MECHANICAL OF MODEL FOR LIQUEFACTION SOIL OF THE ANALYSIS HORIZONTAL DEPOSITS by KWOK WING LEE B.So. (Eng.), University of Hong Kong, I966 M.Sc. (Eng.), University of Hong Kong, I968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1975 In presenting t h i s thesis in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l E n g i n e e r i n g The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Sept. 1975 i ABSTRACT During the development of l i q u e f a c t i o n In a s o i l deposit subjected to v i b r a t i o n there are two processes which work i n opposite d i r e c t i o n s . tendency under cyclic loading The volume compaction causes the pore water pressure to r i s e , and the d i s s i p a t i o n of excess pore water pressure (consolidation) decreases i t . Recently, Martin, Finn and Seed(l975) studied the mechanics .of pore pressure generation of water a s o i l sample subjected to c y c l i c loading and a r e l a t i o n between shear s t r a i n cycles, volume compaction and established. pore water pressure thesis f o r saturated granular under c y c l i c simple shear conditions. hysteretlc compaction, was A material model based on t h i s r e l a t i o n s h i p i s developed i n t h i s a increment stress-strain pore water The model includes relationship, pressure soil rise volume and d i s s i p a t i o n . Using t h i s proposed comprehensive material model, a global mechanical model i s constructed to simulate the l i q u e f a c t i o n (including consolidation) behavior of a thick horizontal motion. deposit when In t h i s way the subjected coupled response, pore water pressure the deposit under numerical techniques discussed i n d e t a i l . rise to horizontal problems and of dynamic consolidation of seismic loading can be analysed. used to solve base such The problems are The response of a t y p i c a l saturated 11 sand d e p o s i t under earthquake l o a d i n g i s determined u s i n g the proposed model and the r e s u l t s show t h a t t h e model can p r e d i c t the v a r i o u s phenomena t h a t s a t u r a t e d sand d e p o s i t s e x h i b i t d u r i n g earthquakes. The g l o b a l model a l s o makes c l e a r the i n f l u e n c e o f p e r m e a b i l i t y p o t e n t i a l o f the s o i l deposit. on the liquefaction ill TABLE OF CONTENTS Page ABSTRACT 1 TABLE OF CONTENTS i i i LIST OF TABLES vl LIST OF FIGURES vil NOTATION xii ACKNOWLEDGEMENTS rvii CHAPTER 1. INTRODUCTION. 1.1 General C h a r a c t e r i s t i c s of S o i l Liquefaction. 1 1.2 Description of Some Liquefaction Case H i s t o r i e s . 2 1.3 Spontaneous Liquefaction, Cyclic Mobility and Liquefaction. 7 1. k Purpose and Scope of Research. 10 CHAPTER 2. REVIEW OF PAST INVESTIGATIONS ON LIQUEFACTION OF SATURATED SANDS. 2.1 Liquefaction P o t e n t i a l Deposit Based on Experience. of a Field 12 2.2 Laboratory Investigation of Liquefaction Using Scaled Sand Models. 18 2.3 Laboratory Investigation of Liquefaction Using Small Sand Samples. 26 2. JJ- Some Methods of Determining Liquefaction P o t e n t i a l of a S o i l Deposit. 38 CHAPTER 3. CHAPTER 4. CHAPTER 5. CHAPTER 6 MODELLING OF SATURATED GRANULAR SOIL UNDER CYCLIC SIMPLE SHEAR CONDITIONS. 3.1 Volume Change C h a r a c t e r i s t i c s During C y c l i c Drained Shear. 3.2 Stress-Strain Under Cyclic Conditions. 3.3 Modelling Of Saturated Granular Soil Under Cyclic Simple Shear. 3.4 Computation o f S o i l Behavior Using the Proposed S o i l Model. Relationship Simple Shear MECHANICAL MODEL FOR THE ANALYSIS OF LIQUEFACTION. 4.1 Introduction. 4.2 F o r m u l a t i o n o f the Problem. 4.3 S o l u t i o n Scheme. APPLICATION OF THE MECHANICAL MODEL A DYNAMIC ANALYSIS. 5.1 A Hypothetical S o i l 5.2 Response Characteristics the S o i l D e p o s i t . 5*3 Magnitude o f Pore Pressure R i s e and i t s E f f e c t on the Dynamic Response of the Deposit. 5.4 Effect of Dissipation. Pore SUMMARY AND CONCLUSIONS. 6.1 Summary. 6.2 Conclusions. Deposit. of Pressure V 6.3 Suggestions Studies. For Further 147 LIST OF REFERENCES 148 APPENDIX I 155 APPENDIX I I 162 APPENDIX I I I 169 APPENDIX IV 173 vi LIST OF TABLES Table 5-1 5-2 5-3 gaffe Data f o r Layer S o i l Deposit Approximation Frequencies and Phase the Simulated Accelerogram to 118 Angles f o r Earthquake S'l-Values f o r Different Layers 130 130 vii LIST OF FIGURES Figure Page 1- 1 Modes of Foundation F a i l u r e 5 2- 1 C r i t i c a l N-Value versus Depth 15 2-2 Zone of Liquefaction 17 2-3 P r o b a b i l i t y of Liquefaction 19 2-4 Critical Acceleration f o r Various 21 Sands 2-5 2-6 E f f e c t of Acceleration Level Resistance to Liquefaction on Effect on of Resistance Surcharge 2k Pressure 25 to Liquefaction 2-7 Idealised F i e l d Loading. Conditions 28 2-8 Idealised T r i a x i a l Test Conditions 29 2-9 C y c l i c Shear Stress Wave Forms 33 2-10 Typical Cyclic Simple Shear Test Data Jk 2-11 T y p i c a l C y c l i c T r i a x i a l Test Data 2-12 Resistance Simple Shear to Liquefaction 35 in 37 e F Line f o r Sand B Method of Potential Evaluating Liquefaction Compaction of Dry Coarse Sand Effect of Compaction vertical Stress E f f e c t of Relative Density Shear S t r a i n on Compaction Incremental Curves Volumetric on and Strain Comparison of Measured and Computed Volumetric S t r a i n Hyperbolic S t r e s s - S t r a i n Curve Equivalent Shear I n i t i a l Loading Modulus D e f i n i t i o n of Equivalent Modulus And Damping Symmetrical Loop for Shear Ratio Hysteretic C h a r a c t e r i s t i c s One-Dimensional S o i l Model Force-Displacement Relationship S l i p Elements of Volume Compaction Caused by C y c l i c Loading 'Reduced' Strain Incremental Volumetric Increments of Volumetric S t r a i n , €. Hyperbolic Stress-Strain Loops (Drained Test) Showing E f f e c t of Hardening Shear Modulus as a Function of C y c l i c Shear S t r a i n and Volumetric Strain Shear Modulus as a Function C y c l i c Shear S t r a i n and Number Cycles of of Computed Damping Ratios Shear Modulus as a Function of Shear S t r a i n and T o t a l Compaction Damping Ratios f o r Monterey Sand One Dimensional Unloading Curves Hyperbolic Stress-Strain Loops (Undrained Test) Showing Softening due to Pore Pressure Rise S t r a i n and Pore Pressure Undrained C y c l i c Shear Test during X Figure Page J 4-1 Idealization of the Response Problem 97 4-2 Approximation by Lumped Mass System 100 4-3 Consolidation Model 107 4- 4 Flow Chart 114 5- 1 Properties of a S o i l Deposit 116 5-2 Input Base Acceleration 120 5-3 Surface Acceleration Response 122 5-4 S t r e s s - S t r a i n Response 5-5 Surface Acceleration Response Curve 5-6 Surface Acceleration 123 from 'SHAKE 1 Program 126 5-7 Simulated Earthquake Accelerogram 5-8 Surface Acceleration 5-10 129 Response (No Pore Pressure) 5-9 124 S t r a i n and Stress Responses f o r Layer 11 (No Pore Pressure) Surface Acceleration and Displacement Responses (Pore Pressure Generated) 131 133 136 xi Figure 5-11 Page Stress, Strain and Pore Pressure Responses f o r Layer 11 5-12 