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Vibration analysis of dry sand models Aoki, Yoshinori 1969

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VIBRATION ANALYSIS OF DPY SAND MODELS by Yoshinori Aoki B tE«, Tokyo Metropolitan U n i v e r s i t y , Japan, .1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A«3c, i n the Department of C i v i l Engineering-We accept t h i s t hesis as conforming to the. required standard THE UNIVERSITY OF BRITISH COLUMBIA March., .1969 i i ABSTRACT Usiwg the shaking t a b l e , two types of tests have been made with 8 feet long by 1-1/2 feet wide and from 1/2 foot to 1 foot high dry Wedson sand models'. One of them was a response study of the h o r i z o n t a l model, from which the a f f e c t s of boundary r e s t r a i n t s * the frequency-response of s o i l layer.and the•dynanic properties of dry sand were studied. The ' measured r e s u l t s with regard to the e f f e c t s of boundary r e s t r a i n t s and the frequency response of s o i l l a y e r agreed with p r e d i c t i o n s by the l i n e a r v i s c o - e l a s t i c theory. The measured shear wave moduli and damping r a t i o also agreed, with those obtained by previous workers. The other type of tests performed was a study of slope s t a b i l i t y using t i l t e d models, i n which the accumulative displacement of slope was induced, by a s i n u s o i d a l base motion, the c r i t i c a l slope angle and the stable slope angle were studied. The measured accumulative displacement agreed with the theory suggested by Goodman and Seed (.1966), It has been found that there are two d i s t i n c t l y d i f f e r e n t c h a r a c t e r i s t i c angles of slope associated with dynamic s t a b i l i t y f o r a slope. These are the c r i t i c a l slope angle and the stable slope angle. Moreover, the stable slope angle i s unique f o r a material and a frequency of the s i n u s o i d a l base motion and independent of the i n i t i a l c ondition of the slope. In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f C i v i l Engineering The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada D a t e TABLE. OF jCpNTBNTS t'a&e CHAPTER 1 INTRODUCTION , 1 CHAPTER 2 DESCRIPTION OF TEST EQUIPMENT AND.THE SOIL TESTED 3 CHAPTER' 3 DYNAMIC RESPONSE OF HORIZONTAL MODELS , , , . , 11 3.1 Introduction « « . 11 3.2 T h e o r e t i c a l Consideration . « « , « . « « « « . 13 3.3 Boundary E f f e c t s . , , « , « « < « . 24 3.4 Frequency-Response Curve , « « , . « < . « • . « 26 3.5 Measurement of Dynamic Modulus « « , , « « , , , 31 3.6 Summary and Conclusion • « < < < < . « < > « . « 45 CHAPTER 4 SLOPE STABILITY DUPING VIBRATION 47 4.1 Introduction « « « « < « « « < « « « « « « « « 47 4.2 Consideration of Boundary E f f e c t s , , « , , , , 48 4.3 Preliminary Tests . , . 51 4.4 Measurement of Accumulative Displacement . . . . 55 4.5 Measurement of Stable Slope Angle • , 65 . 4.6 Measurement of C r i t i c a l Slope Angle . « « . , . 70 4.7 Summary and Conclusion . , , « , . . . . . . . . 74 CHAPTER 5 SUMMARY AND CONCLUSION . « . . . , , . . , . . 76 APPENDIX I SOLUTION FOR VIBRATION OF FINITE MODEL . , , , , 81 " II BOUNDARY EFFECTS ON VIBRATION . . . . . . . . . 88 " I I I BOUNDARY EFFECTS ON PERMANENT DISPLACEMENT . . . 92 " IV THEORY OF ACCUMULATIVE DISPLACEMENT 98 " V SIMILITUDE , . . . « . . . < . . 102 X V LIST OF FIGURES Figure Page 2-1. Photograph of shaking table,, h o r i z o n t a l model and con t r o l system , , . , , . « « , , . 7 2-2 Photograph of shaking table with t i l t e d model . « 7 2- 3 Sketch of c o n t r o l system for shaking table . , . . 8 3- 1 E f f e c t s of v a r i a t i o n i n shear modulus . « . . « « 3-2 Dimension of model . « . « « . . « 20 3-3 Comparison of frequency-response curve . . . . . . 22 3-4 Explanation of h a l f power point method , , « . . « 25 3-5 Explanation of logarithmic decrecent « . . « • • . 25 3-6 Comparison of a c c e l e r a t i o n r a t i o . • . < 27 3-7 Frequency-response of h o r i z o n t a l model . . . « < • 29 3-S • Relation between void r a t i o and a c c e l e r a t i o n applied during v i b r a t i o n « < « . . . . . . « « • « 33 3-9 An example of free v i b r a t i o n record . , . < « • < 34 3-10 An example of resonant frequency . . . • « « . . . 34 3-11 Fundamental frequency . • . . . . . . . . . . . . 39 3-12 Shear wave v e l o c i t y . . . . . . . . . . . . . . . 39 3-13 Shear wave v e l o c i t y of previous workers and t h i s research . . • . . . . . « • < « . 41 3-14 g =i§= curve . . . . 44 3- 15 Logarithmic decrecent . . . < « . . « < . . « « • 44 4- 1 F r i c t i o n measurement test . . . . . . . . . . . . 50 4-2 An example of the (2) type f a i l u r e plane . . . . . 53 4-3 Deformation of a v e r t i c a l marker at the side w a l l for the (3) type f a i l u r e . . . . . . . . . . 54 Displacement and shape of f a i l u r e plane , * • » * * , Mechanism of f a i l u r e plane * * * , , , * * , , * * Condition at f a5.1ure f o r s l i c i n g element . < . . , Permanent displacement i n one cycle * * * * * * * * Stable slope angle , , , , * , * , , , . * , * * , C r i t i c a l slope angle * * * * * * * * * * • • « • Relation between-, slope angle and amplitude of a c c e l e r a t i o n , , , , * * * * , * * * * * * * * , Coordinate of model * * * * * * * * * * * * * * * * E f f e c t of length of model on i t s na t u r a l frequency E f f e c t of width of model , , * , » , * * Force a c t i n g on a f a i l u r e plane * * * * * * * * * * A se c t i o n of the model , . * * , , , * * , , , * * Forces acting on the toe * * * * * * * * * * * * * S i m p l i f i e d f a i l u r e plane , * , * * * * * * , * , , Forces acting on a slidi.ng element , , * , * * , , Acce l e r a t i o n and v e l o c i t y . * , , * , Acce l e r a t i o n and v e l o c i t y when ?A / * * * * LIST OF TABLES Table jjj&SS. 2- 1 Composition of Wedron sand i'4098 10 2~2 Angle of i n t e r n a l f r i c t i o n • « « * * « « • « « « « 10 3- 1 Relation between frequency and ticceleration r a t i o 28 3- 2 Fundamental frequency . « o . . « . . . < < « < < 37 4- 1 F r i c t i o n between sand and p l e x i g l a s s . • « < « < « 50 4-2 L i s t of displacement measurement tests . < « . < < 62 4-3 L i s t of angles of i n t e r n a l f r i c t i o n , . « < « « « 64 10-1 ' Simi l i t u d e f o r dynamic model test (1) • • • • • • 105 10-2 Simi l i t u d e f o r dynamic model test (2) . « » « . . 105 LIST OF SYMBOLS Area of h y s t e r e s i s loop Ac c e l e r a t i o n amplitude of base motion Width of model C o e f f i c i e n t of f r i c t i o n Damping c o e f f i c i e n t Cohesion Equivalent damping c o e f f i c i e n t Depth of f a i l u r e plane Young's modulus Void r a t i o F r i c t i o n force acting on the f a i l u r e plane Confining force at toe F r i c t i o n force acting on the side w a l l E x c i t a t i o n forces i n the d i r e c t i o n i n d i c a t e d by indeces Frequency i n terms cf cps Resonant frequency Shear modulus Acceleration of gravity Thickness of layer Height of the model Bessel function A c c e l e r a t i o n r a t i o to the ac c e l e r a t i o n of gravity (g) Ac c e l e r a t i o n of base motion i n terms of (g) Acc e l e r a t i o n at base, mid-height and top of the model re s p e c t i v e l y i n terms of (g) Y i e l d a c c e l e r a t i o n i n terms of (g) kD Stress r a t i o between v e r t i c a l and h o r i z o n t a l i n s o i l JLJC Wave length of Love wave £ Length of the model or sample M A c c e l e r a t i o n r a t i o ^ Mass n Void r a t i o to t o t a l volume "P Force acting on toe wedge £ * tan ^  k 7? Frequency r a t i o of s e m i - f i n i t e layer to r e s t r i c t e d model P Ratio of wave v e l o c i t y r = ^/uk Equivalent d i s t r i b u t e d force due to toe confining 7 7T- 2 t, ~ti Times at which f a i l u r e s t a r t s and stops r e s p e c t i v e l y t T i m e of maximum v e l o c i t y W Weight of s o i l mass U^w_c*r Displacements U Displacement i n one cycle U$ Amplitude of displacement of base motion V e l o c i t y of d i l a t a t i o n a l wave lfc ' V e l o c i t y of l o n g i t u d i n a l wave of rod ^ V e l o c i t y of Love wave k£ V e l o c i t y of Raleigh wave Us V e l o c i t y of Shear wave Ve l o c i t y of t o r s i o n a l wave ^ C ^ ^ Rectangular coordinate C* Angle of slope &c C r i t i c a l angle of slope ' Stable angle of slope i x F Constant. Roots of J-b(p„)=0 r . Gainnia function r Unit weight of s o i l r Shear s t r a i n Logarithmic decrecent 0 Angle between the surface of slope and f a i l u r e plane at toe 4> Angle of i n t e r n a l f r i c t i o n -Angle of f r i c t i o n on the plane of toe wedge Angle of f r i c t i o n mobilized at i n i t i a t i o n of y i e l d Angle of f r i c t i o n mobilized during s l i d i n g Phase angle Density of mass ;-X Lame's constant Scale f a c t o r of length // Poisson's r a t i o cr Confining pressure P r i n c i p a l stresses Mean normal st r e s s = cos cx •!- sin<?< tar. 0 Damping r a t i o CO Angular frequency Undamped n a t u r a l frequency 1 Scale f a c t o r -for an item i n d i c a t e d by indeces X ACKNOWLEDGE_MS_MT The i n v e s t i g a t i o n reported herein has been supported by funds provided by the National Research Council of Canada, These funds also included f i n a n c i a l support for the w r i t e r . G r a t e f u l appreciation i s expressed f o r t h i s assistance without which the graduate studies and t h i s thesis could not have been accomplished. The w r i t e r wishes to express h i s thanks to Dr. K,D, Liam Finn, Dr. R.G. Campanella and Professor P.M. Byrne f o r t h e i r guidance and constructive c r i t i c i s m during the preparation of t h i s t h e s i s . The t e c h n i c a l assistance supplied by the s t a f f of the C i v i l Engineering Department i s g r a t e f u l l y acknowledged. 1. 2IM>ATIQN..ANAMSIS ..QF, PRY .SAND MODELS. by Aoki CHAPTER 1 Introduction In recent years there have been great advances made i n the i n v e s t i g a t i o n of the dynamic properties of s o i l s and s o i l s t r u c t u r e s . This area of s o i l mechanics, however, i s so nex* and complicated that many aspects are s t i l l l e f t to be studied. For example, Hardin and Richart (1963) extended laboratory i n v e s t i g a t i o n s of wave propagation phenomena i n s o i l but t h e i r samples were tested under a h y d r o s t a t i c confining pressure i n a t o r s i o n a l mode v i b r a t i o n to measure shear modulus. In order to estimate the v e l o c i t y of wave propagation i n the ground, or i n some earth s t r u c t u r e , i n v e s t i g a t i o n s which involve t e s t conditions under plane s t r a i n and simple shear and the e f f e c t s of v a r i a t i o n of i n d i v i d u a l stresses are needed. Although the approach used to a n a l y t i c a l l y determine the response of s o i l structures i s quite, advanced due to the co n t r i b u t i o n of the f i n i t e elemeiit methods of a n a l y s i s , r e l a t i v e l y l i t t l e i s known concerning dynamic properties of s o i l s , i n s i t u . The purpose of t h i s research i s to examine the s u i t a b i l i t y of the new shaking table from an experimental point of view and to obtain some b a s i c data f o r future developments using t h i s kind of apparatus i n s o i l mechanics. Fundamental but broadly d i f f e r e n t studies of model behaviour may provide valuable information concerning dynamic properties 2. of s o i l , at l e a s t f o r the model. Therefore, one of the important points of the i n v e s t i g a t i o n reported herein i s to compare the observed behaviour of the model with those predicted by e x i s t i n g t heories. Dry sand was chosen as the model material f o r t h i s i n i t i a l study because of i t s r e l a t i v e ease oJ: handling. Moreover, one of the purposes of t h i s research i s to i n v e s t i g a t e the dynamic properties of dry sand, which i s a fundamental ma t e r i a l , and to provide data for futur i n v e s t i g a t i o n s i n v o l v i n g i t . The s i z e of the model i s one of the more important factors of model studies which i s associated with boundary e f f e c t . In t h i s research an 8' long by 1,5' wide and 2* high r i g i d container was b u i l t to house the s o i l . The boundary e f f e c t s of the r e s t r i c t i n g dimensions of the model were examined both experimentally and t h e o r e t i c a l l y i n •order to obtain information concerning boundary e f f e c t s on model behaviour f o r future i n v e s t i g a t i o n s , Tests performed i n t h i s research are b a s i c a l l y as follows; (1) the dynamic response of h o r i z o n t a l models ( i n Chapter 3), and (2), the s t a b i l i t y of sand model slope s i n u s o i d a l v i b r a t i o n (Chapter 4), Because t h i s research involves quite broad and d i f f e r e n t aspects throughout the area of dynamic problems of s o i l s and s o i l s t r u c t u r e s , i t i s d i f f i c u l t to review the l i t e r a t u r e i n one chapter. Therefore, i n each corresponding chapter cr r e l a t e d section the pertinent l i t e r a t u r e w i l l be reviewed. 3. CHAPTER 2 DESCRIPTION OF TEST EQUIPMENT AND THE SOIL TESTED 2.1 Description of .test, eiuipmgnt and instrur.imtation., The shaking table and connected instrumentations have already been reported by Finn, Campanella and Aoki (1968). These are as follows. The.photograph i n Fig,2-1 shows the s i n g l e - a x i s shaking table together with a frame supporting a model ready f o r t e s t i n g . Also shown i n the f i g u r e are the hyd r a u l i c loading p i s t o n , c o n t r o l console, and monitoring and recording equipment. The table ( 6 f t , wide by 9 f t , long and 6-3/4 i n , deep ) i s made up of welded aluminum sections with 3/8 i n , plates top and bottom separated by 2 x 6 i n , hollow rectangular tubing i n a g r i d pattern to provide s t i f f n e s s . The e s s e n t i a l feature of the table i s that i t be very s t i f f i n order to transmit uniform base accelerations and be as l i g h t as possible i n order to make optimum use of the loading ram. The tabl e , which weighs about 1000 l b s 0 , i s mounted on a set of 4 h o r i z o n t a l , V - s l o t t e d needle bearings (manufactured by Schneeberger, Switzerland) which allow only one degree of motion. The bearings are mounted on a welded s t e e l frame made up of wide flange sections which, i n turn, i s bolted to the f l o o r . Great care and p r e c i s i o n i s required i n the alignment of the four sets of bearings. The h y d r a u l i c ram i s mounted i n a very s t i f f , a l l welded frame made up of 3/4 i n , plate s t e e l which, i n turn, i s r i g i d l y mounted to the base frame. The ram i s a double-ended hy d r a u l i c p i s t o n with a 2 i n , stroke, a dynamic capacity of 2500 l b s , and a maximum v e l o c i t y of about 17 i n , per sec, at f u l l load, A r e l a t i v e l y small 3 g a l . per min, pump (not shown i n F i g . l ) supplies o i l at 3000 p s i to the ram through 4. an e l e c t r o n i c c o n t r o l l e d servo-valve« The ram, which provides the force to move the tab l e , i s connected to the table through a r i g i d l i n k , A sketch of the co n t r o l system f o r the shaking table i s shown i n Fig.2-'3 S The system i n c l u d i n g e l e c t r o n i c c o n t r o l s , h y d r a u l i c pump and ram were obtained from MTS Systems Corporation, Minneapolis, Minn, Fig,2 shows that the c o n t r o l l e r w i l l accept a voltage vs, time s i g n a l from e i t h e r a function generator, curve follower or magnetic tape. For shaking table operation t h i s input voltage representes a displacement-time command. An LVDT (displacement transducer), which i s housed within the hyd r a u l i c ram also transmits an e l e c t r o n i c s i g n a l i n t o the c o n t r o l l e r which represents the actual movement of the ram and therefore the t a b l e c The command s i g n a l , V^, i s compared to the feedback s i g n a l from the displacement transducer, and the voltage d i f f e r e n c e , - V^, i s transmitted to the e l e c t r o n i c c o n t r o l l e d servo-valve located i n the hydraulic ram. The e l e c t r o n i c valve responds by allowing the ram to move i n such a manner as to make the feedback voltage from the displacement transducer, X^, equal to the. command