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A non-linear dynamic finite element analysis Quong, Wayne 1983

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C . i A NON-LINEAR DYNAMIC FINITE ELEMENT ANALYSIS by WAYNE QUONG B.A.Sc. U n i v e r i t y Of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of C i v i l Engineering We accept t h i s t h e s i s as conforming to £heyrequired standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1983 © Wayne Quong, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree 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 copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) i i ABSTRACT A two-dimensional f i n i t e element method of a n a l y s i s for p r e d i c t i n g the s t r e s s and permanent displacements of earth s t r u c t u r e s to seismic loading i s presented. The i n e l a s t i c behavior of the s o i l i s modelled by an incremental l i n e a r approach i n which the tangent shear modulus i s v a r i e d with the l e v e l of both the shear s t r a i n and the mean normal s t r e s s . The2 shear modulus s t r a i n dependency i s based on hyperbolic r e l a t i o n s h i p s governing i n i t i a l loading and unloading behavior, leading to a h y s t e r e t i c type energy d i s s i p a t i o n . The tangent bulk modulus i s v a r i e d with the l e v e l of the mean normal s t r e s s only, and h y s t e r e t i c e f f e c t s are not considered. The incremental l i n e a r equations of motion of the s t r u c t u r e are solved using the Newmark step-by-step i n t e g r a t i o n procedure in the time domain a l l o w i n g the s t r e s s e s and displacement to be computed. A f t e r each time step the tangent shear and bulk modulus are re-evaluated. H y s t e r e t i c damping as a r e s u l t of the hyp e r b o l i c shear s t r e s s - s t r a i n law i s inherent i n the model. Viscous damping may a l s o be included. The a n a l y s i s i s a p p l i e d to a number of dams and slopes and the earthquake induced displacements are compared with those p r e d i c t e d by a simpler Newmark s i n g l e degree of freedom r i g i d p l a s t i c a n a l y s i s . As w e l l , a comparison i s made with Makdisi's p r e d i c t i o n of deformation of embankments. For a c l a y slope s t r u c t u r e , the o v e r a l l displacements are of s i m i l a r order. For a c l a y dam s t r u t u r e the non-linear f i n i t e element r e s u l t s i n d i c a t e that the Newmark type methods are ove r l y conservative. The more rigorous multi-degree of freedom a n a l y s i s allows the d i s t r i b u t i o n of displacements w i t h i n and on the surface of the embankment to be obtained. i v TABLE OF CONTENTS ABSTRACT LIST OF TABLES LIST OF FIGURES ACKNOWLEDGEMENTS Chapter 1 INTRODUCTION 1.1 P r e l i m i n a r y Remarks 1.2 Scope of Thesis Chapter 2 CRITICAL REVIEW OF CURRENT METHODS IN THE ASSESSMENT OF SEISMIC PERFORMANCE OF EARTH STRUCTURES 2.1 2.2 2.3 Pseudo-static A n a l y s i s Newmark A n a l y s i s Procedure Seed Lee I d r i s s Proceduce Chapter 3 THE FINITE ELEMENT METHOD 3. 1 I n t r o d u c t i o n 3.2 C o n s t i t u t i v e r e l a t i o n s h i p s 3.2.1 Bulk modulus s t r e s s dependency 3.2.2 H y s t e r e t i c h y p e r b o l i c Shear s t r e s s - s t r a i n R e l a t i o n s h i p 3.3 Formulation of s t r u c t u r e s t i f f n e s s , mass and damping matrices. 3.3.1 P a r t i a l S t i f f n e s s matrices 3.3.2 Mass matrix 3.3.3 Damping matrix Chapter 4 NUMERICAL ANALYSIS OF NON-LINEAR DYNAMIC RESPONSE 4.1 4.2 General Equation of motion Page i i v i v i i i x 1 21 44 4.3 Incremental equation of motion 4.4 Step by Step I n t e g r a t i o n 4.5 C o r r e c t i o n f a c t o r 4.6 Procedure f o r s t r a i n r e v e r s a l occurence 4.7 Summary of procedure Chapter 5 APPLICATIONS OF THE NON-LINEAR FINITE ELEMENT METHOD 5. 1 I n t r o d u c t i o n 5.2 S i n g l e Element Comparison with Newmark A n a l y s i s 5.3 Dynamic Response a n a l y s i s of Clay S t r u c t u r e s 5.3.1 Comparison with the Newmark and the Makdisi-Seed A n a l y s i s 5.3.1.1 Clay Slope Comparison 5.3.1.2 Clay Dam Comparison 5.3.2 Comparison of Hyperbolic and E l a s t i c - P l a s t i c Shear s t r e s s - s t r a i n Laws Chapter 6 SUMMARY AND CONCLUSIONS 6.1 Summary 6.2 Conclusions 6.3 Suggestions f o r Further Research REFERENCES APPENDIX I D e r i v a t i o n of element s t i f f n e s s matrix v i LIST OF TABLES Table I S t a t i c and Dynamic S o i l P r o p e r t i e s v i i LIST OF FIGURES Figure 2-1 Figure 2-2 Figure 2-3 Figure 2-4 Figure 3-1 Figure 3-2 Figure 4-1 Figure 5-1 Figure 5-2 Figure 5-3 Figure 5-4 Figure 5-5 Figure 5-6 Figure 5-7 Figure 5-8 Figure 5-9 Figure 5-10, Figure 5-11 Figure 5-12 Figure 5-13 Pseudo-Static method Forces on a S l i d i n g Block I n t e g r a t i o n of E f f e c t i v e A c c e l e r a t i o n Time H i s t o r y T y p i c a l Shear moduli and Damping Ra t i o s for Sands Hyperbolic Shear stress-Shear S t r a i n R e l a t i o n s h i p Hyperbolic R e l a t i o n s h i p under General Loading Program Flowchart A c c e l e r a t i o n Response and Time h i s t o r y for San Fernando Earthquake S i n l g l e F i n i t e Element Model Shear Deformation R e l a t i o n s h i p S i n g l e F i n i t e Element - Newmark ' Displacement Comparison Single F i n i t e Element Model Newmark A n a l y s i s Displacement H i s t o r y Comparison F i n i t e Element G r i d of Clay Slope S t r u c t u r e F i n i t e Element G r i d of Clay Dam Slope S t a b i l i t y A n a l y s i s on Clay Slope S t r u c t u r e Slope S t a b i l i t y A n a l y s i s on Clay Dam Clay Slope Non-linear - Makdisi - Newmark Displacement Comparison Clay Slope Non-linear F i n i t e Element Newmark An a l y s i s Displacement H i s t o r y Comparison Displaced G r i d of Clay Slope S t r u c t u r e Non-linear - Makdisi - Newmark Displacement Comparison (Clay Dam) Figure 5-14 Clay Dam Displacement Time H i s t o r y v i i i Figure 5-15 Figure 5-16 Figure 5-17 Figure 5-18 Figure I-1a Figure I-1b Displaced G r i d of Clay Dam E l a s t i c - P l a s t i c Approximation of Hyperbolic Curve Hyperbolic - E l a s t i c P l a s t i c Displacement Comparison (Clay Slope) Hyperbolic - E l a s t i c - P l a s t i c Displacement H i s t o r y Comparison Q u a d r i l a t e r a l Element Unit Square i x ACKNOWLEDGEMENTS The author wishes to thank h i s primary a d v i s o r , Professor Peter M. Byrne for h i s u n f a i l i n g guidance and encouragement, and for making c r i t i c a l c o n t r i b u t i o n s towards the completion of the t h e s i s . Professor D.L. Anderson was of valuable a s s i s t a n c e during the preparation and review of the t h e s i s . The author wishes to express h i s g r a t i t u d e to h i s parents fo r t h e i r continued support and patience during t h i s p e r i o d . The f i n a n c i a l a s s i s t a n c e provided by a Natural Sciences and Engineering Research C o u n c i l of Canada postgraduate s c h o l a r s h i p i s g r a t e f u l l y appreciated. 1 CHAPTER 1 INTRODUCTION 1.1 P r e l i m i n a r y Remarks The frequent occurrence of d e s t r u c t i v e earthquakes during the past few decades has caused b i l l i o n s of d o l l a r s i n property damage and l o s s of thousands of l i v e s . I t i s t h e r e f o r e , important that earth s t r u c t u r e s l o c a t e d i n an a c t i v e seismic region be designed to withstand s a f e l y the expected earthquake motion for that region. In February 1971, a earthquake measuring 6.6 on the R i c h t e r scale occurred in C a l i f o r n i a . F i f t y - e i g h t people were k i l l e d , over two thousand injured,' and 1500 b u i l d i n g s were damaged beyond safe occupational l e v e l s . This earthquake, s a i d to be the most severe to occur i n C a l i f o r n i a i n the past 80 years, caused an estimated f i v e b i l l i o n d o l l a r s i n damage. Of p a r t i c u l a r g e otechnical importance was the p a r t i a l f a i l u r e and near t o t a l c o l l a p s e of the Lower San Fernando Dam, and downstream movement of the Upper San Fernando Dam. P r i o r to the earthquake the water l e v e l i n the r e s e r v o i r was 35 feet below the c r e s t of the dam. The s l i d e which occurred i n the upstream s h e l l of the Lower San Fernando Dam f o l l o w i n g the earthquake r e s u l t e d i n a free board of only 4 to 5 feet of cracked m a t e r i a l . At the same time, the Upper dam which forms part of the same r e s e r v i o r complex, d i s p l a c e d some 6 feet downstream. For t u n a t e l y t h i s movement d i d not cause a release of water from the r e s e r v o i r of the Upper Dam. I f i t had, the 2 r e s u l t i n g overtopping of what remained of the Lower Dam might have caused considerable damage and l o s s of l i f e (Seed 1979). Fi v e years previous, based on extensive s t a t e of the a r t procedures, a c o n s u l t i n g board, a design agency, and review board had deemed the design of the Lower San Fernando Dam to be adequate against any earthquake to which i t could be subjected. But the margin by which the t o t a l c o l l a p s e of the Lower San Fernando Dam was avoided was uncomfortably s m a l l . Because of t h i s and other recent events with a s s o c i a t e d d i s a s t r o u s consequences, (Osaki 1966, Seed e t . a l . 1966) there was a need for a r e t h i n k i n g of the earthquake r e s i s t a n t design philsophy of earth and rock f i l l dams and the development of a l t e r n a t e a n a l y s i s procedures. Up to t h i s time, the standard method for 40 years or more for e v a l u a t i n g the safety of earth dams or embankment slopes to earthquake forces has been the s o - c a l l e d pseudo-static method a n a l y s i s . In t h i s method the e f f e c t of the earthquake on a p o t e n t i a l s l i d e mass i s represented by an equivalent s t a t i c h o r i z o n t a l f o r c e , determined by the product of a seismic c o e f f i c i e n t and the t o t a l weight of the s l i d e mass. Assuming the earthquake force acts permanently on the slope m a t e r i a l in one d i r e c t i o n only, and thru the c e n t r o i d the mass, conventional slope s t a b i l i t y a n a l y s i s are a p p l i e d to evaluate the f a c t o r of safety against movement. A f a c t o r of safety of l e s s than unity i m p l i e s movement, but because of the t r a n s i e n t and randomly o s c i l l a t i n g nature of earthquake motions, a c o l l a p s e ( i n terms 3 of an excessive displacement c r i t e r i a ) may not occur. In t h i s sense, a f a c t o r of s a f e t y cannot determine adequate seismic performance. I t i s now g e n e r a l l y accepted that the magnitude of r e l a t i v e displacements i s a more l o g i c a l and better c r i t e r i o n than the f a c t o r of safety for assessing the dynamic performance of earth s t r u c t u r e s . The allowable displacements may vary from a few inches to a few yards depending on the f u n c t i o n a l aspects of the s t r u c t u r e s concerned. Newmark (1965) f i r s t proposed a procedure for p r e d i c t i n g permanent deformations i n earth embankments during earthquakes using a s i m p l i f y i n g r i g i d p l a s t i c s i n g l e degree of freedom approach. More r e c e n t l y , dynamic response analyses of s o i l s t r u c t u r e s (Seed e t . a l . , 1973) have been included i n a more elaborate and encompassing procedure i n p r e d i c t i n g displacements. These procedures w i l l be discussed i n Chapter 2. Dynamic response analyses as p r a c t i s e d today had i t s o r i g i n i n the pioneering attempts of Seed and h i s co-workers at the i f n i v e r s i t y of C a l i f o r n i a at Berkeley to p r e d i c t the a c c e l e r a t i o n s i n h o r i z o n t a l s o i l deposits by bedrock earthquake motions. The Seed Approach (Dynamic s t r e s s path Method) was used to evaluate the seismic performance of s t r u c t u r e s founded on s o i l d e p o s i t s . For such s t r u c t u r e s , the performance of the s o i l deposit to act as an adequate foundation base when subjected to earthquake motions and the m o d i f i c a t i o n of the bedrock motion as i t propogates through the s o i l are important 4 c o n s i d e r a t i o n s . I t has been i d e n t i f i e d that a m p l i f i c a t i o n or d e - a m p l i f i c a t i o n (depending on the c h a r a c t e r i s t i c s of the s o i l d eposit) occurs when seismic waves t r a v e l through s o i l . In a d d i t i o n , there i s a tendency for the r e s u l t a n t motion at the top of the deposit to be a f f e c t e d by the n a t u r a l frequency of the dep o s i t . The predominant pe r i o d of the motion at the top of a deposit i s g e n e r a l l y longer than the predominant period of the bedrock motion. The p r e d i c t i o n of seismic response of h o r i z o n t a l l y layered deposits has been w e l l researched (Schnabel et a l , 1972; Str e e t e r et a l , 1974; Finn et a l , 1977). These methods assume that s o i l p r o p e r t i e s vary i n the v e r t i c a l d i r e c t i o n and remain uniform i n the h o r i z o n t a l d i r e c t i o n . These methods vary i n the s i m p l i f y i n g assumptions that are made , the modelling of the n o n - l i n e a r i t y and p o s s i b l e strength l o s s of s o i l during the dynamic l o a d i n g , and i n the methods used to i n t e g r a t e the equation of motion. Often s o i l s t r u c t u r e s cannot be modelled adequately as h o r i z o n t a l l y layered d e p o s i t s , and i t becomes important to consider two or even three dimensions. For example, i n zoned dams, where the c r o s s - s e c t i o n a l shape as w e l l as the s p a t i a l v a r i a b i l i t y of the s o i l p r o p e r i t e s n e c e s s i t a t e a second dimension. Where the t h i r d dimension i s much l a r g e r than the other two and p r o p e r t i e s i n t h i s t h i r d dimension are uniform, two dimensional analyses are adequate. The e f f e c t s of the t h i r d dimension have been considered i n two dimensional analyses by 5 Makdisi (1976). The predominant pe r i o d of the two dimensional cross s e c t i o n was a l t e r e d according to a length of s t r u c t u r e to height r e l a t i o n s h i p . Dynamic response analyses were gener a l i z e d to two dimensions with the development of the computer programs QUAD4, LUSH and FLUSH. Their method of a n a l y s i s i s based on an equivalent l i n e a r e l a s t i c approach. Linear e l a s t i c methods do not allow computation of displacements d i r e c t l y , but are used in conjunction with l a b o r a t o r y and a d d i t i o n a n a l y t i c a l procedures to determine displacements. These procedures w i l l be discussed in greater d e t a i l i n Chapter 2. St r e e t e r proposed the f i r s t true non-linear a n a l y s i s where a Ramberg-Osgood representation of the s t r e s s - s t r a i n behavior of s o i l was used. The a n a l y s i s can only be used i n one dimensional t o t a l s t r e s s a p p l i c a t i o n s . The development of a two dimensional non-linear dynamic a n a l y s i s i s the next l o g i c a l step. The work h e r e i n , i s presented with t h i s aim i n mind. 1.2 Scope of the Thesis The t h e s i s presents a non-linear f i n i t e element dynamic response a n a l y s i s to p r e d i c t deformations during earthquake lo a d i n g s . The f o l l o w i n g basic assumptions are made while formulating the model: 1) Theory of l i n e a r incremental e l a s t i c i t y i s a p p l i c a b l e . 2) S o i l behaves i s o t r o p i c a l l y . 6 3) Plane s t r a i n c o n d i t i o n s p r e v a i l . 4) The earthquake ground motion i s i d e n t i c a l at a l l p o i n t s along the base of the s t r u c t u r e . The s t r e s s - s t r a i n behavior of s o i l under shear loading i s assumed to be a hyperbolic s t r e s s - s t r a i n r e l a t i o n s h i p s i m i l a r to that used by M a r t i n , Finn and Seed (1975). Volume change behavior as proposed by Duncan e t . a l . , (1980) i s followed. The dynamic a n a l y s i s i s c a r r i e d out using an incremental time step approach where for each element the s t r e s s - s t r a i n curve i s followed i n an incremental manner. With t h i s approach, permanent deformations can be evaluated d i r e c t l y . The seismic e x c i t a t i o n i s assumed to be v e r t i c a l l y t r a v e l l i n g shear waves which are defined by a s p e c i f i e d a c c e l e r a t i o n time h i s t o r y at the r i g i d base of the f i n i t e element model. In Chapter 2, a d e s c r i p t i o n of current methods i n assessing seismic i n s t a b i l i t y and p r e d i c t i n g permanent deformations i s given. Their main assumptions, a n a l y s i s procedures and l i m i t a t i o n s are c r i t i c a l l y reviewed. Chapter 3 and 4 deals with the formulation of the non-l i n e a r dynamic f i n i t e element a n a l y s i s . Assumptions, c o n s t i t u t i v e behavior, s t i f f n e s s and damping p r o p e r t i e s of the s o i l m a t e r i a l s , and the method of s o l u t i o n of the equations of motion are presented. The method i s used to evaluate the seismic behavior of some 7 t y p i c a l e a rth s t r u c t u r e s . These r e s u l t s are presented i n Chapter 5. Displacement p r e d i c t i o n s are compared against r e s u l t s obtained from procedures developed by e a r l i e r i n v e s t i g a t o r s . A b r i e f summary, c o n c l u s i o n , suggestions for f u r t h e r research are presented i n Chapter 6. 