Stress-Strain Response f o r Layer 135 11 136 (Pore Pressure Generated) 5-13 Pore Pressure Distribution at 139 Various Times (No Dissipation) 5-14 Pore Pressure Distribution Different k Values for 141 xii NOTATION A Amplitude of v i b r a t i o n A^, A2. A-j Constants f o r the shear modulus function a^ Constants t &2 f o r the increase in shear modulus due to compaction B^, B2, B 3 Constants f o r the shear modulus function bj_, b2 Constants f o r the increase i n l i m i t i n g shear stress due to compaction Cit C2, C3, C4 Constants f o r the volume change function [C ] Damping matrix Cq Equivalent c± Damping c o e f f i c i e n t f o r the i - t h layer c S p e c i f i c heat of a s o l i d D Damping r a t i o e D D Maximum damping r a t i o m a x Relative r D50 viscous damping f a c t o r T n e density maximum p a r t i c l e - s i z e of the smallest f i f t y percent of a s o i l e Void r a t i o e" C r i t i c a l void r a t i o l i n e fi Stress-strain-rate function F S t r e s s - s t r a i n function Shear modulus Maximum shear modulus I n i t i a l maximum shear modulus Maximum shear modulus at the n-th cycle Thickness of a s o i l deposit Thickness of the I-th layer Coefficient of the earth pressure at rest Bulk modulus of water One-dimensional rebound soil modulus of the skeleton S t i f f n e s s matrix S t i f f n e s s value of the i - t h layer Value of permeability Thermal conductivity Mass matrix Mass lumped at the top of the i - t h layer Coefficient of volume compressibility C r i t i c a l standard penetration value Number of cycles to l i q u e f a c t i o n Equivalent number of constant amplitude shear stress cycles Porosity I n e r t i a l force vector In s i t u v e r t i c a l stress Position vector Shear strength Time Time Volume change function Uniformity c o e f f i c i e n t Velocity Energy dissipated per cycle of loading F i n i t e difference approximation to cr w Displacement of the i - t h layer - relative to the base Displacement of the i - t h layer Base displacement Height measured from the bedrock Height of the top of the i - t h layer C r i t i c a l acceleration Constants f o r the numerical integration Damping c o e f f i c i e n t s Shear s t r a i n Hyperbolic shear s t r a i n Unit weight of water Amplitude of a shear s t r a i n cycle Increment i n shear s t r a i n Volume change increment Recoverable component of the volume change increment Compaction compnent of the volume change Increment Strain Volumetric s t r a i n Recoverable component of the volumetric strain Compaction component of the volumetric strain Temperature Mass densityStress Mean normal stress Octahedral normal stress Vertical Stress I n i t i a l v e r t i c a l stress Pore water pressure Minor p r i n c i p a l stress at f a i l u r e E f f e c t i v e stress Shear stress Amplitude of a shear stress cycle C y c l i c horizontal shear stress Equivalent 10 cycle shear stress amplitude that i s developed C y c l i c shear stress amplitude required to xvl cause l i q u e f a c t i o n i n 10 cycles Tj Maximum n a x shear stress that can be applied to a s o i l element Maximum shear stress at the n-th cycle <j>' +1» +2* E f f e c t i v e angle of shearing resistance ^3* 4*4 Constants f o r the volumetric function compaction xvil ACKNOWLEDGEMENTS The author wishes to express his gratitude to his principal advice adviser Dr. W.D. Liam Finn and guidance research. Thanks during are the entire also due Dr. R.G. Campanella, Dr. G.R. Martin their f o r his invaluable course of the to Dr. P.M. Byrne, and Dr. Y. Vaid f o r encouragement and constructive discussion at various stages of the research. The National Research Council of Canada the financial possible. assistance which made provided this investigation This assistance i s g r a t e f u l l y acknowledged. The author deeply appreciated the support and consideration given t o him by his wife during the period of his studies. 1 CHAPTER 1 INTRODUCTION 1.1 GENERAL CHARACTERISTICS OF SOIL LIQUEFACTION. It has relatively been loose observed sandy soil that when deposits saturated are subjected to Imposed by r e p e t i t i v e or c y c l i c shear l o a d i n g , such as those an earthquake, turned into extreme they a case often 'quick' where exhibit or great the liquefied phenomena of b e i n g condition. earthquake and shocks In occur the In an a l l u v i a l r e g i o n , the pore water i n the l i q u e f i e d s o i l which i s under v e r y l a r g e p r e s s u r e may temporary pressure geysers. In generated is be f o r c e d t o the s u r f a c e the less severe cases excess pore original occur when pressure i s d i s s i p a t e d . It masses the o n l y a s m a l l f r a c t i o n of the e f f e c t i v e normal p r e s s u r e and minor s e t t l e m e n t may forming is generally compact vibration. or If accepted contract the s o i l In that volume dry granular when soil subjected i s s a t u r a t e d and drainage prevented then i t i s not d i f f i c u l t to see t h a t the pore will a g r a n u l a r s o i l , the maximum s h e a r i n g build resistance up. For that can be developed on a water to plane, pressure or shear s t r e n g t h , s, i s g i v e n by the f o l l o w i n g e x p r e s s i o n , s = (<r -<r )tan<t>' v w (i.-D 2 where s I s t h e shear s t r e n g t h , 0* i s t h e t o t a l normal p r e s s u r e , v cr $' w i s t h e pore water p r e s s u r e , and, i s the e f f e c t i v e angle o f i n t e r n a l r e s i s t a n c e . As t h e pore p r e s s u r e r i s e s t h e e f f e c t i v e normal s t r e s s , 0^.-0^, decreases causing a decrease i n the shear s t r e n g t h , s . In the extreme case where the v i b r a t i o n a l d i s t u r b a n c e i s o f l a r g e magnitude and r e l a t i v e l y l o n g d u r a t i o n the pore water p r e s s u r e may r i s e t o such an extent t h a t t h e may become zero. The effective condition of normal zero or stress near c o n f i n i n g p r e s s u r e makes t h e s o i l behave l i k e a l i q u i d , little equation describe or no shear r e s i s t a n c e can be developed (1-1) and hence the the phenomenon. tanks and o t h e r b u r i e d surface. term the next 1.2 i s used to floating up to the s l o p e and dam f a i l u r e s caused soil ground by t h i s have been i n numerous i n v e s t i g a t i o n s and they a r e d e s c r i b e d i n section. DESCRIPTION OF SOME LIQUEFACTION CASE HISTORIES. In liquefaction areas. of according to LIQUEFACTION type o f s t r e n g t h r e d u c t i o n i n the s u p p o r t i n g reported since There have been i n s t a n c e s o f sewer objects Foundation, zero the San Francisco phenomena were Earthquake observed The quake o c c u r r e d on A p r i l about 8.2. The f o l l o w i n g r e p o r t o f the earthquake: of 1906 soil i n numerous a l l u v i a l 18, 1906 w i t h a magnitude i s quoted from the official "One o f the most common phenomena i n 3 such a situation was the expulsion of water i n jets from apertures which suddenly appeared