voltage, X^, This primary route of cont r o l i s i n d i c a t e d as a Servo-Loop i n Fig,2-3. The system i s also capable of t r e a t i n g the input command s i g n a l as one of ac c e l e r a t i o n vs, time by i n p u t t i n g a s i g n a l to the c o n t r o l l e r which i s the double i n t e g r a l of the acceleration-time command, (The in t e g r a t i o n i s done e l e c t r i c a l l y ) . Thus, the system always controls displacement within the servo-loop f o r shaking table operation. The actual table accelerations are measured with a K i s t l e r Model 305A/515 Servo-Accelerometer which i s also used i n a secondary feedback loopo The command s i g n a l , table displacement and a c c e l e r a t i o n can be measured by any r e l a t i v e l y high input impedance voltage recording instrument such as an o s c i l l o s c o p e . The response of the shaking table i s a function of the desired amplitude, the response of the sexvo-valve, the pumping capacity of the hydraulic pump, the volume compliance of the h y d r a u l i c ram, the s t i f f n e s s of the ram and table and the mass of the table and model. Tests show that the present shaking system has a resonant frequency of about 17 cps (cycles per second) for a moving mass of about 1000 l b s , and about 15 cps when the moving mass i s increased to about 2700 l b s . At frequencies above 20 cps a c h a r a c t e r i s t i c r o l l o f f occurs. However, i t was possible to use the shaking table at s i n u s o i d a l frequencies as high as 60 cps and s t i l l obtain smooth a c c e l e r a t i o n sine curves by i n c r e a s i n g the input amplitude and operating along the r o l l o f f curve. For the model studies reported herein a r e l a t i v e l y r i g i d container was b u i l t to house s o i l models, This container i s 2 f t . high by 8 f t , long and 1-1/2 f t , wide with p l e x i g l a s s sides (see F i g . l ) , The base was made of 3/4 i n , plywood with a layer of sand bonded to the upper plywood surface with epoxy to provide f r i c t i o n a l r e s i s t a n c e for the sand models, A K i s t l e r Model 505A/515 accelerorceter was mounted beneath the plywood base to measure base ac c e l e r a t i o n s . The container was designed so i t could be t i l t e d f o r s t a t i c t e s t i n g , as well as fixed at a given t i l t angle up to 40° at i n t e r v a l s of 5 degrees and tested under dynamic conditions, (See Fig,2-2), Instrumentation In order to measure the response of the s o i l models during base accelerations i t was necessary to measure accelerations within the s o i l mass. The e s s e n t i a l requirements of such an accelerometer are that i t be very small and have the same unit weight as the s o i l . The highly s e n s i t i v e and accurate accelerometers used to measure base motions were too large and too heavy, as are the presently a v a i l a b l e s t r a i n gage accelerometers which are normally used f o r seismic studies. P i e z o e l e c t r i c accelerometers appear to be small enough to be embedded i n the s o i l . Unfortunately, the smaller the ac.celerom.eter the greater i s i t s f u l l s c a l e a c c e l e r a t i o n and the poorer i s i t s low frequency response thereby, reducing i t s s e n s i t i v i t y i n the 0 to l g and DC to 10 cps range. For example, a t y p i c a l miniature p i e z o e l e c t r i c accelerometer might have a frequency response from 5 to 10,000 cps and a range of lOOOg's, Fortunately, some p i e z o e l e c t r i c accelerometers are extremely s e n s i t i v e and have a r e s o l u t i o n of ,01 g's or b e t t e r . Thus, i t was decided to obtain miniature accelerometers from several manufacturers and determine t h e i r s u i t a b i l i t y f o r embedment i n s o i l i n seismic t e s t i n g . In order to check the c a l i b r a t i o n and response of accelerometers the K i s t l e r 505A/515 servo-accelerometer was used as the primary reference (same type used on t a b l e ) . This extremely s e n s i t i v e instrument has an o v e r a l l accuracy of about ,01% i n c l u d i n g l i n e a r i t y and h y s t e r e s i s which represents ,005g. I t also has a f l a t frequency response of DC to 500 cps and a f u l l scale range of + 50g, A K i s t l e r model 504A charge Amplifier was used with a l l of the following p i e z o e l e c t r i c accelerometers, CEC Model 4-274-001, weight 6 gm, u n i t weight about 450 pcf, KISTLER Model 803A , weight 20 gm, unit weight about 430 pcf, METRIX Model 502 , weight 6 gm, u n i t weight about 3.30 pcf, WILCOXON Model 127, weight 1-3/4 gm, u n i t weight about 185 pcf. Of these 4 tested, the Matrix, which was judged best, had the highest s e n s i t i v i t y , gave the most noise-free signals at low accelerations and frequencies above about 1 cps, and had a u n i t weight very close to that of s o i l . The Wilcoxon was also quite good and had the 7. Fig.2 - 2 . PHOTOGRAPH OF SHAKING TABLE WITH TILTED MODEL, •' 3o00p.s.i PlSPLACEM&ST TBED&QCK ~\ TABLE „ ACCELERATION TZBOBflCk Z _ 11 \ACCBL o scope Fla 2-3 SKETCH OF CONTROL SYSTEM] FOR SHAKIAjtr TABLE advantage of being several times smaller than the Metrix, although i t was not as s e n s i t i v e . The CEC and K i s t l e r were judged l e a s t s u i t a b l e f o r low frequency - low a c c e l e r a t i o n t e s t i n g . For a l l of the t e s t ^ reported herein only the Metrix and Wilcoxon accelerometers were used, 2-2, Description of the s o i l tested, The models were comprised of Wedron sand, which i s clean, w e l l rounded, uniform, medium s i l i c a sand and almost i d e n t i c a l to Ottawa sand 20-30, The composition of the' sand i s shown i n Table 2-1, Five samples were tested by vacuum t r i a x i a l r e s t i n order to determine the angle of i n t e r n a l , f r i c t i o n of the sand. The r e s u l t s are shown i n Table 2-2, In t h i s t e s t d i s t i n c t d ifferences between the peak strength and the r e s i d u a l strength were not observed up to 6-8% of the a x i a l s t r a i n . During the experiment of the dynamic response of h o r i z o n t a l models the r e l a t i o n s h i p between the average void r a t i o of models and the maximum amplitude of the a c c e l e r a t i o n applied during v i b r a t i o n has been obtained (see Fig,3-8 i n Chapter 3). From these r e s u l t s , the void r a t i o s of models tested were 0,47 and 0,65 f o r dense models and loose models r e s p e c t i v e l y . 7a Me 2-f C0F7m5/T/D,V Of' UfeDROA/ S/)MO #4090 % Ret on U.S. A/o. 20 2.2 30 38-6 4-0 S7.6 so /.4 " 70 O. Z Tah/e 2-2 AMfrlZ OF /AJTFRA/AL FK/CT/Otf Wold ratio Con-f. pressure 3x/ai •far ^ snaK .SOB 2.5 P**1 47-5 ° . S/o S-O ' r a L 6 % /O.O 38-7° 3 % 2.S /*« 36. <?° 4- % . 773 s . o ^ 35. 6° B % 11. • CHAPTER 3 DYNAMIC RESPONSE OF HORIZONTAL MODELS 3-1„ Introduct i pn In general, s o i l s are not l i n e a r l y e l a s t i c m a t e r i a l , but i t i s -4 known that under small amplitude of s t r a i n , say les s than 10 , the Kelviiv-Voigt model which consists of a simple l i n e a r e l a s t i c spring and pure viscous damping assembled p a r a l l e l to each other may s a t i s f a c t o r i l y explain the dynamic behaviour of some s o i l s . For the Kelvin-Voigt model, three parameters are enough to describe the dynamic, behaviour of s o i l s , which are, f o r instance, the sh v e l o c i t y , V 9 the d i l a t a t i o n a l wave v e l o c i t y , V 9 and the equivalent s c damping r a t i o £, or e i t h e r the shear modulus, G, or Young's modulus E, Poison's r a t i o , u,and £ , Methods of measurement of dynamic properties of s o i l s which have been reported can be c l a s s i f i e d as follows: (1) F i e l d or i n - s i t u measurement - Resonant method. - Direct measurement of wave v e l o c i t y , (2) Laboratory test - Resonant method, - Free v i b r a t i o n method, - Direct measurement of wave v e l o c i t y , - Amplitude r a t i o and phase s h i f t measurement. Because of the s i z e r e s t r i c t i o n of the te s t model and the type of instrumentation used, only the laboratory resonant and free .vibration methods were used f o r the i n v e s t i g a t i o n reported h e r e i n . These methods are associated with the frequency equation which involves only the wave v e l o c i t y or dynamic modulus as an unknown f a c t o r i f the damping 12. r a t i o i s small enough to be ignored. I f the n a t u r a l frequencie3 are measured f o r s o i l models or s o i l structures.whose frequency equation i s known, the wave v e l o c i t y or the dynamic modulus can be ca l c u l a t e d from the frequency equation by s u b s t i t u t i n g the measured frequency. In the resonant method* resonant frequencies are found by changing the frequency of e x c i t i n g motion, The free v i b r a t i o n method i s based on the f a c t that models of structures v i b r a t e i n t h e i r fundamental mode during free v i b r a t i o n , ((therefore an e x c i t a t i o n impulse or an i n i t i a l disturbance i s applied to the model, and the r e s u l t i n g frequency of free v i b r a t i o n i s measured. Hardin and Richart (1963), for example, have used the laboratory resonant method and Zeevaert (1967) used the laboratory free v i b r a t i o n method. Further d e t a i l s with regard to the frequency equation of the model tested herein are discussed i n the next s e c t i o n . Other methods of measurement of dynamic properties of s o i l s are discussed elsewhere, f or example, Jones (1958), Barnhard (1959) and Martin and Seed (1966) have discussed f i e l d or i n - s i t u measurement methods,. At MIT (Taylor and Whitman (1954)), the method of d i r e c t measurement of wave v e l o c i t y has been developed and Seli'g and Vey (1965) and S t a l l et a l (1965) have also used t h i s method, Chae (1968) has discussed the amplitude r a t i o and phase s h i f t measurement method. This chapter concerns i t s e l f with the study of the dynamic response of h o r i z o n t a l models, whose purpose i s . t o observe the behvaiour of s o i l models and compare i t with e x i s t i n g theories and to in v e s t i g a t e the dynamic modulus of s o i l s through the model study and compare i t with those obtained by other i n v e s t i g a t o r s , The following experiments were c a r r i e d out: (1) study of boundary e f f e c t s on the response of models to s i n u s o i d a l base motion (Section 3-3), (2) study of the frequency-response curve of the h o r i z o n t a l s o i l model to s i n u s o i d a l base motion (Section 3-4). (3) measurement of dynamic modulus of dry sand by the free v i b r a t i o n method and by the resonant method with h o r i z o n t a l s o i l models (Section 3-5), 3-2, T h e o r e t i c a l consideration,, Before the experimental r e s u l t s are discussed, r e l a t e d theories of the dynamic response of h o r i z o n t a l s o i l models on which the analysis of t h i s research i s based w i l l be b r i e f l y presented here, A s o l u t i o n of the response of a h o r i z o n t a l model to a s i n u s o i d a l base motion, i n which boundary e f f e c t s due to the l i m i t e d length and width of the model were taken i n t o account, w i l l also be presented. There are four fundamental types of waves which propagate through the ground. These are as follows: Shear wave D i l a t a t i o n a l wave Raleigh wave Love wave Body waves — Surface waves-The shear wave involves no volumetric change and the movement of s o i l p a r t i c l e s i s perpendicular to the d i r e c t i o n of the propagation, and the v e l o c i t y i s given by 3-1 14. where p stands f o r the mass density of s o i l s . In the d i l a t a t i o n a l wave the movement of s o i l p a r t i c l e s i s back and f o r t h , i n the d i r e c t i o n of propagation, therefore compressions and extensions are involved. Its v e l o c i t y i s given by 2J2 =f(?\ * £-)/[> 3-2 where ?\ = /-r^X' -2/J) • The Raleigh wave consists of p a r t i c l e movements of e l l i p t i c a l shape which i s on the plane perpendicular to the surface and p a r a l l e l to the d i r e c t i o n of propagation. The v e l o c i t y i s given by IT* = r V7 3-3 where r i s ,a constant which depends cn Poison's r a t i o , u , for example u » 0.2 r » 0,911 . • y =» 0.3 r = 0.928 y = 0.4 r » 0,942 y « 0.5 r = 0.955 The Love wave i s sometimes referred to as h o r i z o n t a l shear wave SR, The movement of s o i l p a r t i c l e s i s h o r i z o n t a l and normal to the d i r e c t i o n of the propagation. This wave ex i s t s only i f the shear wave v e l o c i t y of the surface l a y e r i s less than that of the lower l a y e r . The v e l o c i t y of the Love wave depends upon i t s wave length and the longer the wave length the greater the v e l o c i t y . The maximum v e l o c i t y of the Love wave i s equal to the shear wave v e l o c i t y of the lower l a y e r and the minimum i s equal to the shear wave v e l o c i t y of the surface l a y e r . If the h o r i z o n t a l s o i l model i s assumed to be.an i n f i n i t e l a y e r , i t s response to a.horizontal base motion can be analyzed as a pure shear-wave propagation problem i n a semi-infinite.media. The equation of motion for v i b r a t i o n of a s e m i - i n f i n i t e h o r i z o n t a l layer with uniform shear modulus, G, subjected at i t s base tc a h o r i z o n t a l motion, ii , i s g r dt ^dtf*  r u 3 3-4 where c i s a damping c o e f f i c i e n t . The s o l u t i o n of t h i s equation f o r s i n u s o i d a l base motion, i . e . u •= a s i n ut i s g S 3-5 where YnC^i-COS^O-Ji- 3-6 H i s the thickness of l a y e r , £ ~ /^ft-i an V f ?4 - **"'Tl3tf 3-9 In f a c t , models tested here were neither s e m i - i n f i n i t e l a y e r s , nor with uniform shear modulus, > hese two conditions;, that i s , the f a c t s that the length and width of the model were s i g n i f i c a n t l y small compared with i t s depth and that the shear modulus of the modo.l increases with depth, are quite d i f f i c u l t to account f o r at the rtame time. Therefore, the one which l e a s t a f f e c t s the analysis w i l l be ignored. If the model i s assumed to be a s e m i - i n f i n i t e layer and i t s sheax, modulus increases with depth by 3-10 then the equation of motion i s ( I d r i s s and Seed, 1966) 3-11 If p^. , the steady-state s o l u t i o n f o r s i n u s o i d a l base motion of equation 3-11 i s ^ ^ y ; = J> YnCy)- , »->-2, 3-12 and 3-13 where ./3„ = roots of J^ = 0 » J-l a B e s s e l function of the f i r s t kind of order «= -£> and p = gamma funct i o n . Also, /-^ 3- 1 5 17 / c m ) 3-17 s and 0 are constants re l a t e d to by / Q 0 - 0 - r 2 5 = O In order to examine the e f f e c t of the v a r i a t i o n of the shear modulus of models, the fundamental frequency and mode shape of a 1 f t . deep layer were ca l c u l a t e d by using solutions f o r both a s e m i - i n f i n i t e layer with uniform shear modulus, equations (3-8) and (3-5), and f o r a semi-i n f i n i t e l ayer whose shear modulus increases with depth by equations (3-10), (3-15) and (3-12). Results are 'shown i n Fig«3-1« The assumed shear, moduli which were obtained by extrapolating from Hardin and Richart's equation (see Eq.3-27) are shown i n Fig,3-1, Unit weight of the s o i l , /" = 113 psf damping r a t i o , ^ 0.02 and ft f 1/2 were assumed. From Fig,3-1 ( a ) , i t can be seen that the fundamental frequency of the laye r with uniform shear modulus G - 2,20xl0~* psf, i . e , 1520 p s i , corresponds to that of the lay e r whose -shear modulus va r i e s with depth by r\ •• G = (2.76x10^),y^/2,psf. Moreover, i t i s found from Fig,3-l(b) that, i f a uniform shear modulus i s chosen i n such a way that the chosen uniform shear modulus gives the same fundamental frequency as that c a l c u l a t e d by the equation which takes the v a r i a t i o n of shear modulus into account, mode shapes calculated by both equations agree with each other. Therefore, e f f e c t s of v a r i a t i o n of the shear modulus with depth are not very important i f G - f(y^/2) and can be overcome by taking a s u i t a b l e average as a uniform shear-modulus, On the other hand, e f f e c t s on the v i b r a t i o n of models due to the r e s t r i c t e d length and width of the model compared with i t s depth are as follows. These e f f e c t s w i l l be c a l l e d "boundary e f f e c t s on the IS. I 6/ 62 -2,7Sx-/C!' ?.8x/cf ._<> FOR. SHEAR H0DULU5 &.'l/£A/ •By ef. (3-/SJ © FOfr UMFORM SHER MODULUS e%- (3-8) •75 /.o CbJ PTPOE SHBAf? MO DULL/5 <j-/t/£,</ Br C£o e?. (3-/2) •© FOR V/J/FO/ZM SH£Z rroDuu/s ^=2.2x/ot(pJF) ef(3-£) {"= //3 PCF J % .