8 CHAPTER 2 CRITICAL REVIEW OF CURRENT METHODS IN THE  ASSESSMENT OF SEISMIC PERFORMANCE  EARTH STRUCTURES The seismic performance of earth embankments and dams have received considerable a t t e n t i o n i n recent years. There are c u r r e n t l y s e v e r a l methods being using i n engineering p r a c t i c e for assessing the seismic s t a b i l i t y and p r e d i c t i n g deformations of earth s t r u c t u r e s . By way of i n t r o d u c t i o n to l a t e r chapters i n t h i s t h e s i s , these methods w i l l be examined h e r e i n . The assumptions made i n the formulation , a n a l y s i s procedures, and l i m i t a t i o n s of each method w i l l be c r i t i c a l l y reviewed. 2.1 Pseudo-static Method In t h i s method the randomly o s c i l l a t i n g i n e r t i a forces caused by an earthquake i s represented by an equivalent s t a t i c h o r i z o n t a l force and a conventional slope s t a b i l i t y a n a l y s i s i s a p p l i e d to evaluate the f a c t o r of safety against c o l l a p s e . The equivalent s t a t i c h o r i z o n t a l force i s determined by m u l t i p l y i n g the design seismic c o e f f i c i e n t by the t o t a l weight of the p o t e n t i a l s l i d e mass. The method i s i l l u s t r a t e d i n Figure 2-1. The e f f e c t i v e n e s s of the method depends upon, amongst other t h i n g s , the s e l e c t i o n of a design seismic c o e f f i c i e n t value. Values for the seismic c o e f f i c i e n t i n the design of earth dams worldwide have v a r i e d from .05 to .20 . The standard North American p r a c t i c e i s to use seismic c o e f f i c i e n t values of 0.05, FIG. 2-1 PSEUDO-STATIC METHOD 10 0.10, 0.15, i n areas of low, medium and high s e i s m i c i t y . The s e l e c t i o n of a seismic c o e f f i c i e n t value seems l a r g e l y based on past experience. P r i o r to 1971, very few dams designed i n accordance with these p r i n c i p l e s , have been subjected to very strong earthquake shaking. For t h i s reason, there had been no r e a l f i e l d experience i n which to base the adequacy of the pseudo-static method f o r assessing seismic s t a b i l i t y . The pseudo-static approach f a i l e d to p r e d i c t the slope f a i l u r e s of the Lower and Upper San Fernando Dam (Seed, 1979). The primary cause of f a i l u r e was the build-up of pore water pressures i n the embankment and the l o s s of strength r e s u l t i n g from these pore pressures. The seismic i n s t a b i l i t y which can occur i n loose cohesionless s o i l s depends upon; the peak ground a c c e l e r a t i o n and frequency content of the earthquake motion, i n i t i a l s t r e s s s t a t e of the s o i l , r e s i d u a l pore water pressures. These f a c t o r s cannot be represented i n any r a t i o n a l way by a equivalent s t a t i c h o r i z o n t a l f o r c e . Seed (1979) suggests that the pseudo-static method can only be used with any assurance for s o i l m a t e r i a l s that do not s u f f e r s i g n i f i c a n t strength or s t i f f n e s s l o s s during dynamic l o a d i n g . Under these c o n d i t i o n s , f a c t o r s of safety against movement i n the range of 1.0 to 1.2 are considered adequate f o r seismic s t a b i l i t y . 2.2 Newmark A n a l y s i s Procedure Newmark (1965), and Seed (1966) have both c r i t i c i z e d the concept of f a c t o r of sa f e t y as a means of assessing the probable 11 performance of an earth dam during an earthquake. A f a c t o r of sa f e t y of l e s s than u n i t y i n a s t a t i c a n a l y s i s i s not acceptable as i t imp l i e s c a s t a s t r o p h i c displacements. However f a c t o r s of sa f e t y of l e s s than u n i t y are acceptable i n dynamic analyses, since the earthquake forces acts f o r a short time and a l t e r n a t e i n d i r e c t i o n , only small displacements may occur and these may be q u i t e acceptable. The pseudo-static method does not provide a p r e d i c t i o n on the magnitude of displacements. Therefore i t cannot be viewed as an adequate method for e v a l u a t i n g seismic performance. In the Rankine Lecture of 1965, Newmark f i r s t o u t l i n e d the basic elements of a procedure for p r e d i c t i n g the p o t i e n t i a l deformations of an embankment slope due to earthquake f o r c e s . I t was assumed that slope f a i l u r e would be i n i t i a t e d and outward movement would begin to occur i f the i n e r t i a forces on a p o t e n t i a l s l i d e mass were large enough to overcome the y i e l d r e s i s t a n c e and that the movement would stop when the i n e r t i a forces were reversed. Newmark proposed that the movement of s l i d e mass along i t s f a i l u r e surface could be adequately modelled by the movement of a r i g i d block on a i n c l i n e d plane (Figure 2-2). By computing an a c c e l e r a t i o n at which the i n e r t i a f o rces become s u f f i c i e n t l y high to cause s l i d i n g to begin and i n t e g r a t i n g the e f f e c t i v e a c c e l e r a t i o n on the r i g i d block i n excess of t h i s y i e l d a c c e l e r a t i o n as a f u n c t i o n of time for the duration of the earthquake motion (Figure 2-3), v e l o c i t i e s and u l t i m a t e l y displacements of the r i g i d block could be determined. Newmark (1965) presented a chart for computing such 1 2 FIG. 2-2 FORCES O N SLIDING BLOCK 13 4 Time FIG. 2-3 INTEGRATION OF EFFECTIVE ACCELERATION TIME HISTORY 1 4 displacements. In the development of the c h a r t , the displacements were computed using four earthquake motions normalized to a maximum a c c e l e r a t i o n of .5g and a maximum v e l o c i t y of 30 i n / s e c . The narrow s c a t t e r i n the data i n d i c a t e s that the earthquake motions used have e s s e n t i a l l y the same number of s i g n i f i c a n t p u l ses. This may not be n e c e s s a r i l y true for other earthquakes. For a conservative estimate of the permanent deformation of an embankment slope, the equations proposed by Newmark (and shown on the chart to give an upper bound f i t to the data) may be used. The maximum v e l o c i t y and a c c e l e r a t i o n values are the values appropriate to the design earthquake motion. The maximum r e s i s t a n c e c o e f f i c i e n t i s defined as the value of the h o r i z o n t a l seismic c o e f f i c i e n t which w i l l give a f a c t o r of safety equal to u n i t y i n a pseudo-static a n a l y s i s of the embankment. The r e s i s t a n c e c o e f f i c i e n t can be considered the y i e l d a c c e l e r a t i o n of the s l i d e mass. Evidence had been presented that s l i p i n dense sands, when subjected to uniform a c c e l e r a t i o n s , occurs along a t h i n surface zone. The s l i d i n g mass may be considered analogous to a block r e s t i n g on a i n c l i n e d plane. Shaking t e s t s performed by Goodman and Seed (1966) on small scale embankments of dry dense sands, demonstrated the v a l i d i t y of the fundamental p r i n c i p l e s of the Newmark approach. Refinements to allow for the v a r i a t i o n s i n a c c e l e r a t i o n throughout the embankment and s l i d e mass were proposed by Seed and Martin (1966), Ambraseys and Sarma (1967), and Seed and 15 Makdisi (1978). On the basis of two dimensional response a n a l y s i s on embankments subjected to a given earthquake a c c e l e r a t i o n time h i s t o r y , average induced a c c e l e r a t i o n time h i s t o r i e s for a number of p o t e n t i a l s l i d i n g masses were computed by Seed and M a k d i s i . Deformations are determined using the average induced a c c e l e r a t i o n s time h i s t o r i e s and Newmark's procedure. Design curves presented by Seed and Makdisi show good agreement with Ambraseys and Sarma r e s u l t s . The determination of displacement values using a Newmark approach i s s t r a i g h t f o r w a r d enough. A complex multi-degree of freedom system i s presented by a simple s i n g l e degree of freedom model. But i n doing so, there a r i s e s the problem of i n t e r p r e t a t i o n of r e s u l t s . Where does t h i s displacement occur? I t can be seen e i t h e r as, the h o r i z o n t a l movement or the s l i d i n g movement along a s l i p surface of the f a i l u r e mass. In any case, the Newmark approach gives some idea of the average displacement. I d e a l l y , maximum displacement as w e l l as a d i s t r i b u t i o n of displacements w i t h i n an embankment i s d e s i r e d . 2.3 Seed Lee I d r i s s A n a l y s i s Procedure Seed and h i s co-workers at the U n i v e r s i t y of C a l i f o r n i a , Berkeley have developed a comprehensive dynamic a n a l y s i s procedure for p r e d i c t i n g deformations i n earth s t r u c t u r e s . The method i n i t i a l l y proposed by Seed (1966) and l a t e r has undergone refinements (Seed e t . al.,1973), endeavours to account f o r dynamic forces induced by the earthquake and the 1 6 e f f e c t of s t i f f n e s s l o s s due to dynamic c y c l i n g . The method has been used with reasonable success i n back c a l c u l a t i n g the response of a number of earth dams (Seed e t . a l . , 1975). Commonly r e f e r r e d to as the Seed Dynamic s t r e s s path Appproach, the procedure i s summarized by the f o l l o w i n g steps: a) Sele c t appropriate cross s e c t i o n of earth s t r u c t u r e to be used i n a n a l y s i s and model with a f i n i t e element g r i d . b) Determine appropriate s t a t i c and dynamic p r o p e r t i e s of the s o i l to used i n s t a t i c and dynamic f i n i t e element a n a l y s i s. c) Determine the s t a t i c s t r e s s e s which e x i s t e d before the earthquake. I n s i t u s t r e s s e s are evaluated using a convention s t a t i c f i n i t e element a n a l y s i s . d) Sele c t a design earthquake time h i s t o r y . e) Compute the dynamic shear s t r e s s time h i s t o r i e s f o r s e l e c t e d elements w i t h i n the cross s e c t i o n using a two-dimensional computer response a n a l y s i s . f) Subject r e p r e s e n t a t i v e samples of the s o i l s t r u c t u r e m a t e r i a l to the combined e f f e c t s of i n s i t u s t a t i c s t r e s s e s and the superimposed dynamic shear st r e s s e s and determine t h e i r e f f e c t s i n terms of development of p o t e n t i a l s t r a i n s . g) From the knowledge of the s o i l deformation and 18 shown i n Figure 2-4. These curves are used i n an i t e r a t i v e procedure known as the equivalent l i n e a r method. In t h i s method, successive l i n e a r problems are solved u n t i l the modulus and damping corresponds to the average dynamic shear s t r a i n ( u s u a l l y taken as 65 percent of the max shear s t r a i n ) . The f i n a l l i n e a r s o l u t i o n with s t r a i n -compatible s o i l p r o p e r t i e s i s taken as an approximation to a true non-linear response. The equivalent l i n e a r s o l u t i o n corresponds to the assumption of an e l l i p t i c h y s t e r e s i s loop for c y c l i c l o a d i n g . C u r r e n t l y , the computer programs QUAD4, LUSH and FLUSH based on the equivalent l i n e a r method, are used i n two dimensional dynamic response a n a l y s i s . QUAD4 and LUSH are the most fr e q u e n t l y used programs for s o i l s t r u c t u r e s such as slopes and earth dams. FLUSH i s commonly used for the s o l u t i o n of dynamic s o i l s t r u c t u r e s i n t e r a c t i o n problems such as the response of embedded nuclear reactor s t r u c t u r e s to earthquake l o a d i n g . Since the f i n a l i t e r a t i o n with s t r a i n compatible s o i l p r o p e r t i e s i s purely e l a s t i c , the permanent deformations caused by earthquake shaking cannot be computed d i r e c t l y from t h i s type of a n a l y s i s . As i n a l l e l a s t i c analyses the f i n a l deformations return to zero. To circumvert t h i s shortcoming, a laboratory procedure has been developed (steps f to h) to p r e d i c t s t r a i n s from computed dynamic shear s t r e s s h i s t o r i e s . While the computed e l a s t i c s t r a i n s bear no r e l a t i o n to s t r a i n s i n the f i e l d they are used for d e r i v i n g the s t r a i n compatible s o i l FIG. 2-4 TYPICAL SHEAR MODULI AND DAMPING RATIOS FOR SANDS 20 p r o p e r t i e s . Stresses obtained from the a n a l y s i s using these s t r a i n compatible p r o p e r t i e s are assumed to be rep r e s e n t a t i v e of s t r e s s e s i n the ground. As i n d i c a t e d i n step f, the computed s t r e s s e s are used to estimate permanent deformations. Herein l i e s a se r i o u s i n c o n s i s t e n c y i n the Seed procedure; stresses computed are considered accurate while the s t r a i n s are not, obviously there i s a one to one correspondence. There are two other p o s s i b l e shortcoming to t h i s method. F i r s t , equivalent l i n e a r methods may overestimate the response of earth s t r u c t u r e due to a pseudo resonance e f f e c t . This can occur when the predominant period of the earthquake motion c o i n c i d e s with the n a t u r a l p e r i o d of the earth s t r u c t u r e . In pure l i n e a r e l a s t i c systems, resonance i s a r e a l and p o s s i b l e event. I t cannot occur i n m a t e r i a l s , where the s t i f f n e s s p r o p e r t i e s are h i g h l y non-linear s t r a i n dependant. Second, i t i s not uncommon that computed s t r e s s e s exceed the dynamic strengt h r e s i s t a n c e of the s o i l . This cannot a c t u a l l y happen as at most the dynamic s t r e s s can equal the dynamic r e s i s t a n c e . The u l t i m a t e aim of a dynamic response a n a l y s i s i s to be able to p r e d i c t permanent deformation. This can only be done d i r e c t l y by a non-linear a n a l y s i s i n the time domain. A non-l i n e a r f i n i t e element a n a l y s i s i s presented i n the f o l l o w i n g two two chapters. 