i n the f l a t lying grounds. The water was usually thrown into the a i r f o r several feet; i n some cases i t was reported to be as much as 20 feet, and the e j e c t i o n continued f o r several minutes a f t e r the earthquake. The continuance of the e j e c t i o n a f t e r the shock indicates that an e l a s t i c stress has been generated i n the saturated ground, which thus found r e l i e f In the expulsion ... The of contained water water i n I t s passage to the surface brought considerable quantities of fine sand, which from up its p r e v a i l i n g l i g h t bluish gray colour was evidently derived from considerable depth. the On the flood p l a i n of the Salinas r i v e r , sand was recognized by the people of the neighbourhood to be the same as that of a stratum of sand pierced by wells at a depth of 80 f e e t . " Usually the sand brought up i n t h i s way formed funnel-shaped sand craters on the ground surface. abundance of saturated a l l u v i a l f l a t s with such an craters gave Indication of the size of the area which l i q u e f i e d as well as of the s e v e r i t y of the l i q u e f a c t i o n problem. not many structures were b u i l t i n such areas at the The earthquake and which may the time of l i t t l e or no damage to structures due to s o i l l i q u e f a c t i o n was reported. flows, Fortunately, have been Numerous cases of earth- due to l i q u e f a c t i o n , were also recorded. Liquefaction of a saturated damage to buildings and sandy deposit causing structures occurred i n the Nllgata 4 The quake took place at around 13 Earthquake of 1964. on June 16, 1964 i n the Niigata and Yamagata area i n Japan. less than 7.5 The magnitude of the quake was Scale and effects i n the Hi enter Intensity i n the Niigata City was around 8 i n the the Modified M e r c a l l i Scale. the hours of the A d e s c r i p t i o n and analysis quake have been published i n a d e t a i l e d report compiled by the E d i t o r i a l Committee of "General on the Niigata Earthquake observation was that structures(up to Osaki(1968) 1964,"(1968). of modern 1964) the Shinano r i v e r buildings, bridges and b u i l t on the a l l u v i a l formation around were damaged more readily than others. a whole damage i n the superstructure, i n d i c a t i n g severe foundation f a i l u r e . upheaval Moreover, f o r t h i s type of of soli overturning predicted by theory, shown as other reckoned that two t h i r d s of the damaged buildings much usual Report An i n t e r e s t i n g In these areas Just s e t t l e d , t i l t e d or overturned as without of in on the the opposite side of t i l t i n g or conventional figure 1-la, foundation was deformation pattern s i m i l a r to figure 1-lb contrast to the failure settlement not observed but a was and failure obtained. tilting of In heavy superstructures as described, l i g h t buried structures, such as concrete s e p t i c tanks, 'floated' during the quake and remained protruding above ground surface a f t e r the disturbance In the light of these data, Osaki reasoned that the s o i l at such s i t e s behaved l i k e a l i q u i d and foundation Also an soil area ceased. liquefied near the under same he the concluded that the earthquake v i b r a t i o n . location, but on which no (a) (b) Fig. 1-1 Modes of (after Foundation Osaki, 1968 ) Failure 6 superstructures were built, geysers erupted during the which continued for a mud volcanoes earthquake and and these temporary outbursts, while ' a f t e r the tremor passed away, f i n a l l y died down giving r i s e to sand c r a t e r s . Not only was liquefaction under loose earthquake could also l i q u e f y when the quake were sandy both large. Intensity sand. large to and duration of the Seed(1972) reported that i n the San 1971, medium dense sand foundation beneath induced susceptible v i b r a t i o n s , medium dense sand Fernando Earthquake of February 9, Plant deposit the l i q u e f a c t i o n of a the Jensen Filtration l a t e r a l movements i n the f i l l above the As a r e s u l t of t h i s , extensive damage was done to the s i t e as well as to structures which were under construction. Failures caused r e s t r i c t e d to foundations flows by of soil liquefaction structures. are Numerous not earth- and slope f a i l u r e s have occurred during earthquakes due to l i q u e f a c t i o n of granular s o i l masses or seams or lenses similar material. Seed(1968) of i n his Terzaghi Lecture gave a large number of examples of slope f a i l u r e s or s l i d e s r e s u l t i n g from s o i l liquefaction failure of each case. dynamic analysis, Seed and he examined the mechanism of Using a recently developed method of et al (1975) reconstructed the mechanics of s l i d i n g of the upstream part of the embankment of the Lower Fernando Dam, which moved some 70 feet or more into the reservoir. found liquefaction They developed near that an the base extensive zone of of the embankment and 7 t h e i r conclusion was confirmed by the sand bolls, cracks filled observed phenomena with l i q u e f i e d sand i n the s l i d e mass and spreading of the embankment s o i l 250 feet beyond toe of the of the embankment. 1.3 SPONTANEOUS LIQUEFACTION, CYCLIC MOBILITY AND LIQUEFACTION. With regards to the term controversy on i t s usage. liquefaction there is a There has been no d e f i n i t i o n f o r i t i n terms of stress or s t r a i n nor has there been any i n terms of other phenomenological parameters. Castro(l969) Hazen(1920) was Allen "liquefies" the i n h i s Ph.D. t h e s i s pointed out that the first one to use the word to describe flow f a i l u r e s and he continued to use term i n association with flow s l i d e s i n h i s t h e s i s . wrote : "Liquefaction or flow f a i l u r e o f sand Is caused by substantial reduction of i t s shear sand actually acting within compatible flows the with spreading sand the However, Terzaghi and He become reduced strength. mass of out u n t i l the shear stresses so shear Peck(l968) The used small that strength the term they are of the sand". spontaneous l l q u e f a c t i o n to describe t h i s phenomenon o f flow f a i l u r e which occurred shock. mostly i n loose fine sand when subjected to mild They suggested that the metastable structure formed by the sand grains was responsible f o r t h i s kind of behaviour. It was because of such a metastable structure of the sand 8 aggregate that the f a i l u r e process could take place i n a short time. Indeed, the term spontaneous very liquefaction i s very appropriate i n describing t h i s type of flow f a i l u r e s . Several researchers i n the f i e l d of e.g.. Seed soil et a l at the University of C a l i f o r n i a and Finn et a l at the University of B r i t i s h Columbia, c a r r i e d loading undrained tests out cyclic on t r i a x i a l as well as simple shear specimens of granular material and observed the dynamics, In general that pore pressure i n the specimens continued to r i s e , leading to a decrease i n the e f f e c t i v e confining (or normal) pressure, and f i n a l l y reached a stage where large deformations under constant cyclic "liquefaction" was phenomenon. This stress also used is in quite resistance of the s o i l decreased cycles of little or no However, stress until on so that suitable with dilative samples term with this since the number of f a i l e d when i t offered large of deformation. the sand response at specimen the peak an additional cycle f o r very of shear loose and stress l i q u e f a c t i o n can increase the maximum shear s t r a i n by fifteen per cent. response suggestion of some of shear f o r example very large s t r a i n s cannot develop abruptly i n f a i r l y dense sand, although loose The increasing at a l l to the denseness there are varying degree of shear connection i t finally resistance depending amplitude. occurred near ten to In view of the above described d i l a t i v e samples, Castro(1969)• Dr. A. Casagrande, following the referred to the process of 9 the gradual development of pore straining as cyclic pressure mobility and with cyclic reserved shear the term l i q u e f a c t i o n f o r the f a i l u r e of s o i l under shear when the loss of strength due to r i s e i n pore pressure was not regained (due to contractive response) even at large s t r a i n s . It i s well-known that v i b r a t i o n causes compaction i n a granular s o i l mass. cyclic In other words, I t can stresses, e s p e c i a l l y granular s o i l mass always strain (contraction). shear cause Each an though f o r constant amplitude stresses, increase additional s t r a i n always causes an a d d i t i o n a l be said acting -that on a i n volumetric cycle of stress or Increment i n contraction stress or s t r a i n the increment i s a decreasing function of the number of cycles. I f the s o i l mass i s saturated and drainage (volume change) prevented, a. portion of the confining (or normal) pressure a c t i n g on the s o i l skeleton would an incremental be transferred to the pore water, causing rise i n pore pressure. Since the volume of s o i l always decreases when drained, as described pore pressure above, w i l l always r i s e i n the undrained case. the As a r e s u l t there i s always the p o s s i b i l i t y of making the e f f e c t i v e confining pressure zero by continuing the process the stresses or strains. word mobility does cycling Though the term c y c l i c mobility describes the c y c l i c nature of the the of not softening of the material l i k e process quite adequately give the notion of the gradual the word liquefaction does. It may be more suitable to use the term c y c l i c l i q u e f a c t i o n J 10 Terzaghl and Peck(1968) In the recent e d i t i o n of t h e i r book, " S o i l Mechanics In Engineering Practice", added new section on 'Liquefaction under Reversals of Stress or Strain* to describe the behavior of the r i s e i n pore in a soil failure. Include mass due to cyclic loading pressure which may lead to Thus the broad usage of the word ' l i q u e f a c t i o n ' flow failures, a to with sharp r i s e i n pore pressure and c a t a s t r o p h i c a l l y large strains (spontaneous liquefaction), as well as limited f a i l u r e , with incremental or continual r i s e i n pore under pressure and comparatively smaller strains ( l i q u e f a c t i o n reversals of generalization. As w i l l be used i n this More discussion stress or strain), is not an over- a matter of f a c t , the word l i q u e f a c t i o n broad sense throughout this thesis. of the s i m i l a r i t i e s of these behaviors will follow i n the coming chapters. 1.4 PURPOSE AND In subjected SCOPE OF RESEARCH the to liquefaction c y c l i c loading The volume causes the described above, and the pore compaction water dissipation (consolidation) decreases comprehensive material model has been t h i s coupled problem of l i q u e f a c t i o n . and of a soil deposit v i b r a t i o n there are two processes which work i n opposite d i r e c t i o n s . pressure process Seed(1975) studied tendency under pressure to r i s e , as of excess it. presented pore As water yet, to no explain Recently, Martin, Finn the mechanics of pore water pressure 11 generation relation of a s o i l sample subjected between pore water research, shear pressure a model was for simple shear conditions has been relationship that includes r e l a t i o n , volume compaction, dissipation. Using this established. liquefaction constructed a thick horizontal Liquefaction may deposit then water proposed this on rise comprehensive constructed subjected investigated to and material to simulate behavior as this stress-strain pressure consolidation) when be based hysteretlc pore (including In granular s o i l under c y c l i c model, a global mechanical model i s the a s t r a i n cycles, volume compaction and increment material to c y c l i c loading and base of a motion. the cummulatlve consequences of the following i n t e r a c t i n g processes: 1. Dynamic response to base motion, 2. Increase i n volumetric compaction s t r a i n due to cyclic straining, 3. Pore water pressure generation as a r e s u l t of ( 2 ) , and, 4. Dissipation of excess pore water pressure (consolidation). Numerical techniques for the solution of the liquefaction problem are developed and an example problem i s solved showing that the model i s capable of simulating the of a soil deposit as well as giving permeability or c o e f f i c i e n t of consolidation potential. actual the on behaviour influence of liquefaction 12 CHAPTER 2 . REVIEW OF PAST INVESTIGATIONS ON LIQUEFACTION OF SATURATED SANDS 2.1 LIQUEFACTION POTENTIAL OF A DEPOSIT BASED ON FIELD EXPERIENCE. As Earthquake, early 1906, as i t was after observed the the saturated alluvial sand that flats aggregates the main constituent of the s o i l deposits. at Francisco that the places that were most susceptible to s o i l l i q u e f a c t i o n were where loose to very loose San I t was were recognised time that these loose sands were responsible f o r the liquefaction of the soil earthflows. However, descriptions of the soils deposits apart as from that were well as these most f o r many qualitative susceptible to l i q u e f a c t i o n , the values of the s o i l parameters, i f ever known at that time, were not reported. Florin field They and investigations studied the Ivanov(196l) into the liquefaction published soil some of t h e i r liquefaction problem. behavior of r e l a t i v e l y loose sandy s t r a t a which were around eight to ten metres (around feet) i n depth vibration. using explosives as an 30 energy source f o r Throughout the i n v e s t i g a t i o n they used a charge consisting of 5 kilograms of ammonite buried at a depth of 4.5 13 meters (about 15 feet). They observed that a f t e r a b l a s t , the loose saturated sandy s o i l deposit surface settlements after liquefied with large re-consolidation whereas i n dense sands no evidence of l i q