= o.oZ 7:P£GU&VCy oT EXC/TAT/O// oJ=SOQoJ 1.0 2.0 3.0 4.D P/5PLACEMEA/T PAT/0 UMTH RE5PECT TO BASE p. 3-f EFFECTS OF. MRIAT/OA/ //V WEAK M0PUW5 v i b r a t i o n hereafter. Taking boundary e f f e c t s on the v i b r a t i o n into account; the more general equation of motion of a h o r i z o n t a l layer subject at i t s base to a h o r i z o n t a l s i n u s o i d a l motion has been solved i n Appendix I, The. steady-state part of the s o l u t i o n i s and l/cz, 5/» J?^sm%^ sJr>-£^. 3 . 1 9 64- a? si»(^£ + &?rr) 3-20 3-21 where p.q.r, » 1,3,5 , ^ p ^ r ~ phase angle, k£ » d i l a t a t i o n a l wave v e l o c i t y ^J-2LLiJ3L t • }fs a shear wave v e l o c i t y j t a u c * Y « b, and h are dimensions of the model as shown i n Fig,3-2, Assumptions made i n t h i s s o l u t i o n are as f o l l o w s : (1) At each boundary, bottom, both ends and s i d e s , the r e l a t i v e movements between s o i l p a r t i c l e s and boundaries are zero, (2) The d i r e c t i o n of e x c i t a t i o n and the d i r e c t i o n of s o i l movements i s u n i a x i a l ( i n the d i r e c t i o n of y i n Fig,3-2) and h o r i z o n t a l , (3) Shear modulus i s uniform throughout the model. The assumption (1) looks v a l i d from the observation i n tests described i n the next section as shown on Fig,3~6, provided the amplitude of s t r a i n i s small. Namely, r a t i o s of the a c c e l e r a t i o n of s o i l s adjacent to boundaries with respect to the base are almost Unity, For assumption (2) i t has been shown by the technique of the f i n i t e element analysis of plane s t r a i n condition that when the model i s F/j 3-2 D I M E A / S / O A / S O F M O D E L 21. excited by a u n i a x i a l h o r i z o n t a l base motion the predominant motion of s o i l p a r t i c l e s i s h o r i z o n t a l and the v e r t i c a l movement i s quite small . and n e g l i g i b l e . Assumption (3) has already been shown to be s a t i s f i e d , namely i f a s u i t a b l e average shear modulus i s chosen, the equation f o r the model with a uniform shear modulus w i l l give a reasonable approximation. I t i s i n t e r e s t i n g to note that equation (3-18) i s no longer for a simple shear v i b r a t i o n but consists of shear v i b r a t i o n s i n two planes, which are h o r i z o n t a l and v e r t i c a l , and a d i l a t a t i o n a l wave, a l l i n the d i r e c t i o n of the e x c i t a t i o n . In order to examine the boundary e f f e c t s on the v i b r a t i o n , frequency response curves i n terms of a c c e l e r a t i o n r a t i o at the center of the surface with respect to the base were c a l c u l a t e d from both the equation of s e m i - i n f i n i t e layer with uniform shear modulus, equation (3-5), and the equation of the f i n i t e model, equation (3-18), and these, the curve (1) and the curve (2) r e s p e c t i v e l y , are shown on Fig,3-3, In t h i s f i g u r e , the negative r a t i o means that when the base moves towards a p o s i t i v e d i r e c t i o n , the surface moves towards an opposite or negative d i r e c t i o n . This happens, f o r example, at mode shapes of odd numbers f o r a simple h o r i z o n t a l shear v i b r a t i o n , Two curves, curve (1) and (2) i n Fig.3-3 show quite d i f f e r e n t shapes, These dif f e r e n c e s are as follows; ( i ) fundamental frequencies are d i f f e r e n t ; 63 cps f o r a s e m i - I n f i n i t e l a y e r , curve (1), while 108 cps was obtained f o r the f i n i t e layer of a given dimension, curve. (2), This i s because the boundary r e s t r a i n t s cause a higher s t i f f n e s s and therefore a higher fundamental frequency, ( i i ) the maximum response a c c e l e r a t i o n r a t i o s are d i f f e r e n t , more than 20 for the magnification f a c t o r f or curve (1) while l e s s than 15 for curve (2), This i s because the boundaries r e s t r i c t the s o i l movement. 22. ^ <5> V, ^ ^ ( i i i ) the shape of the curves i t d i f f e r e n t , with a more complicated shape f o r curve (2) than that f or curve (1), This i s , as mentioned previously, because equation (3-18) involves not only the shear v i b r a t i o n i n the h o r i z o n t a l plane but also the shear v i b r a t i o n i n the v e r t i c a l plane and the d i l a t a t i o r . a l v i b r a t i o n i n the d i r e c t i o n of the e x c i t a t i o n motion. For example, peak "a" on curve (2), which i s given by equation (3-18), corresponds to the peak "a"' on curve (1) which i s given by equation (3-5), and "b" on curve. (2) corresponds to "b" 1 on curve (1). Chese are the fundamental frequency and the second resonant frequency, r e s p e c t i v e l y associated with the h o r i z o n t a l shear v i b r a t i o n . On curve (2) between "a" and "b" there are two d i s t i n c t peaks which cannot be found on curve (1); these are resonant frequencies associated with the 3rd and 5th d i l a t a t i o n a l v i b r a t i o n i n l o n g i t u d i n a l d i r e c t i o n . I n c i d e n t l y , mode shapes of even number do not contribute to the v i b r a t i o n of the model, because those mode shapes are symmetric, hence forces i n the model due to these modes of v i b r a t i o n cancel out each other and have no e f f e c t on external forces. As discussed previously the boundary e f f e c t s on the v i b r a t i o n are much more s i g n i f i c a n t than the e f f e c t of the v a r i a t i o n of shear modulus with depth. Therefore, the r e s t r a i n i n g boundary e f f e c t s w i l l be taken into account and equation (3-18) w i l l be used i n the analysis hereafter. Boundary e f f e c t s are evaluated and discussed in.more d e t a i l i n Appendix I I , Another point to be discussed here i s the determination of damping r a t i o from experimental r e s u l t s . The equivalent damping r a t i o can be determined from experiment by various methods (see H a l l and Richart 1963), Of these methods only those applicable to t h i s research w i l l be discussed. If the frequency-response curve i s obtained for forced v i b r a t i o n the equivalent damping r a t i o , ^ , i s given by where ^ f i s the width of frequency at the amplitude of. •—= » 0,707 times the maximum amnlitude, U - ,as shown i n F i n , 3 - 4 , and f i s the J max' T 1 o resonant frequency, Martin and Seed (1966), f o r example, have applied t h i s i n t h e i r f i e l d t e s t data. In connection with the free v i b r a t i o n of the model, the equivalent damping c h a r a c t e r i s t i c i n terms of logarithmic decrecent, £ , i s given by ^ - i r ^ i t ^ 3-23 and the damping r a t i o , ^ , i s given by J - 3-24 where u's are amplitudes of motion at nth and (n-l-m) th cycle as shown on Fig,3-5, Using t h i s method, H a l l and Richart (1963) have determined the damping c h a r a c t e r i s t i c s of sand, 3-3, Boundary e f f e c t s on the v i b r a t i o n of the model. In order to examine the v a l i d i t y of the s o l u t i o n (3-18) of the equation of motion of the model which takes boundary e f f e c t s into account, several preliminary t e s t s were performed. The experimental procedure i s as follows. About 700 lbs of sand was placed i n the container to a height of 6 inches. Then-the sand was compacted by v i b r a t i o n i n t o a h o r i z o n t a l model, While the model was vib r a t e d by a base motion of constant a c c e l e r a t i o n and frequency, the a c c e l e r a t i o n d i s t r i b u t i o n throughout the surface of the model was measured by moving an accelerometer from point to point. Then a c c e l e r a t i o n £}j 3-<r FXPL4AI/)TlOAl Of LOGARITHMIC VBCRSCEA/T. 26. r a t i o s at the surface with respect to the base were c a l c u l a t e d . Measured amplitude r a t i o s are shown i n Fig,3-6 f o r a frequency of 50 cps. Also shown are t h e o r e t i c a l values, which agree f a i r l y c l o s e l y with measured values. The t h e o r e t i c a l values were ca l c u l a t e d by equation (3-18) with E = 1300 p s i , u - 0,3 and 5 = 3 % , Also a c c e l e r a t i o n r a t i o s adjacent to both sides and end walls were almost u n i t y , which was assumed i n the s o l u t i o n , 3-4, Frequency-response curve of a h o r i z o n t a l model. In order to obtain the frequency-response curve of a h o r i z o n t a l model, a couple of tests were done with 1 foot high models. The experimental procedure i s as follows. Two accelerometers were placed i n the model which had been compacted by v i b r a t i o n , one (metrix model 502) at the center of the surface and the second (Wilcoxon model 127) at* the center of the mid-height, namely 1/2' from the surface. A f t e r the accelerometers were placed, a v i b r a t i o n with small a c c e l e r a t i o n was applied so that the sand and embedded accelerometers reached a stable configuration r e l a t i v e to each other, • Lhen the model was vibrated by a s e r i e s of s i n u s o i d a l base motions of various frequencies from 7,5 cps to 54 cps. Accelerations at three points of the model, i«e, at the surface, k , at the mid-height, k « aud at the base, k, , were measured t * nr " b* at each frequency of base motions (see Table 3-1), The a c c e l e r a t i o n r a t i o at the surface with respect to the base, , and the a c c e l e r a t i o n at the mid height with respect to the base M2= » were ca l c u l a t e d and p l o t t e d against frequency i n Fig.3-7, Pronounced magnifications of the r a t i o f o r both M^ and M^ were noted at a frequency of approximately 48 cycles per second. From these r e s u l t s dynamic moduli and the equivalent damping r a t i o can be c a l c u l a t e d . S u b s t i t u t i n g the length of the model £ <* 8', and the HE/G-h'T OP DSO'SB l\Z£OR0/J — y£ PT FPEQUEA/CY = 52?cpsC^JNUSDID/U-') •- /.ao • . /, 20-/.<$0-A 60 A-7Z <P) TrtEOP£7?C4/~ P>ISTR/3UrtOA/ OE ACCELEfcqTf OA/ RAT/o. AT SUREQCE WITH RESPECT TO T ^ S B 28, Tah/e 3-/ ' fiEL/)770AI' 3E7WE&A7 FREQU&VCYAMD ACC£L£RAT/OA/_ FAT/D -fee P S ) r.9) &b(3) fc„ (3) Mr M2 A/OTG-7,5 .39 .36 .36 .925 . 9J5 . 4o . 412 •3? .38 . ?3 .905 /3.5 . 7D . 72 .67 . 93 /7. 9 .70 .83 • 76 .73 . 9^5 . 86 23. Z .72 . 7B .74 .76 . 45 .975 274 .32 .34 .32 .34 . 942 /.OO 333 - 74 ./B ./~7 .79 .945 /.o5 37. O . /O ./Z ./3 .78 /. 08 A. 5 bad l/a/u€ 38.5 - 73 ./5 . /5 ./6 /. 00 /.o7 3?.Z ./z ./3 ./3 . 73 / . OO /. OO 40.0 . O0 • o? .o9 . 70 /. O0 /. // a/.o .07 .08 .085 . 70 5 /. 06 /.3f 42.0 . /O . /Z .'25 .75 y. 04 f.25 Sine C6£) 42.5 . // . // ./2 . , 6 * /.o? /. 45 /n/'st>ehai.'£. 1 43S .72 ./25 ./35 . 76 /.oB /.28 44.5 .21 .25 .30 2. 20 A 6 45. 5 .24- .34 . 4-3 . 6o*~ /.26 /. 77 416.5 • .21 . 30 .ar .62* A 37 2.06 A poor-47S . 76 .22 .32 ,44 A&5 7. OO 40.7 . 08 .09 .74 . 20 /.55 2222 zo.o .2Z .30. . 38 . 46 A 27 7. 53 bad 52.5 ./?5 •26 .30 7.#2 fa/>/e /fc/ejt^-frA 54*. O ./6 .30 . 20 •.'£>* . a . 535 W-S-4* a/7 (zveraja 29. F,g 3~7 •7=RE0UJ£A7CY PZ5pDf/SB OT HDR/ZOMTAL SO/^L MODEL. 30, width b = 1-1/2' into equation (3-21), and s e t t i n g p.q, and r •? 1, Let us assume y = 0,3., i . e . Then '3-25a . HE = S u b s t i t u t i n g f = 48 cpf and h => 1 f t , ° o ' and s u b s t i t u t i n g f3 = J^d J f. " 113^°^ and g => 32.2 feet/sec. The shear wave v e l o c i t y or modulus i s r e l a t i v e l y low compared with other data (see Fig,3-14) and with previous i n v e s t i g a t o r ' s r e s u l t s . Further d e t a i l s w i l l be discussed i n the next section together with other r e s u l t s . If the logarithmic decrecent equation (3-22) i n the previous section i s applied to the r e s u l t s i n Fig,3-7, the equivalent damping r a t i o i s evaluated as follows, since M = 2,26 then 0,707 M = 1.60, •' • max max 31. A f « 5.7 and £ ~- 48 pcf, o The damping r a t i o , 6%, i s s l i g h t l y high compared with other data (see Fig,3-15)c The order of the damping r a t i o obtained here, however, i s quite reasonable. At the resonant frequency the maximum r e l a t i v e displacement between the surface and the base i s given by Since the height h = 1 fo o t , the average s t r a i n i s p = 4x'O* (0.004%) The magnitude of s t r a i n of 4x10 i s , i n general, small enough to consider the model as a l i n e a r v i s c o - e l a s t i c m a t e r i a l , which i s one of the b a s i c assumptions of the a n a l y s i s . The shape of the frequency-response curves i n Fig,3-7 are s i m i l a r to curve (2) i n Fig.3-3 which i s calculated by equation (3-18) ( f o r a f i n i t e layer) rather than the curve (!) by equation (3-5) ( f o r semi-f i n i t e l a y e r ) . Namely, r i g h t a f t e r the resonant frequency a marked r o l l o f f occurs and the amplitude r a t i o goes below unity, therefore i f G = 285 p s i i s s u bstituted i n t o equation (3-18) the t h e o r e t i c a l curve agrees with observed curve i n Fig,3-7, 3-5, Measurement of dynamic modulus. Using h o r i z o n t a l models a s e r i e s of tests were c a r r i e d out i n order to measure the dynamic properties of the model. The void r a t i o s of the model were changed i n the range of 0,47 to .68, and 6" high models and 9" high models were tested. Two types of t e s t s , the f i r s t the free v i b r a t i o n method and the second the resonant method, were done se q u e n t i a l l y 32, with an i d e n t i c a l model. The procedure f o r the preparation of models 'was as follows. The sand was weighed when i t was poured i n t o the container so that, the t o t a l weight of the model would be known. Then the sand was placed i n a h o r i z o n t a l layer by v i b r a t i o n of amplitudes of a c c e l e r a t i o n i n the range of 0,3 - l o 0 g and a frequency of 15 cps. A f t e r that, the height of the l a y e r was read by the scales attached to the p l e x i g l a s s w a l l of the container at eight points and an average was taken. Knowing the t o t a l weight, the height, the area, and the s p e c i f i c gravity of s o i l s the average void r a t i o was c a l c u l a t e d . The next model, having a d i f f e r e n t void r a t i o , was prepared by v i b r a t i o n of a d i f f e r e n t amplitude of a c c e l e r a t i o n , while the sand l a y e r was being dug around over the e n t i r e model i n order that the model would be uniform, A unique r e l a t i o n s h i p between the average void r a t i o of the model and the applied amplitude of -acceleration, during v i b r a t i o n has been obtained. I t i s p l o t t e d on Fig,3-3, This r e l a t i o n s h i p has been obtained f o r the v i b r a t i o n of a frequency of only 15 cps, however Greenfield and Misiaszek (1967) have reported that the predominant f a c t o r of the void r a t i o of the v i b r a t i o n a l sand mass i s the amplitude of a c c e l e r a t i o n of v i b r a t i o n and the void r a t i o i s almost independent of the frequency of the v i b r a t i o n . A f t e r a h o r i z o n t a l model was prepared, an accelerometer (Metric model 502) was embedded at the center of the model surface, then a v i b r a t i o n was made so that the sand and embedded accelerometer would reach a stable configuration r e l a t i v e to each other. The free v i b r a t i o n method has been made by the f o l l o w i n g procedure, A square wave having a frequency of p r e c i s e l y 2,5 cycles per second and an amplitude of displacement of 0,02" was applied, and both the 33. F>?3-B RELAT/OA/ BE7WEBA/ MD RA770 AA/P ACCELEMT/OA/ APPL/ED PUPJMfr V/3RS7'/OA/ 35. e x c i t a t i o n square wave and the response a c c e l e r a t i o n at the surface of the model were recorded by an o s c i l l o g r a p h . An example of these o s c i l l o g r a p h records i s shown i n Fig,3-9. From the o s c i l l o g r a p h records, the frequency of the free v i b r a t i o n of the model was determined, using the period of the e x c i t a t i o n square wave as a time reference. From these records, the logarithmic dacrecent was also obtained by taking the decrement of the amplitude within several cycles and using equation 3-23. Another t e s t by the resonant method..was c a r r i e d out i n turn using the'..same model as used for the free v i b r a t i o n method. The test procedure i s