21 CHAPTER 3 THE FINITE ELEMENT METHOD 3.1 I n t r o d u c t i o n Several mathematical techniques have been developed to evaluate the seismic performance of earth embankments, and s t r u c t u r e s founded on s o i l d e p o s i t s . The f i n i t e element method has been commonly used i n geotechnical engineering problems due to the ease and accuracy with which geometry and varying s o i l p r o p e r t i e s can be modelled. More r e c e n t l y , the method has been used to evaluate the e f f e c t s of n o n l i n e a r i t y in s t r e s s - s t r a i n s o i l behavior, i n i t i a l s t a t i c s t r e s s e s , and boundary c o n d i t i o n s . As the f i n i t e element method i s the only a v a i l a b l e technique which w i l l allow a rigorous assessment of the n o n - l i n e a r i t y of s o i l , t h i s method w i l l adopted i n t h i s t h e s i s . The f i n i t e element method has been developed as a consequence of the advent of high-speed d i g i t a l computers and has been extremely s u c c e s s f u l i n s o l v i n g many s t a t i c and dynamic problems i n continuum mechanics. I t s a p p l i c a t i o n to s t a t i c analyses of e l a s t i c continua has been described by Wilson and Clough (1962) and i t s extension to dynamic analyses by Clough and Penzien (1975). This method w i l l be b r i e f l y o u t l i n e d here. The f i n i t e element method may be described as a numerical d i s c r e t i z a t i o n proceduce whereby continuum i s i d e a l i z e d as an assemblage of d i s c r e t e elements. The numerical technique allows an i n f i n i t e degree of freedom system to be transformed to a 22 f i n i t e degree of freedom system. Proper modeling of the system i n terms i n element s i z e and s e l e c t i o n w i l l permit accurate p r e d i c t i o n s of displacements, s t r e s s e s , and s t r a i n s of the a c t u a l continuum. The displacement formulation of the f i n i t e element method i s employed h e r e i n . This assumes an i n t e r n a l displacement d i s t r i b u t i o n w i t h i n an element i n terms of the nodal displacements such that c e r t a i n required c o n d i t i o n s on c o m p a t i b i l i t y and completeness are s a t i f i e d . Once the displacement f i e l d has been assumed for any element of p r e s c r i b e d geometry and once the c o n s t i t u t i v e p r o p e r t i e s of the elements are determined, i t i s p o s s i b l e with the a i d of the v i r t u a l work theorem to derive the s t i f f n e s s matrix of the element. The s t i f f n e s s matrix represents the s t i f f n e s s p r o p e r t i e s a s s o c i a t e d with the displacements at the nodes of the element. The s t i f f n e s s matrix of the e n t i r e continuum i s obtained by the proper a d d i t i o n of i n d i v i d u a l element s t i f f n e s s matrices by the d i r e c t s t i f f n e s s method. S o l u t i o n of the s t i f f n e s s equations s a t i s f i e s e q u i l i b r i u m ( i n a g l o b a l sense.) The advantage of t h i s d i s c r e t e mathematical formulation i s that for the dynamic problem the e q u i l i b r i u m of the system may be expressed by a set of ordinary d i f f e r e n t i a l equations rather than a set of p a r t i a l d i f f e r e n t i a l equations, while i n a s t a t i c problem the p a r t i a l d i f f e r e n t i a l equations are reduced to a set of a l g e b r a i c equations. I f the displacements of a l l nodal p o i n t s i n the complete assemblage are designated by the vector 23 {r} , the corresponding nodal forces by the vector {R}, and the s t i f f n e s s matrix of the e n t i r e system by the matrix [K] ; the s t a t i c e quilbrium equation may be expressed i n the form [K] {r}= {R} (3-1) The f i n i t e element method has many advantages over other numerical techniques. D i f f e r e n t m a t e r i a l p r o p e r t i e s can be pre s c r i b e d from one element to the next and/or can have varying p r o p e r t i e s w i t h i n the element themselves. Any displacement, s t r e s s , or coupled boundary c o n d i t i o n can be handled, and the boundaries can be very i r r e g u l a r i n shape. I t can be shown that for e l a s t i c systems by using elements with properly s e l e c t e d displacement f u n c t i o n s , the f i n i t e element method converges to the exact s o l u t i o n as the number of elements used to model a system increases. This i n d i c a t e s that any d e s i r e d degree of accuracy can be obtained. The governing c o n s t r a i n t here being the computational costs a s s o c i a t e d with the increased numerical operations performed on l a r g e r s t i f f n e s s matrices and v e c t o r s . Two dimensional systems may be represented by elements of various shapes. The simplest, the three node constant s t r a i n t r i a n g l e (CST), has been used e x t e n s i v e l y by e a r l i e r i n v e s t i g a t o r s . Displacements w i t h i n t h i s element are assumed to vary l i n e a r l y through the element. This l i n e a r displacement f u n c t i o n r e s u l t s i n constant s t r a i n and s t r e s s w i t h i n the element, hence i t s name CST. Higher order displacement 24 displacement fu n c t i o n s for t r i a n g l e s , and q u a d r i l a t e r a l elements have been introduced i n t o s o i l dynamic analyses by Finn and M i l l e r (1971), and Seed (1969) r e s p e c t i v e l y . The plane q u a d r i l a t e r a l isoparametric element i s used i n the present a n a l y s i s . The plane q u a d r i l a t e r a l element possesses eigh t degree of freedom, namely two t r a n s l a t i o n a l degrees of freedom at each of the four corner nodes. The term 'isoparametric' i s derived from the use of the same i n t e r p o l a t i o n functions to map the q u a d r i l a t e r a l element shape as are used to define displacements w i t h i n the element. The q u a d r i l a t e r a l element need not be rectangular but can be of any a r b i t r a r y shape. This feature allows i r r e g u l a r shaped s t r u c t u r e s to be modelled without d i f f i c u l t y . As w e l l , the four node element may be s p e c i f i e d such that the l o c a t i o n of any two adjacent nodes i s the same, making the element t r i a n g u l a r . The s t i f f n e s s matrix of a t r i a n g u l a r element, obtained t r e a t i n g i t as a four node element and f o l l o w i n g the procedure for isoparametric elements, reduces to that of a CST. Or i n other words, by s p e c i f y i n g the node number of the t h i r d node to be the same of the node number of the f o u r t h node, the element i s a c t u a l l y a CST. From t h i s r e s u l t , q u a d r i l a t e r a l s and/or CST's can be used i n the a n a l y s i s . The formulation of the s t i f f n e s s matrix f o r plane q u a d r i l a t e r a l isoparametric element i s presented i n Appendix I. The dynamic f i n i t e element a n a l y s i s presented i s t h i s t h e s i s c o n s i s t s of three major parts 25 a. Modelling the c o n s t i t u t i v e behavior of s o i l m a t e r i a l s . b. Formulation of s t i f f n e s s , mass and damping matrices c. S o l u t i o n of the equations of motion Parts a and b w i l l be discussed i n the remaining sections of t h i s chapter. Part c w i l l be discussed i n chapter 4. 26 3.2 C o n s t i t u t i v e r e l a t i o n s h i p s A fundamental assumption made i n the a n a l y s i s i s that s o i l behaves i s o t r o p i c a l l y . This allows two-dimensional s t r e s s -s t r a i n behavior to be described by two e l a s t i c parameters during each increment. In the present a n a y l s i s , the shear modulus G and the bulk modulus B were s e l e c t e d as the e l a s t i c parameters for the f o l l o w i n g reasons. Dynamic loading i s e s s e n t i a l l y due to v e r t i c a l l y propagating shear waves. The shear waves induce dynamic shear s t r e s s e s and shear deformations on the deposi t . Therefore, shear deformations make up a great part of the o v e r a l l displacements. Bearing i n mind that the ' f a i l u r e c o n d i t i o n ' should be based on the magnitude of r e s i d u a l displacements that can be t o l e r a t e d . A proper s t r e s s - s t r a i n law should c o n t r o l shear deformation i n order that a p r e d i c t i o n on f a i l u r e can be made. Based on t h i s argument, i t was decided that shear modulus be taken as one of the e l a s t i c parameters, which can be reduced i f near f a i l u r e c o n d i t i o n s occurs using a simple hyperbolic model. During dynamic loading a s o i l element often experiences near f a i l u r e conditon, i e . , reaches near maximum shear strength. During t h i s c o n d i t i o n , higher shear deformation can occur without an accompanying increase i n volumetric s t r a i n as the bulk modulus remains constant. This behavior i s observed i n lab o r a t o r y t e s t s on c l a y and sand. In a s t r e s s - s t r a i n model, E and v can be used as the v a r i a b l e s . And from e l a s t i c i t y , we 27 know, B= E (3-2) 3(1-2*) Near the f a i l u r e c o n d i t i o n , where E i s reducing, i f v i s kept constant, from equation (3-2) B w i l l reduce, r e s u l t i n g i n an increase i n volumetric s t r a i n . This can be avoided by varying v to keep B constant (Byrne et a l 1982). I t i s s i m p l i e r to use the bulk modulus d i r e c t l y as one the e l a s t i c parameters. As w e l l , r e l a t i o n s h i p s f or determining bulk modulus values have been w e l l developed from extensive i n v e s t i g a t i o n s by Duncan e t . a l . (1980). In order to a r r i v e at expressions d e s c r i b i n g the s t r e s s -s t r a i n behavior of an i d e a l i z e d s o i l element under general loading i t i s assumed that h y p e r b o l i c shear stress-shear . s t r a i n r e l a t i o n s h i p s i m i l a r to that used by M a r t i n , Finn and Seed (1975), and that volume change behavior as proposed by Duncan et. a l . (1980) i s followed. Tangent values of shear modulus i s v a r i e d with the l e v e l of both the shear s t r a i n and the mean normal s t r e s s . Values of bulk modulus i s v a r i e d with the l e v e l of mean normal s t r e s s only. These r e l a t i o n s h i p s are discussed in the f o l l o w i n g s e c t i o n s . 28 3.2.1 B u l k Modulus s t r e s s d e p e ndency S t u d i e s by Duncan e t . a l . , (1980) have shown t h a t t h e volume change b e h a v i o r of most s o i l s c a n be m o d e l l e d r e a s o n a b l y a c c u r a t e l y by a s s u m i n g t h a t t h e b u l k modulus of t h e s o i l v a r i e s w i t h c o n f i n i n g p r e s s u r e , and i s i n d e p e n d e n t o f t h e p e r c e n t a g e of s t r e n g t h m o b i l i z e d . V a l u e s o f B have been f o u n d t o i n c r e a s e w i t h i n c r e a s i n g c o n f i n i n g p r e s s u r e and i n terms of t h e mean nor m a l p r e s s u r e c an be a p p r o x i m a t e d by t h e e q u a t i o n , B= K b - P a ( ^ f (3-3) Pa' where, B = t a n g e n t b u l k modulus Kb = b u l k modulus p a r a m e t e r m = b u l k modulus e x p o n e n t o m= mean n o r m a l e f f e c t i v e s t r e s s P a = a t m o s p h e r i c p r e s s u r e , i n c l u d e d t o have a n o n - d i m e n s i o n a l e q u a t i o n , e x p r e s s e d i n u n i t s c o n s i s t e n t w i t h o m and B. The t a n g e n t b u l k modulus i s d e f i n e d by B= Ag, + Ao 2 + Ap 3 (3~4) 3Ae v where Aa,, A a 2 , and A a 3 a r e t h e c h a n g e s i n t h e v a l u e s of t h e p r i n c i p a l s t r e s s e s , and Ae v i s t h e c o r r e s p o n d i n g change i n v o l u m e t r i c s t r a i n , w hich f o r t h e p l a n e s t r a i n c o n d i t i o n i s , Ae v = Ae x + Ae y (3"5) a l t e r n a t i v e l y e q u a t i o n (3-4) can be w r i t t e n , 29 B= (3-6) where, Ao m i s the change i n mean normal s t r e s s . As w i l l be discussed i n greater d e t a i l i n chapter 4, the non-linear response of earth s t r u c t u r e to seismic motions i s solved using the Newmark step-by-step i n t e g r a t i o n procedure. This procedure computes displacements and s t r a i n s at the end of consecutive time i n t e r v a l s At assuming the the m a t e r i a l p r o p e r t i e s have been held constant during the increment. At the end of a time i n t e r v a l the values of bulk moduli must be updated for the e f f e c t s of s t r e s s change. The bulk modulus for an s o i l element at any time can be determined as f o l l o w s : 1. At time T, the bulk modulus B from equation 3-3 as i s known. 2. During time step At using bulk modulus values c a l c u l a t e ev at time T + At. 3. At time T + At c a l c u l a t e the new bulk modulus B. 30 3.2.2 H y s t e r e t i c h y p e r b o l i c Shear s t r e s s - s t r a i n  Relat i o n s h i p A comprehensive survey of the f a c t o r s a f f e c t i n g the shear * moduli of s o i l s and expressions for determining t h i s property have been presented by Hardin and Drnevich (1970). In t h e i r study, an e m p i r i c a l equation was presented to determine the values of maximum shear modulus G m a x • Their equation i s as fo l l o w s : G m a x = 320.8-Pn (2.973-e) 2- (OCR)° f o m j ' / ; (3-7) (1+e)  (Pc7' where, G m a x = Maximum shear modulus e = voi d r a t i o OCR = o v e r c o n s o l i d a t i o n r a t i o a = parameter that depends on the p l a s t i c i t y index of the s o i l a m = mean normal e f f e c t i v e s t r e s s . For c l a y s , Seed and I d r i s s used an equation of the form, Gmax= (S 0 )• (constant) (3-8) where Su i s the undrained shearing strength of the c l a y Laboratory and i n - s i t u t e s t data performed by sever a l i n v e s t i g a t o r s have found the constant value to vary from 1000 to 3000. For the i n i t i a l l oading phase the hyper b o l i c shear s t r e s s -s t r a i n r e l a t i o n s h i p formulated by Kondner and Zelasko (1963) to 31 model t h e r e s p o n s e of g r a n u l a r s o i l i n s i m p l e s h e a r i s u s e d . The same h y p e r b o l i c r e l a t i o n s h i p i s u s e d f o r c l a y s , as c y c l i c t r i a x i a l t e s t s p e r f o r m e d by I d r i s s e t . a l . ( 1 9 7 8 ) on c l a y s h as shown t h i s b e h a v i o r . The e f f e c t o f t h e s t a t i c s h e a r s t r e s s i s i n c l u d e d . F o r t h e assumed i s o t r o p i c m a t e r i a l , i n g e n e r a l two d i m e n s i o n a l l o a d i n g , t h e i n i t i a l r e s p o n s e up t o the f i r s t r e v e r s a l t o g i v e n by t h e e q u a t i o n , and i s shown i n F i g u r e 3 - 1 . The s t a r t i n g p o i n t on t h e s t r e s s - s t r a i n c u r v e i s (0, r 5 t ). T = G mox Tmax + TST ( 3 ~ 9 ) 1 + (T~ -y "max /mox  T u l t 1 'st Rf where, T = s h e a r s t r e s s Tmax = m a x s h e a r s t r a i n r 0| t = t h e s h e a r s t r e n g t h o f t h e s o i l T S T = t h e s t a t i c s h e a r s t r e s s of t h e s o i l Rf = the f a i l u r e r a t i o The - s i g n and + s i g n a r e a p p l i e d t o l o a d i n g i n t h e p o s i t i v e and th e n e g a t i v e d i r e c t i o n r e s p e c t i v e l y . 7 m a x i s t h e s e c o n d i n v a r i a n t o f s t r a i n , a s d e t e r m i n e d by, Tmax =1 ) ( T * y ) 2 + U " e y ) 2 ( 3 - 1 0 ) where, c x= t h e normal s t r a i n i n t h e x - d i r e c t i o n e y= t h e normal s t r a i n i n t h e y - d i r e c t i o n 7 = t h e s h e a r s t r a i n i n t h e x-y p l a n e . Tmax i s g i v e n t h e same s i g n as 7 x y . 32 Shear strain FIG. 3-1 HYPERBOLIC SHEAR STRESS-SHEAR STRAIN RELATIONSHIP 3 3 For unloading and r e l o a d i n g , the Masing ( 1 9 2 6 ) type of h y s t e r e t i c behavior has been used by Lee ( 1 9 7 7 ) , i n the development of a one-dimensional dynamic e f f e c t i v e s t r e s s a n a l y s i s f o r saturated sand d e p o s i t s . B r i e f l y described here, the Masing c r i t e r i o n assumes that i f the i n i t i a l l oading curve, or skeleton curve, which i s described by equation ( 3 - 9 ) can be represented by, T = f ( 7 m a x > r s t ) where rst i s set to zero ( 3 - 1 1 ) Then the unloading or rel o a d i n g curve can be obtained from, where ( 7 r , r r ) i s the l a s t r e v e r s a l point i n the s t r e s s - s t r a i n p l o t . Geometrically equation ( 3 - 1 2 ) means that the unloading and r e l o a d i n g branches of a h y s t e r e t i c loop are the same skeleton curve with both the s t r e s s and s t r a i n s c a l e s increased by a f a c t o r of two and the o r i g i n t r a n s l a t e d to the r e v e r s a l p o i n t . The tangent modulus a f t e r s t r e s s r e v e r s a l i s equal to G m a x . Further to t h i s Lee assumes t h a t , i f the s t r e s s - s t r a i n curve described by equation ( 3 - 1 2 ) i n t e r s e c t s an extension of the skeleton curve, the s t r e s s - s t r a i n path f o l l o w s the skeleton curve u n t i l there i s a r e v e r s a l of loading again. I f the s t r e s s s t r a i n curve i n t e r s e c t s the curve of the previous load c y c l e , the s t r e s s - s t r a i n path then f o l l o w s the l a t t e r s t r e s s - s t r a i n curve. 2 - f ( 7 . ( 3 - 1 2 ) The s t r e s s - s t r a i n model re q u i r e s that previous s t r e s s r e v e r s a l p o i n t s for each s o i l l a yer or element be recorded, in 34 o r d e r t o d e t e r m i n e w h e t h e r i n t e r s e c t i o n o f t h e s k e l e t o n o r p r e v i o u s p r e v i o u s u n l o a d i n g o r r e l o a d i n g s t r e s s s t r a i n c u r v e s h a v e o c c u r r e d . B e c a u s e v e r y l i t t l e e x p e r i m e n t a l work h a s b e e n p e r f o r m e d on s a n d s o r c l a y s t o v e r i f y t h e L e e ' s a s s u m p t i o n s , i t may be more r e a s o n a b l e t o s i m p l i f y t h e r e l o a d i n g a n d u n l o a d i n g b e h a v i o r . I n t h e p r e s e n t a n a l y s i s , i f t h e i n i t i a l l o a d i n g a s d e s c r i b e d by e q u a t i o n ( 3 - 9 ) i s r e p r e s e n t e d b y , r = f ( 7 m w , ± r , t ) ( 3 - 1 3 ) T h e n i t i s a s s u m e d t h a t t h e u n l o a d i n g a n d r e l o a d i n g c u r v e c a n be o b t a i n e d f r o m , * = f <7max " 7 r , ± T r ) ( 3 " 1 4 ) C o n s i d e r a s o i l e l e m e n t b e i n g s t r a i n e d f r o m i t s u n d e f o r m e d s t a t e t o a s h e a r s t r a i n l e v e l 7 A , w i t h c o r r e s p o n d i n g s h e a r s t r e s s T a . The e l e m e n t i s now u n l o a d e d f r o m t h a t p o i n t . The s t r e s s - s t r a i n d i a g r a m i s shown i n F i g u r e 3 - 2 . C u r v e OA i s g i v e n by ( 3 - 9 ) w h i l e c u r v e AB i s g i v e n by e q u a t i o n ( 3 - 1 4 ) , w h e r e t h e r e v e r s a l p o i n t ( 7 r , r r ) i s ( 7 A , T a ) . E q u a t i o n ( 3 - 1 4 ) h a s t h e e f f e c t o f r e s e t t i n g t h e o r i g i n o f t h e s t r e s s - s t r a i n c u r v e a t t h e r e v e r s a l p o i n t ( 7 A , r A ) , a n d t h e s t r e n g t h o f t h e s o i l e q u a l t o t h e v a l u e Tui t + r A . Upon a n o t h e r r e v e r s a l a t p o i n t B t h e new s t r e n g t h a s y m p t o t e i s e q u a l t o T u ) t + r B . T h i s s i m p l i f i e d h y p e r b o l i c u n l o a d i n g a n d r e l o a d i n g i s e s s e n t i a l l y h y s t e r e t i c i n n a t u r e . 