u e f a c t i o n could be found and l i t t l e or no surface settlement was observed. loose Also, i n the case of a sandy deposit l i q u e f a c t i o n always occurred within a few seconds of f i r i n g the explosives. experiments they As a result of these were able to e s t a b l i s h a c r i t e r i o n f o r s o i l l i q u e f a c t i o n using the magnitude of settlement due to b l a s t i n g as a key v a r i a b l e , namely:1. I f the average settlements (due to the blast created by the explosion of the 5 kg of ammonite described above) i n a radius of 5 meters (16 (around 3.5 inches), no feet) was less than 8 to 10 cm provision against liquefaction should be made, and, 2. I f the r a t i o of settlements between successive blasts was greater than 1:0.6 l i q u e f a c t i o n of the stratum would be very l i k e l y to occur. It was also reported that t h i s blasting provided quite suffers the following reliable results. of standard However, i t disadvantages:- 1. Blasting has to be done investigation method which may on the p a r t i c u l a r site under be quite objectionable i n some cases, 2. Uncertainty involved when applying the c r i t e r i o n to s o i l s with properties differ from those described i n the 14 literature, 3. Magnitude and, and duration of v i b r a t i o n a l disturbance cannot be taken into account. Koizumi(I966) examined the boring data f o r the conditions in the Niigata area and a f t e r the 1964 before Niigata earthquake i n an e f f o r t to study the changes density due to the earthquake. some in soil From standard penetration N- values f o r about 20 locations i n the N i i g a t a area that soil he noticed loose sand s t r a t a were compacted by the earthquake while other not-so-loose sand s t r a t a were loosened. Based on these observations he reasoned that f o r the sandy s o i l at Niigata area a c r i t i c a l void r a t i o as a function of pressure should e x i s t . A curve was proposed c r i t i c a l standard penetration values, N , cr overburden to relate which were used as a measure f o r the c r i t i c a l void r a t i o at various depths, depth of overburden as shown i n Figure 2-1. N__ deposits with N-values above Niigata. given or below However, the loosening or compaction would occur i f the sand subjected to the same quake again. volume contraction occurred i n an earthquake the be the result N c r for values, deposit were Whenever compaction or saturated sandy deposit pore water pressure would r i s e causing the e f f e c t i v e stress i n the s o i l to would as would not s u f f e r any change i n density when subjected to an earthquake s i m i l a r to that of during with Thus, according to Koizumi, a sand deposit having the same N values by the decrease. Liquefaction when the e f f e c t i v e stress approached a N-Value 0 10 20 30 UO 50 • -N ig.2-1 Critical N-Value c r versus Depth (after Koizumi, 1966) 16 zero o r near zero v a l u e . also be this the N Consequently, curve cr could viewed as a c r i t e r i o n f o r s o i l l i q u e f a c t i o n . criterion, liquefaction Osaki(1968) ' determined the Using zone of of" sand d e p o s i t s i n h i s study i n o r d e r t o r e l a t e the depth and t h i c k n e s s o f l i q u e f i e d s o i l s t r a t a t o the amount o f damage t o s u p e r - s t r u c t u r e s . a deposit and the N cr The a c t u a l N-value curve curve were p l o t t e d i n the same graph, and t h e zone o f l i q u e f a c t i o n was o b t a i n e d from the intersection, the c h a r a c t e r i s t i c s at a general potential of of p u b l i s h e d s e v e r a l papers on o f l i q u e f i e d sands i n an attempt t o a r r i v e criterion a points 2-2. a and b, as shown i n F i g u r e Kishida(1964,1969,l970) for level f o r accessing deposit f o l l o w i n g items determined and the he the occurrence liquefaction suggested t h a t the of l i q u e f a c t i o n of a d e p o s i t :1. Earthquake i n t e n s i t y around V to VI according t o the Japanese M e t e o r o l o g i c a l Agency I n t e n s i t y S c a l e . 2. E f f e c t i v e overburden p r e s s u r e l e s s than 2 kg/sq.cm. 3. R e l a t i v e d e n s i t y l e s s than 15%, 4. Saturated. 5. Coarse-grained s o i l w i t h 0.074<D^<2.0 mm. 6. S o i l g r a i n s a r e f a i r l y uniform w i t h U.<10. Q From was o b t a i n e d . item number 3 a curve o f N-values versus The curve, which has a s i m i l a r N curve, Nrty. curve which was o b t a i n e d by Koizumi i s a l s o cr is shown i n F i g u r e 2-3. usage as depth the In the same f i g u r e the plotted for 17 N-Value Fig.2-2 Zone of Liquefaction (after Osaki, 1968) 18 comparison. It i s quite obvious that these curves are applicable to the Niigata Use of these curves to predict the l i q u e f a c t i o n deposit located elsewhere would potential the Nevertheless, preliminary these c r i t e r i a are stage of site LABORATORY INVESTIGATION OF quite useful i n v e s t i g a t i o n so that a d i r e c t i o n f o r d e t a i l e d s i t e i n v e s t i g a t i o n may 2.2 relating the s o i l properties, as well as those r e l a t i n g to the base rock motion. in of be quite dubious since the dynamic stresses developed depend on many parameters to 1964. deposit subjected to the earthquake of be decided. LIQUEFACTION USING SCALED SAND MODELS. As dam Two early as 1936 Casagrande(l936) sections to demonstrate the model dam sections, one stability Water side sandy slopes. tanks mounted l e v e l on the upstream side of the dam was kept at a small distance below downstream of loose sand and one of dense, were b u i l t of ordinary beach sand and placed i n casters. constructed model the base was the crest while model on covered with a pervious The tanks were shaken and i t was the filter blanket to keep the l i n e of seepage away from the face of slope. on the observed that the loose sand model l i q u e f i e d ( f a i l u r e of flow s l i d e type) readily However, no while the dense model remained i n t a c t . d i r e c t l y useful numerical data were obtained testing. from the very model N-Value 0 10 2.0 \/ ^ N / / U0 — - Low Possib ility /\ Y 30 \ \ y / / \ High Possibil ty \> / s \ \ N. / / . «—s\ " \\ X / v // ^ / ^\ N Relative Density 'X / X ^ s dium \ P Dssibility' Fig. 2-3 Probability (after of Liquefaction Kishida, 1969) 20 Maslov(1957) and F l o r i n and Ivanov(l96l) model t e s t i n g , as well as f i e l d t e s t i n g , f o r the of saturated sandy s o i l masses. performed liquefaction Complete d e t a i l s of t h e i r models and experimental procedures have not been published that accessment of boundary e f f e c t s cannot be made. their already generated by, seismic Maslov suggested the existence °crit» w n o s e of a had been disturbances. critical acceleration, value depended on the s o i l density and type, and i f the maximum acceleration of of c * would or vibrational be (re-consolidation) concerned process established a criterion water a comparison pressure between