as follows. The displacement was chosen as a command s i g n a l of the shaking table system i n t h i s test because i t was quite d i f f i c u l t to operate at high frequency by the a c c e l e r a t i o n command s i g n a l , while i t was possible to operate by the displacement command s i g n a l , While the frequency of the s i n u s o i d a l base motion was gradually increased from a low enough value, the o s c i l l o s c o p e which was connected to both accelerometers at the base and the surface of the model was c a r e f u l l y observed. When the resonant frequency was recognized on the o s c i l l o s c o p e , the frequency was recorded. In f i g u r e 3-10 an example of the records at a resonant frequency i s shown, Within a very narrow band resonant frequencies were recognized at which the a c c e l e r a t i o n r a t i o at the surface of the model with respect to the base of the container reached as high as 12, Fundamental frequencies measured by both the free v i b r a t i o n method and the resonant method are shown i n Table 3-2 and Figure 3-11, with void r a t i o of models. Figure 3-11 appears to be very scattered, e x p e c i a l l y at low void r a t i o , however, considering the fa c t that the scale of the frequency coordinate i s large and that the range of s c a t t e r i s about 15%, i t i s not too bad. Void r a t i o s shown i n Table 3-2 and on 36. the v e r t i c a l coordinate i n Fig,3-11 are i n i t i a l average void r a t i o s of the model. Because the maximum amplitude of a c c e l e r a t i o n of the applied s i n u s o i d a l base motion was at most 0,20, as recognized from Fig,3-8, the void rc.tio of the model would hardly change during the t e s t . Therefore these void r a t i o s can be considered as those during the t e s t s . Figure 3-11 shows that the fundamental frequeiicies of models were approximately 140 cycles per second for the dense 6" high model, 130 cycles per second f o r the dense 9" high model and 90 cycles per second f o r loose models, and that the fundamental frequencies f o r the 9" high model were s l i g h t l y lower than those for the 6" high model at the same void r a t i o . Using Equation 3-26, shear wave v e l o c i t i e s were c a l c u l a t e d and p l o t t e d i n Figure 3-12 with void r a t i o of models. This Figure shows that the average shear wave v e l o c i t y of the model increases as the void r a t i o decreases and the height of the model increases, and that those values were 160~230 and 200~290 feet/sec. f o r 6" high models and 9" high models r e s p e c t i v e l y . These r e s u l t s as discussed above can be i n t e r p r e t e d from the following f a c t s ; the fundamental frequency, f Q , f o r a h o r i z o n t a l s o i l l a y e r i s given by (see Eq,3-8) i , e , the fundamental frequency i s i n v e r s e l y proportional to the height of the model. Also, according to Hardin and Richart (1963), the shear wave v e l o c i t y of s o i l s , L7s } decreases 3 with void r a t i o and i s proportional to /10 power of the confining pressure, <r , as shown in-Eq,3-27, Therefore the r e s u l t s shown i n Fig,3-11 shows that the e f f e c t of the height of the model on the fundamental frequency was greater than that of the shear wave v e l o c i t y which i s associated with void r a t i o and confining pressure, Because the confining pressure, i s l a r g e r i n the higher model, r e s u l t s shown i n TM 3-2 FWSOA/lBA/mL FR£QL/BA/£Y a) /?&4J!>s)a/7i'- Method-far 6" rf/jh /tfodeT? 7 2 U5> e. i 0 (CPS) (FEBT) V5}J> P-3/-4 . 60s /2?^0 .53& /.7o5 2/6—B 4./5 $•2.0^52.5 • 56> A35-6 . 521 A 63 H?^9 4./0 $53-55.8 7-2-2 .55 /34~5 .578 /.67 J*4~5 4.09 54.1^55.(0 7-3-/ .525 /4a-f .5// A 655 232-4 4.0$ 573 ~ 57.1 7-3-2 . -57 /.42~"5 .505 A 64 23>^5 4.0S $7.5-518.0 J?) /-he*, irzlrczfyow Srte-tfpJ -/or 6>" -&2g7? *H0<P7&£ &-/<?-/ . 68 90.2 .562 /•76 /S0S 4,26 37.2 P-20-2 .5-0 A3B .50/ /.63 225 4.02 56.Q P-JO-3 .455 W.2 .555 /.74 /22.5 27.22 4D.B P~zb-4- • 595 s27.5 /.70 2'7. 4./4 62. 7 D-20-5 • 55 /3S .5/9 A 67 J230 4.o9 56.2 D-13-J • 63 s/5 • 545 /. 72 /9B 4.20 47- O 27-13-2 .495 /33 .499 /. 63 2/& 4.o2 54.2 .4.27. A39 ' .489 A67 224 3.97 56.S • 55 /32 . 5/7 A &7 220 4.09 53.1 72-3D-2 .5/ /30 .7o5 A 64 273 4-05 52. 5 77-3/-/ • 49 /37 . 498 A 625 2/3 4.00 53.2 P-3/-2 .4g /3B .495 A 62 224 3.W 567 Z>-3/-3 .65 /o4-.5 .552 /. 74 /&2 4.22 4.3. f Z>-?/-4 .6o5 / J>7. 5 . 536, A 7#$ 2/8 4'5 52.5 J--2-' .76 /2B . 521 A.68 2/5 4./0 57.0 7-2-2 . 55 /3? . 57Z /.67 233 4.69 53.5 J--3-/ .525 /37 . $/o A657 2/7 4.05 58. O 7-3-2 .7! /45 . 5o5 /.64 235 4-oS 77.8 7-3-? . <Z77 /4/.7 . 49S A 6 2 237 3.99 53.5 C) P&j-o/iait /rje/fiod -for rf" d/jh MO. e CCPS) 2 Vh 17* IS A 7-3^4 .470 .1/9 2.02 24o ^  2£r8 4.8/ 49-3^ $35 7-9-1 . 630 92 . 80 2./2 /9S S.ol 38.4 7-9-2 . 53 S 2 0 . 175 2.09 5. DO 4-9-sro.2 7-/1-/ .53 S2S .776 2.o? 2&Z 5.O0 $-25 J - / / - ? .525 s40 . 704 2.o5 287 4.39 S&.T 7-/1-3 .50 s35 . 736 2.o4 275 4,87 $6.5 7-//-4 .485 /4/~-243 .729 2.o3 4.34 590^ <T9.8 <d) Free u/brn-f/o/) weft/ad -Toy <?" fi/yb mpc/el 7-3-4 .490 /28 .7; 9 2.o2 258 4.81 53.5 7-9- f . 630 S8.7 .Boo /SB 5.07 37.2 7-tt-f .380 yo7 • 775 2.0? 224 5.00 44.8 J--/1-Z .525 /30 .744 2.o5 267 4.89 54.5 7-//-3 • 50 .736 2.o4 286 4?87 S£.6 7- //- 4 .485 /a3 .778 2.03 2 90 4.84 60.O /&££ WBt?GT/0/L> METHOD, £ " AfOPEL " 9 "MOOB-L. R&SOMA/T METHOD , rtOOZL „ *? " MODJrL O o e P CO O O Q 9o /OO •f. (CP.?) /30 7^ 3-/f 7=C/A/OAM&VfAL FREtQUBA/CY /40 3-/z S H Z A R mva \/&iocf7Y 40. Fig*3-12 are reasonable f o r the range of low void r a t i o s . I t i s i n t e r e s t i n g to note that for high void r a t i o s , the e f f e c t of height of the model on the shear wave v e l o c i t y i s not d i s t i n c t , and both the 6" high mode; and 9" high model give the same shear wave v e l o c i t y f o r a given r a t i o , A s u b s t a n t i a l number of laboratory tests have been c a r r i e d out to measure the dynamic moduli of s o i l s , Hardin and Richart (1963) have reviewed these previous works f u l l y . From these works the shear wave v e l o c i t y of various dry sands i s p l o t t e d i n Figure 3-13, with the' r e s u l t s of t h i s s e r i e s of t e s t s . According to Hardin and Richart (1963) the shear wave v e l o c i t y of Ottawa dry sand f o r small amplitude of s t r a i n may be given by = (//9 -S6 e ) 3 - 2 7 where (r~ i s the confining pressure l e s s than 2000 psf. Harding and Richart used c y l i n d r i c a l specimens tested by the resonant method i n t o r s i o n a l v i b r a t i o n under h y d r o s t a t i c confining pressures. In Figure 3-13 equation 3-27 was extrapolated to very small c o n f i n i n g pressures, although the range of confining pressures of t h e i r experiments was 500~2000 psf f o r t h i s equation, A r a i and Umehara (1966) have c a r r i e d out a s e r i e s of very s i m i l a r t e s t s to those reported here. About 17' long by 5' wide models, whose heights were 1,25' f o r loose state and 1,3', 2,0' and 2,7' for dense s t a t e , were tested by a shaking table using s i n u s o i d a l base motion, A r a i and Umehara's r e s u l t s are also p l o t t e d i n Fig,3-13, The agreement of r e s u l t s with previous works i s very good except that the value for e » ,68 as recognized i n Fig,3-13, 41. HARD/A/ AVO R/&/ART 0$£3), DTTAU/A SAA/D , DRY. o-o-o- JIDA £rR/$D£-JD QUART? SAAJD, 6'= CTA/^A/OU/A/; uT=/4.g% ,<r.TO Vf. oT Sp&c/H&v _ k//t30/J AW M/JULER (/?<£2), MED.F/ME, CtEA-A/ SAA/O €=.7/ uT^ /3.2jg fj tfYVROSTAT/C • Q ARAI A/v>£> UrtEHfr&A OV£6)y 7?VE ErRA//j£D PAY, (T7Cue- 7V u/T. OE SPBCIM&A/ @ RESULTS of Trf/S RBSEARCH OB7A/A/EP 7/J 7Hf5 5ECT/OA/ © RESULTS OT TH/S RESEARCH OT3TflUVE& FROM FZE&U&A/CY- f?ESPO/JSe CURVE '7v'f 3-/3 SHEAR WAVE t/E-LOUTY oT- FXE-WOUS U/ORKER5 AA/D T/I'/S RESEARCH 42. In Fig,3~12, wave v e l o c i t i e s are also p l o t t e d f o r the model calculated from equation 3-27 for 0~- « 27.5 and 48 psf, which corresponds to the overburden pressures at the mid-height of 6" and 9", r e s p e c t i v e l y . I t i s recognized from the fi g u r e that shear wave v e l o c i t i e s i n the range of void r a t i o s l e s s than 0.6 agree reasonably with the equation 3-27, but f o r void ra.-.ios greater than 0,6 the wave v e l o c i t i e s obtained here are much less than those calculated by the equation 3-27, One possible reason f o r this discrepancy i s presumably the f a c t that the (T i n the equation 3-27 i s a hy d r o s t a t i c confining pressure i n the range of 500 p s f ~ 2000 psf, while i n t h i s t e s t the confining pressure i s due to the over-burden pressure of the sand i t s e l f and i s very small. As mentioned i n the previous s e c t i o n , i t i s found on Figure 3-13 that the shear wave v e l o c i t y (108 feet/sec,) obtained from the frequency response curve (Figure 3-7) was much lower than the others (230^-280 feet/sec.) Fig,3-12. One possible reason i s presumably the magnitude of s t r a i n . I t i s known that the shear wave v e l o c i t y decreases as the amplitude increases, Drnevich, H a l l and Richart (1967) have reported some tes t r e s u l t s regarding the e f f e c t s of amplitude of v i b r a t i o n on the shear modulus of sand, Zeevaert (1967) has tested some Ottawa sand specimens by the free v i b r a t i o n method, i n which he used rather high amplitude of s t r a i n because of the nature of h i s apparatus, and obtained s l i g h t l y low shear wave v e l o c i t i e s . In these t e s t s , c y l i n d r i c a l specimens were tested In t o r s i o n a l mode and the s t r a i n was expressed i n terms of radian, therefore i t i s d i f f i c u l t to compare d i r e c t l y . I t i s i n t e r e s t i n g to note that A r a l and Umehara (1966) have obtained a shear wave v e l o c i t y c f about 570 feet/sec. In dense state and 400 feet/sec, in. loose state i n the small s t r a i n range (5x10 , but i n both cases i t decreased continuously as the shear s t r a i n increased and at a s t r a i n of lx.10-3 i t reached as low as 100 feet/sec. The amplitude 43. of t -train at the resonant frequency i n the frequency-response curve was 4>0x10 , while i t was around 4,0x10 i n the free v i b r a t i o n method, and those of the resonant method varied i n each t e s t but were i n the range of around 1.3x10 Therefore, the shear wave v e l o c i t y obtained from the frequency-response curve can be smaller than that obtained by other methods 4 Another reason may be the shape of the base motion. As in d i c a t e d i n Table 3-1, the shape of the base motions which were supposed to be s i n u s o i d a l was not quite so, e s p e c i a l l y i n the range of high frequencies, Therefore, i f the response spectrum of the base, motion which was ac t u a l l y applied to the model was d i f f e r e n t from that of a s i n u s o i d a l base motion, then i t i s possible that the measured frequency response curve w i l l be d i f f e r e n t from that predicted by a theory, i n which a si n u s o i d a l motion was assumed. In Figure 3-14 dimensionless qu a n t i t i e s — = I—3L. are p l o t t e d against void r a t i o , e. I t cannot be sa i d d e f i n i t e l y because the data are not good or extensive enough, but i t looks as though / <5~ 1rf) i s constant f o r a given void r a t i o i n the range of low void r a t i o as f a r as the data obtained here i s concerned. It follows that an average shear modulus of dry sand l a y e r i s proportional to i t s thickness. This means the d i s t r i b u t i o n of the shear modulus of a dry sand layer i s also l i n e a r with respect to the depth. According to Hardin and Richart (1963) predominant factors of the shear wave v e l o c i t y of dry sand are void r a t i o , e, confini n g pressure, cr , and s t r a i n amplitude, f , and other factors such as grain s i z e and shape and frequency have minor e f f e c t s , and the dynamic shear modulus i s 11~12% higher than, s t a t i c , . © O n O e FX££ V/&mr/PA>/iE7rtODj <L '' A70DBL O * <f" /10OSL-o a o O O ES3 O © a ea O 30 5£ - A 55 6o &j-3-/4 e- J 2 L §.3 s I-' A/&&TOF weogoA/ MOP&L ^  6 waves o Cl •5 -6 -7 yo/D RAT/Oe Fty 3-/S- LO$-A/?/77irifC DEOZECBAfT As i s shown on Figure 3-15, the logarithmic decrecent observed i n t h i s s e r i e s of tests was i n the range of 0,1~0,2, No s i g n i f i c a n t e f f e c t of void r a t i o of the model was observed, Thesewalues give an equivalent damping r a t i o i n the range of 0,02-^-0.03, i , e , 2-^3%, Although the frequency and confining pressure are quite d i f f e r e n t , the order of the logarithmic decrecent agrees with Hardin and Richart's r e s u l t s (1963), A r a i and Umehara (1966) have, obtained a value of the damping r a t i o less than 0,1 f o r the small s t r a i n and 0.4 at s t r a i n of _3 1x10 . According to Hardin (1966) , the logarithmic decrecent increases as the amplitude of s t r a i n increases and the c o n f i n i n g pressure decrease 3-6. Summary and conclusion. Using the shaking table and a s o l u t i o n of the equation of motion of a f i n i t e s o i l l a y e r subject at i t s base to a h o r i z o n t a l motion, a s e r i e s of t e s t s x^ere c a r r i e d out i n order to observe the behaviour of the h o r i z o n t a l dry sand model during v i b r a t i o n and to measure the dynamic p r o p e r t i e s . The t e s t i n g technique used herein has the following advantages: (1) The stresses i n the model are overburden pressure and k condition o xtfhich represent the stress conditions of s o i l i n s i t u , (2) The predominant v i b r a t i o n of the model i s a h o r i z o n t a l shear v i b r a t i o n x^hich i s an important component of the earthquake motions of the ground, and which governs the v i b r a t i o n mode of dams, (3) Because the model i s considerable l a r g e , the e f f e c t s of detectors are small, Conclusions obtained are as follows: (1) The behaviour of the model durin.g v i b r a t i o n agreed with that predicted by the e l a s t i c theory, and the s o l u t i o n f or the equation of 46. motion of the model which takes boundary e f f e c t s i n t o account was confirmed to be v a l i d . But the f a c t that the sand has a non l i n e a r modulus, namely that the shear modulus depends on the amplitude of s t r a i n , has been observed, (2) For the void r a t i o l e s s than ,6, the obtained shear wave v e l o c i t i e s agreed with those extrapolated from r e s u l t s by -Hardin and Richart (1963), taking overburden pressure as confining pressure, but for the void, r a t i o greater than ,6 the r e s u l t s were quite d i f f e r e n t . Therefore, when the shear wave v e l o c i t y of the ac t u a l ground or some s o i l structures are estimated, the value of the k w i l l be very important, * o ' (3) If the void r a t i o of a dry sand layer i s a constant and i s given, then the shear modulus of the layer i s l i n e a r l y p r o p o r t i o n a l to i t s depth. This conclusion i s n o t . d e f i n i t e because of the scattered data and i n s u b s t a n t i a l number of data with respect to height of models,, (4) Damping r a t i o of dry sand under low confining pressure and frequency i n the range of 100~140 cps was around 2~-3%, ' 47. CHAPTER.4 STABILITY OF MODEL SAND SLOPES DURING VIBRATION 4T1. Introduction The s t a b i l i t y during an earthquake of an earth or rock f i l l dam, embankment, cu t t i n g of the e x i s t i n g ground or even e x i s t i n g n a t u r a l ground slope, i s one of the most important f i e l d s of s o i l mechanics. Slope s t a b i l i t y during an earthquake involves quite d i f f e r e n t mechanisms depending on the materials comprising the slope, which i s associated with the r o l e of pore water pressure, The f i r s t mechanism deals with slopes comprised of dry cohesionless material i n which the pore pressure does not a f f e c t the s t a b i l i t y of slopes, iThe second deals with slopes comprised of saturated f i n e sand i n which l i q u e f a c t i o n phenomenon becomes an important f a c t o r . The l a s t i s cohesive slopes. The mechanism of f a i l u r e and the strength c h a r a c t e r i s t i c s of these materials are completely d i f f e r e n t from one another. The convetitional method f o r the p r e d i c t i o n of the s t a b i l i t y of slopes comprised of dry material i s fundamentally the same as that of^ the s t a t i c slope s t a b i l i t y , except that h o r i z o n t a l i n e r t i a forces are taken into account. In p r a c t i c e , however, t h i s method involves the following • d i f f i c u l t procedures: (1) determination of the design acceleration of the earthquake (2) determination of the design strength of the material during the earthquake The s l i c e method by Mononobe and Takata (1936) (reviewed by Seed and Martin (1966)) and the f i n i t e element method of the response analysis of s o i l structures (Clough and Chopra (1966) , Finn (1966b) , Finn and Khanua (1966), I d r i s s and Seed (1967) and I d r i s c (1968)), have made 48, great contributions to the determination of design a c c e l e r a t i o n . Unfortunately, there has not been much progress i n the whole problem of slope s t a b i l i t y during earthquakes because r e l a t i v e l y l i t t l e i s known concerning the dynamic c h a r a c t e r i s t i c s of s o i l s . Although Seed (1966) (1968) has developed t e s t i n g methods and reported some r e s u l t s , other extensive i n v e s t i g a t i o n s are required i n t h i s area. The purpose of t h i s chapter i s to observe the mechanism of f a i l u r e of dry sand slopes and to in v e s t i g a t e the dynamic strength c h a r a c t e r i s t i c s of dry sand and to compare the r e s u l t s with e x i s t i n g t h e o r i e s . A f t e r the e f f e c t s of the sides and ends of the model cn the slope s t a b i l i t y were examined (section 4-2) a s e r i e s of preliminary tests were made i n order to get an idea of the behaviour of the model (section 4-3). Then the measurements of the accumulative displacement due to the base motion were made (section 4-4) , I t was found i n the preliminary tests that there were two d i s t i n c t c h a r a c t e r i s t i c slope angles of the dry sand f o r a given amplitude of a c c e l e r a t i o n . Therefore, two s e r i e s of tests concerned with these slope.:angles, which w i l l be defined i n each corresponding s e c t i o n , was made (se c t i o n 4-5, 4-6), 4-2, Consideration of the boundary e f f e c t s . Because the container has a r e s t r i c t e d width and length, these e f f e c t s should be examined i n order to determine i f the problem i s one of plane s t r a i n and i n f i n i t e slope. The consideration in section 3-3, Chapter 3, i s concerned with the boundary e f f e c t s on e l a s t i c v i b r a t i o n , which was associated with "forces due to e l a s t i c s t r a i n caused by the boundary r e s t r i c t i o n s , but i n this s ection e l a s t i c deformation w i l l be neglected and only the f r i c t i o n force between the sand and side walls and the confining force at the toe. of the model w i l l be considered, 49. Therefore, i n t h i s s e c t i o n , large i r r e v e r s i b l e deformations are taken into account while i n Chapter 3 the r e l a t i v e ' deformation between sand • and bourdaries were assumed to be zero 0 A few tests were made i n order to measure the f r i c t i o n between the p l e x i g l a s s side plates and the sand* The test procedure was as follows; a p l e x i g l a s s p l a t e , l ' square x 1/2" t h i c k , was embedded v e r t i c a l l y i n a h a l f - f o o t high sand l a y e r , under the conditions shown i n Table 4-1* Then the force required to p u l l the p l a t e out of the sand l a y e r was measured by weights through a p u l l y and a piece of wire attached to the top of the p l a t e , as shown i n Fig*4-1. Results are shown i n Table 4-1, In dynamic t e s t s , the plate was p u l l e d during v i b r a t i o n of an amplitude of a c c e l e r a t i o n of 0,4g and a frequency of 10 cycles per second ( s i n u s o i d a l ) , ^ f ^ 0 * s a P r°duct °f the f r i c t i o n c o e f f i c i e n t and the r a t i o ' o f l a t e r a l pressure to overburden earth pressure, assuming a l i n e a r l a t e r a l earth pressure d i s t r i b u t i o n with depth, Using the values of f r i c t i o n shown i n Table 4-1 the f r i c t i o n force between the p l e x i g l a s s walls of the container and the sand has been evaluated. It shows that the e f f e c t of the side f r i c t i o n w i l l be l e s s than 1% of the t o t a l f r i c t i o n force acting on the f a i l u r e plane when 1/4" deep f a i l u r e plane i s assumed, and even for 1-1/4" deep f a i l u r e plane the e f f e c t w i l l be only a few per cent. Thus, these errors can be ignored. Another component of the boundary- e f f e c t i s a toe confining force due to the r e s t r i c t e d length of the model. This e f f e c t has also been evaluated i n Appendix I I I , I t shows that the toe c o n f i n i n g force w i l l be l e s s than a few per cent of the t o t a l f r i c t i o n force acting on the f a i l u r e plane i f 1/4" thick f a i l u r e plane i s assumed, and i t w i l l be 50. K 7u6fe 4-1 FfZ/CT/OA/fiETweEA/ SA/SD A/SP PLEXl£rt-AS5 RWCE mueiReo Q fa SJa.&'c , -£oos--c sa/>4 Py/wm/c c/eos-e Sa/7#7 2.$/^ 3.3/ /bs 6.07^ 6.42 '*s p./z —a /4 A 37 ~ 0 . 4 O (2.23 ~-<?, 24 FAEXIG-JLASS 7*f' . u/e/^ f-tT \ v , fry <z-/ f7?/C7?&A7 r~/£ASui?EMEA7T TBST 51. several per cent for 1-1/4" thick f a i l u r e plane, In conclusion, the boundary e f f e c t s due to r e s t r i c t e d width and length of the model are so small i n terms of percentage of the forces that they can be ignored and the analysis can be represented as a pl^ne s t r a i n and i n f i n i t e slope problem, 4-3, Preliminary t e s t s . In order to obtain a q u a l i t a t i v e idea about the behaviour of the dry sand model slope during induced v i b r a t i o n , a s e r i e s of preliminary tests have been c a r r i e d out. The basic procedure of the t e s t s was as follows; about 700 l b s , of sand was placed i n the container to a compacted height of 6 inches. The sand was compacted by v i b r a t i o n with an amplitude of a c c e l e r a t i o n of 1.0'g and a frequency of 10 cycles per second i n the h o r i z o n t a l model, Various kinds of markers, f o r example transverse l i n e s on the surface, v e r t i c a l l i n e s on the p l e x i g l a s s side w a l l o r . t h i n columns i n s i d e the model, were used i n order to measure displacement or to observe the f a i l u r e plane. Dyed sand, which was made of the same sand as the: model by means of dying with blue ink, was used. The appropriate markers were placed a f t e r the model was compacted, then the whole mode was compacted again by v i b r a t i o n having an amplitude of a c c e l e r a t i o n of around ,7-^--.8g, i n order that the marker sand i t s e l f and the sand surrounding the marker, which was disturbed when the markers were placed, would have the same consistency as the remaining part of the model. A f t e r the model was prepared, i t was t i l t e d to a proposed angle (see Figure 2-2), Then various s i n u s o i d a l h o r i z o n t a l motions were applied by changing the amplitude of a c c e l e r a t i o n and the frequency. The slope angles of the model were also changed by 52. t i l t i n g . The behaviour of the model during v i b r a t i o n , the mode.-; of f a i l u r e and the f a i l u r e plane developed were observed« Then, a f t ^ r v i b r a t i o n was stopped and the model was taken down to the h o r i z o n t a l , the accumulative displacements were measured. The modes of the slope f a i l u r e of dry sand which have been observed' i n t h i s s e r i e s of tests are as follows; (1) progressive type of f a i l u r e ; s t a r t i n g with a l o c a l i z e d motion of p a r t i c l e s on the surface, •< the f a i l u r e propagates up the slope and outward throughout the e n t i r e surface of the slope. This type of f a i l u r e has been observed during tests on a steep, dense slope subjected to small amplitudes of a c c e l e r a t i o n , (2) f a i l u r e with d i s t i n c t shear planes; along a f a i l u r e plane or several f a i l u r e planes, a mass of material moves down as a u n i t . This type of f a i l u r e has been observed during the te s t s i n which a dense slope i s subjected to accelerations of large amplitudes, A d i s t i n c t c h a r a c t e r i s t i c of t h i s type of f a i l u r e i s to have a s i n g l e or compound f a i l u r e plane. In Figure 4-2 an example of the type (2) kind of f a i l u r e i s shown, (3) f a i l u r e xcithout shear planes; the displacement i s l a r g e s t at the surface and decreases gradually with depth, therefore there i s no d i s t i n c t f a i l u r e plane (see Figure 4-3), This type of f a i l u r e has been observed during te s t s on loose sand slopes, Bustamate (1965) has • reported types (1) and (2) modes of f a i l u r e but not mode (3), Another i n t e r e s t i n g point to be discussed regarding r e s u l t s of the preliminary tests i s the f o l l o w i n g ' t e s t . In order to get the shape of the f a i l u r e plane i n s i d e the model and examine the e f f e c t s of f i n i t e boundaries, a te s t was made by using 32 v e r t i c a l markers not only at the side walls but also i n s i d e the model. Measuring the displacement of the marker columns i n s i d e the model by gradually digging the sand around 4 3; 8 vb CN I Uj Si N I F/j 4-3 VBFDRtfffnOA/ OF l/ERT/CAL. .P/A.QKER AT THE SfPE K//1LL f&g 7H£. F3J TYPY miVPE 55.' the marker, displacements i i i s i d e the model were measured and the shape of the f a i l u r e plane i n s i d e the model was obtained. Results are shown i i . Figure 4-4, Test conditions are as follows; the slope angle <X =» 15°, the applied amplitude of a c c e l e r a t i o n , k = 0,73g the frequency S of applied motion, f - 10 cps and the number of cycles applied, N ~ 26 cycles . In. t h i s t e s t two f a i l u r e planes, the upper f a i l u r e plane at a depth of about 1/4" and the deeper f a i l u r e plane at a depth of about 1-1/4", were observed. These are denoted by "1st" and "2nd" r e s p e c t i v e l y i n Figure 4-4, Shaded areas on Figure 4-4 are considered as f a i l u r e zones, i n which the marker column p o s i t i o n could not be confirmed. From t h i s f i g u r e i t i s recognized that the e f f e c t of side f r i c t i o n a f f e c t s the shape of the deeper f a i l u r e plane, n>amely the depth of the deeper ("2nd" i n f i g u r e ) f a i l u r e plane adjacent to the side w a l l i s shallower than at the center (see Figure 4-4(b)), the. magnitude of displacement, however, i s almost constant throughout a section (see Figure 4-4(c), s e c t i o n d^-d^, or e^-e^), I t was also found that the magnitude of displacement at a l l sections i s almost the same, for example f o r the upper f a i l u r e plane ("1st" on the figure) i t i s about 5 inches and f o r deeper f a i l u r e plane ("2nd" on the figure) about 3-1/4". From these facts i t can be s a i d that the assumption of plane s t r a i n and i n f i n i t e slope i s v a l i d f o r the model tested as had been t h e o r e t i c a l l y concluded i n the previous s e c t i o n . 4-4. m Measurement of accumulative disjda_cejnent, A very important concept concerning the s t a b i l i t y of slopes comprised of dry sand during v i b r a t i o n has been suggested by Newmark (1963) and (1965) and a s e r i e s of r e l a t e d experiments have been reported by Seed and Goodman (1964) and (1966), As described p r e v i o u s l y , past p r a c t i c e a) /.OA/Sr/TUDWAL. SHAPE- Op PA/llSSE PAA/E 6 56. S/.OPE Amt-E, <*^/£-° fiQCEl E/SAT/OA/j • 7Z$3 tt£Q(/E/JCy, -f^/pcpS CYCIES Xl///, //=2&cycles t>) Tm.VSyEtfSS D15TPJ6Ur/OA/ Of- PJ5PLACEH&A/T b> C, A, e, C) Tf?A/JSVe#SE SHAPE &E F4ILL/PE plA//Z SerflM b,~bz 3-' <urt jerif 5-ecfy'on di ~d? 7 4-4 V/5PLA)CEM£A/r /WD sHAV op miURB PLAve 57. i n the design of slopes against earthquake forces has usually involved the simple computation of a f a c t o r of safety against s l i d i n g when an i n e r t i a - f o r c e , generated by the earthquake ground motions, i s included i n the s t c t i c s t a b i l i t y a n a l y s i s . During an earthquake, the i n e r t i a , forces may be s u f f i c i e n t l y large to drop the f a c t o r of safety f o r b r i e f periods cf time and permanent displacements may occur. The movements, however, w i l l be arrested when the magnitude of a c c e l e r a t i o n decreases or i s reversed. The o v e r a l l e f f e c t of a s e r i e s of large but b r i e f i n e r t i a forces may w e l l be an accumulative permanent displacement of a se c t i o n of the embankment, but once the ground motions generating the i n e r t i a forces have ceased, no further deformation w i l l occur unless there has been a marked loss i n strength by the deformations. Thus, s i g n i f i c a n t displacement may have occurred but the f a c t o r of safety of the se c t i o n a f t e r earthquake may be approximately the same as i t was before the earthquake. Therefore, the e f f e c t s of earthquakes on slope s t a b i l i t y should be assessed i n terms of the deformation they produce rather than the minimum f a c t o r of safety developed, I f the concept of y i e l d a c c e l e r a t i o n , k^, i s introduced, the deformation of slopes during earthquake can be evaluated as i s shown i n Figure 4-5, The concept of y i e l d a c c e l e r a t i o n i s defined to be the maximum ac c e l e r a t i o n which the slope can r e s i s t without any permanent deformation. In other words, an ac c e l e r a t i o n greater than the y i e l d a c c e l e r a t i o n induces permanent deformation i n the slope. I t i s derived as follows; consider t h e . s i m p l i f i e d model as shown i n Figure 4-6, I f the a c c e l e r a t i o n i s applied i n the d i r e c t i o n of angle G above the h o r i z o n t a l , the forces are re l a t e d as follows; by W = w 4_1 J/nC^-ci) 5in(qo°-t-&-f-cX- 4) ACCB1.-ERATIDA/ J-4-S MECHANISM Op P/SPLACEflEA/T < <5ooDH/W AMD SEED^ iq6£,) COA/0T/O,V AT FA/LUXE FOR - s i / P / A / ^ ELEMENT 59. i . e . ^ Sin (J,-*) • 4_2 where 0 i s angle of f r i c t i o n , W i s weight of the mass r e s t i n g on the slope, and CX i s angle of the slope, has i t s maximum value when 6 0 - (X and y i e l d s sin (0-a ), If 9 » 0 (when seismic force i s h o r i z o n t a l anJ at r i g h t angle to gravity) = tan (4 ^ 3 This equation i s usually applied i n the analysis of dynamic slope s t a b i l i t y . I t should be noted that the r e s u l t which Finn (1966a) has developed for y i e l d a c c e l e r a t i o n , considering the v i s c c - e l a s t i c e q u i l i b r i u m i n the material comprising the slope, shows that equation 4-3 can be applied f o r cohesionless material v/ithcut l o s i n g any g e n e r a l i t y , and i f the material has some cohesion component, the y i e l d a c c e l e r a t i o n i s given by where c i s magnitude of cohesion, P i s unit weight of s o i l , y^ i s the depth at which down slope v e l o c i t y reaches a maximum and 7^^7cos&-ts/>ic(Tan<fi, In appendix IV, the permanent, displacement of the slope i n one cycle induced by a s i n u s o i d a l displacement was t h e o r e t i c a l l y estimated, where the angle of i n t e r n a l f r i c t i o n of the sand was assumed to be f u l l y developed with no progressive f a i l u r e because i t has been found that the sand tested herein has no d i s t i n c t d i f f e r e n c e between the peak strength and the r e s i d u a l strength. The permanent displacement i n one c y c l e , u, i s given by 60 . where f - frequency i n terms of cycles per second, ^~=^(cos& + sincr tan 0), k i s the amplitude of a c c e l e r a t i o n of a h o r i z o n t a l base motion —1 k " and 1 ~ 7T - 2 s i n _v_ « A product of the square of the frequency k and the permanent displacement i n one c y c l e , f^u , i s independent of the frequency. In order to measure the accumulative displacement of the model and to compare r e s u l t s with equation 4-5, a s e r i e s of t e s t s was made. The procedures for the preparation of the model were exactly the same as those c a r r i e d out f o r the preliminary tests described i n the Section 4-3, except that f i v e dot type markers were used. Five small, about 1/4" diameter, dot type markers were placed at 1 f t , i n t e r v a l s along the center l i n e on the surface of the model. A f t e r the model was t i l t e d to a slope angle of 15 degrees, a s i n u s o i d a l v i b r a t i o n was applied which had a frequency of 10 cycles per second and a p a r t i c u l a r amplitude of a c c e l e r a t i o n which was changed for each t e s t . The number of cycles applied during v i b r a t i o n was counted. A f t e r the v i b r a t i o n was stopped, displacement of the markers was measured and an average was taken. This accumulative displacement was divided by the number of cycles applied and the permanent displacement i n one cycle was obtained. Results i n terms of the permanent displacement in"one c y c l e , u , and the products 2— of the square of the frequency and the displacement i n one c y c l e , f u , are p l o t t e d against the amplitude of a c c e l e r a t i o n i n Figure 4-7, The measured r e s u l t s are l i s t e d i n Table 4-2, The t h e o r e t i c a l values c a l c u l a t e d by Equation 4-5 are also shown on Figure 4-7. The value on the v e r t i c a l coordinate i n F i g ,4 - 7 , i . e . k = ,51g and x - 0 , was taken N TLiWe 4-2 l/ST OF D/SpiAc&MEA/T . /7E6SUJ?EME*/T TEST 7EST A/0. 