35 FIG. 3-2 HYPERBOLIC RELATIONSHIP UNDER GENERAL LOADING 36 The shear modulus at any time, i s the value of the tangent of the s t r e s s - s t r a i n curve at the s t r e s s - s t r a i n point corresponding to that time. The tangent shear modulus i s c a l c u l a t e d by e v a l u a t i n g the d i f f e r e n t i a l of the equation d e s c r i b i n g the i n i t i a l or a f t e r r e v e r s a l hyperbolic curve, whichever i s appropriate at that i n s t a n t i n time. 3.3 Formulation of s t r u c t u r e s t i f f n e s s , mass and damping  matrices 3.3.1 P a r t i a l S t i f f n e s s M a trices An element s t i f f n e s s matrix i s , for a given geomtry, a l i n e a r f u n c t i o n of the terms i n the s t r e s s - s t r a i n matrix [D], ( {6o}= [D]{6e} ). In general the D matrix i s a f u l l 6 by 6 matrix, with 36 independent terms. For s t a b i l i t y , the element s t i f f n e s s matrix must be p o s i t i v e d e f i n i t e , t h i s r e q u i r e s the [D] matrix to be as w e l l . As mentioned e a r l i e r , the present a n a l y s i s considers i s o t r o p i c s o i l behavior under the r e s t r i c t e d but p r a c t i c a l case of plane s t r a i n . Thus only 6 of the 21 independent terms(which may be expressed i n terms of two m a t e r i a l parameters) are relevant i n the present study. The i s o t r o p i c plane s t r a i n l i n e a r e l a s t i c r e l a t i o n between s t r e s s and s t r a i n can be w r i t t e n as: {a}= [D]{e} (3-15) For an increment change t h i s can be w r i t t e n as: {Aa}= [D]{Ae} ' (3-16) where 37 {Ae} t h e i n c r e m e n t a l s t r a i n v e c t o r (A£ x ,A£ y ,A7 x y) {Aa} i s t h e i n c r e m e n t a l s t r e s s v e c t o r (Ao x ,Aa y ,Ar x y) F o r t h e p l a n e s t r a i n c o n d i t i o n t h e [d] m a t r i x i s commonly w r i t t e n a s , ~\-v v 0 f D l = E v \-v 0 (3-17) 0 0 1-2* ( 1 + * ) ( 1 - 2 * ) where D= Young's modulus and *= P o i s s o n ' s r a t i o . F o r t h e s i m p l e i s o t r o p i c c a s e where t h e r e i s o n l y one i n d e p e n d e n t modulus E, t h e Young's modulus w i t h a c o n s t a n t P o i s s o n ' s r a t i o (Duncan e t . a l . 1980), n o n - l i n e a r a n a l y s i s programs d i d n o t do any more t h a n m e r e l y change t h e D m a t r i x by m u l t i p l y i n g by a c o n s t a n t . The s t i f f n e s s m a t r i x of an e l e m e n t i s commonly w r i t t e n a s : [k] = f [ B ] T [ D ] [ B ] d A r ~Area (3-18) [B] i s t h e d i s p l a c e m e n t m a t r i x f o r t h e i s o p a r a m e t r i c e l e m e n t , A p p e n d i x I . L i k e w i s e , i t c a n be seen t h a t t h e new s t i f f n e s s m a t r i x a t e a c h l o a d s t e p i s o b t a i n e d m e r e l y by m u l t i p l y i n g by a c o n s t a n t . A s l i g h t l y more complex c a s e i s i s o t r o p i c c o n d i t i o n s w i t h v a r y i n g P o i s s o n * s r a t i o and Young's mo d u l u s . The new [D] c a n n o t be o b t a i n e d d i r e c t l y by a m u l t i p l i c a t i o n of t h e o l d [ D ] , From 38 equation (3-18), each element s t i f f n e s s matrix must be regenerated at each load step. For the type of a n a l y s i s undertaken, t h i s would be a q u i t e expensive process. E and v are not independentterms of the [D] matrix: we cannot w r i t e , [D]= E dp dE + yfdp Id? We can, however, w r i t e , [D]= B 'dD ' + G 'dD ' dB dG (3-19) where G and B are the shear and bulk modulus, and, dD dB 1 1 0 4/3 -2/3 -2/3 4/3 0 0 (3-20),(3-21) The above matrices are commonly r e f e r e d to as the c o n s t i t u t i v e patterns for the G, B model. Since the element s t i f f n e s s matrix i s a l i n e a r f u n c t i o n of the terms i n D, and since the terms i n D are a l i n e a r f u n c t i o n of the independent moduli G & B, i t f o l l o w s that the element s t i f f n e s s matrix (for a given geometry) i s a l i n e a r f u n c t i o n of the independent moduli G & B. Thus we can w r i t e , for the G-B model, [k]= B[S R ] + G[S G] (3-22) Where G and B are tangent values at the beginning of an increment and evaluated according to c o n s t i t u t i v e r e l a t i o n s h i p s developed i n chap 3.2. 39 S B i s the element ' p a r t i a l s t i f f n e s s matrix' f o r B= 1 and G= 0, and S G i s element p a r t i a l s t i f f n e s s matrix for B= 0 and G= 1. S B and S G are obtained from: [ S R ] = f [B] •/Area dD dB [B]dA r (3-23) [ S r J - f [B] •'Area dp dG [B]dAr (3-24) For isoparametric q u a d r i l a t e r a l s , the i n t e g r a l s (3-23) and (3-24) have to be evaluated n u m e r i c a l l y . The procedure i s o u t l i n e d in Appendix I . The advantage of the p a r t i a l s t i f f n e s s approach i s that the shear and bulk s t i f f n e s s matrices f or each element need be evaluated j u s t once. At each load step, an element s t i f f n e s s matrix i s obtained by f i r s t m u l t i p l y i n g t h e i r p a r t i a l s t i f f n e s s matrices by t h e i r r e s p e c t i v e modulus values and adding the two matrices, as i n d i c a t e d by equation (3-22). I t should be noted that for isoparametric elements the s t r a i n s and ther e f o r e the st r e s s e s vary over the element and consequently the moduli are not constant throughout the element. I f t h i s was taken i n t o c o n s i d e r a t i o n the p a r t i a l s t i f f n e s s approach would not be v a l i d and the re-generation of the element s t i f f n e s s matrix at each step would be required. Instead the stres s e s and s t r a i n s at the element c e n t r o i d were taken as being r e p r e s e n t a t i v e of the values i n the e n t i r e element and the 40 moduli appropriate only to the c e n t r o i d was used to describe the s t r e s s - s t r a i n behavior of the e n t i r e element. This assumption allows p a r t i a l element s t i f f n e s s matrices to be represented i n t h e i r present form, equations (3-23) and (3-24). 3.3.2 Mass Matrix I t i s p o s s i b l e to develop a mass matrix 'for each f i n i t e element which i s c o n s i s t e n t with the adopted displacement i n t e r p o l a t i o n f u n c t i o n and mass d i s t r i b u t i o n w i t h i n the element (Cooke 1975). The r e s u l t i n g mass matrix i s l i k e i t s element s t i f f n e s s matrix i n that i t i s banded and possesses coupling p r o p e r t i e s . A lumped mass approximation , where the mass of the elements i s assumed to be concentrated at the nodal p o i n t s , leads to a diagonal mass matrix and no c o u p l i n g . When the diagonal mass matrix i s used, Lsymer (1979) has observed that the r o t a t i o n a l i n e r t i a of the i n d i v i d u a l elements i s overestimated, which r e s u l t s i n an underestimation of the highest n a t u r a l frequencies of the system. On other hand, the c o n s i s t e n t mass matrix leads to an overestimation of the same order of magnitude of the highest n a t u r a l frequencies. For s i t e response problems where the response i s governed g r e a t l y by the lower n a t u r a l frequencies e i t h e r approach for d e r i v i n g a mass matrix may be s u i t a b l e . Good r e s u l t s have been shown by Penzien (1969) when using the lumped mass approximation. I t has the advantages of savings i n computer storage and computation time required f o r matrix c a l c u l a t i o n s . 41 For the present a n a l y s i s , o n e - t h i r d of the mass of each t r i a n g u l a r element and one-fourth of the mass of each q u a d r i l a t e r a l element are lumped at t h e i r r e s p e c t i v e nodes. The mass of any one node i s the sum of the c o n t r i b u t i o n s of i t s surrounding elements to that p a r t i c u l a r node. The matrix [M] i s as f o l l o w s : [M] = m. 0 0 0 0 0 — 0 0 m2 0 0 0 0 - 0 0 0 m3 0 0 0 - 0 0 0 0 m0 0 0 - 0 0 0 0 0 m5 0 - 0 0 0 0 0 0 m6 - 0 0 0 0 0 0 0 0 - m (3-25) where m i s the mass of the node a s s o c i a t e d with the i t h degree of freedom, and n i s the t o t a l number of degrees of freedom. 3.3.3 Damping Matrix When v i b r a t i o n a l energy i s being t r a n s m i t t e d through a m a t e r i a l medium, a p o r t i o n of i t s energy i s d i s s i p a t e d i n t e r n a l l y due to a number of mechanisms. One of these i s a viscous type of damping. This d i s s i p a t i o n of energy causes a decrease i n the amplitude of v i b r a t i o n and can be broadly termed 'material damping'. In the equivalent l i n e a r analyses by Seed 42 (1969) and Lsymer (1969) where non-linear h y s t e r e t i c m a t e r i a l i s represented by a l i n e a r v i s c o - e l a s t i c model, m a t e r i a l damping has to be introduced a r t i f i c i a l l y through a frequency dependent viscous type damping. In t r u l y non-linear a n a l y s i s , as i s i n the present a n a l y s i s , where a h y s t e r e t i c s t r e s s - s t r a i n r e l a t i o n s h i p i s used, a r t i f i c i a l viscous damping i s no longer needed to model the d i f f e r e n t types of m a t e r i a l damping. Damping i s introduced i n t o the present a n a l y s i s to describe the a c t u a l viscous e f f e c t s due to the presence of water i n the s o i l g r a i n s . Studies by Finn e t . a l . , (1979) have shown that some degree of viscous damping i s required to s t a b i l i z e systems at r e v e r s a l p o i n t s , where there are abrupt changes i n modulus values. Damping expressions introduced by Rayleigh to produce o r t h o g o n a l i t y i n modal s u p e r p o s i t i o n s o l u t i o n s to dynamic s t r u c t u r e response, have been used i n equivalent l i n e a r analyses by Seed (1960). A Rayleigh type damping expression (although the o r t h o g o n a l i t y feature i s not important i n the present a n a l y s i s ) i s used to model viscous e f f e c t s i n the present a n a l y s i s . The f o l l o w i n g r e l a t i o n s h i p i s used for each element; [c] e= a [m] e + b [ k ] e (3-26) in which [ c ] e , [m] e and [ k ] e are the damping, mass and s t i f f n e s s matrices r e s p e c t i v e l y for element e. The element s t i f f n e s s matrix corrresponds to i t s value at time= 0. The parameters a and b are given by: 43 a = Xg'to, (3-27) b = X/w, (3-28) The value of X e expressed i n percentage of c r i t i c a l damping represents the damping r a t i o for element e. The parameter C J, i s equal to the fundamental frequency of the system and i s evaluated by the a n a l y s i s at time t=0 (or based on G m a x ). The complete damping matrix [C] of the e n t i r e s t r u c t u r e i s obtained from the i n d i v i d u a l element damping matrices by the d i r e c t s t i f f n e s method. Given the form of equation (3-26), element damping matrices and th e r e f o r e the s t r u c t u r e damping matrix i s assumed to remain constant during dynamic l o a d i n g . 44 CHAPTER 4 NUMERICAL ANALYSIS OF NON-LINEAR DYNAMIC RESPONSE 4.1 General In t h i s chapter the numerical technique used to solve the f i n i t e element modelling of the non-linear dynamic reponse of an earth s t r u c t u r e to earthquake motions i s presented. In many p r a c t i c a l cases a s t a t e of plane s t r a i n can be assumed so that for a n a l y s i s purposes three dimensional s t r u c t u r e s can i d e a l i z e d by a f i n i t e element system representing the cross s e c t i o n of the s t r u c t u r e . Modelling of the cross s e c t i o n by a f i n i t e element system of t r i a n g l e s and q u a d r i l a t e r a l s r e q u i r e s , the formulation of c o n s t i t u t i v e s t r e s s s t r a i n behavior of the s o i l m a t e r i a l , and the d e r i v a t i o n of the s t i f f n e s s and damping p r o p e r t i e s . This has been o u t l i n e d i n chapter 3. In earthquake response analyses of l i n e a r s t r u c t u r e s , many e a r l y i n v e s t i g a t o r s have used the mode-superposition method, (Wilson (1962) ,Clough and Penzien (1975)). This method inv o l v e s the s o l u t i o n of the c h a r a c t e r i s t i c value problem represented by the free v i b r a t i o n response of the system, followed by the transformation of the displacements to the mode shapes of the system. This procedure uncouples the response of the system, so that the response of each mode may be evaluated independently of the others. The second method of dynamic a n a l y s i s i s c a l l e d the step-by-step method, and involves the d i r e c t numerical i n t e g r a t i o n of the e q u i l i b r i u m equations i n t h e i r o r i g i n a l form. 45 The main advantage of the mode su p e r p o s i t i o n method i s that the response of a system may be obtained with good accuracy by co n s i d e r i n g only a few of the lower normal modes, while i n the step-by-step method a l l g e n e r a l i z e d coordinates must be r e t a i n e d , Penzien (1969). On the other hand, the eva l u a t i o n of the c h a r a c t e r i s t i c value problem and transformation to the mode shapes are major computational problems not required i n the step-by-step method. Recent i n v e s t i g a t o r s (Seed and I d r i s s 1969) have used an equivalent l i n e a r v i s c o e l a s t i c i t e r a t i v e method to approximate the non-linear s o l u t i o n . The dynamic response solved i n the frequency domain i s based on the assumption of l i n e a r s t r u c t u r a l behavior, cannot be used for a non-linear system. The step-by-step i n t e g r a t i o n method, on the other hand can be a p p l i e d to non-linear systems. In t h i s approach, the response i s c a l c u l a t e d for a short time increment At, s t a r t i n g with known c o n d i t i o n s , to evaluate the c o n d i t i o n s at a l a t e r time. The incremental l i n e a r nature of the system i s considered by assuming l i n e a r behavior throughout each successive time step, and by making proper m o d i f i c a t i o n s to the l i n e a r p r o p e r t i e s p r i o r to each step. In the present study, where the s t i f f n e s s p r o p e r t i e s of the s t r u c t u r e behave non-1inearly, being s t r a i n and s t r e s s dependent, the step-by-step i n t e g r a t i o n method i s used. The complete response i s obtained by using the known displacement, v e l o c i t y and a c c e l e r a t i o n at the end of one time 46 i n t e r v a l as the i n i t i a l c o n d i t i o n s for the next i n t e r v a l . The process i s continued step-by-step from i n i t i a l s t a t i c c o n d i t i o n s to the completion of seismic motions. The dynamic e q u i l i b r i u m c o n d i t i o n i s s a t i s f i e d at the beginning and end of each time i n t e r v a l . The equations of motions for non-linear s t r u c t u r e s , together with t h e i r s o l u t i o n by a step-by-step i n t e g r a t i o n procedure w i l l be presented i n the f o l l o w i n g s e c t i o n s . 4.2 Equation of motion As mentioned p r e v i o u s l y i t i s assumed that the earthquake ground motion i s i d e n t i c a l at a l l p o i n t s along the base of the s t r u c t u r e . S p a t i a l v a r i a t i o n s i n the ground motion are not considered i n the present a n a l y s i s . The dynamic equation of motion of nodal p o i n t s above the the r i g i d base for the f i n i t e element system when subjected to earthquake ground motion can be expressed i n the matrix form: D e t a i l s of the d e r i v a t i o n i s given i n Zienkiewicz (1971) and Newmark and Rosenblueth (1971). The matrices are defined as f o l l o w s : [M] = the mass matrix ( s e c t i o n 3.3.2) [M] {x} + [C] {x} + [K] {x.