obtained section, for l i q u e f a c t i o n based on the amount of settlement due to b l a s t i n g at the s i t e . made with the of a l i q u e f i e d s o i l stratum, though, as discussed i n the previous they generated. for various s o i l s i s shown i n Figure 2-4. c r i t The work of F l o r i n and Ivanov was mainly consolidation the f o r the p a r t i c u l a r s o i l mass no excess pore water pressure graph that (vibrational) disturbance was less than the required A However, e f f o r t s seemed to have been focused on the d i s t r i b u t i o n and d i s s i p a t i o n of excess pore water pressure that so They also the d i s t r i b u t i o n of excess pore theoretically and that obtained through model t e s t i n g and seemingly good agreement was arrived at. However, neither of these investigators attempted to access the amount of excess water pressure generated function of the s o i l parameters and the acceleration duration of the v i b r a t i o n a l disturbance. as a l e v e l and 21 POROSITY (Vo) Fig. 2-4 Critical A c c e l e r a t i o n Sands. (After for Various Masloy. 1957) 22 Yoshlml(1967) placed loose saturated sand i n a r i g i d 50cm box, 25cm x 25cm x i n dimension, and applied h o r i z o n t a l v i b r a t i o n to the box i n an e f f o r t e f f e c t on sand deposits. put over the sand to simulate the An impervious f l e x i b l e membrane was i n the box so that a surcharge applied by means of a i r pressure on the membrane. two stages, pressure rose an initial in a liquefaction stage compaction near-hyperbolic when failure consisted stage i n which pore fashion occur could be Using such a set up, he observed that the l i q u e f a c t i o n process of seismic and in a a sudden short time. Although quantitative r e s u l t s were not d i r e c t l y applicable to field problems conditions, describe because qualitative the behavior of the difference extrapolations i n boundary could of a sand stratum be made to located at depth and sandwiched between impervious l a y e r s . Recently, Finn saturated et al(1970,1971) sands performed on samples. The size of sample used was approximately by 18 Inches by 7 inches and input was along the using reported relatively the d i r e c t i o n longest side of minimize, i f not eliminate, the boundary the sample by the container. of tests large sand 72 inches base motion the sample so as to effects Imposed An electro-mechanical on servo- controlled shake table was used as a source f o r imparting base motion to the sample. table i s that The advantage of using such a shake various c o n t r o l l e d input formats can be used. The input can be e i t h e r a c c e l e r a t i o n , v e l o c i t y or displacement 23 controlled and the wave form sinusoidal, or records. that triangular of the square input or can be irregular Test equipments and procedures were sample uniformity was within reasonable earthquake developed sinusoidal base input the ratio acceleration amplitude to that of the base points so tolerance of +2% Lee(1973). v a r i a t i o n i n void r a t i o , see e.g., Emery, Finn and For either of the input response at various within the sample was also measured i n several samples and was shown to be close to unity when the frequency of input was l e s s than 10 hz. constant amplitude Numerical results were obtained for sinusoidal base motion input r e l a t i n g the amplitude of acceleration to the number of cycles of v i b r a t i o n to cause l i q u e f a c t i o n , N , as shown i n Figure 2-5. E f f e c t of T i n i t i a l e f f e c t i v e normal (or surcharge) pressure, o-', on N V Jj was also examined and the r e s u l t i s as shown i n Figure The general conclusion that can 2-6. be obtained from these model tests can be summarized as follows :1. High degree of saturation of the sand model i s required. 2. The looser the sand the shorter the time (or smaller number of cycles) to l i q u e f a c t i o n . 3. The higher the i n i t i a l normal e f f e c t i v e confining stress) the pressure longer (or the Initial time (or larger number of cycles) to l i q u e f a c t i o n . 4. The (or larger the acceleration input the shorter the time smaller number of cycles) to l i q u e f a c t i o n . It should be pointed out that these conclusions agree 0.5 0.U h o I z 0.3 o I— < DC LU _l 0.2 LU O O < 0.1 0.0 10 Fig. 2-5. 1000 100 N, Effect to CYCLES of TO LIQUEFACTION Acceleration Liquefaction, . I 10000 Level (after Finn on Resistance et al, 1971) 8.0 ~ 6.0 Q. LU O < U.0 o or 3 CO 2D 10000 10 N Fig. 2-6. Effect to CYCLES L of TO Surcharge Liquefaction. LIQUEFACTION Pressure (After Finn on Resistance et al, 1971) ro 26 extremely well with f i e l d observation. In addition to t h i s , model t e s t i n g provides a basis f o r evaluating the e f f e c t of single variable on the l i q u e f a c t i o n p o t e n t i a l while from observations the effect of a single variable is a field seldom presented i n a clear-cut manner. In passing, mention should be made that i n most, not a l l , model t e s t i n g the r e s u l t s obtained are useful as a stepping stone f o r p r e d i c t i n g f i e l d behavior. of material (or s o i l ) behavior predict the if behavior is I f any theory proposed to describe or of an a c t u a l structure (including s o i l deposit), the theory should be able to predict the behaviour of the model with better accuracy since, 1. the loading on the model can be controlled and measured, 2. the boundary conditions of the model are known, and, 3. the material in the model can be made more homogeneous and i s o t r o p i c than any f i e l d deposit i s l i k e l y to be. In order liquefaction to behavior arrive of at some sandy m a t e r i a l , samples under presumably uniform stress or should not be 'theory* overlooked. testing strain for the of small conditions These are described i n the next section. 2.3 LABORATORY INVESTIGATION OF LIQUEFACTION USING SMALL SAND SAMPLES During an earthquake a typical soil element in a 27 deposit Is deformed by a system of dynamic stresses as a r e s u l t of earthquake waves passing through The most significant the soil medium. set of dynamic stresses are considered generally to be'caused by the propagation of shear waves the bedrock to the ground surface. For a h o r i z o n t a l s o i l deposit on a horizontal bedrock formation system can t h i s dynamic be i d e a l i z e d as shown i n Figure 2-7. the s o i l element i s subjected to stresses, cr^ and K <r^ , where normal stress K 0 0 and the 'at r e s t * principal cr^ i s the i n i t i a l e f f e c t i v e 0 Q i s the c o e f f i c i e n t of earth pressure at Q During the earthquake the dynamic simple shear system as developed stress i n Figure 2-7b i s imposed r e s u l t i n g i n the stress system as shown i n Figure 2-7c. been stress Initially, rest. shown from to apply Test procedures have a uniform stress system s i m i l a r or equivalent to that of Figure 2-7c and three o f the most useful and commonly employed