7/OV20A/T/4-A7o. OECYCIES TOTAL Z77SPL4CEM&/T 8 A S E ACC. APPLIED PJSP16CB4EA/J W 6A/B CYCLE . $34- &J 2. 3 O'nches) . 07-37(7»c7jes » 2. . <790 30 4.2(7 ./4/5 - 3 . 652 3f 6./P • 7 99 " 4 . 7 0 S 24 6-40 . 22 D « 5 . - 7 S 8 32 S.35 • 26 f " 6 .8S4 26 7.90 . 3 0 4 '< 7 . 930 20 7-'f . 3ZS AA/4L£ OT- SLOPE J c2=/jr 63. from the r e s u l t of the measurement of the c r i t i c a l angle of slope (see Fig,4-8 i n the next s e c t i o n ) . From t h i s f i g u r e i t i s found that the y i e l d a c c e l e r a t i o n , k^, or the mobilized angle of i n t e r n a l f r i c t i o n , 0, depends on the amplitude of ac c e l e r a t i o n applied. For small accelerations 0 was as high as 42 degrees; then 0 decreased with inc r e a s i n g amplitude of a c c e l e r a t i o n . For amplitudes of a c c e l e r a t i o n greater than 0,6, 0 reached a r e s i d u a l value of 27 degrees. I t i s i n t e r e s t i n g to note that, f o r amplitudes of ac c e l e r a t i o n l e s s than 0,6, the r e s i d u a l angle of i n t e r n a l f r i c t i o n was not reached, even though t o t a l displacemeixt was as high as 2,3 inches which i s large enough to reach the r e s i d u a l angle of i n t e r n a l f r i c t i o n i n the ordinary shear t e s t . From t h i s evidence the f a i l u r e mechanism on the f a i l u r e plane i s presumably as follows; u n t i l the amplitude of a c c e l e r a t i o n of base motion reaches the y i e l d a c c e l e r a t i o n , k^, of the model, no permanent displacement w i l l occur,bbut once the amplitude of applied a c c e l e r a t i o n exceeds k^ f o r some period of time, the mobilized angle of i n t e r n a l f r i c t i o n decreases and reaches i t s r e s i d u a l value. A f t e r the motion has ceased or reversed i t s d i r e c t i o n , however, the void r a t i o of the sand on the f a i l u r e plane becomes denser than that during f a i l u r e and the angle of i n t e r n a l f r i c t i o n increas-es again. This procedure i s repeated f o r each cycle of base motion. Therefore, the observed angles of i n t e r n a l f r i c t i o n are average values i n one c y c l e , which i s presumably a function of the displacement i n one cycle rather than t o t a l displacement, .For example, f o r the amplitude of ac c e l e r a t i o n 0.53g (see Table 4-2) the angle of i n t e r n a l f r i c t i o n did not reach the r e s i d u a l value because the displacement i n one c y c l e , 0,064 inches, was not large enough even though the t o t a l displacement was 23 inches. In the preliminary te s t the deeper f a i l u r e plane, about 1-1/4 inches, was observed beside the t h i n f a i l u r e plane, which i s shown i n Fig.ire 4-4, The same analysis as discussed previously was applied to t h i s deep f a i l u r e plane. In the instance when more than two f a i l u r e planes occur i n the model, equation 4-5, however, cannot be applied except to the shallowest f a i l u r e plane, because the f r i c t i o n force transmitted from the upper f a i l u r e plane a f f e c t s the displacement of the lower f a i l u r e plane, i herefore, assuming the sand has r i g i d p l a s t i c properties and the model comprised of s l i c e s of sand, and following Penzien (1960) or Finn and Byrne (196S) , the mobilized angles of i n t e r n a l f r i c t i o n i n the model shown i n Figure 4-5 were c a l c u l a t e d from i t s measured displacement with the use of the computer. Results are as follows; along the deeper f a i l u r e , i , e , 1-1/4 inches deep, the mobilized angle of f r i c t i o n during f a i l u r e was 32 degrees. The angle of i n t e r n a l f r i c t i o n , which must be mobilized i n order to r e s i s t the applied amplitude of a c c e l e r a t i o n , 0,725g, was calculated at j u s t below the actual f a i l u r e plane where any permanent displacement has not taken place. The r e s u l t was 49 degrees. These angles of i n t e r n a l f r i c t i o n are l i s t e d i n Table 4-3 with r e s u l t s of the vacuum t r i a x i a l t e s t . TABLE 4-3. LIST OF ANGLES OF INTERNAL FRICTION COSJD/T/OA/S Cafai/tzted -fto/n displacement nt s/z/if/ice. -fa/fore, plane ffn~o 27° '/ at deep 7-al/ure p/a/ie 07j=.l 32° — — • / * at" lasf J?e/t>ur -Me deeper -/nu'/ut-e t>fa/ie Vacuum '-/rt'axiaf T W - tr% = p s L e^-^o 4-75° <r, =/o. p s i e = -53 3S.7" m - '?.5*si e = . 64 03 = 5. p s i e -.77 35.6° According to Finn (1966) , i n dense f r i c t i o n a l material the angle of f r i c t i o n , 0 , mobilized during s l i d i n g down the slope w i l l be l e s s s than the angle of f r i c t i o n , 0 „ mobilized at; i n i t i a t i o n of y i e l d at-° ' mB J the l a t t e r includes the e f f e c t of d i l a t a t i o n and the material moving down a slope i s l i k e l y to occur at the c r i t i c a l void r a t i o or constant volume. From these postulations the r e l a t i o n s h i p between angles of i n t e r n a l f r i c t i o n i n Table 4-3 can be explained. The f i r s t two values, i . e . 27° and 32°, correspond to 0- , Moreover, 27° may be representative of the c r i t i c a l void r a t i o under almost zero c o n f i n i n g pressure at 'the surface of the model and 32° may be representative of the c r i t i c a l void r a t i o under about 0,1 p s i , and confining pressures of the t r i a x i a l test specimen were much greater than those on the f a i l u r e plane of the model. Therefore, the angles of i n t e r n a l f r i c t i o n c a l c u l a t e d from displacement of the model should be l e s s than the value obtained from * o . the t r i a x i a l t e s t , 49 corresponds to 0^  which agrees with the maximum angle of i n t e r n a l f r i c t i o n obtained by the t r i a x i a l t e s t of dense specimens and low confining pressure, considering the f a c t that the void r a t i o of the model, 0,47, was lower than that of the t r i a x i a l t e s t specimen, ,50 and the confining pressure of the model, . — 0.1 p s i , was l e s s than that of the t r i a x i a l specimen 2,5 p s i , 4-5. Measurement of stable__slope_._an_gle_.. As mentioned previously, i t has been found that there are two d i s t i n c t c h a r a c t e r i s t i c slope angles of dry sand f o r a given amplitude of ac c e l e r a t i o n of base motion. These are denoted by "the c r i t i c a l slope angle a " and the stable slope angle, ". The c r i t i c a l slope angle C • s i s defined as the maximum slope angle which can r e s i s t a given amplitude of a c c e l e r a t i o n of base motion without any permanent deformation. In 66. other words the slope-whose slope angle i s greater than the c r i t i c a l slope angle undergoes permanent deformation for a given base acceleration. The yielc acceleration, k^, the angle of internal f r i c t i o n mobilized at i n i t i a t i o n of yield of the slope, 0 , and the c r i t i c a l slope angle are, in general, describing the same phenomenon, and these are looking at the phenomenon from different aspects. A l l of them can be connected in one equation, which is T ^ y = -tan (<j>„ - CYc) The stable slope angle, however, is a different concept from those described above. It i s defined as follows; when a dry sand slope i s subjected to a base motion whose amplitude of acceleration i s greater than the yield acceleration of the slope, some permanent displacement takes place and the angle of slope decreases gradually u n t i l i t reaches a particular angle of slope, after which permanent displacements are no longer induced unless the amplitude of acceleration i s increased. This angle of slope at which the slope f i n a l l y comes to rest i s referred to as the stable slope angle. A couple of static t i l t i n g tests were made. The ssad was compacted by vibration in a horizontal layer then i t was t i l t e d gradually u n t i l failure took.place. The mode of failure in this test was the progressive type of failure which has been described in section 4-3 as the f i r s t type of mode of failure. The angle of the slope at which the failure started and the angle of slope at which the slope settled were measured. At 37 degrees the failure started and kept going u n t i l the slope came to rest at an angle of 30 degrees, 37 degrees of slope angle can be considered to be a good example of the c r i t i c a l angle of slope under zero amplitude of acceleration, and 30 degrees of slope angle as an example of i t s stable slope angle, also for zero acceleration amplitude. 67. In order to measure the stable slope angle two different kinds of test were carried out. The test procedure of one of the testt i s as follows; a slope of loose sand was built into the horizontal container as indicated in the sketch in Fig,4-8, Its i n i t i a l angle was 30 deg.-ees which was the repose angle of the sand, i.e. the static stable angle of slope, nhen the model slope was subjected to sinusoidal vibration of an amplitude of acceleration of 0,05g and a frequency cf 10 cycles per second. The vibration was continuously applied u n t i l the slope reached the stable angle of the slope. After the permanent movement of sand particles ceased.the angle of slope was measured. Then the amplitude of acceleration of the vibration was increased to O.lg, the model slope was again vibrated u n t i l i t reached the. stable angle, keeping the frequency constant, and the stable slope angle was measured. This procedure was continued up to ,6g amplitude of acceleration and a dozen of the stable slope angles were obtained, as shown in Figure 4-8, This test, however, involves the following problem; this series of tests was carried out sequentially with only one i n i t i a l model, therefore the conditions of each model (or void ratio) were controlled by the previous test. The results might be connected with each other as shown by the solid dots and curve in Figure 4-8, Another type of test was made to examine i f the stable slope angles obtained were unique and independent of the starting conditions, iThe test procedure is as follows; about 700 pounds of sand was placed in the container to a compacted height of 6 inches. It was compacted by vibration of an amplitude of acceleration of l,0g and a frequency of 10 cycles per second. Then the model was t i l t e d to a given angle of slope. 2 models were started from the slope angle of 30 degrees and 3 models were from 15 degrees, as shown in Figure 4-8. For each model a sinusoidal VJBBZotJ SfiUP STGPTE-P U ' l T H UPfSPITlOfiJ \ \ 1 _ l • I I I I .2 .4 .3 AO /)MPL/7VP£ OT ACCBUE&iT/Ofi/, f? (JJ smeu? S L O P E M f r L e 69. v i b r a t i o n of the indic a t e d amplitude cf a c c e l e r a t i o n shown In-F.!.g,4-3 was applied, at a constant frequency of 10 cycles per second, Th,*. v i b r a t i o n was continued u n t i l the slope s e t t l e d to i t s stable angle,, then the angle of slope was measured (indicated as open-circle i n Fig«4-8)« The f i r s t thing to be found from Figure 4-8 i s the i n d i c a t i o n that the stable slope angle may be independent of i n i t i a l conditions of the slope, such as the density and the i n i t i a l slope angle, and a function only of the applied amplitude of a c c e l e r a t i o n and perhaps the material comprising the slope. This i s presumably because, during the f a i l u r e induced by v i b r a t i o n , the i n i t i a l , conditions of the slope w i l l be completely destroyed and new conditions such as the void r a t i o , the slope angle and the arrangement of sand p a r t i c l e s , are formed, which are j u s t stable f o r the amplitude of applied a c c e l e r a t i o n , Lherefore, the void r a t i o s of the points along the curve of the stable slope angle'in Figure 4-8 are a l l d i f f e r e n t . In t h i s s e r i e s of tests the frequency of the v i b r a t i o n was kept constant, therefore the e f f e c t of frequency on the stable slope angle has been l e f t f o r future i n v e s t i g a t i o n s . In some tests the stable slope angle was measured along the sand surface 2 inches each side w a l l as well as along the center l i n e of the slope. According to these r e s u l t s the e f f e c t of the side f r i c t i o n was seen, namely, the stable slope angle along the. l i n e s close to the . side walls were somewhat steeper than along the center l i n e of the ' slope. The angles p l o t t e d on the. Figure 4-8 are a l l those measured along the center l i n e of the slope. 70. 4-5. Measurement of the c r i t i c a l slope angle. In order to measure the c r i t i c a l angle of slope, which was defined in the previous section, a series of tests was made. The testing procedure is as follows; the model, which was compacted by vibration at an amplitude of acceleration of l.Og and a frequency of 10 cycles per second in a 6 inch high horizontal layer, was t i l t e d to a given angle. Then, the amplitude of sinusoidal acceleration was gradually increased at a constavit while thecsurface of the slope was carefully observed. When a permanent displacement of sand particles on the slope was f i r s t recognized, the amplitude of acceleration of the applied sinusoidal vibration was recorded. These amplitudes of acceleration are plotted against the angle of slope in Figure 4-9, As is shown i n Figure 4-9 during this series of tests, two different kinds of behaviour of the slope have been observed. The f i r s t is the progressive type of failure which has been described in section 4-3, This type of failure was observed for slope angles greater than 20 degrees. The second i s as follows; the sand particles on the slope surface underwent an irreversible movement, but the movement stopped after they moved down a l i t t l e , and any change in the slope angle was hardly recognizeable, In the second type of behaviour, the slope presumably reached a different void ratio, or a different arrangement of sand particles on the slope surface which was made more stable by the sand particles moving down a l i t t l e . The permanent deformation of the slope in this series of tests was slightly different from the usual failur e , because the failure took place within the thickness of a few sand particles right on the surface of the slope. Therefore, an appropriate method of analysis must be sought. Because i t was, however, very d i f f i c u l t to describe the motion AS? —O TEST RBSUL 7"5, J=/OcpS THEORETICAL l/ALUE Za/7(43 ') NOI/&V&UT OE SiRFME PARTICLE: PPOtrfBSSED CAUSED COMPLETE PA/LURE AA/D A MARKED FL4TT&VTA/£T OE THE.SLOPE 71. MOl/BMSVrOE SURFACE PARTICLES 370PP&P AFTER OA/L-Y 3L/&ttT MCmEME*/T /ODHJ/J SLOPE DBA/SB H/BDRO// S/}AM) COMPACTED &YV/3RAVOAJ ' HoertOAWAL. Ar -ft" p=/OCp5 2 .