} = {R} (4-1 ) [C] the damping matrix ( s e c t i o n 3.3.2) [K] the s t i f f n e s s matrix ( s e c t i o n 3.3.2) 47 {x} = vector of nodal displacements r e l a t i v e to base {R} = i n e r t i a force vector For the lumped mass system the i n e r t i a force vector {R} i s (R) = -{M^xb - {M\% (4-2) where, {M}x = ( m , 0 m 2 0 - - - m 0 - - m 0 ) (4-3) and {M}Y = (0 m, 0 m2 - - - 0 m - - 0 m ) (4-4) {M}x and {M}Y are the vectors of the mass of the nodes as s o c i a t e d with the x and y degrees of freedom r e s p e c t i v e l y (m i s the mass of the node associated with the i t h degree of freedom which i s in the x d i r e c t i o n ) . Therefore, {M}x + {M}Y = diagonal of the mass matrix [M], x D and % are r e s p e c t i v e l y the h o r i z o n t a l and v e r t i c a l components of the ground a c c e l e r a t i o n . The s t i f f n e s s matrix for each f i n i t e element during any time i n t e r v a l of the step by step method i s obtain by the procedure described i n chapter 3.3 and i n appendix I . The complete s t i f f n e s s matrix [K] of the e n t i r e s t r u c t u r e i s obtained from the i n d i v i d u a l element s t i f f n e s s matrices by the d i r e c t s t i f f n e s s method. In a s i m i l a r manner the s t r u c t u r e damping matrix i s obtained. These matrices are of order N x N, 48 where N i s the number of degrees of freedom. Since the matrices are banded and symmetric, only matrices of s i z e N x M need be considered, where M i s the h a l f bandwidth of the s t r u c t u r e . This g r e a t l y reduces the computer storage requirements. 4.3 Incremental equation of motion Equation (4-1) must be s a t i s f e d at every i n s t a n t i n time. Let T = t + At, where At i s a small time i n t e r v a l then, [M] t{x} t+ [C] t {x}t + [K] t {x} t = {R}t (4-5) [M] T{x} T+ [C] T{x} T+ [ K ] T { x } T = {R} T (4-6) The s u b s c r i p t r e f e r s to the i n s t a n t of time at which the p a r t i c u l a r q u a n t i t y takes on i t s value. Since the mass matrix i s constant matrix, [M] T = [M] t = [M] so t h a t , [M] T{x} T - [Mj t{x} t = [ M ] T ( { x } T - { x } t )= [M]{Ax} (4-7) where, ( { x } T - {x} t)= {Ax} T (4-8) and the damping matrix i s assumed to remain constant throughout the a n a l y s i s , [ C ] T { x } T - [ C ] t { x } t = [ C ] T ( { x } T - { x } t )= [C] {Ax} T (4-9) where, 49 ({x} T - {x} t )= {Ax} T (4-10) However, the [K] matrix depends on the l e v e l of shear s t r a i n i n the s o i l , as a r e s u l t of the non-linear s t r e s s - s t r a i n r e l a t i o n , and thus equations s i m i l a r to (4-7) and (4-9) can not be obtained for t h i s matrix i n a s t r a i g h t forward manner. There are two ways to a r r i v e at expressions s i m i l a r to (4-9) using approximation methods. A crude method i s to replace [ K ] T with [K] , so t h a t , [ K ] T { x } T - [ K ] t { x } t = [ K ] t ( { x } T - { x } t ) = [K] f{Ax} (4-11) where, U x } T - {x} t )= {Ax} T The tangent s t i f f n e s s p r o p e r t i e s defined at the beginning of the time i n t e r v a l are used. During any time i n t e r v a l the [k] matrix i s changing as the s t i f f n e s s p r o p e r t i e s change with s t r a i n . The [K] matrix should represent some average s t i f f n e s s p r o p e r t i e s during the time i n t e r v a l . This can be only done by i t e r a t i o n because the displacement at the end of the time increment depend on these p r o p e r t i e s . As w i l l be explained i n chapter 4.3, i t e r a t i o n i s not always d e s i r e d . For t h i s reason the approximate method i s used i n the present a n a l y s i s . Subtracting equation (4-5) from equation (4-6) and t a k i n g note of equations (4-7), (4-9) and (4-11), gives the incremental form of the equation of motion f o r the time i n t e r v a l s t a r t i n g at time t , which i s as f o l l o w s : 50 [M] {Ax} T + [C] {Ax} T + [K] t{Ax} T= {AR}T (4-12) Mat r i x equation (4-12) which i s a set of second order d i f f e r e n t i a l equations, may be reduced to a recurrence equation i f an assumption i s made regarding the v a r i a t i o n of the a c c e l e r a t i o n of each node w i t h i n the time i n t e r v a l At. To t h i s end, numerical procedures developed by Newmark (1959) can be a p p l i e d to the equation so that unknown displacements { x } T , { x } T , { x } T at time T, can be expressed i n terms of known displacements {x} t ,{x} t ,{x} t at time t . 4.4 Step by Step I n t e g r a t i o n In Newmark's 0 method of step by step i n t e g r a t i o n two parameters, a and 0 are used so that the v e l o c i t y and displacement at time T can be expressed i n terms of the a c c e l e r a t i o n , v e l o c i t y and displacement at time t , and of unknown a c c e l e r a t i o n at time T. The equations for v e l o c i t y and displacement at time T are as f o l l o w s : (x} T = {x} t + (1-a)At{x} t + aAt{x} T (4-13) {x} T = {x} t + At{x} t + (.5-/3) ( A t ) 2 { x } t + 0 ( A t ) 2 { x } T (4-14) At i s r e f e r e d to as the 'time step' of i n the i n t e g r a t i o n . Newmark (1959) proposed f o r a u n c o n d i t i o n a l l y stable i n t e g r a t i o n procedure that a = 1/2 and 0 = 1/4. This 51 corresponds to a constant average a c c e l e r a t i o n method of i n t e g r a t i o n . When values of a = 1/2 and /3 = 1/6 are used, a l i n e a r v a r i a t i o n of a c c e l e r a t i o n i s assumed over the time increment. The l i n e a r v a r i a t i o n of a c c e l e r a t i o n method was proposed by Wilson and Clough (1962). Both approaches have been incorporated i n t o the computer program and are options a v a i l a b l e to the user. The l i n e a r v a r i a t i o n of a c c e l e r a t i o n method may lead to an i n s t a b i l i t y of the s o l u t i o n . This i n s t a b i l i t y i s u s u a l l y dependent on the s i z e of the time step At used i n the i n t e g r a t i o n , m a t e r i a l p r o p e r t i e s , and s i z e of the f i n i t e element g r i d . The Wilson 8 method which provides s t a b i l i t y i n the s o l u t i o n (Wilson et a l , 1973) has been w r i t t e n i n t o the computer program. If the f o l l o w i n g s i m p l i f y i n g expressions are used, {a} t = At{x} t (4-15) {b}» = At{x} t +.5(At)Mx} t (4-16) then equation (4-13) and equation(4-14) can w r i t t e n as: {Ax} T= {a} t + aAt {Ax }T . (4-17) {Ax} T = {b} t + 0 ( A t ) 2 { A x } T (4-18) Equations (4-17) and (4-18) are s u b s t i t u t e d i n t o equation (4-12). T r a n s f e r r i n g a l l forms a s s o c i a t e d with known values to the r i g h t hand side leads to the f o l l o w i n g equation, [D] T{Ax} T = [ P ] T (4-19) 52 where, [D] T = [M] + aAt[C] + /3(At) 2[K] (4-20) and [ P ] T = (R} T ~ (R), - [C] (a} t - [Kj t{b} (4-21) Equation (4-19) i s equivalent to a s t a t i c increment e q u i l i b r i u m r e l a t i o n s h i p and may be solved by matrix i n v e r s i o n and m u l t i p l i c a t i o n . Consequently, with incremental displacements {Ax} T and v e l o c i t i e s {Ax} T obtained from equations (4-17) and (4-18). This numerical a n a l y s i s procedure includes two s i g n i f i c a n t assumptions: (1) the a c c e l e r a t i o n i s assumed to vary i n some described manner, (as determined by the values of the a and 0 ), during the time step and (2) the s t i f f n e s s p r o p e r t i e s of the s t r u c t u r e remain constant during any time step and are equal to i t s values at the beginning of the time step. Neither of the two assumptions are s t r i c t l y c o r r e c t but can be viewed as good approximations when the time step At i s chosen to be sma l l . Because the s t r u c t u r e damping matrix i s assumed to remain constant f o r the duration of the a n a l y s i s , inherent e r r o r s a r i s i n g from assumption (2) are not present i n the modeling of damping p r o p e r t i e s . {Ax} T = [ D ) f M P l T (4-22) 53 As discussed i n Chapter 3.3, the s t i f f n e s s property of the s o i l i s determined by the tangent shear modulus which i s v a r i e d with the l e v e l of shear s t r a i n and mean normal s t r e s s , and by the tangent bulk modulus which i s v a r i e d with the mean normal s t r e s s only. Element shear s t r a i n and mean normal s t r e s s values c a l c u l a t e d at the end of the time step are used to compute moduli according to t h e i r c o n s t i t u t i v e r e l a t i o n s . The [K] matrix for the next time step i s based on these newly c a l c u l a t e d moduli. In t h i s way the i n e l a s t i c behavior of s o i l i s modelled by an incremental l i n e a r approach. In non-linear problems true convergence seldom occurs, i n that the incremental forces a p p l i e d are not e q u i l i b r a t e d by the incremental s t r e s s e s . In order to resolve t h i s i n c o nsistency an er r o r c o r r e c t i o n i s incorporated i n t o the a n a l y s i s This i s discussed i n the next s e c t i o n . 4 . 5 Er r o r C o r r e c t i o n A tangent s t i f f n e s s method has been used to formulate the incremental f i n i t e element matrix equation (4-12). With t h i s method there are b a s i c a l l y three approaches that can be taken. 1. Accept the s t r e s s e s : Accept {6c}= [D L]{6e}, where [D L] i s the tangent s t r e s s -s t r a i n matrix used f o r the l a s t i t e r a t i o n , see equation (3-20). C a l c u l a t e the new [D L] based on the s t r e s s l e v e l a t t a i n e d . In t h i s approach e q u i l i b r i u m i s maintained 54 throughout the a n a l y s i s , however the true s t r e s s - s t r a i n curve i s not followed. The r e s u l t i n g i n c onsistency between the computed s t r a i n and exact s t r a i n s i s accepted. 2. Accept the s t r a i n s : Accept s t r a i n s computed from nodal displacements obtained by the step by step i n t e g r a t i o n procedure and c a l c u l a t e the new tangent modulus based the the s t r a i n l e v e l . E q u i l i b r i u m i s v i o l a t e d , but the true s t r e s s - s t r a i n curve i s followed. 3. Accept the s t r a i n s and apply c o r r e c t i o n f o r c e s : The true curve i s followed, and any e r r o r i n e q u i l i b r i u m at the end of the time step i s c a l c u l a t e d and a p p l i e d at the beginning of the next time step. In t h i s way any e r r o r s i n e q u i l i b r i u m do not accumulate. A l l the above methods can b e n e f i t from m u l t i p l e i t e r a t i o n s . For example, from the i n i t i a l tangent moduli used for the f i r s t i n t e g r a t i o n of the time step, the tangent moduli at the end of the time step can be obtained. I t e r a t i o n i s then performed on the time step using the average of the two tangent moduli. This approach may be described as a step-secant approach. While i t e r a t i n g i s d e s i r a b l e i n terms of improving the accuracy of the incremental approach, the number of a d d i t i o n a l i t e r a t i o n s at each time step would p r o p o r t i o n a l l y increase the cost of the a n a l y s i s . This may prove to be expensive when a large number of elements i s i n v o l v e d . (Although there i s a trade o f f here in that one can g e n e r a l l y use l a r g e r time increments i f i t e r a t i o n s are used.) For t h i s reason, i t e r a t i o n i s not considered i n the 55 present a n a l y s i s . Approach (3) has the advantage of f o l l o w i n g the a c t u a l s t r e s s - s t r a i n curve while at the same time s a t i s f y i n g e q u i l i b r i u m . In t h i s sense, and when m u l t i p l e i t e r a t i o n s are not performed, approach (3) i s the most accurate method. This approach has been b u i l t i n t o the a n a l y s i s and i s discussed here i n greater d e t a i l . The s t r a i n s determined i n each i n t e g r a t i o n are accepted as the true s t r a i n s and the s t r e s s - s t r a i n r e l a t i o n s are used to determine the 'true r e s t o r i n g ' force [ K ] T {x} T . However these r e s t o r i n g forces do not n e c e s s a r i l y s a t i s f y the equlibrium equation (4-3). An set of a r t i f i c i a l ' e x t e r n a l ' f o r c e s , {P }, defined by the f o l l o w i n g can be a p p l i e d to the system so that e q u i l i b r i u m i s r e s t o r e d , {P e r r }= {R}T - [M]{x} T - [C] { x } T - [ K ] T { x } T (4-23) The {P e r r ) i s added to {P} T for the next time increment of the a n a l y s i s . A l l terms i n the above equation are computed i n a s t r a i g h t forward manner. The manner i n which the term [ K ] T { X } T i s evaluated deserves comment. A l l other terms i n the above equation are computed i n a s t r a i g h t forward manner. A basic assumption of the f i n i t e element method i s that a l l forces are transmitted at the nodal p o i n t s . The t o t a l force at any one node i s the sum of the c o n t r i b u t i o n s of i t s surrounding elements to that p a r t i c u l a r node. The r e s t o r i n g forces [K] y {x} T then, are nodal forces representing the sum of the s t r e s s e s of the elements. The s t a t e of s t r e s s w i t h i n an element can be 56 represented by a s t a t i c a l l y equivalent system of nodal point f o r c e s . From the s t r e s s e s {a}, nodal forces are obtained by, As was the d e c i s i o n e a r l i e r i n ev a l u a t i n g the s t i f f n e s s matrix f o r an element, Chapter 3.3., the str e s s e s and s t r a i n s at the c e n t r o i d of the element are taken to be re p r e s e n t a t i v e of the average s t r e s s e s and s t r a i n s i n the element. In t h i s case, the s t r e s s vector can be removed from w i t h i n the i n t e g r a l s i g n , as shown above. The t o t a l force at any one node i s the sum of the c o n t r i b u t i o n s of i t s surrounding elements to that p a r t i c u l a r node. The r e s t o r i n g forces [ K ] T { x } T then, are nodal forces representing the sum of the. st r e s s e s of the elements. The manner i n which element s t r e s s e s are c a l c u l a t e d i s described below. From the accepted s t r a i n s the corresponding s t r e s s e s can be determined as f o l l o w s : 1. C a l c u l a t e incremental s t r a i n s : (4-24) {Ae} ; = [B; ]{Au}i (4-25) where, {Au}| i s the vector of incremental nodal displacements a s s o c i a t e d with element i . [B i ] i s the value of the s t r a i n displacement matrix element i , at i t s c e n t r o i d , see Appendix I . 57 {Ae } jthe incremental s t r a i n vector (A£ ,&ZY,&yY) for element i . 2. C a l c u l a t e increment s t r e s s e s : {Aa},= [ D L ] { A e}j (4-26) where, [D L] i s the s t r e s s - s t r a i n matrix, based on tangent shear and • bulk moduli, of element i , during the i n t e r a t i o n time step, see equation (3-20). {Ao}j i s the incremental s t r e s s vector (Aa x ,Aa y ,Ar x y ) for element i . 