l o w i n g c o n c l u s i o n s are drawn from the r e s u l t s of the i n v e s t i g a t i o n p r e s e n t e d i n the p r e c e d i n g c h a p t e r s : 1. The h i g h e r dynamic h o r i z o n t a l s t r e s s e s induced i n the upper p a r t of the s l o p i n g c o r e , c o u p l e d w i t h the e x i s t i n g s t a t i c t e n s i l e s t r e s s e s i n t h i s r e g i o n , combine to make t h i s type of c o r e l e s s d e s i r a b l e f o r e a r t h dams i n areas s u b j e c t t o s e i s m i c a c t i v i t y . The dynamic h o r i z o n t a l s t r e s s e s induced i n the upper c o r e by the earthquake i n c r e a s e as the core i s made more f l e x i b l e . ' 2. There a r e no t e n s i l e s t a t i c s t r e s s e s i n the c e n t r a l core dam and the dynamic h o r i z o n t a l s t r e s s e s i n the core are v e r y 57. low. However, the dynamic shear s t r e s s e s a re c o n s i d e r a b l y h i g h e r i n the c e n t r a l c o r e than i n the s l o p i n g c o r e , p a r t i c u l a r l y i n the upper l e v e l s of the c o r e . 3. The dynamic shear s t r e s s i s lower i n the c o r e , both c e n t r a l and s l o p i n g , than a t c o r r e s p o n d i n g p o i n t s i n the homogeneous dam. I t i s a f f e c t e d o n l y to a v e r y s m a l l degree a t p o i n t s o u t s i d e the c o r e , depending on the f l e x i b i l i t y of the c o r e . 4. The v a r i a t i o n of the a c c e l e r a t i o n s w i t h h e i g h t i n the dam i n d i c a t e s t h a t , i n the s e i s m i c c o e f f i c i e n t method of d e s i g n , a c o e f f i c i e n t w hich i n c r e a s e s w i t h e l e v a t i o n s h o u l d be used. 5. The a c c e l e r a t i o n s are h i g h e r i n the s l o p i n g core dam and, hence, a h i g h e r s e i s m i c c o e f f i c i e n t would be r e q u i r e d than f o r the c e n t r a l c o r e dam. 6. The f i n i t e element method i s seen to be s e n s i t i v e to i r r e g u l a r i t i e s i n the s u b d i v i s i o n of the dam i n t o f i n i t e e l e m ents. A s y m m e t r i c a l s u b d i v i s i o n s h o u l d be used when p o s s i b l e . t 7. In a comparison between the f i n i t e element method and the shear wedge a n a l y s i s , i t i s seen t h a t o n l y the f i r s t mode appr o x i m a t e s to a shear mode. A s t a t e of pure shear e x i s t s o n l y a t the c e n t e r - l i n e of the dam, and the shear s t r e s s i s 58. not u n i f o r m on a h o r i z o n t a l plane through the dam, an as s u m p t i o n made by the shear wedge approach. i 8. The v e r t i c a l earthquake a c c e l e r a t i o n m o d i f i e s o n l y the v e r t i c a l dynamic s t r e s s and has a i . a g l i g i b l e e f f e c t on the dynamic shear s t r e s s and h o r i z o n t a l dynamic s t r e s s . 9. The Modal P a r t i c i p a t i o n F a c t o r , which can be e a s i l y d e t e r m i n e d b e f o r e the dynamic a n a l y s i s i s r u n , i s shown to be a v e r y u s e f u l guide i n s e l e c t i n g the number of modes which s h o u l d be used to a c h i e v e the d e s i r e d degree of a c c u r a c y . 