<£ .6 •& - AMPUTUPB OP ACCBLERATWA/j £ (?) AO f/j 4-9 CPJT/CAL SLOPE AA/EFLT: 72, of the p a r t i c l e s i n such a condition, equation 4-3, i , e 0 ky = tan(0-A'), was.applied. In Figure 4-9 t h i s r e l a t i o n f o r 0 = 43° i s also p l o t t e d . I t i s found that the observed curve In Figure 4-9 agrees f a i r l y welil with the r e l a t i o n between k and & f o r 0 » 43° except i n the range of small slope angles. I t i s i n t e r e s t i n g to note that 0 « 43° obtained from dynamic te s t s was greater than 0 = 37° which has been obtained from the s t a t i c t i l t i n g t e s t s . The curve of the c r i t i c a l angle of slope i n Figure 4-9 depends on the void r a t i o of the slope and. the arrangement of the sand p a r t i c l e s on the surface of the slope, while the curve of the stable slope angle i n Figure 4-8 i s independent of these conditions of the i n i t i a l slope. In other words, i f other models with higher i n i t i a l void r a t i o are tested, the curve of the c r i t i c a l slope angle w i l l come below the curve shown i n Fig.4-9, Though the curve i n Figure 4-8 i s a l s o a c r i t i c a l ' slope angle s i n c e , i f the amplitude of a c c e l e r a t i o n i s increased from the point of the curve permanent displacement w i l l occur, each point on the curve of the stable slope angle i n Figure 4-8 has d i f f e r e n t conditions. I t i s i n t e r e s t i n g to superimpose these two curves,i,e, the curve i n Figure 4-8 and the curve i n Figure 4-9, But when these.two curves are superimposed, the part of the curve of the c r i t i c a l slope angle which comes below the curve of the stable slope angle has no s i g n i f i c a n c e from the p r a c t i c a l point of view. In other words, for t h i s part of the c r i t i c a l slope angle curve i f an angle of slope exceeds the c r i t i c a l slope angle no s i g n i f i c a n t permanent displacement takes place unless that angle exceeds the stable slope angle. In Figure 4-10 these two curves are superimposed, • They are-divided i n t o three zones, A, B and C, I f the r e l a t i o n between an amplitude of a c c e l e r a t i o n and a slope angle i s located i n the zone A, 73. o I :—-«— 1 1 —t a <? -^5" AtfPl/T(/D£ &P ACCEiapATfOA/y A ^ 7V^? ^-/O tfBLAT/OA/ BS-7WBBU /}//&££: P^AAJ AMPJ-(TL/D£'ACCELERA770A/ 74, the slope i s quite safe. I f i t i s located i n the zone C, permanent deformations w i l l n e c e s s a r i l y take place and the slope w i l l f i n a l l y s e t t l e on a point on the curve of the stable slope angle. I f the r e l a t i o n between an amplitude of a c c e l e r a t i o n and a slope angle i s located i n the zone B, the slope i s safe unless any pulse h i t s the curve of the c r i t i c a l angle of slope, but even i f only one pulse h i t s the curve of the c r i t i c a l angle of slope the f a i l u r e i s trigge r e d , and i t w i l l not stop u n t i l the angle of slope reaches a point on the curve of the stable slope angle, Bustamate (1965) has reported that i f the angle of slope i s steeper than the angle of repose of s o i l i n loose condition, the f a i l u r e w i l l not stop, even, when the earthquake motion has ceased, u n t i l the angle of slope reaches the angle of repose i n loose condition. This condition i s describing the hatched zone i n Figure 4-10, since, the angle of repose of s o i l i n loose conditions corresponds to the i n t e r s e c t i o n of the curve of the stable slope angle at the v e r t i c a l coordinate, namely stable slope angle. 4-6, Summary and conclusion, . From the s e r i e s of the dynamic slope s t a b i l i t y tests with a dry sand model, the following d i s c o v e r i e s have been made: (1) The boundary e f f e c t s of the model on the slope s t a b i l i t y were not very s i g n i f i c a n t . (2) The angle of i n t e r n a l f r i c t i o n mobilized during s l i d i n g down a slope i s l i k e l y to be a function of the magnitude of the permanent displacement i n one cycle rather than t o t a l displacement, For a large enough displacement i n one c y c l e , say 0.15 inches, the r e s i d u a l angle of i n t e r n a l f r i c t i o n mobilized during s l i d i n g was 27° at 1/4 inches deep f a i l u r e plane f o r the angle of slope 15 degrees. 75, (3) In dense f r a c t i o n a l materials the angle of f r i c t i o n mobilized during s l i d i n g down the slope was much l e s s than the angle of f r i c t i o n mobilized at i n i t i a t i o n of y i e l d . These were 32 degrees and 49 degrees r e s p e c t i v e l y at 1-1/4 inches deep f a i l u r e plane. (4) For the c r i t i c a l angle of slope, the r e l a t i o n of k = tan(0-<tf) i s l i k e l y to Le ap p l i c a b l e . The value of 0 c a l c u l a t e d from the r e s u l t s of dynamic slope s t a b i l i t y tests was greater than the angle of repose obtained by the s t a t i c t i l t i n g t e s t . These were 43 degrees and 37 degrees r e s p e c t i v e l y , 76. CHAPTER 5 SUMMARY AND CONCLUSIONS In osder to in v e s t i g a t e the fundamental dynamic properties of dry sand through model studies and. to compare, the behaviour of models during v i b r a t i o n with e x i s t i n g theories, the following two types of experimental' studies were performed; (1) the dynamic response of h o r i z o n t a l models and (2) the s t a b i l i t y of model sand slopes during v i b r a t i o n . A l l tests were c a r r i e d out on the shaking table using dry Wedron sand models housed i n an 8 foot long by 1-1/2 foot wide r i g i d container. Conclusions obtained are as follows: •5-1.. Dynamic response of h o r i z o n t a l models, (i) The t e s t i n g technique used herein has seve r a l s i m p l i f y i n g advantages, namely the stresses i n the model were overburden pressure with k condition and the predominant motion of the model was i n o . h o r i z o n t a l shear v i b r a t i o n , ( i i ) The behaviour of the model during v i b r a t i o n agreed with that predicted by e l a s t i c theory, and the s o l u t i o n for the equation of motion of the model which takes boundary e f f e c t s i n t o account was found to be applicable to the observed behaviour, ( i i i ) ' For a void r a t i o less than 0,6, the obtained shear wave v e l o c i t i e s agreed with those extrapolated• from, r e s u l t s by Hardin and Richart (1963), Experimental r e s u l t s tend to suggest that f o r a dry sand layer at a given constant void r a t i o the shear modulus of. the la y e r may be l i n e a r l y proportional to i t s depth. 77. (iv) Damping r a t i o of dry sand under low confining pressure and a frequency of v i b r a t i o n i n the range of 100 140 cps was measured to be about 2 3%. 5-2. S t a b i l i t y of model sand slope duving v i b r a t i o n . (I) The angle of i n t e r n a l f r i c t i o n mobilized along slippage plaices at r e l a t i v e l y shallow depths below surface of slopes during v i b r a t i o n may be a function of the magnitude of permanent displacement i n one cycle rather than t o t a l displacement. For a large enough displacement i n one c y c l e , say 0.15 inches, the r e s i d u a l angle of i?iternal f r i c t i o n mobilized during s l i d i n g was found to be about 27° at 1/4 inches deep f a i l u r e plane. ( i i ) The angle of f r i c t i o n mobilized during s l i d i n g down the slope was much less than the angle of f r i c t i o n mobilized at i n i t i a t i o n of y i e l d . These angles were 32° and 49° r e s p e c t i v e l y , along 1-1/4 inches deep f a i l u r e planes. ( i i i ) For the c r i t i c a l angle of slope, the r e l a t i o n of k => tan(0- ) i s l i k e l y to be ap p l i c a b l e . iThe value of 0 c a l c u l a t e d from the re s u l t s of dynamic slope s t a b i l i t y tests was greater than the angle of repose obtained by s t a t i c t i l t i n g t e s t , or 43° and 37° re s p e c t i v e l y , (iv) I t was found that i t i s very dangerous to design a slope with an angle of slope greater than the repose angle of the material comprising the slope i n i t s loose condition. 78. REFERENCES A r a l , R, and Umehara, Y e (1966) "V i b r a t i o n of Dry Sand Layer". (Summary i n E n g l i s h ) , Japanese National Conference on Earthquake Engineering, 1966, Bemhard, R,,K, (1959) "On Determination of Dynamic C h a r a c t e r i s t i c s of S o i l I n s i t u " , ASTM Special Technical P u b l i c a t i o n No,254, 1959, Bastamante, J.L. (1965). "Dynamic Behaviour of Noncohesive Embankment Model", Proc. of 3rd World Conf, on Earthquake. Engineering, 1965, Chae Yan Suk (1968), " V i s c o e l a s t i c Properties of Snow and Sand", Journal of Engineering Mechanics D i v i s i o n , ASCE, No,EM6, Dec, 1968, Clough, R,W, (1958) "Earthquake Resistance of Rock F i l l Dam", Transaction of ASCE, Vol,123, 1958, pp.792. Clough, R,W. and Chopra, A.K, (1966) ."Earthquake Stress Analysis i n Earth Dam", Journal of Engineering Mechanics D i v i s i o n , ASCE, Vol.92, No.EM2, A p r i l , 1966, p,147. Drnevich, V,P.,'Hall, J.R.Jr,, Richart, F,E« (1967) " E f f e c t s of Amplitude of V i b r a t i o n on the Shear Modulus of Sand", Proc. 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Greenfield, B»J, and Misiaszek, E,T, (1966) "Vibration-Settlement C h a r a c t e r i s t i c s of Four Gradation Ottawa Sand", Proceedings of Japan Earthquake Engineering Symposium, Oct,1966, Tokyo, H a l l , J.R, and Richart, F,E, (1963) " D i s s i p a t i o n of E l a s t i c Energy i n Granular S o i l s " , Journal of Soil-Mech. and Foundations D i v i s i o n , ASCE, Vol,89, No,SM6, November 1963, Hardin, B,D, (1965) "The Nature of Damping i n Sand", Journal of the S o i l Mech, and Foundations D i v i s i o n , ASCE, Vol,91, No,SMI, January 1965 0 Lee, Tang-Ming (1963) "Method of Determining Dynamic Properties of V i s c o - E l a s t i c S o l i d Employing Forced V i b r a t i o n " , Journal of Applied Physics, Vol,34, No,5, May 1963, Martin, G.R, and Seed, H,B, (1966) "An Investigation of the Dynamic Response C h a r a c t e r i s t i c s of Bon Tempe Dam, C a l i f o r n i a " , Report No.TE-66-2 to State of C a l i f o r n i a Department of Water Resources, 1966, Monohobe, N, and Takata, A, (1936) "Seismic S t a b i l i t y of the Earth Dam", Proceedings 2nd Congress on Large Dams, Washington, D,C,, 1936, Vol,IV. Morgenstern, N,R, and P r i c e , V,E, (1965),"The Analysis of the S t a b i l i t y of General Surfaces", Geotechnique, XV, No.l, P.79, March 1965, Newmark, N.M, (1963) "Earthquake E f f e c t s an Dams and Embankments", presented at the ASCE S t r u c t u r a l Engrg, Conf, San Francisco, C a l i f o r n i a , October 1963, Newmark, K,M, (1965) "Earthquake E f f e c t s on Dams and Embankments", Geotechnique, Vol,15, No,2, June-1965, Penzien, J (1960) " E l a s t o - p l a s t i c Response of Ide a l i z e d Multi-Storey Structure Subject to a Strong Motion Earthquake", Proceedings of 2nd World Conf, on Earthquake. Engrg,, Japan 1960, Hardin, B.O, and Richart, F,E, (1963) " E l a s t i c Wave V e l o c i t i e s i n Granular S o i l s " , Journal of the S o i l Mechanics and Foundation D i v i s i o n , ASCE, Vol,89, No.SMI, February 1963. Hatanaka, M, (1955) "Fundamental Consideration on the Earthquake Resistance Properties of Earth Dams", B u l l e t i n No,11, Disaster Preventive Research I n s t i t u t e , December 1955, I d r i s s , I.M, (1968) " F i n i t e Element Analysis for the Seismic Response of Earth Dams", Journal of S o i l Mechanics and Foundation D i v i s i o n , ASCE, Vol.94, No.SM3, May 1968, pp.617. 80, I d r i s s , I.M, and Seed, K.B, (1967) "Response of Earth Banks During Earthquake", Journal of S o i l Mechanics and Foundation D i v i s i o n , ASCE, Vol.93, No.SM3, May 1967, p,61. I d r i s s , I.M, and Seed, I I ,B. (1968) "Seismic Response of Horizontal Layers", Journal of S o i l Mechanics and Foundations D i v i s i o n , ASCE, Vol,94, No,SM4, July 1968. I i d a , K, (1938) "The V e l o c i t y of E l a s t i c Wave i n Sand", B u l l e t i n Earthquake Research I n s t i t u t e , Tokyo Imperial Univ., Vol.16, 1938, Jones, R. (1958) " I n - s i t u Measurements of the Dynamic Properties of S o i l by V i b r a t i o n Method", Geotechnique, Vol.VIII, 1958, Roscoe, K.H, (1968) "Rocha's Approach to S i m i l a r i t y Conditions i n S o i l Mechanics Model Test", The Journal of Stress A n a l y s i s , Vol,3, K o . l , January 1968, Seed, H,B. (1966) "A Method f o r Earthquake Resistant Design of Earth Dams", Journal of S o i l Mechanics and Foundations D i v i s i o n , ASCE, Vol.92, No.SMI, January 1966. Seed, K.B. (1968) "Landslide During Earthquake Due to S o i l L i q u e f a c t i o n " , Journal of S o i l Mechanics and Foundations D i v i s i o n , ASCE, Vol , No.SMS, September 1968. Seed, K.B, and Clough, R.W. (1963) "Earthquake Resistance of Sloping Co're Dam", Journal of S o i l Mechanics and Foundations D i v i s i o n , ASCE, Vol,89, No.SMI, February 1963. Seed, K.B, and Goodman, R.E. (1964) "Earthquake S t a b i l i t y of Slopes of Cohesionless S o i l " , Journal of S o i l Mechanics and Foundations D i v i s i o n , ASCE, N0.SM6, November 1964. Seed, I I ,B, and Martin, G.R., (1966) "The Seismic C o e f f i c i e n t i n Earth Dam Design", Journal of S o i l Mechanics and Foundations Da*.vision, ASCE, No.SM3, May 1966, pp.25, S e l i g , E.T. and Vey, E.E, (1965) "Shock Induced Stress Wave Propagation On Sand", Journal of S o i l Mechanics and Foundations D i v i s i o n , ASCE, No.SM3, May 1965. S t a l l , R.D, et a l (1965) "Shock Wave i n Granular S o i l " , Journal of S o i l Mechanics and Foundations D i v i s i o n , ASCE, No,SM , Jul y 1965, Taylor, D,W, and Whitman, R,V. (1954) "The Behaviour of S o i l s under Dynamic Loading", 3rd F i n a l Report on Laboratory Studies, Report to O f f i c e of the Chf, of Engrs,, Dept, of C i v i l and Sanitary Engrg,, S o i l Mechanics Lab., AFSWP-118, 1954. Wilson, S.D, and M i l l e r , R.P. (1962) "Discussion of Foundation V i b r a t i o n " , by F.E, Richart, J r . , Transaction, ASCE, Vol.127, Part I, 1962, Zeevaert, L, (1967) "Free V i b r a t i o n Torsion Tests to Determine the Shear Modulus of E l a s t i c i t y of S o i l " , Proc, of 3rd Panamerican Conf, on S o i l Mech. and Foundation Engrg,, Venezuela 1967, Vol I, p p . I l l , 81, Appendix- I - Solution f o r 3-dimensional mvibration. The governing equations of the v i b r a t i o n of a continuum with i n t e r n a l damping are as follows: where y\ = E^/^uX/~2//). , s/zCf+f)j E = Young's modulus, jU - Poisson's r a t i o , e ~ volumetric s t r a i n , yz- ^ . -/- &*~ -r , u, v and w - displacement i n the d i r e c t i o n of x, y and z r e s p e c t i v e l y , as shown on Figure => density ; Fx(t) , Fy(t) and Fz(t) are e x c i t a t i o n body forces i n . the d i r e c t i o n of x, y and z r e s p e c t i v e l y . I f the e x c i t a t i o n i s only i n x - d i r e c t i o n , Fy(t) and v can be ignored without l o s i n g any generality and, moreover, i t has been coil firmed by the f i n i t e element method that w can be ignored i f the e x c i t a t i o n force i s applied only i n x d i r e c t i o n . Let ISC j Us and 2 ^ be ^t2(r)/f> t/^"/j° a n c ^ r e s P e c t l v e l y > then the equations 6-1 are reduced to *'yp - ? ^ - & ' ^ 6 ' z The s o l u t i o n of the homogeneous equation of (6-2) can be obtained by the separation of v a r i a b l e s . Let u (x,y,z,t) = X(x) , Y(y), Z(z),T(t) and s u b s t i t u t i n g i n t o (6-2) y i e l d s D i v i d i n g through by' X.Y.Z.T and arranging z(ur) E;j 6~( COORD/A/ATE OF MODEL 83, Then j£<™.JJLCLTL_tf„M*jfe--^ (6-4) anc, V~7dX _ _ j 2 jj; cf^Y whereii and j are constants. Then the equation (6-2) can be divided into the following four eauations + <*>'-{') Z .0 (6-5) and The solutions f o r these equations are as follows* For the f i r s t equation and f o r fourth equation V(xJ = C sJn-L X-h C COS-4? XL Let us apply the boundary condition with regard to the x d i r e c t i o n , that i s C'=0 and --£r / = .-. j = - ^ ^ then hence And -for the third egctaf/o/o Y(yj = Vsin HEI^ + p ' ^ ^ E Z T y Let us apply the boundary condition with regard to the y d i r e c t i o n , that i s then D'~o and J/j5=Jlb = ?7? . ' . - f i ^ / : = LT3 * • *. * hence and f o r the second equation is* i/s Let us apply the boundary conditions with regard to the & d i r e c t i o n , that i s £/^D— O at the base then E ' - O and f.^u ) ~ 7Z.