3. C a l c u l a t e t o t a l element s t r e s s e s : For any element, s t r e s s e s at time T are equal to the summation of the incremental s t r e s s e s up to time T. They are obtained as f o l l o w s : ( a } T i = {a} t i + {Aa} T i (4-27) where, T = t + At {a} f i s the t o t a l s t r e s s (o x , oY , r x y) , at time t { A a } T i s the incremental s t r e s s for the l a s t time step 4.6 Procedure for s t r a i n r e v e r s a l occurrence A complete d e s c r i p t i o n of the s t r e s s - s t r a i n r e l a t i o n s h i p under general loading and unloading has been given i n Chapter 3.2.2. On unloading or r e l o a d i n g of a s o i l element, the shear stress-shear s t r a i n behavior of the m a t e r i a l i s governed by a 58 newly defined curve, equation (3-14), where i t s o r i g i n i s l o c a t e d at the shear s t r e s s shear s t r a i n r e v e r s a l p o i n t , ( 7 , , T R ) , see Figure 3-2. For two reasons i t i s important that the a n a l y s i s can determine when s t r a i n r e v e r s a l occurs: (1) upon r e v e r s a l a new s t r e s s - s t r a i n branch i s followed, and (2) there i s a d i s c o n t i n u i t y of shear modulus at any s t r a i n r e v e r s a l l o c a t i o n . Shear s t r a i n , as defined i n Chap 3.2.2 i s r e - w r i t t e n here for convenience, The rate of shear s t r a i n , d i f f e r e n t i a t i n g equation (4-29) with respect to time, i s defined by, 7 m a x (4-28) 7 , m a x ( 7 X y ) ( 7 x y ) + U x ~ C y H e x ~ f y ) (4-30) m a x Based on displacement and v e l o c i t y values {x} and {x}, element s t r a i n s and shear s t r a i n rates are c a l c u l a t e d by, {e}; = [ B j ] {u}j (4-31) where, {u}| i s the vector of nodal displacements a s s o c i a t e d with element i . i s the s t r a i n vector ( £x , i - y , 7 x y ) for element i . and, (4-32) where, 59 {u}j i s the vector of nodal v e l o c i t i e s a s s o c i a t e d with element i . {e}j = i s the rate of s t r a i n vector ( £ x , J-y ,7xy) for element i . S t r a i n r e v e r s a l can be defined as a decrease from the most recent |7 m a x | value, see Figure 3-2. A l o c a l maximum or minimum in the value of shear s t r a i n occurs when 7 m a x i s equal to zero. Therefore, s t r a i n r e v e r s a l occurs i n an element when there i s a change i n sign i n the rate of shear s t r a i n 7 m Q X from one time step to the next time step. Let T = t + At S t r a i n r e v e r s a l occurs when, 7 -7mri* < 0 (4-33) 'max, 'maxj where, the s u b s c r i p t r e f e r s to the i n s t a n t time at which the p a r t i c u l a r q u a n t i t y takes on i t s value. Now that a c r i t e r i o n for s t r a i n r e v e r s a l has been e s t a b l i s h e d , one f u r t h e r point should be considered. Since a r e v e r s a l point i s c h a r a c t e r i z e d by a d i s c o n t i n u i t y i n modulus, i t may be d e s i r a b l e to reduce the time i n t e r v a l At of the a n a l y s i s with subsequent i t e r a t i o n when they occur, i n order to obtain a be t t e r accuracy when tu r n i n g these ' p o i n t s ' . I d e a l l y the time step of the a n a l y s i s should be reduced s u f f i c i e n t l y in order to land r i g h t on them. With e a r l i e r i n v e s t i g a t i o n s by Salgado (1980) on a one degree of freedom, e l a s t i c - p l a s t i c system, t h i s was e a s i l y achieved. With multi-degree of freedom 60 systems, where more than one s t r a i n r e v e r s a l ( i n d i f f e r e n t elements or l a y e r s ) may occur w i t h i n a time i n t e r v a l and not n e c e s s a r i l y at p r e c i s e l y the same time, and where s t r a i n r e v e r s a l i n one part of the system and may l a t e r i n the same time i n t e r v a l cause s t r a i n r e v e r s a l i n another part of the system, cost c o n s i d e r a t i o n s make t h i s approach i m p r a c t i c a l . A reasonable approach which does req u i r e landing on r e v e r s a l p o i n t s would be to make a p r e d i c t i o n regarding the segments of the time i n t e r v a l f or which s t r a i n r e v e r s a l has and has not occurred. Within the time i n t e r v a l of s t r a i n r e v e r s a l , t h i s can be approximated by assuming a l i n e a r v a r i a t i o n i n a c c e l e r a t i o n . The time of r e v e r s a l can be expressed as, At, • (TW T> 'At ( 4 - 3 4 ) ' Tmaxj — ^maxt and, A t 2 = At - At, ( 4 - 3 5 ) where, At, i s the time segment 'before' s t r a i n r e v e r s a l A t 2 i s the time segment ' a f t e r ' s t r a i n r e v e r s a l This approximation can be used on s t r a i n r e v e r s a l elements, with i t e r a t i o n performed f o r the time i n t e r v a l segment corresponding to the s t r a i n reversed element with the shortest At, time i n t e r v a l . A f t e r the i n t e g r a t i o n f or t h i s time segment has been completed, s t r a i n reversed modulus i s assigned to the appropriate element, and the i n t e g r a t i o n i s performed on the remaining p o r t i o n of the time i n t e r v a l . The s t r a i n r e v e r s a l check procedure and subsequent segmenting of A t 2 f u r t h e r i n t o 61 two smaller segments i f r e v e r s a l occurs, i s performed on A t 2 . This type of approach has been cost e f f i c i e n t when one dimensional s o i l l a y e r systems (max degree of freedom= 20, halfbandwidth= 2) have been analysed, with the l i m i t i n g of the s i z e of At, and A t 2 to At/10, (Lee, 1977). This approach can be e x o r b i t a n t l y expensive f o r t y p i c a l l y large degree of freedom systems that r e s u l t from f i n i t e element modelling. A crude approximation to t h i s approach can be made by es t i m a t i n g , on s t r a i n r e v e r s a l ( s ) , whether r e v e r s a l occurred c l o s e r to the beginning or the end of the time i n t e r v a l . As determined from equation (4-34), i f , At, < .5«At, assume s t r a i n r e v e r s a l at t i f , At, > .5«At, assume s t r a i n r e v e r s a l at T If At, < ,5-At for any s t r a i n reversed elements, s t r a i n reversed moduli are assigned to these elements and the complete time step At i s repeated. Any a d d i t i o n a l s t r a i n reversed elements are assumed to have reversed at the end of the time step and are assigned the appropriate moduli. When s t r a i n r e v e r s a l occcurs c l o s e r to the end of the time i n t e r v a l , the s t r a i n reversed moduli are not assigned to these elements u n t i l the next time step step. I t e r a t i o n i n t h i s case i s not necessary. With t h i s crude approach m u l t i p l e s u b i t e r a t i o n s of any time step are avoided, and at most, a time step i s repeated once. This procedure to handle the occurrences of s t r a i n r e v e r s a l has been adopted i n the present a n a l y s i s . 62 4.7 Summary of procedure A b r i e f o u t l i n e of s o l u t i o n scheme fo r any given time increment i s given as f o l l o w s : 1. Based on the current values of 7 , T , at time t , the tangent shear modulus G and tangent bulk modulus B are c a l c u l a t e d as described i n chapter 3.2.1 and 3.2.2 respect i v e l y . 2. The matrix [K] i n equation (4-20) and (4-21) i s then updated, see chapter 3.3. 3. C a l c u l a t e {a} t and {b} t equations (4-15) and (4-16), s u b s t i t u t e i n t o equation (4-21) to set up [P] and set up [D] 4. Solve for {Ax} Tusing equation (4-22), and subsequently {Ax} T.and {Ax} T from equations (4-17) and (4-18) 5. A c c e l e r a t i o n , v e l o c i t y and displacements of the nodes at the end of the increment are obtained from: {x} T = {x} t + {Ax} T {x} T = {x} t + {Ax} T {x} T = {x}, + {Ax} T 6. Based on displacement and v e l o c i t y values at the end of increment {x} T and {x}T, element s t r a i n s and shear s t r a i n rates are c a l c u l a t e d by, {e} = [B|]{u} (4-31) and, {e} = [B;]{u} (4-32) 7. Check to see i f there are any s t r a i n r e v e r s a l s . S t r a i n r e v e r a l i s determined by, 63 a change i n sign of: If there are s t r a i n r e v e r s a l s and they occur at the beginning of the time step, the i n t e g r a t i o n i s repeated for the time step, by repeating steps 1 to 8. 8. C a l c u l a t e s t r e s s e s according to the appropriate s t r e s s - s t r a i n law. 9. In order to b r i n g the s t r e s s - s t r a i n point c l o s e r to the a c t u a l s t e s s - s t r a i n curve, an a r t i f i c i a l ' e x t e r n a l ' f o r c e , {P e r r }, defined by equation (4-23), {Perr }= {R) T " [M]{x} T - [C] {X }T - [ K ] T { x } T i s c a l c u l a t e d and added to {P} i n step 3 for the next increment. When step 9 has been completed, the a n a l y s i s for one time increment i s f i n i s h e d . The e n t i r e process may be repeated for the next time i n t e r v a l . The process can be c a r r i e d out co n s e c u t i v e l y for any d e s i r e d number of time increments; thus the complete response h i s t o r y can evaluated for the non-linear system. A computer program has been w r i t t e n based on t h i s s o l u t i o n scheme and the o v e r a l l flow chart i s shown i n f i g u r e 4-1. 64 Read i n Data I n i t i a l i z e Values S t a r t of do loop Assign BULK and SHEAR Modulus for each Element at each time step B u i l d Master STIFFNESS and DAMPING Matrices [M] & [C] at each time step Set up Dynamic eqns [D] & {R} redo time step Solve for DISP.,VEL.,ACC. C a l c u l a t e STRAINS at the c e n t r o i d of each Element Are there any s t r a i n Reversals Yes 65 No C a l c u l a t e Stresses according to S t r e s s - S t r a i n Law Store S t r e s s and S t r a i n s i f re q u i r e d Yes More Increments No STOP FIG 4-1 PROGRAM FLOWCHART 66 CHAPTER 5 APPLICATIONS OF THE NON-LINEAR FINITE ELEMENT METHOD 5.1 General The non-linear dynamic response a n a l y s i s developed i n e a r l i e r chapters was a p p l i e d to a number of s o i l s t r u c t u r e systems. The same s t r u c t u r e s were a l s o analysed using Newmark type methods to estimate movements so that a comparison could be made on the p r e d i c t i o n of permanent displacements. If comparison of r e s u l t s are to be meaningful, the analyses should be based on s i m i l a r s t i f f n e s s and damping c h a r a c t e r i s t i c s . In the Newmark a n a l y s i s , (Chapter 2.2) the s o i l i s assumed to behave i n a r i g i d p l a s t i c manner. To c l o s e l y match the r i g i d p l a s t i c behavior, the non-linear f i n i t e element method employs a l i n e a r e l a s t i c - p l a s t i c shear s t r e s s - shear s t r a i n law. This i s done by s e t t i n g the Rf value i n the hyperbolic r e l a t i o n s h i p equal to 0.01 or l e s s . M a t e r i a l damping in the form of p l a s t i c deformation i s inherent i n both approaches. In the f i n i t e element a n a l y s i s viscous damping must be included f o r numerical reasons. A nominal but small value of 2 percent of c r i t i c a l damping i s used to s t a b i l i z e the s o l u t i o n at t u r n i n g p o i n t s of s t r e s s - s t r a i n curves. The Makdisi-Seed procedure f o r p r e d i c t i n g displacements i s a l s o performed. The matching of s t i f f n e s s and damping c h a r a c t e r i s t i c s cannot be s t r i c t l y done i n t h i s case, as the dynamic response of earth s t r u c t u r e s i s evaluated using an equivalent l i n e a r method. While comparison may not be as v a l i d , Makdisi and Seed have 67 shown good agreement with the r i g i d block analogy. In view of t h i s , t h e i r approach i s included i n the comparative study. A h y p o t h e t i c a l c l a y slope s t r u c t u r e and c l a y dam were used i n the study. Each has a height of 150 f t . and slopes of 2 to 1. The y i e l d a c c e l e r a t i o n value of the two d i f f e r e n t s t r u c t u r e s , obtained from pseudo-static analyses (see Chapter 5.3.1) are the same. The Newmark a n a l y s i s i m p l i e s that given the same value of y i e l d a c c e l e r a t i o n , the p r e d i c t e d displacements of the two s t r u c t u r e s w i l l be the same. This cannot be s t r i c t l y c o r r e c t as there i s no c o n s i d e r a t i o n taken for geometrical d i f f e r e n c e s . The i n f l u e n c e of the remaining s o i l mass of the s t r u c t u r e on the s l i d e mass i s not i n h e r e n t l y considered i n the Newmark a n a l y s i s . The displacement f i e l d s may be s i g n i f i c a n t l y d i f f e r e n t . In order to i n v e s t i g a t e the p o s s i b l e d i f f e r e n c e s , the non-linear f i n i t e element method i s a p p l i e d to the two d i f f e r e n t s o i l s t r u c t u r e s . In a l l the methods of a n a l y s i s , the San Fernando earthquake February 9, 1971 N21E component recorded at Lake Hughes S t a t i o n 12 i n C a l i f o r n i a was used as the d i g i t i z e d r i g i d base motion. The a c c e l e r a t i o n time h i s t o r y and a c c e l e r a t i o n response spectra are shown on Figure 5-1. The earthquake had a maximum a c c e l e r a t i o n of .35g and a predominant period of .18 sec. The Lake Hughes record was obtained on rock and i s an appropriate free f i e l d motion to use when the base of an earth s t r u c t u r e re st on rock. The most s i g n i f i c a n t motions occur i n the f i r s t 15 seconds of the record. As a large percentage of the 6 8 FIG. 5-1 ACCELERATION RESPONSE SPECTRA AND TIME HISTORY FOR SAN FERNANDO EARTHQUAKE 69 permanent deformation should occur w i t h i n t h i s time, (for s t r u c t u r e s subjected to t h i s motion) a duration of 15 seconds i s used i n the analyses. The time i n t e r v a l of the d i g i t i z e d earthquake record i s used as the time step (At= .02) of the dynamic f i n i t e element analyses. As a p r e l i m i n a r y v e r f i c a t i o n of the non-linear method, a s i n g l e q u a d r i l a t e r a l f i n i t e element comparison with Newmark's procedure i s presented i n the next s e c t i o n . 5.2 Si n g l e element comparison with Newmark a n a l y s i s A s i n g l e q u a d r i l a t e r i a l f i n i t e element r e s t i n g on a r i g i d base with assigned s t a t i c and dynamic s o i l p r o p e r t i e s are shown in Figure. 5-2. Nodes 3 and 4 are allowed to t r a n s l a t e i n the h o r i z o n t a l d i r e c t i o n only. The element behaves i n a l i n e a r e l a s t i c - p l a s t i c manner to shear as noted i n the e a r l i e r s e c t i o n of t h i s chapter. The shear s t r e s s - shear s t r a i n c h a r a c t e r i s t i c s of the element are shown i n Figure 5-3. The s t a t i c shear s t r e s s Qst can be considered a ' s t a t i c b i a s ' , whereby on shaking, the accumulation of displacement w i l l tend to be i n the d i r e c t i o n of the s t a t i c f o r c e . The formulation of the mass matrix assumes the mass of the element i s lumped at the nodes, see Chapter 3.3.2. The f i n i t e element under these c o n d i t i o n s i s e s s e n t i a l l y a lumped mass, spr i n g dashpot mechanical model, where the s p r i n g and dashpot replace the shear s t i f f n e s s and viscous damping of the element r e s p e c t i v e l y . The movement of the lumped masses at the top nodes of the s t a t i c l y T in 10 f t . — 1 0 0 p c f Qy= 5 0 - p s f Gmax=3000psf FIG. 5-2 SINGLE FINITE ELEMENT MODEL FIG. 5-3 SHEAR DEFORMATION RELATIONSHIP 71 biased f i n i t e element should be very s i m i l a r to the s l i d i n g in the Newmark analogy of a r i g i d block on a i n c l i n e d plane. Therefore a v a l i d comparison can be made. The s i n g l e f i n i t e element connected to a r i g i d base i s subjected to the San Fernando earthquake motion scaled to a maximum a c c e l e r a t i o n of .50g. The maximum displacement of the top nodes f o r d i f f e r e n t s t a t i c shear values i n the f i n i t e element method are compared with displacements obtained by Newmark (1965) i n Figure 5-4. The Newmark p r e d i c t i o n of displacements were computed f o r a range of r e s i s t a n c e c o e f f i c i e n t "N" values using the scaled San Fernando earthquake motion. The procedure i s described i n Chapter 2.2. The re s i s t a n c e c o e f f i c i e n t corresponds to the y i e l d a c c e l e r a t i o n of the f i n i t e element. The y i e l d a c c e l e r a t i o n i s r e l a t e d to i t s s t a t i c shear s t r e s s and was computed as f o l l o w s : N was defined by Newmark as being the c o e f f i c i e n t of a c c e l e r a t i o n a c t i n g i n the proper d i r e c t i o n to cause s l i d i n g (or i n t h i s case y i e l d i n g of the f i n i t e element), hence: ( Q y - Q s t )L= NW (5-1 ) where Q y y i e l d s trength (50psf) L Length of element (10ft) N Newmark's c o e f f i c i e n t W weight of the top h a l f of the element (2.5X10.X100 l b / f t 3 ) Q s t s t a t i c shear s t r e s s 10. 5 • • 1 -• LEGEND _ D » S F E ^ • Newmark - D -• — • • _ • • • • • • • B 1 1 1 r. 0.0 .10 .20 .30 .40 N _ Max Resistance Coeff A Max Earthquake Acc F1G.5-4 SINGLE FINITE ELEMENT NEWMARK DISPLACEMENT COMPARISON 73 T y p i c a l time h i s t o r i e s of the earthquake induced displacements computed using the two procedures are shown i n Figure 5-5 f o r comparison. 1 0 . 0 Time sees FIG. 5-5 SINGLE FINITE ELEMENT MODEL NEWMARK ANALYSIS DISPLACEMENT HISTORY COMPARISON A s t a t i c bias of 43.75 l b s i s used i n the f i n i t e element method, corresponding to the Newmark N value of .025. As expected the displacement accumulates with time and the maximum values occurs at the end of the shaking p e r i o d . I t may be seen that the a n a l y s i s presented herein p r e d i c t s displacements that are i n 74 good agreement with the simpler Newmark r i g i d p l a s t i c model. 5.3 Dynamic Response of Clay S t r u c t u r e s The f i n i t e element re p r e s e n t a t i o n s of the cross sections of the c l a y slope s t r u c t u r e and c l a y dam are shown on Figure 5-6 and Figure 5-7 r e s p e c t i v e l y . The s t r u c t u r e s are assumed to rest on a r i g i d rock base. The dynamic response procedure requires that the pre-earthquake s t r e s s e s be determined beforehand. The s t a t i c a n a l y s i s i s performed using the f i n i t e element program SOILSTRESS developed by Byrne (1981). For each s t r u c t u r e , the same f i n i t e element g r i d was used for both the s t a t i c and dynamic a n a l y s i s . The non-linear nature of s t a t i c shear s t r e s s - s t r a i n behavior i s modelled assuming the Duncan and Chang (1970) h y p e r b o l i c r e l a t i o n s h i p . A secant modulus as determined by the hyperbolic r e l a t i o n s h i p i s used i n a one step l i n e a r e l a s t i c i t e r a t i v e approach. According to t h i s method, the l i n e a r problem i s solved u n t i l agreement i s obtained between the s t r a i n s used to compute secant moduli i n each s o i l element and the s t r a i n s developed using these assumed secant moduli. S t a t i c and dynamic s o i l parameters used for the a n a l y s i s of the c l a y s t r u c t u r e s i n l a t e r s e c t i o n s of t h i s chapter are shown in Table I . For saturated undrained normally consolidated c l a y s , Duncan 1980 suggested E ; ( i n i t i a l Young's modulus) i s roughly equal to 600Su. Assuming a Poisson r a t i o .50, the i n i t i a l s t a t i c shear modulus Gj i s equal to 2 0 0 S U . Laboratory 80 160ft Scale FIG. 5-6 FINITE ELEMENT GRID OF CLAY SLOPE STRUCTURE I I 1 0 80 160 ft Scale F I G . 