10. From an a p p r a i s a l of the f i n d i n g s of t h i s i n v e s t i g a t i o n , i t i s c o n c l u d e d t h a t the use of a c e n t r a l type core i n areas of e a r t h q u a k e a c t i v i t y would l e a d to s a f e r s e i s m i c d e s i g n of e a r t h dams. F u r t h e r r e s e a r c h on t h i s t o p i c i s n e c e s s a r y . The e f f e c t of v a r y i n g s i d e s l o p e s , h e i g h t of dam, core p l a c i n g and c o r e d i m e n s i o n s , as w e l l as the response of the dam when u n d e r l a i n by v a r i o u s types of f o u n d a t i o n s , s h o u l d be s t u d i e d . The dynamic r e s p o n s e of the dam under earthquakes of the h i g h e s t and l o w e s t f r e q u e n c i e s to be expected s h o u l d be i n v e s t i g a t e d , w i t h p a r t i c u l a r emphasis on resonance e f f e c t s . 59. BIBLIOGRAPHY d o u g h , R. W. , ( 1 9 6 5 ) , " S t r e s s A n a l y s i s " , E d i t o r s O.C, Z i e n k i e w i c z and G . S . H o l l i s t e r , Ch.7 pp.85-119, W i l e y , New Y o r k , 1965. C l o u g h , R. W., and Woodward, R. J . , ( 1 9 6 7 ) , " A n a l y s i s o f Embankment S t r e s s e s and D e f o r m a t i o n s " . J o u r n a l of the S o i l M e c h a n i c s and F o u n d a t i o n D i v i s i o n , ASCE, SM4, J u l y , 1967 S h e r a r d , J . L., ( 1 9 6 7 ) , " E a r t h q u a k e C o n s i d e r a t i o n s i n E a r t h Dam D e s i g n " . J o u r n a l of the S o i l M e c h a n i c s and F o u n d a t i o n D i v . , ASCE, SM4, 1967 . F i n n , W. D. L i a m , ( 1 9 6 6 ) , " S t a t i c and S e i s m i c B e h a v i o u r of an E a r t h Dam". S o i l M e c h a n i c s S e r i e s No. 5, U n i v e r s i t y o f B r i t i s h C o l u m b i a . F i n n , W. D. L i a m , and Khanna, J . , ( 1 9 6 6 ) , "Dynamic Response of E a r t h Dams". P r o c e e d i n g s , 3rd E a r t h q u a k e Symposium, U n i v e r s i t y of Ro o r k e e , I n d i a . Mdnonobe, N. , T a k a t a , A., and Matumura, M., ( 1 9 3 6 ) , " S e i s m i c S t a b i l i t y of the E a r t h Dam". i / v . j . . , . . . j . v- b J w w..— - ~ * «- ~ " • T r a n s a c t i o n s , v o l . 4, 2nd C o n g r e s s on L a r g e Dams, W a s h i n g t o n , D . C , 1936. H a t a n a k a , M., ( 1 9 5 5 ) , " F u n d a m e n t a l C o n s i d e r a t i o n s on the E a r t h q u a k e R e s i s t a n t P r o p e r t i e s o f the E a r t h Dam". B u l l e t i n No. 11, D i s a s t e r P r e v e n t i o n R e s e a r c h i n s t i t u t e , K y o t o U n i v e r s i t y , K y o t o , J a p a n , December, 1965. Ambraseys, N. N., ( I 9 6 0 ) , "The S e i s m i c S t a b i l i t y of E a r t h Dams". P r o c e e d i n g s , v o l 2j 2nd World C o n f e r e n c e on E a r t h q u a k e E n g i n e e r i n g , J a p a n , 1 9 60. 60. 9 I s h i z a k i , H., and Hatakeyama, N., (1962), " C o n s i d e r a t i o n s on the V i b r a t i o n a l Behaviour of E a r t h Dams". B u l l e t i n No. 52, D i s a s t e r P r e v e n t i o n Research I n s t i t u t e , Kyoto U n i v . , K y o t o , Japan, Feb., 1962. 10 C l o u g h , R. W. , and Chopra, A n i l K., (1966), "Earthquake S t r e s s A n a l y s i s i n E a r t h Dams". J o u r n a l , E n g i n e e r i n g Mechanics D i v . , ASCE, EM2, A p r i l , 1966. 11 Chopra, A n i l K. , (1967), "Earthquake Response of E a r t h Dams". J o u r n a l , S o i l Mechanics and Found D i v . , ASCE, SM2, March, 1967. 