--0 at the surface then 77 Us -2 hence A l l constants i n the expression of X(x), Y(y), and Z(z) can be represented by A and B, then the complete s o l u t i o n of homogeneous equation i s •e "(Asintof^pi 13 costd//-ytJ ( 6-6a ) where Let us take a s i n wt as e x c i t a t i o n , and express t h i s in:terms g \ of t r i p l e Fourie S e r i e s , i . e . f g r r i Z t> 2 J •hhtn or f ^ ' ^ % r - : < ' >:i,r. .... <6-7) In order to f i n d the p a r t i c u l a r i n t e g r a l , l e t us s u b s t i t u t e f t r into the o r i g i n a l d i f f e r e n t i a l equation • - U/ ZZ S / 7 J ^ ^ ^ ' y ^ jrTp (<W + HnyCos bit) 86. Making t h i s r e l a t i o n i d e n t i c a l hy equating the c o e f f i c i e n t of EJ'xi,n 5/'"jJiE t c z e r o f or each p, q and r , and s u b s t i t u t i n g u> from equation (6- 6 b ) t pgr * ' * The c o e f f i c i e n t of coscot should be zero or and regarding the c o e f f i c i e n t of or (6-9) 2 ju>tont. Hp%h T (to r ) ^ f „ - f P Z y = o From (6-8) S u b s t i t u t i n g into (6-9) 7=w (6-8) then (j jU>(Ajf,th) -f- (kJx- U)rfr) Now the complete s o l u t i o n of the equation of forced v i b r a t i o n of the model with uniform shear modulus and i n t e r n a l damping i s ready, namely (6-10) The constants A and 33 are to be determined by i n i t i a l condition, pqr pqr Only steady state v i b r a t i o n i s of i n t e r e s t here. Then the expression of steady state v i b r a t i o n i s etc*. - Z 212 tfo-Zf-x sm^u sfo-tg-s '^^3 p f r * b J J-*> p%r-7j or Pit- t? J -2-;v (6-11) where <Pp<ir = Lav f (6-12) (6-13) Appendix I I . EfJ[g.c_tg...flj: Boundary Restraints on V i b r a t i o n . E f f e c t s of the r e s t r i c t e d length of the model compared with i t s depth and e f f e c t s of the r e s t r i c t e d width w i l l be evaluated and discussed here, also the preferable width of the model f o r the experiment concerning the v i b r a t i o n of the h o r i z o n t a l s o i l layer i s suggested. Resonant frequencies of the model can be evaluated by equation (6-13) i n Appendix I, i . e . R gives the r a t i o of the frequencies between a model having r e s t r i c t e d length and one with a s e m i - i n f i n i t e l a y e r . In Figure 7-1 R i s plotted "against ^ f which shows that i f the length and depth of a model are 8' and 1/2* r e s p e c t i v e l y , i , e , = , the erro r i n the fundamental frequency w i l l be less than 3% i f s e m i - i n f i n i t e layer analysis i s used, but i f the depth of the mode i s increased to 1', i . e . % = £ } t then the e r r o r w i l l increase to more than 10% and become quite s i g n i f i c a n t . Also, the error increases as Poisson's r a t i o and mode number increase. E f f e c t s of the r e s t r i c t e d width on the fundamental frequency and the response amplitude of the model were calculated and .shown i n Figure 7-2, In t h i s c a l c u l a t i o n , the length and depth of the model were f i x e d as 8' and 1/2' re s p e c t i v e l y and dynamic properties of the s o i l as stated on the. Figure were assumed, Figure 7-2a shows the d i s t r i b u t i o n of the acc e l e r a t i o n r a t i o on the transverse center l i n e at the surface with respect to the base for several values of b/h. 90. c?J D&TP/BVT/OAA e>TAccBA.E-pAr/CH/ /?A)T/0 8.00 AO /•3 A2 /•/ ACCEIERA T/OAV RA77P AT^Lf?FAC& WITH RESPECT TV 3 =. S'f-eef J = 0.03 \p- S0CP5 -2^  -21-Si §2 / 3 ° 1 I 1 /<?£> -//O /OP b ) rVA/DAMEMTAL PREQuEA/cr FiX /AJF/MFE MOTH AMP S^eL^ FtP fEFf/ JA/T/AJI TEIAYER"^ /£> // 7yf 7-2 £FF£^T cOF k/WFH OF MODEL I t i s found on the Figure that the 1-1/2' wide by 1/2' high model, i . e . *Vh = 3 t There are no regions where the ac c e l e r a t i o n d i s t r i b u t i o n i s uniform, but i f ^/h i s increased up to about 7, tl.e a c c e l e r a t i o n i n the middle t h i r d of the width w i l l be steady and not a f f e c t e d by side r e s t r a i n t . This f i g u r e shows that the maximum response increases as ^ /h increases. This i s because every curve has been ca l c u l a t e d f o r 50 cycles/sec, as a frequency of e x c i t a t i o n , however the na t u r a l frequency of the model decreases as the width of the model increases, as shown i n Figure 7-2b, the r a t i o of t%j„ , namely the r a t i o of frequency to the na t u r a l frequency of the model, i s not constant f o r each curve,, Therefore, the shape of the curve i s importaiit rather than the maximum amplitude. As mentioned pre v i o u s l y , ^/h r a t i o of a model greater than 7 i s preferable to carry out t e s t s i n v o l v i n g the response of s o i l l a y e r s . This value of /h reduces also the deviation of the fundamental n a t u r a l frequency from that of i n f i n i t e l a y e r , as i s seen i n Figure 7-2b, It i s i n t e r e s t i n g to note that according to Hatanaka (1955), wlfea the length of a dam i s about four times the height, the influence of end r e s t r a i n t has n e g l i g i b l e e f f e c t on the nat u r a l frequency of v i b r a t i o n and the magnitude of response i n the c e n t r a l region, hence the use of an analysis based on the assumption of i n f i n i t e length can be considered s u f f i c i e n t l y accurate f o r a l l p r a c t i c a l purposes. 92 Appendix III... Boundary E H e c t s on Permanent Displacement. There are two d i f f e r e n t types of boundary e f f e c t s on the r e s u l t s of s t a b i l i t y tests of model sand slope, the f i r s t i s the side f r i c t i o n and the second i s the end co n f i n i n g . Side E f f e c t s . From the t e s t of the measurement of f r i c t i o n s between the p l e x i g l a s s plate and the sand, the products of the f r i c t i o n c o e f f i c i e n t , and the c o e f f i c i e n t of h o r i z o n t a l earth pressure, C rk , are as follows r ' f o' (see section 4-2), S t a t i c , dense - C rk » .372 ~ ,396 - ,384 ' f o dynamic, dense CJc =» ,231 ~- ,244 =*• ,238 • * f o Consider a section of model as i s shown i n Fig,8-1, In f a c t , the f a i l u r e plane was not s t r a i g h t but a curve as shown i n Figure 3-1 (see Figure 4-4), The assumption of h o r i z o n t a l s t r a i g h t l i n e w i l l give a s l i g h t l y l a r g e r r e s i s t a n c e , but would not make a large d i f f e r e n c e Then, a s t r a i g h t l i n e i s assumed here as the f a i l u r e plane. Then the forces a c t i n g on the f a i l u r e plane are shown i n Figure 8-2, On the other hand, the f r i c t i o n force on the w a l l s , Fw, i s F« = i rtQ*. *z = rt*c,A0 (8-D S u b s t i t u t i n g the values for E = 1,5', f ra 113 psf and C^k ,238, the f r i c t i o n force acting on the f a i l u r e plane, F^, i s p^ '= /.5*//3 Acos&J-ao& = /70 ficos<?<teo<fi (8-2) SURFACE • ACTUAL FlAA/5 ASSUMEP FAILURE PLO/JE ~/f 8 - / A SECTION OF THE-MODEL. I t P~/J S~Z POPXES ACTfA/fT OA/ PA71 L/J?E PIP/7B and 7v= //3 x 0.2386* «= z&f*1 (8-3) Then the percentage of Fw with respect to F (8-4) /TO co5tX-fo/l<p' For p a r t i c u l a r values of cx and 0 , say of ~ 15 and 0 = 30 (8-5) /?o* .466X.S-77 S u b s t i t u t i n g h ~ 1-1/4 inches and 1/4 inches, the e f f e c t s i n terms of the percentage of forces are 3% and 0,6%, These are small enough to be neglected. End co n f i n i n g . I f the force acting on the toe of the f a i l u r e plane i s assumed as i s shown i n Figure 8-3, then the following r e l a t i o n i s obtained from the force polygon. where F i s the confining force at the tow, 9 i s an angle between the surface of the slope and the f a i l u r e plane, 0 i s the angle of i n t e r n a l f r i c t i o n , 0' i s the angle- of f r i c t i o n on the toe wedge as i s shown, in"Figure 8-3, f- tan ^  k,!-7 i s the total-weight of the toe wedge, Vf -f- fiw (8-6) Co S(0r <£-<£>') and k i s the c o e f f i c i e n t of earthquake. Then (8-7) 95. Hence the component of the F £ p a r a l l e l to the surface of the slope, P, which i s of i n t e r e s t , i s given by p=([V+ &W) — ; ;— <8 B<> But then •w= ^rhhcota = opt a fin/= £££ cote (8-9) nence ' ' p. /JTT t0sc*+*+*j The P expressed by the equation 8-9 has a minimum value at the minimum value of 9, i n other words f a i l u r e plane w i l l be a shape of wedge provided that other conditions are homogeneous. However, the r e s u l t s obtained were not so. This i s presumably clue to other e f f e c t s , such as side f r i c t i o n . For deeper, 1-1/4", f a i l u r e plane, 9 was around 5°, and s u b s t i t u t i n g (X =» 15°, k •- 0,725, f - 113 pcf, h - 1-1/4", 0 » 35°, then cf = 36° and assuming 01 » 0, i t follows —0 19x8? P = 6,95 x 1,32 — - — g ™ < 0 , The P turns out negative, t h i s means the wedge s l i d e s by i t s e l f . This value i s , however, the minimum resistance because the l a r g e r the k the l a r g e r the E i n Figure 3-8(c) and i t leads the less F , The a c c e l e r a t i o n equal to y i e l d a cceleration leads the maximum resistance which has to be taken i n t o account. In t h i s case ky = tan (0-#) = tan 20° = .364, Hence /TTT = /TT7TTT =. /.o£ 7if .B-'S FORCES ACT/A/tr OA/ THE TOE: 97. The length of this wedge was 1.5' as shown in Figure 8-4, then assuming the toe confining force uniformly distributes along entire failure 2 plane, i.e. 6-1/2 feet (see Figure 8-4)', then 2.0/6,5' =» 0.31 l b s . / f t , , While the f r i c t i o n force acting on the failure plane is ^ = cos* i a n O. /of * O- 9U * O. 7 = 9 2 ( 8_j 0) The error w i l l be . 0.3/ E r r o r = T/TCx/ao = - ^  • ^ 4-%> (8-11) If the same magnitude of .9 is assumed for the shallow failure plane (h = 1/4"), the error w i l l be 1/5 of 4%, namely less than 1%, because the error is proportional to the depth of the failure plane (see Equation 8-11, 8-9 and 8-10). Appendix IV. Thjgojry. of Accumulative ^ Displacement, The permanent displacement of the slope per one cycle induced by sinusoidal vibration can be calculated as follows; simplifying the problem as shown in Figure 9-1, an aquation of dynamic equilibrium is (see Goodman a7;id Seed (1966)) d l < L . ^ f&tj- fy") (9-1) where ^(ui - J ( cosa! y stoc* -fa/7 <fi) (9-2) = -fan (<t>-ct ) ( 9 " 3 ) k^ is referred to as the yield acceleration, g i s the acceleration of gravity, o( is the angle of slope and 0 is the angle of f r i c t i o n . As shown in Figure 9-2, the displacement of the slope per one cycle i s given by integrating (9-1) twice from the starting point of displacement, t^, to the termination of displacement, t^. And assumin k(t) - k ^ s i n LO t and equating ky and k(t) , where ko is the amplitude of acceleration of an excitation base, motion., and tm is the time when the velocity i s maximum. If the jp (u) i s independent of (t) with i n i t i a l condition (t) - 0 at t « t^, then • •• t / W = f[-&(u,seot,-cosu>t)i--fijH/-t)] (9-5) Then integrating again u " J umeft = / cos«Jt,-cos<Jif&/-tj] dt . = //- - A . £ , - ^ trS/nu) I,) CO = - ^ L ^ - t / ; z . th en 100. But t^. is given by t/(t) - 0, i,e, from the equation 9-5 Oct) = '^{-^-Cecseot, - Ccscji) + Ct,-t)]=0 tos coi, - u>su)'t-+ /mi,-iot) - o 3 f}€nc€ where ^(~^~) is a modification factor depending upon the value of JL ' If ky/ko « 1, t = 2t - t , . then 2 m l - 2Ctm~t,)--^Cr>- 2s;ro-'^L) (9-8a) 7/-|^J = ( 7 J ( 9 - 8 b ) This condition, is shown in Figure 9-3, Since ^ co&co~t/ t equation (9-6) is reduced to or 101. Ffc? 9-2 /)CC&L&RAT/OA/ AMD VBLO/L/Ty 102. Appendix V. Similitude. Clough (1958) has shown the similarity relationship of the dynamic model test of s o i l structure, which.is as follows; five types of forces are of importance in v. the analysis, these are: 1, Dead weight of material, 2, Inertia forces due to earthquake acceleration, 3, Forces associated with elastic deformations in the structure. 4, External force, " . 5, Forces connected with the ultimate strength of the structure. And the scale factor for each force i s : dead weight force ^ ^ ^ ^ inertia force = M m *"> = 4^  (10-2) elastic force q = JOk? = . ^ (10-3) €p Gjp fip &p <q-p cohesion force = (r^1)" ^ (10-4) where suffix m and p represent the items cf model and prototype respectively, ; scale factors for each item is indicated by i t s surface, /*; unit weight, V; volume, L; specific length, A «• Lm/Lp; linear'scale ratio, M; mass, k; acceleration, £ ; strain, G; shear modulus, Aj area, C; cohesion. In order to gei similarity, a l l scale factors of the forces should equal. At f i r s t , equating (10-1) and (10-2) i t follows On T i _ _ „ far* . _ /_ ST** _ ~> / tp ) tfien « fK (10-5) This i s a very important r e l a t i o n s h i p i n connection with the dynamic model t e s t . Then, equating (10-1) and (10-3) and l e t - f ^ -*~p because geometric s i m i l i t u d e should hi maintained as follows; F i n a l l y , equating (10-1) and (10-4), i t follows, frn \ 3 _ Cm ~ 2 —— A ~ -JF-A. (10-7) -p  v > p These r e l a t i o n s are shown i n Table 10-1, The angle of i n t e r n a l f r i c t i o n has not been discussed, but i t obviously should remain constant, since i t i s of dimensionless value Clough did not discuss f u r t h e r , but these s i m i l i t u d e s are for rather pseudo s t a t i c a n a l y s i s . According to the dimension a n a l y s i s , the equation of v i b r a t i o n of the continuous media i s given by where cOn I undamped natural frequencies of the structure ^ ; damping r a t i o I n order to s a t i s f y - t h e s i m i l a r i t y r e l a t i o n s h i p between model and prototype, a l l terms i n equation (10-8) should be constant regardless of the model or prototype, The l e f t hand side of (10-8) gives the f i r s t item of Table 10-1, The f i r s t terra of the r i g h t hand side of the equation (10--8) gives Ctin)? ^  (UJ)P r.(V*)p\\ (10-9) since U)„ - Xt») f-yr- - I(n) * w n e r e ^ I s shear wave velocity, i f only the shear vibration i s considered and TfaJ is a constant depending on the mode number of vibration* The equation (10-9) means that, i f the same material as the prototype is used for the model, the scale factor of time is not /X , but just X t and for arbitrary material the scale factor of time is given by equation (10-9), For a special case, in which the similitude becomes most simple, i f the model material is chosen in such a way that the scale factor of modulus of ri g i d i t y i s X a s shown in Table 10-1, in other words the scale factor of shear wave velocity is ] / , the scale factor of time is reduced to /X » which agrees, with Table 10-1, provided that the /!/ are the same. Similarly, the third and fourth term of the right hand side gives • M — ) = ( * ) ^ivjx ' m I u>7x Jp then (10-10) ftp '(KJ?X)p (V,)p S\ and f _ • ' (10-11) In order to obtain the relation of equation 10-11, since f= — — - c /? . the internal damping coefficient, C , has to hold the following relation, •tSLzL - _A? (& fi" (U>)m I (10—12) Cp = hr» rp (^i)p ^ fp ( U s ) p * t 105. TaBk /O-/ Stf/L/TcVDE FOR PYA'AM/C MOPBL TE5T (/J SCALE FACTOR X ?• T/sr> e •3- dcc-ef-emt'/hn / 4. MoJuCxs ofhW'f1/ X S~. fif/nfcrnaP-f/-;ct/o>r / f 6. (//ictm/'/oec/sAearstrc/yTi 7\ 7ai>/e ZO-2 SM/L/TUP£ R)R DYA/AMIC/VODE-L 7t^ST(2) ITEMS SCAIB FACTOR i 2 . Damp/'/ij -facfor /A/ (LS)& / 3. T/'/ne (U~s)p -7 «A)/n ^ <? A Therefore i f fy* - 1 and ^^/(l/^ & o r ^%-p^'/\ t the scale factor of the damping coefficient of the material is — - = -=L= (10-13) In table 10-2 the scale factors for arbitrary model material are shown. Usually, the ideal material which satisfies a l l similarity relations is very d i f f i c u l t to find, and then some of them are ignored. Villich condition should be taken depends upon the condition of greatest importance to the test. If any resonant effect is not of interest and only the failure or the strength of the structure Is of interest, then Table 10-1 should be considered as the f i r s t p riority. For the response analysis, however, the table 10-2 should be considered to be Important, As Seed and Clough (1963) have suggested, in the similitudes discussed above, the effect of pore water pressure was not taken into account. Roscoe (1968) has discussed the similitude regarding pore water pressure. The gravity force of the solid substance is given by ft (/-/>) 1/ (10-14) where n is the void ratio to total volume. Letting be the scale factor for the unit weight of the solid material, i.e. (tt)m = (S*Jp a n c* ^lf~ be the stress scale factor, then from equation (10-14) ^--^ns-~^r  (10-15) Likewise scaling the uplife of the solid phase, (f (l-n)V where is unit weight of the pore liquid. / _ " m (10-16) Where ^ i s the scale factor for the unit weight of the pore liquid, and scaling the self weight of the liquid phase, / f ^ l A requires that 7 ^ <M>-^f- (10-17) From equation (10-15) and (10-16), i t is evident that 7, = (10-18) and from equations (10-16) and (10-17) (10-19) Now the bulk unit weight, f , of a saturated media is given by (10-20) 103, and i f ^ , the scale factor for the bulk unit weight is defined by fm =76C (10-21) th en, provided n = n_^  and y = rj^, i t follows that ( 1 0- 2 2 ) If equations (10-19). and (10-22) are satisfied, the requirements (10-15), (10-16) and (10-17) a l l reduce to the equation These three equations, (10-18), (.10-22) and (10-23) specify the conditions that the material must satisfy for similarity to prevail. 

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