5 - 7 FINITE ELEMENT GRID O F CLAY D A M -si CO 7 7 and f i e l d s t u d i e s have shown that f o r the same s o i l under the same c o n d i t i o n s , maximum dynamic shear modulus G m a x i s greater than the s t a t i c equivalent G ; by a f a c t o r of 4 to 5 . A c c o r d i n g l y , the dynamic maximum shear modulus i s assigned the value l O O O S u . This i s i n agreement with the Seed e t . a l . , ( 1 9 6 9 ) f i n d i n g s , where la b o r a t o r y r e s u l t s have shown G m a x ranges 1 0 0 0 to 3 0 0 0 S u . The shear strength Su i s assumed constant (therefore making G m a x constant) throughout the c l a y slope and dam s t r u c t u r e s . In the dynamic a n a l y s i s , the i n i t i a l value of bulk modulus i s c a l c u l a t e d from the Gma% value and an assumed Poisson r a t i o equal to . 4 5 . TABLE I S t a t i c and Dynamic S o i l P r o p e r t i e s A n a l y s i s Shear Strength l b / f t 2 Unit Wt. l b / f t 3 Poi sson Ra t i o F a i l u r e Ratio Shear modulus parameters STATIC 2 9 4 0 . 1 2 5 . . 4 9 0 . 9 n= 0 Kg= 2 7 8 . ( G i / S ^ 2 0 0 . ) DYNAMIC 2 9 4 0 . 1 2 5 . . 4 5 . 0 1 e l a s - p l a s 1 . 0 hyperbolic G m a x / S u = 1 0 0 0 . In the non-linear method, the s t a t i c s t r e s s e s obtained from the SOILSTRESS program are assumed to be the i n i t i a l s t r e s s state f or the dynamic a n a l y s i s . F i n i t e elements are assigned a s t a t i c b i a s on t h e i r shear s t r e s s - shear s t r a i n curve equal to t h e i r s t a t i c shear s t r e s s . The i n i t i a l shear modulus i s the 78 G m a x value. I t has been observed i n the f i e l d that seismic loading of e a r t h s t r u c t u r e s cause l o n g i t u d i n a l v e r t i c a l t e n s i l e cracks. The presence of a t e n s i l e crack allows free movement i n a d i r e c t i o n away and perpendicular to the crack plane. This free h o r i z o n t a l movement corresponds to free volume change. The approach i n which t e n s i l e cracks are modelled i s l i m i t e d by the i s o t r o p i c assumptions of the present a n a l y s i s . T e n s i l e crack behavior may only be p r o p e r l y modelled i n an a n i s o t r o p i c manner, . where Young's modulus i s reduced i n the d i r e c t i o n of free movement and i s kept at the o r i g i n a l value i n the d i r e c t i o n of the crack plane. For an i s o t r o p i c m a t e r i a l , t h i s e f f e c t can be achieved by s o f t e n i n g of the s t i f f n e s s moduli G and B. While a v e r t i c a l t e n s i l e crack may allow h o r i z o n t a l movement at the crack l o c a t i o n w i t h i n an element, shear s t r e s s can be c a r r i e d throughout the i n t a c t remainder of the element. Therefore i t would not be s t r i c t l y c o r r e c t to reduce G throughout the e n t i r e element. As w e l l , reducing G to s i g n i f i c a n t l y low values would a c t u a l l y model a l i q u i f i e d c o n d i t i o n , which i s not the i n t e n t . An attempt to incorporate t e n s i l e crack behavior i n t o the a n a l y s i s was c a r r i e d out by reducing only the bulk modulus values, i n elements where t e n s i l e s t r e s s occur. S t a r t i n g from the i n i t i a l s t a t i c s t r e s s s t a t e , i f during shaking the minor p r i n c i p a l s t r e s s o3 becomes t e n s i l e , the bulk modulus i s assigned a value of 1 percent of i t s i n i t i a l value. The bulk modulus of the f a i l e d element i s l e f t at the reduced value for the remainder of the earthquake l o a d i n g . 79 5.3.1 Comparison with the Newmark and  the Makdisi-Seed A n a l y s i s The non-linear dynamic response of a c l a y slope s t r u c t u r e and dam subjected to the San Fernando earthquake motion i s compared against r e s u l t s obtained from procedures developed by both Newmark, and Makdisi-Seed. Newmark's r e s i s t a n c e c o e f f i c i e n t "N" i s the value of the h o r i z o n t a l seismic c o e f f i c i e n t determined from a pseudo-static conventional slope s t a b i l i t y a n a l y s i s which w i l l give a f a c t o r of safety equal to u n i t y . F a i l u r e s u r f a c e , and s o i l p r o p e r t i e s used i n the slope s t a b i l i t y analyses of the c l a y s t r u c t u r e s are shown i n Figure 5-8 and Figure 5-9. The r e s i s t a a c e c o e f f i c i e n t (or y i e l d a c c e l e r a t i o n ) values are c o i n c i d e n t l y equal to O.lOg for the two c l a y s t r u c t u r e s . Computations of displacements using Newmark's procedure are performed using the San Fernando earthquake scaled to maximum a c c e l e r a t i o n values ranging from 0.4 to 1.Og., f o r the determined y i e l d a c c e l e r a t i o n value of 0.1Og. Refinements to allow for v a r i a t i o n i n a c c e l e r a t i o n throughout the slope and s l i d e mass as p r e s c r i b e d by the Makdisi-Seed a n a l y s i s are made as f o l l o w s . The response of the s t r u c t u r e s when subjected to the San Fernando earthquake i s evaluated using the dynamic f i n i t e element program QUAD4. The a n a l y s i s employs the use of shear modulus reduction and damping r a t i o s curves which are b u i l t i n t o the computer program. No attempt i s made to modify these curves. The s o i l elements are FIG. 5-8 SLOPE STABILITY ANALYSIS ON CtAY SLOPE STRUCTURE 4. Center of Failure Circ le FIG. 5 - 9 SLOPE STABILITY ANALYSIS ON CLAY DAM CO 82 are assigned the same G m a x value of lOOOSu. The time h i s t o r y of the average a c c e l e r a t i o n of the s l i d e mass obtained from the dynamic response a n a l y s i s i s used to estimate permanent diplacement of the s l i d e mass. The average a c c e l e r a t i o n i s the weighted average of the a c c e l e r a t i o n values of the nodes w i t h i n the s l i d e mass as given by, a ( t ) = Zntj . X ; (t) Im j where mj and Xj are the mass and a c c e l e r a t i o n at node i . The summation i s taken over a l l the nodes w i t h i n the s l i d e mass. The displacement for both the Newmark and Makdisi-Seed approaches are determined by i n t e g r a t i n g the e f f e c t i v e a c c e l e r a t i o n i n excess of the y i e l d a c c e l e r a t i o n value (O.lOg) for the duration of the earthquake motion. In the f i n i t e element method each mode w i t h i n the domain d i s p l a c e s a d i f f e r e n t amount. In order to compare with the Newmark and Makdisi-Seed methods an average of the f i n a l displacements of the nodes w i t h i n the s l i d e mass i s used. 5.3.1.1 Clay Slope Comparison The r e s u l t s from the Newmark type methods and the non-l i n e a r f i n i t e element a n a l y s i s of the c l a y slope are i n good agreement as shown i n Figure 5-10. T y p i c a l displacement time h i s t o r i e s computed using the Newmark a n a l y s i s and non-linear f i n i t e element method under the same earthquake c o n d i t i o n s are shown i n Figure 5-11. The non-linear f i n i t e element 10B-• LEGEND ° A • N FE _ • Newmark 5 h A M-S A • A a A • .1.0 .20 .30 N _ Max Resistance Coeff A Max Earthquake Acc FIG 5-10 CLAY SLOPE NON-LINEAR FINITE ELEMENT-MAKDISI SEED-NEWMARK DISPLACEMENT COMPARISON 84 N j r~*-^NFE J 0 5 1 0 15 Time sees FIG 5-11 CLAY SLOPE NON-LINEAR FINITE ELEMENT NEWMARK ANALYSIS DISPLACEMENT HISTORY COMPARISON 85 displacement h i s t o r y shows an some u p h i l l motion which the Newmark a n a l y s i s ignores, and my be one reason why the Newmark displacements are l a r g e r . For the c l a y slope s t r u c t u r e , the t y p i c a l displacement p a t t e r n as shown by the d i s p l a c e d g r i d i n Figure 5-12, i s of t r a n s l a t i o n or s l i d i n g of the e n t i r e embankment along a h o r i z o n t a l s l i p plane. The c i r c u l a r arc f a i l u r e does not seem apparent. The y i e l d a c c e l e r a t i o n obtained from a pseudo s t a t i c a n a l y s i s i s f o r the f a i l u r e surface with the minimum f a c t o r of s a f e t y . Therefore the y i e l d a c c e l e r a t i o n fo r any other f a i l u r e surface ( i n t h i s case of a s l i d i n g block) would be somewhat higher value. A higher y i e l d a c c e l e r a t i o n value would r e s u l t i n lower Newmark p r e d i c t e d displacements. From the formulation of the dynamic f i n i t e element a n a l y s i s , the forces on a s t r u c t u r e caused by the r i g i d base a c c e l e r a t i o n i s represented by i n e r t i a loads at the free nodes, (Chapter 3). I n e r t i a f orces for any given row of elements should be greatest for the row of elements j u s t above the r i g i d base. With t h i s c o n s i d e r a t i o n , the p r e d i c t e d behavior of s l i d i n g along the bottom row of elements (Figure 5-12) i s reasonable. As w e l l , h o r i z o n t a l s l i d i n g of embankment slopes have been observed by Seed (1979). The r e s u l t s of the non-linear f i n i t e element a n a l y s i s i n d i c a t e that i n general, t e n s i l e s t r e s s e s f i r s t occur i n elements l o c a t e d at the top of the embankment, and progress v e r t i c a l l y downwards from there. This i s i n agreement with f i e l d o bservations, where v e r t i c a l t e n s i l e cracks are known to Displacements Magnificied 30 times FIG. 5-12 DISPLACED GRID OF CLAY SLOPE STRUCTURE 00 CO 87 occur i n s e i s m i c l y loaded slopes and embankments, (Seed e t . a l . , 1973). With these t e n s i l e c r a c k s , there i s p o t i e n t a l for a d d i t i o n a l deformations. As observed i n Figure 5-12, the t e n s i l e f a i l e d elements i n the upper s e c t i o n of the slope s t r u c t u r e do not r e s u l t i n a d d i t i o n a l r e l a t i v e displacement of these elements. Displacement, as noted p r e v i o u s l y , i s mainly a t t r i b u t e d to the shear movement i n the bottom row of elements. The i s o t r o p i c assumptions i n the treatment of t e n s i l e cracks i n the present a n a l y s i s may account f o r the lack of agreement between observed and p r e d i c t e d behavior. Though, i t should be noted that w i t h the development of t e n s i l e c racks, any s t a t i c b ias present may be reduced. Therefore i n an a n i s o t r o p i c a n a l y s i s , a d d i t i o n a l displacements where t e n s i l e cracks develop may not be as s i g n i f i c a n t , as there i s an accompanying reduction i n s t a t i s b i a s . O v e r a l l , deformations may not be s u b s t a n t i a l l y d i f f e r e n t than that p r e d i c t e d i n the present a n a l y s i s . There i s another important c o n s i d e r a t i o n . S t a t i c g r a v i t y loads of the s t r u c t u r e during the earthquake loadings i s c a r r i e d by dynamically softened s o i l m a t e r i a l . Under these c o n d i t i o n s , the earthquake loading w i l l cause a d d i t i o n a l s t a t i c displacements. While t h i s may e x p l a i n the observed f i e l d displacements, the s t a t i c e f f e c t has not been formulated i n t o the a n a l y s i s . For t h i s reason, agreement may not be p o s s i b l e . 5.3.1.2 Clay Dam Comparison I t i s observed i n Figure 5-13 that the non-linear method p r e d i c t s markedly lower displacements for the c l a y dam than does LEGEND D A • N FE n Newmark A M-S • A • A • A • : i 2 i_2 .10 .20 .30 N _ Max Resistance Coeff A Max Earthquake Acc FIG 5-13 CLAY DAM NON-LINEAR FINITE ELEMENT -MAKDISI SEED-NEWMARK DISPLACEMENT COMPARISON 89 the Newmark or Makdisi-Seed method. As i n d i c a t e d by the displacement h i s t o r y i n Figure 5-14, the non-linear f i n i t e element method p r e d i c t s that the dam s l i d i n g mass undergoes s i g n i f i c a n t negative displacements. The negative displacements can be explained i n t h i s way. I t i s reasonable to assume that during shaking, outward movement of a s l i d e mass on the other embankment slope should occur. The outward movement of a s l i d e mass on one slope w i l l allow movement of the other s l i d e mass i n i t s inward d i r e c t i o n , as r e s i s t a n c e to movement i n that d i r e c t i o n i s g r e a t l y reduced due to temporary f a i l u r e of the outwarding s l i d e mass. I f t h i s i s the case, there i s a tendency for o v e r a l l outward displacement of the s l i d e mass to be reduced. A t y p i c a l displacement p a t t e r n of the c l a y dam a f t e r earthquake loading Figure 5-15, shows net outward movement of the each dam slope. In addition., lower displacement estimates may be expected as the o v e r a l l s t a t i c b i a s on the c l a y dam i s zero. The t o t a l of the shear s t r e s s e s on the dam slopes are e s s e n t i a l l y equal and opposite. Permanent displacements should be lower than i n the c l a y slope s t r u c t u r e , where there i s a net s t a t i c b i a s . The Newmark type estimates of permanent displacement are ove r l y conservative under these c o n d i t i o n s . Displacements for symmetrical r e s i s t a n c e (Newmark, 1965) are an order of magnitude lower than for unsymmetrical r e s i s t a n c e . Comparison with symmetrical displacements may be more a p p l i c a b l e i n the c l a y dam where net s t a t i c bias i s zero. VI w \A AA ^ 1 \f\rJ\ A ft 1 V v • • v 1/ 1 a m Q x = - 7 5 9 0 5 10 15 Time sees FIG. 5-14 CLAY DAM DISPLACEMENT TIME HISTORY Displacements Magnificied 30 times FIG. 5-15 DISPLACED GRID OF CLAY DAM 92 5.3.2 Comparison of Hyperbolic and E l a s t i c - P l a s t i c Shear s t r e s s - s t r a i n Laws As o u t l i n e d i n Chapter 3, i n shear, s o i l behaves i n a hy p e r b o l i c and h y s t e r e t i c manner. The e l a s t i c - p l a s t i c shear law, while used for comparative purposes e a r l i e r i n t h i s chapter, i s a simple approximation to the hyperbolic r e l a t i o n s h i p . The e l a s t i c - p l a s t i c curve approximation to the hyp e r b o l i c curve i s shown i n Figure 5-16. Using the hyperbolic r e l a t i o n s h i p , (where Rf= 1.0) the dynamic a n a l y s i s w i l l c l o s e r model and p r e d i c t the displacements. For t h i s reason, the c l a y slope s t r u c t u r e i s re-analysed, and comparison of the simpler and a c t u a l r e l a t i o n s h i p i s made. The hy p e r b o l i c and e l a s t i c - p l a s t i c p r e d i c t i o n s of displacement are shown on Figure 5-17. As might be expected the two s t r e s s s t r a i n law r e s u l t s agree q u i t e c l o s e l y for the large disturbances where t h e i r s t i f f n e s s are s i m i l a r , while f o r small disturbances the displacements p r e d i c t e d by the hyperbolic shear law are greater which r e f l e c t s t h i s laws l e s s e r s t i f f n e s s at small shear s t r a i n s . T y p i c a l displacement time h i s t o r i e s are shown on Figure 5-18. Shear strain FIG. 5-16 ELASTIC PLASTIC APPROXIMATION OF HYPERBOLIC CURVE 10 o LEGEND O Hyperbolic • Elastic-Plastic O .10 .20 . .30 N Max Resistance Coeff A Max Earthquake Acc FIG. 5-17 HYPERBOLIC ELASTIC PLASTIC DISPLACEMENT COMPARISON CLAY SLOPE 9 5 6.0 _ 3.0 c to E Hyperbolic ^ \ /Elastic VPlastic amax= - 7 5 9 0 5 10 15 Time sees FIG. 5-18 HYPERBOLIC ELASTIC PLASTIC DISPLACEMENT HISTORY COMPARISON (CLAY SLOPE) 96 CHAPTER 6 SUMMARY AND CONCLUSIONS 6.1 Summary A two-dimensional f i n i t e element method of a n a l y s i s for p r e d i c t i n g the s t r e s s and permanent displacements of earth s t r u c t u r e s to seismic loading i s presented. The i n e l a s t i c behavior of the s o i l i s modelled by an incremental l i n e a r approach i n which the tangent shear modulus i s v a r i e d with the l e v e l of the shear s t r a i n . The shear s t r e s s s t r a i n r e l a t i o n was modelled by hype r b o l i c loading and unloading curves leading to a Masing type of energy d i s s i p a t i o n , and by the more c l a s s i c a l e l a s t i c - p l a s t i c c o n s t i t u t i v e law. The equations of motion of the s t r u c t u r e are solved using the Newmark step-by-step time i n t e g r a t i o n procedure. At each time step the tangent shear and bulk modulus are evaluated. H y s t e r e t i c damping as a r e s u l t of the s t r e s s - s t r a i n loading and unloading curves i s inherent i n the model. Viscous damping may a l s o be included. The non-linear dynamic response a n a l y s i s was a p p l i e d to a number of s o i l s t r u c t u r e s . The same s t r u c t u r e s were a l s o analysed using Newmark type approaches so that a comparison could be made on the p r e d i c t i o n of permanent displacements. In order that comparison be meaningful, the non-linear a n a l y s i s i s based on an e l a s t i c - p l a s t i c shear law. In order to c l o s e r model and p r e d i c t displacements a c l a y slope was re-analysed using the 97 hyp e r b o l i c shear r e l a t i o n s h i p . While the Newmark methods gives a s i n g l e value estimate of permanent displacements, the move rigorous multi-degree of freedom a n a l y s i s i s d e s i r a b l e as the d i s t r i b u t i o n of displacements w i t h i n the s t r u c t u r e s can be obtained. 