12 Seed, H. B. , (1966), "A Method of 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 , S o i l Mechanics and Found a t i o n s D i v . , ASCE, SMI, J a n u a r y , 1966. 13 Seed, H. B., (1967), "Slope S t a b i l i t y D u r i n g E a r t h q u a k e s " . J o u r n a l , S o i l Mechanics and F o u n d a t i o n s D i v . , ASCE, SM4, J u l y , 1967. 14 Seed, H. B., and M a r t i n , G., (1966), "The S e i s m i c C o e f f e c i e n t i n E a r t h Dam D e s i g n " . J o u r n a l S o i l Mechanics and Fo u n d a t i o n s D i v . , ASCE, SM3, May, 1966. 15 I d r i s s , I . M., and Seed, H. B., (1967), "Response of E a r t h Banks D u r i n g E a r t h q u a k e s " . J o u r n a l , S o i l Mechanics and Fo u n d a t i o n s D i v . , ASCE, SM3, May, 1967. 16 Newmark, N. M., (1965), " E f f e c t s of Earthquakes on Dams and Embankments". Geot e c h n i q u e , Vol.XV, No. 2, June, 1965. 17 Ambraseys, N. N., (1960), "On the S e i s m i c Behaviour of E a r t h Dams". P r o c e e d i n g s , V o l . 1, 2nd World Conf. on Earthquake E n g i n e e r i n g , J a p a n , 1 9 60. 61. 18 Faddeeva, V. N . , (1959), " C o m p u t a t i o n a l Methods of L i n e a r A l g e b r a " , (the Square Root Method p.31), Dover P u b l i c a t i o n s , New York, 1959. 19 W i l s o n , E. L., (1963), " F i n i t e Element A n a l y s i s of Two- D i m e n s i o n a l S t r u c t u r e s " . Report No. 63-2, 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 , 1963. 20 W i l s o n , E. L. and Clough, R. W., (1962), "Dynamic Response by Step-by-Step M a t r i x A n a l y s i s " . Symposium on the Use of Computers i n C i v i l - E n g i n e e r i n g , P o r t u g a l , O c t o b e r , 1962. 62. i A P P E N D I X I 6 3 . DESCRIPTION OF FINITE ELEMENT METHOD j ' The f i n i t e element method of s t r e s s a n a l y s i s i s a p o w e r f u l e x t e n s i o n of m a t r i x s t r u c t u r a l a n a l y s i s p r o c e d u r e s f o r o b t a i n i n g d i g i t a l computer s o l u t i o n s to problems of the continuum. A g e n e r a l d e s c r i p t i o n of the method has been g i v e n by Clough ( 1 ) . In p r a c t i c e , the continuum may be comprized of non-homogeneous, a n i s o t r o p i c , and n o n - e l a s t i c m a t e r i a l s . Non-homogeneity and s i m p l e forms of a n i s o t r o p y i n t r o d u c e no d i f f i c u l t y i n t o the f i n i t e element method of a n a l y s i s ; n o n - e l a s t i c i t y i s approximated by m u l t i - l i n e a r e l a s t i c i t y f o r s t a t i c a n a l y s i s . However, dynamic a n a l y s i s i s s t i l l r e s t r i c t e d to l i n e a r e l a s t i c b e h a v i o u r . The a c t u a l continuum i s i d e a l i z e d as an assemblage of d i s c r e t e elements or segments connected a t the nodes. Any shape of element may be used p r o v i d e d a s t i f f n e s s m a t r i x , g i v i n g the r e l a t i o n s h i p between the n o d a l f o r c e s and n o d a l d i s p l a c e m e n t s , i s a v a i l a b l e f o r the element. R e c t a n g u l a r and t r i a n g u l a r elements are commonly used. The b o u n d a r i e s of a dam a r e most e a s i l y f o l l o w e d by t r i a n g u l a r elements which are used* h e r e i n . G e n e r a l l y , the more r e f i n e d the d i s c r e t i z a t i o n of the r e g i o n the more a c c u r a t e the r e s u l t s . Because of l i m i t a t i o n s of computer s t o r a g e and problems of m a i n t a i n i n g 64. a c c u r a c y i n the s o l u t i o n of l a r g e s e t s of e q u a t i o n s , i n p r a c t i c e , the f i n e n e s s of s u b d i v i s i o n v a r i e s w i t h a n t i c i p a t e d s t r e s s g r a d i e n t s . In r e g i o n s of h i g h s t r e s s g r a d i e n t a r e l a t i v e l y f i n e r s u b d i v i s i o n i s used; i n r e g i o n s of low s t r e s s g r a d i e n t s , a c o a r s e r s u b d i v i s i o n . If {F} are the n o d a l f o r c e s on each element and {r} the n o d a l d i s p l a c e m e n t s then the element s t i f f n e s s m a t r i x , [ k ] , i s d e f i n e d by the e q u a t i o n {F} = [ k ] { r > (1) The element s t i f f n e s s m a t r i x used h e r e i n i s determined on the a s s u m p t i o n of a l i n e a r v a r i a t i o n of d i s p l a c e m e n t s over the element. T h i s assumption ensures c o m p a t i b i l i t y of d i s p l a c e m e n t s a l o n g the edges of c o n t i g u o u s elements. The s t i f f n e s s m a t r i x [ K ] , f o r the e n t i r e s t r u c t u r e i s o b t a i n e d by s u p e r i m p o s i n g the a p p r o p r i a t e s t i f f n e s s c o e f f i c i e n t s of the i n d i v i d u a l elements s u r r o u n d i n g each node. The n o d a l f o r c e - d e f o r m a t i o n r e l a t i o n s are then g i v e n by IR} = [K]{r> (2) i n w h i c h {R} i s the m a t r i x of the n o d a l f o r c e s . Nodal f o r c e s due to . g r a v i t y a r e o b t a i n e d by lumping o n e - t h i r d of each element w e i g h t a t the nodes of the elements. For d i s t r i b u t e d a p p l i e d l o a d s , such as s u r f a c e l o a d s , s t a t i c a l l y e q u i v a l e n t c o n c e n t r a t e d l o a d s are a p p l i e d a t the a p p r o p r i a t e nodes. 65. The matrix [K] i s a symmetric band matrix and, f o r the order of equations used h e r e i n , equations (2) are most c o n v e n i e n t l y solved by the Cholesky method (Faddeeva (18)). For systems of very high order an i t e r a t i v e method such as that d e s c r i b e d by Wilson (19) i s d e s i r a b l e to reduce round- o f f e r r o r s . The s t r e s s e s , o, i n the elements are obtained by {o} = [S]{r> (3) i n which [ S ] , the s t r e s s t r a n s f o r m a t i o n matrix, i s determined by the assumed displacement p a t t e r n and the m a t e r i a l p r o p e r t i e s . Nodal s t r e s s e s are obtained by averaging the s t r e s s e s i n elements around each node. This procedure l o s e s accuracy at the boundaries and f o r more accurate r e s u l t s the e x t r a p o l a t i o n procedure suggested by Wilson (19) may be used. SEISMIC ANALYSIS The seismic behaviour of a dam subjected to a base a c c e l e r a t i o n , a ( t ) , may be s t u d i e d by c o n s i d e r i n g the base to be at r e s t and the dam to be acted upon by i n e r t i a f o r c e s , R ^ ( t ) , given by R f ( t ) = - M ± a ( t ) (4) The mass i s obtained by lumping at node i o n e - t h i r d of the masses of a l l elements surrounding node i . L e t t i n g the displacement