6.2 Conclusions The conclusions reached from t h i s research are as f o l l o w s : 1) The Newmark methods and the non-linear f i n i t e element method for p r e d i c t i n g s i n g l e value estimates of permanent displacements of the c l a y slope s t r u c t u r e are i n good agreement. However, the non-linear method p r e d i c t s movement along a h o r i z o n t a l f a i l u r e surface which i s d i f f e r e n t than the Newmark assumed c i r c u l a r arc f a i l u r e and more i n keeping with observed movements. 2) For p r e d i c t i o n of displacement of dam slopes, the Newmark methods are o v e r l y c o n s e r v a t i v e . 3) The simpler e l a s t i c - p l a s t i c shear law and the a c t u a l h y p e r b o l i c r e l a t i o n s h i p gives displacement estimates that are of the same order. 6.3 Suggestions for Further Research Inco r p o r a t i n g a pore pressure model i n t o the a n a l y s i s w i l l allow the e v a l u a t i o n of the very important problem of 98 l i q u e f a c t i o n p o t e n t i a l . The e f f e c t i v e s t r e s s a n a l y s i s can be used to p r e d i c t displacements f i e l d s of saturated cohesionless s o i l s t r u c t u r e s . T e n s i l e crack behavior may only be properly modelled i n an a n i s o t r o p i c manner, where Young's modulus i s reduced i n the d i r e c t i o n of free movement (away and perpendicular to the crack plane) and i s kept at the o r i g i n a l value i n the d i r e c t i o n of the crack plane. S t a t i c g r a v i t y loads c a r r i e d by dynamically softened s o i l during earthquake loading may cause a d d i t i o n a l displacements. The e f f e c t should be modelled and included i n the a n a l y s i s . If p o s s i b l e , c o r r e l a t i o n study with observed f i e l d data should be c a r r i e d out. 99 BIBLIOGRAPHY 1. Byrne P.M., and Janzen W., 1981, S o i l s t r e s s : A Computer  Program f o r Nonlinear a n a l y s i s of Stresses and Deformations i n S o i l , S o i l Mechanics S e r i e s NO. 52, Department of C i v i l Engineering, U n i v e r s i t y of B r i t i s h Columbia. 2. Clough, R.W., and Penzien, 1975, Dynamics of Structures ,McGraw-Hi11, Kogakusha, L t d . 3. Cook R.D., 1974, Concepts and A p p l i c a t i o n s of F i n i t e  Element A n a l y s i s , John Wiley & Sons. 4. D i b a j , M., and Penzien, J . , 1969, Nonlinear Seismic  Response of Earth S t r u c t u r e s , Earthquake Engineering Research Center U n i v e r s i t y of C a l i f o r n i a , Berkeley, C a l i f o r n i a , January 1969. 5. Duncan, J.M., Byrne, P.M., Wong, R.S., and Mabry, P. 1980, Strenth, S t r e s s - S t r a i n and Bulk Modulus Parameters  for F i n i t e Element Analyses , Report No. UCB/GT/80-10 to Nati o n a l Science Foundation. 6. Duncan, J.M., and Chang, C.Y., 1970, Non-Linear A n a l y s i s  of Stress and S t r a i n i n S o i l s , Journal of the S o i l Mechanics and Foundations D i v i s i o n ASCE, V o l . 96, No. SM5, pp. 1629-1653. 7. Finn, L.W.D., M a r t i n , G.R., and Lee, M.K.W., 1979, Coparison of Dynamic Analyses for Saturated Sands , pp. 473-491. 8. Finn, L.W.D., and M i l l e r R.I.S., 1971, Dynamic A n a l y s i s of Plane Non-Linear Earth S t r u c t u r e s , 9. Finn, L.W.D., Byrne P.M., and Martin G.R, 1978, Seismic  Response and L i q u e f a c t i o n of Sands , JSMFD, ASCE, V o l . No. GT8, August 1978 pp. 841-856. 10. Goodman, R.E., and Seed, H.B., 1966 Earthquake-Induced  Displacements of Sand Embankments , Journal of the S o i l Mechanics and Foundations D i v i s i o n , ASCE, V o l . 92, No. SM2, March 1966. 11. Hardin, B.O., and Drnevich, V.P., 1972, Shear Modulus and Damping i n S o i l s : Design Equations and Curves , JSMFD, ASCE, V o l . 98, No. SM7, pp. 667-692. 12. I d r i s s , I.M., Dobry, R., and Singh, R.D., 1979, Nonlinear  Behavior of Soft Clays During C y c l i c Loading , Journal of Geotechnical Engineering D i v i s i o n , ASCE, V o l . No. GT12, Dec. 1979, pp 1427- 1447. 100 13. I d r i s s , I,M., and Seed, H.B., 1974, Seismic Response by  V a r i a b l e Damping F i n i t e Elements , Journal of the Geotechnical Engineering D i v i s i o n , ASCE, V o l . No. GT1, January 1974, pp. 1-13. 14. Lee, M.K., 1977 Dynamic E f f e c t Stress Response A n a l y s i s , Ph.D. Thesis The F a c u l t y of Graduate S t u d i e s , Department of C i v i l Engineering, The U n i v e r s i t y of B r i t i s h Columbia. 15. M a k d i s i , F . I . , 1976, Performance and A n a l y s i s of Earth  Dams during Strong Earthquakes , D i s s e r t a t i o n for Ph.D. in Engineering , Graduate D i v i s i o n , U n i v e r i t y of C a l i f o r n i a , Berkeley, C a l i f o r n i a , 1976. 16. M a k d i s i , F . I . , and Seed, H.B., 1978, S i m p l i f i e d Procedure  for E s timating Dam and Embankment Earthquake Induced Pisplacements , Journal of the Geotechnical Engineering D i v i s i o n , ASCE, V o l . 104, No. GT7, 1978, pp. 849-867. 17. Massing, G., 1926, Eiqenspanningen and Verfestigung beim  Messing , Proc. 2nd I n t e r n a t i o n a l Cong. Applied Mech., 1926 pp. 332-335. 18. Newmark, N.M., 1965, E f f e c t s of Earthquakes on Dams and  Embankments , Geotechnique, London, England, Vol XV, No. 2, 1965. 19. Salgado F.M, 1981, Seismic Response of Ret a i n i n g  S t r u c t u r e s , Masters Thesis, The Fac u l t y of Graduate Studies, Department of C i v i l Engineering, The U n i v e r s i t y of B r i t i s h Columbia, 1981. 20. Sarma, S.K., 1975, Seismic S t a b i l i t y of Earth Dams and  Embankments , Geotechnique, Lond, England, Vol 25, No. 4, 1975, pp. 743-761. 21. Seed, H.B., 1966, A Method f o r Earthquake R e s i s t a n t Design  of Earth Dams , Journal of the Geotechnical Engineering D i v i s i o n , ASCE. no. SM1 1966. 22. Seed, H.B., 1979, Considerations in the Earthquake  R e s i s t a n t Design of Earth and Rock F i l l Dams , Geotechnique, London, England, No. 3, 1979, pp. 215-263. 23. Seed, H.B., and I d r i s s , I.M., 1970, S o i l Moduli and  Damping Factor for Dynamic Response A n a l y s i s , Report No. EERC 70-10, Earthquake Engineering Research Center, Berkeley, C a l i f o r n i a . 24. Seed, H.B., M a k d i s i , F . I . , Lee, K.L., and I d r i s s , I.M., 1973, A n a l y s i s of the S l i d e s i n the San Fernando Dams  during the Earthquake of February 9, 1971 , Earthquake Engineering Research Center Report No. EERC 73-2 101 U n i v e r i t y of C a l i f o r n i a , Berkeley, June. 25. Seed, H.B., M a k d i s i , F . I . , and DeAlba, P., 1975, Performance of Earth Dams During Earthquakes , Jour n a l of the Geotechnical Engineering D i v i s i o n , ASCE, V o l . 104, No. GT7, 1975. 26. Wilson, E.L., and Clough, R.W., 1962, Dynamic Response by  Step-by-Step Matrix A n a l y s i s , Symposium on Use of Computers i n C i v i l Engineering, Libson, P o r t u g a l , October 1 962. 27. Zienkiewicz O.C., 1979, The F i n i t e Elemenet Method , Tata McGraw-Hill P u b l i s h i n g Company L i m i t e d . 102 APPENDIX I Formulation of the Element S t i f f n e s s Matrix  for Plane Isoparametric Q u a d r i l a t e r a l Elements Consider a q u a d r i l a t e r a l having eight degrees of freedom namely u and v at each of the four corner nodes i . An element that has s t r a i g h t sides but i s otherwise of a r b i t r a r y shape Figure I-1a may be considered as a d i s t o r t i o n of a 'parent' rectangular element, Figure I-1b. A mapping f u n c t i o n i s expressed as, N , 0 N 2 0 N 3 0 N , 0 0 N, 0 N 2 0 N 3 0 N , y i x 2 y o where, (1-1) N i = (1 - - 7?), N 2 = ( 1 + j) ( 1 - 1) 4 4 N 3 = (l+jMl+n)-, N 4 = ( 1 - £) ( 1 + T ? ) 4 4 (1-2) This mapping r e l a t e s a u n i t square i n the £TJ coordinates to p o i n t s i n the q u a d r i l a t e r a l i n xy coordinates whose s i z e and shape are determined by the eight nodal coordinates x i , y i , x 2 , y 2 , . . . . y < , . u 1 2 x2 X FIG I - l a QUADRILATERAL ElEMENT V 0,1 i.o € FIG I-lb UNIT SQUARE 1 04 The axes £7} are i n general not orthogonal. They are orthogonal f o r a rectangular element, i n which case they are merely dimensionless forms of rectangular c e n t r o i d a l c o o r d i n a t e s . The 2 by 2 rectangular element, for which we may wr i t e £= x and 77= y, i s a convenient s p e c i a l case to r e f e r when studying the f o l l o w i n g development. Displacements w i t h i n the element are defined by the same i n t e r p o l a t i o n f u n c t i o n s as used to define the element shape and hence the name isoparametric. Thus, {f}= {u v}= [N]{u, v, u 2 ... v„} = [N]{d} (1-3) where [N] i s the rectangular matrix of the N equations as i n d i c a t e d i n the equation (1-1 ), and {d} i s the vector of nodal displacements. The displacements u and v are d i r e c t e d p a r a l l e l to x and y, and not p a r a l l e l to the £ and 17 axes. Note that f o r any point i n the unit i n £77 coordinates, equation 1-1 gives a point (x,y) i n the r e a l coordinates and equation 1-3 w i l l give the displacements u and v i n terms of the nodal displacements. Thus the displacements of every point (x,y) i n the r e a l coordinates i s defined i n terms of the nodal d i splacements. Steps taken to formulate the element s t i f f n e s s matrix according to the approach o u t l i n e d by Cooke (1975) are d e t a i l e d in the remainder of t h i s appendix. Because i t i s impossibly tedious to w r i t e the shape functions i n terms of x and y, the formulation w i l l be c a r r i e d out i n terms of the isoparametric 105 coordinates £77. Equations ( 1 - 1 ) , ( 1 - 3 ) are r e w r i t t e n for convenience i n the form, x= IN, Xj U= IN;Ui 4 v= ZN;v (1-4) A r e l a t i o n s h i p between d e r i v a t i v e s i n the two coordinates, from the chain r u l e of d i f f e r e n t i a t i o n , i s as f o l l o w s : (1-5) where commas denote p a r t i a l d i f f e r e n t i a t i o n . From equation (1-4) [J] i s the Jacobian matrix, < k ( )•* = [ J ] ( )., -< ).y [J]= N M N2, N i . . N 2 , N 3 . £ N 3 ,n x i y i x 2 y 2 x 3 y 3 x„ y. (1-6) where, for example, N = " (1 - T?) (1-7) Using the inverse r e l a t i o n of equation (1-4) and l e t t i n g [J*]= [ J ] " 1 , d i f f e r e n t i a l s i n the xy system may be w r i t t e n as d i f f e r e n t i a l s i n the system as, J l 1 J 1 2 0 0 u . £ U,y J 2 1 J 2 2 0 0 U A •< • = 0 0 J i 1 J 1 2 V,y 0 0 J 2 1 J 2 2 < -(1-8) 106 The strain-displacement r e l a t i o n may be w r i t t e n as ( e } = €y • — V J 1 0 0 0 0 0 0 1 0 1 1 0 U,y • V,* ^ V,y (1-9) and, from equation (1-4), - — « 0 0 0 0 — Ni.K i=3 --A - I U i V i u 2 v 2 u 3 v 3 (1-10) S u b s t i t u t i o n of equation (1-8) i n t o equation (1-9), and t h e r e a f t e r s u b s t i t u t i o n of equation (1-10) i n t o the r e s u l t , y i e l d s the r e l a t i o n {e}= [B]{d}. The d e r i v a t i o n of the element s t i f f n e s s matrix using the p r i n c i p l e of s t a t i o n a r y p o t i e n t a l energy i s w e l l described by Cooke (1975), and Zienkiewicz (1979) and i s obtained by the i n t e g r a t i o n of the expression j f ^ B ] 1 [D] [B]dxdy. Where [D] i s the e l a s t i c i t y matrix, and [B] i s as e s t a b l i s h e d above, expressed i n £TJ coordinates. The e l a s t i c i t y matrix [D] i s determined from equation (3-20), and the c o n s t i t u t i v e patterns for the G, B model, equations (3-21), (3-22). For the present a n a l y s i s the s t i f f n e s s matrix i s expressed i n terms of a shear and bulk, p a r t i a l s t i f f n e s s matrices, see Chap 3.3.1 for further d e t a i l s . The element ' p a r t i a l ' s t i f f n e s s matrices are obtained by i n t e g r a t i o n i n the $77 coordinate system using the f o l l o w i n g 107 transformation r e l a t i o n s h i p . [k] = //[B] T[D][B]dxdy [ B ] T [ D ] [ B ] d e t [ J ] d £ d 7 j (1-11) The determinant of [J] i s a m a g n i f i c a t i o n f a c t o r that y i e l d s area dxdy from area d £ d 7 j , and which i s a f u n c t i o n of p o s i t i o n with the element. Having expressed a l l matrices i n terms of £ and rj, the i n t e g r a t i o n can be c a r r i e d i n £ 7 ? coordinates. However, i n general the i n t e g r a t i o n cannot be performed e x a c t l y due to the complexity of the polynomials i n £ and r\ that appear i n the denominator of [ J * ] . Hence, numerical i n t e g r a t i o n i s commonly used. The Gauss method of numerical e v a l u a t i o n of the d e f i n i t e i n t e g r a l i s used i n the present a n a l y s i s . In two dimensions the quadrature formula i s obtained by i n t e g r a t i n g f i r s t with respect to one coordinate and then with respect to the other, where f ( £ , 7 j ) represents the expression [B] [D] [B]det [ J ] . The i n t e g r a l i s evaluated as the summation of values of the fu n c t i o n at s e l e c t e d sampling p o i n t s , each m u l t i p l i e d by an appropriate 'weight' W . The Gauss method lo c a t e s the sampling p o i n t s so that for a given number of them, greatest accuracy i s obtained. Sampling p o i n t s are l o c a t e d symmetrically with respect to the center of the element. In general, the Gauss [k] = LWi f ( £ , 7 ? ) drj = LEW, Wj f ( £ , 7 ? ) f U , 7 ? ) d £ d T j = /ZW, f ( £ , 7 7 ) dr? (1-12) 108 quadrature using 'n' p o i n t s i s exact i f the i n t e g r a l i s a polynomial of degree 2n-1 or l e s s . In using n p o i n t s , the given f u n c t i o n f ( £ , T ? ) i s e f f e c t i v e l y replaced by a polynomial of degree 2n-1. A sequence of s o l u t i o n s to a problem may be generated by using s u c c e s s i v e l y f i n e r meshes of elements. The sequence may be expected to converge to the c o r r e c t r e s u l t i f the assumed element displacement f i e l d s s a t i s f y the c r i t e r i a of; i n v a r i a n c e , c o n t i n u i t y of displacements w i t h i n elements, r i g i d body modes, constant s t r a i n behavior, and interelement c o m p a t i b i l i t y . The v a l i d i t y of the isoparametric element i s examined with regard to convergence by Cooke, with the element displacement f i e l d s shown to s a t i s f y the above c r i t e r i a . But as w e l l , with numerically i n t e g r a t e d elements, there i s one a d d i t i o n c o n s i d e r a t i o n . That i s , convergence to the c o r r e c t s o l u t i o n i s p o s s i b l e only i f the numerical i n t e g r a t i o n of the element s t i f f n e s s i n t e g r a l i s exact. This c o n d i t i o n i s relaxed and can be r e - s t a t e d ; numerically i n t e g r a t e d elements y i e l d convergence toward c o r r e c t r e s u l t s as the mesh i s r e f i n e d i f numerical i n t e g r a t i o n i s adequate to evaluate the element area e x a c t l y . This statement can be explained by noting t h a t , from the s t a t i o n a r y p o t e n t i a l energy d e r i v a t i o n , formation of a s t i f f n e s s matrix i s e s s e n t i a l l y the same as i n t e g r a t i o n of a s t r a i n energy expression. As a mesh i s r e f i n e d and a constant s t r a i n c o n d i t i o n comes to p r e v a i l i n each element, the s t r a i n energy expression f o r an element assumes the form, (1-13) 109 Hence, i n the l i m i t the s t r a i n energy of the s t r u c t u r e i s c o r r e c t l y assumed i f the volume of each element i s c o r r e c t l y assessed. Examination of the Jacobian determinant y i e l d s the number of Gauss p o i n t s needed to obtain the area of the p a r t i c u l a r elements. For a plane quadratic element d e t [ J ] contain terms £ 3 and T j 3 , hence a 2 by 2 Gauss r u l e (four p o i n t s ) i s the minimum that can be accepted. I t can be shown (Cooke 1975) that W for each Gauss point i s 1.0 and the Gauss point values are + 1/ 3.= +.57735... Thus equation (1-11) can be w r i t t e n , [k] = ( 1 . ) ( l . ) f ( ^ 1 , T ? 1 ) + (l.)(1 . ) f U 2,7?,) + (1 . )(1 .)f Ui ,i?2) + (1.)(1 - . ) fU 2,»?2) (1-14) where 7j t = .57735..., and £ 2 , T J 2= -.57735... 

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