of node i be r^ the equation of motion f o r node i 66. becomes M.r, C i * i K i r i -• R t ( t ) (5) i n w hich i s the v i s c o u s damping, the a p p r o p r i a t e s t i f f n e s s and the d o t s i n d i c a t e d i f f e r e n t i a t i o n w i t h r e s p e c t to time. In m a t r i x form, the e q u a t i o n s of motion f o r the s t r u c t u r e are [M]{r} + [ C ] { r } + [K]{r> = { R ( t ) } (6) i n w hich [M] i s the mass m a t r i x [C] the v i s c o u s damping m a t r i x , [K] the s t i f f n e s s m a t r i x , and R ( t ) the l o a d m a t r i x . In expanded form, R ( t ) i s g i v e n by R ( t ) = - > 3 h ( t ) M n M n a v ( t ) (7) i n w hich a, ( t ) and a ( t ) are the h o r i z o n t a l and v e r t i c a l h v a c c e l e r a t i o n components of the e a r t h q u a k e . The undamped f r e e v i b r a t i o n mode shapes [ <j> ] and the c o r r e s p o n d i n g n a t u r a l f r e q u e n c i e s of v i b r a t i o n { w } a r e n f i r s t d e t e r m i n e d by s o l u t i o n of the u s u a l c h a r a c t e r i s t i c v a l u e problem 2 - % [M]U n} + [K]U n> = 0 (8) 67. The n o r m a l c o o r d i n a t e s of the s y s t e m , Y , may t h e n by r e l a t e d t o the n o d a l c o o r d i n a t e s by fr} = [<b]{Y} " (9) The e q u a t i o n s o f m o t i o n may now be r e d u c e d t o n n o r m a l mode e q u a t i o n s Y + 2Ana) Y + i o 2 Y = _2 (10) n n n n n M x n i n w h i c h M = U } [M]{4> }, ?n = U } A { R ( t ) } n n n n n andX = % of c r i t i c a l damping i n the n t h mode. I t i s assumed n t h a t the damping i s s u c h t h a t the damping m a t r i x has the o r t h o g o n a l i t y p r o p e r t y {A } [ C ] U > = 0 m 4 n The n o r m a l e q u a t i o n s (10) a r e s o l v e d f o r Y n u s i n g s t e p - b y - s t e p m a t r i x a n a l y s i s method of i n t e g r a t i o n ( W i l s o n and C l o u g h (20)). Then u s i n g e q u a t i o n s (9) the dynamic d i s p l a c e m e n t s {r} a r e d e t e r m i n e d a t d i s c r e t e i n t e r v a l s of t i m e . A p p l y i n g e q u a t i o n s (3) the dynamic e l e m e n t s t r e s s e s and f i n a l l y t he dynamic n o d a l s t r e s s e s a r e d e t e r m i n e d . 68.. MODAL PARTICIPATION FACTOR • The modal p a r t i c i p a t i o n f a c t o r (M.P.F.) d i s c u s s e d Chapter 3.8 were c a l c u l a t e d as f o l l o w s 1. For h o r i z o n t a l base motion o n l y , the modal p a r t i c i p a t i o n f a c t o r i n the n t h mode i s g i v e n by T M.P.F. = 1 M 0 M 2 *n • • > • • 0 M n 2. For v e r t i c a l base motion o n l y T 0 M1 M.P.F. = 1 M n M (12) (13) 3. For combined h o r i z o n t a l and v e r t i c a l base motion T M.P.F. = 1 M n • M l M 2 • n • • M„ n M n k 4 (IA) 69. A P P E N D I X II t MODE 4 (T = 0.419 sec) FIG. 22 MODE SHAPES I TO 4 - DAM WITHOUT CORE FIG. 23 MODE SHAPES 5 TO 8 - DAM WITHOUT CORE MODE 9 (T = 0.293 sec ) MODE 10 (T =0.286 sec ) FIG. 24 MODE SHAPES 9 AND 10 - DAM WITHOUT CORE FIG. 25 MODES I TO 4 - DAM WITH C E N T R A L CORE FIG. 26 MODES 5 T O 8 - DAM WITH C E N T R A L CORE FIG. 27 MODES 9 AND 10 — DAM WITH CENTRAL CORE Cn XJi ON FIG. 28 MODES I TO 4 — DAM WITH SLOPING CORE FjG. 29 MODES 5 TO 8 — DAM WITH SLOPING CORE 

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