ones are :1. C y c l i c Loading T r i a x i a l Test - This i s quite the s i m i l a r to conventional t r i a x i a l t e s t though a c y c l i n g deviator load i s used load. instead of the unl-directional The c y l i n d r i c a l saturated s o i l specimen i s f i r s t consolidated under an ambient pressure of further deviator drainage from the the between P 0 ^ D a: a n d °o~ D A0 then, with % sample prevented, a c y c l i n g a x i a l load i s applied so that <r +/ <T o e r i axial o d i c a l stress l • y F o r varies f u l ly saturated s o i l specimens the e f f e c t i v e stress applied can be represented should as shown i n Figure 2-8a and b. It be noted that the actual f i e l d conditions are not a. Fig. 2-7. Idealised b. Field Loading c. Conditions. CO 29 Fig. 2 - 8 Idealised Triaxial Conditions. Test 30 exactly duplicated since the actual principal stress system rotates approximately +40° p e r i o d i c a l l y while that of the laboratory test conditions fluctuates by +90°. The i n i t i a l stress systems are also d i f f e r e n t ; element the soil i n the f i e l d i s K -consolidated while the sample o i n the t r i a x i a l apparatus i s i s o t r o p i c a l l y consolidated. 2. C y c l i c Simple Shear Test - In t h i s test a s o i l sample rectangular cross-section with sides. rigid dimensionally is It then one- t e s t i n g drainage i s prevented and a c y c l i n g shear stress to the vertical consolidated During applied a confined i n a 'shear box' stress of <T^. is under is of top and 0 bottom of the sample. Moreover, two sides of the 'shear box' rotates i n such way so that the uniform stress or s t r a i n condition as shown i n Figure 2-7c stress i s duplicated c l o s e l y . conditions in the simple resembles those acting on a s o i l test data a obtained can be shear element applied Since test in the closely the field d i r e c t l y to f i e l d conditions without any c o r r e c t i o n . 3. C y c l i c Torsion Test - The set up test is similar of equipment in this to that of c y c l i c t r i a x i a l t e s t . But the c y c l i c shear stresses are applied through the use torsional load applied to both ends of the c y l i n d r i c a l sample while the a x i a l load i s kept constant. and Li(1972) used this method of shear test apparatus that Ishihara applying deformation to the samples and they designed torsion of a shear triaxial can be used to run 31 tests with constant confining pressure as well with no lateral strain. the * second test. type the cyclic triaxial two test corresponds to the simple shear However, more c o r r e l a t i o n of t h i s test other tests It was claimed that the f i r s t type of tests corresponds t c while as with the i s needed before the test data can be applied to f i e l d conditions since the usage of average stress and s t r a i n to represent the varying stress and s t r a i n Inside the c y l i n d r i c a l specimen under t o r s i o n needs more substantiation. solved the an of 1 field Ishibashi and 1974) Sheriff varying stress and s t r a i n f i e l d cylindrical ingenious 'donut-like strain Recently problem Inside a s o l i d using fields specimen under sample c r o s s - s e c t i o n . sample i n order to achieve a torsion by They used a uniform shear and consequently they were able to compare t h e i r test r e s u l t s with those obtained from the triaxial and the simple shear apparatus based on the actual stress and s t r a i n values. Investigations of sand l i q u e f a c t i o n using the above mentioned types of laboratory test have been made by Seed and Lee(1966), Peacock and Seed(1968), Finn et al(1971), Ishlhara and Ll(1972) and Ishibashi and Sherif(1974). Most of the tests performed involved c y c l i c shear stress, x^ t during the test the use as shown of in a constant Figure 2-9. amplitude Usually the pore water pressure, cr , c y c l i c s t r a i n , 7 t and the c y c l i c stress, T , were continuously monitored by 32 using electrical transducers and chart recroders. records of simple shear and t r i a x i a l t e s t s are Figures 2-10 and 2-11. portion the test where the pore pressure of as Typical shown It can be seen that during the the amplitude of c y c l i c s t r a i n developed i s in initial r i s e s gradually, almost zero (of the order decimal of a percent) but becomes increasingly large (+15 magnitude to 20^) towards l i q u e f a c t i o n f a i l u r e when the pore pressure r i s e s more r a p i d l y . It sands that behaved in this way and l i q u e f a c t i o n could be described in was a found their that most resistance to simplified manner by using the following variables :N: number of stress cycles to cause l i q u e f a c t i o n . Dt r e l a t i v e density, defined as L r D = where max e ** * " max a r :- x 100% 6 (2-1) min e i s the current void r a t i o of the sand sample and a n d e min a r e t n e v o i d ratios densest state r e s p e c t i v e l y . used to indicate the at the loosest This variable i s state of and generally denseness of the sand specimen. td ! c y°ll c s n e a stress amplitude, r and, C"y : the i n i t i a l e f f e c t i v e v e r t i c a l (or confining) s t r e s s . Q It was and the combined found that the c y c l i c stress amplitude, initial and effective (T /crJ ) A rs can vertical be (j-' . vo as a stress, considered T » d can be single 33 Fig. 2 - 9 Cyclic Shear Wave Forms Stress TEST 58 e = 0.617 2.0n cc «f nr f. r n f. r=Ml Hi h n I I i i 11 II I;II :.| i l l M i l l : i i l i I i.l.uhll • i.i'i t.111y11U11' 1 UJ < rfflllhl — UJ nflWi'r¥rjr»r r rr i r e w ! PORE i.oH * cc ~" PRESSURE i ri M i i i ii i i i i i i i i i ii i • i i i i i i i ii i i i i i i ii I i i i i i 11 I M i lr iff 1 2 UJ=> 1 ce ni 3 Z (O < UJ X o- SHEAR 1 cc 1 5 -10 STRAIN x I z < — cc sS cc •10 < SHEAR cc e 0.4- <o £ 2> ii 0.2- » n fi ii 05 —X 0^ o p z < P I 0.2 £ S 0.4 b£ < c/) . 2-10 II II II II II M II II II ii III II III II III II STRESS M A II II l| n M II M I ! ii ii ii Ii ii ii ii ii iii ii ii H II II II II II II II n in II it II i II ii i II il II II II v y II II II II H II II vhi ii ii H i H n n II i II II n ii ,i M ii II II II H II vHy A A A II II II II n n n II II II L-15 ni A.. Ml III ii :i ii i! H II n II II II ii H H II n II II II III H i n n ii II II II II II i i i II n A ft I! II II u uuuv uy nnJ U U SHEAR II ii FRICTION Typical Cyclic II H II Ii II II it I 11 ., II H II H H yv u w II II A: fl II I UJ X V i ii v y =0.03 K g / c m ' Simple Shear Test Data (after Finn et al, 1971) 36 Independent to present variable. Finn et al(197l) used a summary plot the t e s t r e s u l t s r e l a t i n g N_ , D and L r ) in (T./(J' d a vo more e f f i c i e n t way as shown in' Figure 2-12, where each curve represents a constant (T,/o" ) value. It can be seen from o. vo f this figure by N , that the resistance to l i q u e f a c t i o n , as measured decreases as T :- 1. the r e l a t i v e density of the sample becomes smaller, 2. the c y c l i c shear amplitude becomes larger, and, 3 . the i n i t i a l effective vertical (or confining) stress becomes smaller. Although the c y c l i c t r i a x i a l t e s t imposes a stress condition on the sand sample d i f f e r e n t from that i n the Finn et al(l971) found that i f due consideration i s field, given to i n i t i a l e f f e c t i v e normal pressure acting on the 45°-plane the in the t r i a x i a l t e s t specimen c o r r e l a t i o n between simple shear test and triaxial test is the cyclic possible. In addition, the recent development of the novel Torsional Simple Shear Device by Ishibashi and Sheriff 1974) makes to degree run liquefaction confinement. AT^/<7^ , tests with varying Moreover, they found that the the octahedral normal stress, versus the number of liquefaction al(1971). since it of possible lateral plot of the r a t i o of maximum change i n the shear stress to ct et it cycles to bears close resemblance to that obtained by Finn This c o r r e l a t i o n has a n o n - t r i v i a l implication indicates that the l i q u e f a c t i o n of sand produced i n the laboratory i s not an a r t i f i c i a l , or apparatus-dependent. 37 (07') = 2.0 0 X- w Kg/cm Marked 1 for Edch Line .75 .21 I5 a> I V\ V .70 \.34 V I0 \ .65 o < y.50 \ .60 o > .55 .50 10 CYCLES Fig. 2 - 1 2 Simple 100 TO Resistance Shear 1000 LIQUEFACTION to 10000 -T1, Liquefaction in (After Finn et al, 1971) 38 phenomenon and the data obtained i s good f o r a p p l i c a t i o n to the analysis of actual soil structures. The method of adapting l i q u e f a c t i o n data obtained In the laboratory to f i e l d problem i s discussed i n the following s e c t i o n . 2.4 SOME METHODS OF DETERMINING LIQUEFACTION POTENTIAL OF A SOIL DEPOSIT. In sections 2.1 and 2.2 of t h i s chapter, some of the methods of determining l i q u e f a c t i o n p o t e n t i a l were discussed, for example :- 1. Florin and Ivanov(l96l) - using settlement due to blasting at the s i t e as a key v a r i a b l e , 2. Osaki(l968) - using the c r i t i c a l standard penetration values, N , as a c r i t e r i o n , c r 3. Kishlda(1969) - using the 75% relative density as a c r i t e r i o n f o r the occurrence of l i q u e f a c t i o n , and, 4. Maslov(l957) level, c^Q-pif - using the concept of c r i t i c a l acceleration to determine the p r o b a b i l i t y of having a liquefaction, and they are not repeated concerned with the a p p l i c a t i o n here. The following i s mainly of laboratory test data to f i e l d problems. Castro(I969) following the suggestion of Prof. A. Casagrande used the concept of c r i t i c a l void r a t i o to determine the l i q u e f a c t i o n p o t e n t i a l series of stress-controlled of a sandy soil. A t r i a x i a l undrained t e s t s , which .3 were termed R t e s t s , were performed on representative from the field for various 39 samples void r a t i o or r e l a t i v e density values and at d i f f e r e n t i n i t i a l consolidation pressures, c r ^ . For samples that developed continuous pore pressure r i s e at constant a x i a l load, (SPONTANEOUS LIQUEFACTION), the values of the effective against minor principal 2-13. Figure a ^ f * the void r a t i o of the sample, e^, through the points and a e~p l i n e a^ stress, was w e r A curve was drawn obtained as shown For a s o i l element with consolidation pressure 2-13, have a higher l i q u e f a c t i o n p o t e n t i a l than an element of the same s o i l with <r^ and e^, represented by point the h o r i z o n t a l distance AC reduction minor pressure f In and void r a t i o e^, represented by point A i n Figure would e^ plotted e in effective and BC indicate principal r i s e , at l i q u e f a c t i o n . B, since the amount of stress, or pore A s o i l element with (r*^ and represented by point D, would not develop continuous pressure pore r i s e at constant loading and i s said to be d i l a t i v e . However, i n t h i s method of determining l i q u e f a c t i o n potential there are a number of factors not being taken into account and two of the important ones are :1. Magnitude and duration of the disturbance that cause l i q u e f a c t i o n , and, 2. E f f e c t of a further stress reversal on the sample that exhibit d i l a t i v e behavior. Nevertheless, the method i s a good guideline f o r recognizing deposits with metastable spontaneous l i q u e f a c t i o n . sand aggregates which would develop 0*7 41 Seed determining Idriss(1967.1971) and the liquefaction s u b j e c t e d t o earthquake horizontal shear proposed potential forces. of a method f o r soil deposit The time h i s t o r y response o f stresses, T , at various depths was xy e v a l u a t e d u s i n g dynamic a n a l y s i s . Based on the irregular 10 time h i s t o r y s t r e s s response the amplitude o f an e q u i v a l e n t cycles o f constant c y c l i c shear s t r e s s was each depth so t h a t a p l o t o f e q u i v a l e n t then determined a t 10-cycle shear stress amplitude, T ^ Q , v e r s u s depth was 2-14, From the l a b o r a t o r y c y c l i c simple shear t e s t curve AB. o b t a i n e d as shown i n F i g u r e data, o r c y c l i c t r i a x i a l t e s t d a t a m u l t i p l i e d by a c o r r e l a t i o n f a c t o r , the shear s t r e s s amplitude t h a t caused l i q u e f a c t i o n i n 10 cycles, T^Q» was p a r t i c u l a r values of D curve CD determined as shown i n F i g u r e 2-14 To w i t h i n which simplify potential the Seed T d| >T 0 was is no» »S«» e procedure and each depth using of Idriss(197l) obtained. determined z a The zone o f by the t o z^ i n F i g u r e determining proposed that cyclic shear 2-14. instead of 10 s t r e s s , T,, , dlO approximation was used t o determine the e q u i v a l e n t N , o f constant c y c l i c shear o f amplitude, x . eq av the constant amplitude depth liquefaction approximating the s t r e s s response by an e q u i v a l e n t of the and cr^ a t the p a r t i c u l a r depth and the r l i q u e f a c t i o n , i f t h e r e i s one, interval at rt cycles a direct cycles , However, b a s i c method of d e t e r m i n i n g l i q u e f a c t i o n p o t e n t i a l was not changed. At f i r s t glance, it would seem that conclusions 42 Stress Depth FIG.Z-14 Method of Evaluating Liquefaction Potential (After Seed et a l , 1971) 43 about l i q u e f a c t i o n p o t e n t i a l based on the c r i t i c a l void r a t i o concept contradicts that using c y c l i c test to data. According the c r i t i c a l void r a t i o concept an increase i n the i n i t i a l consolidation pressure would according increase while reduced. However, closer examination reveals that in the two cases. the latter liquefaction potential 'liquefaction p o t e n t i a l ' was to the the p o t e n t i a l i s the term used to convey d i f f e r e n t meanings In actual fact when the initial consolidation pressure i s increased, a. the critical void ratio concept Indicates a higher p o t e n t i a l of f a i l u r e i n the spontaneous l i q u e f a c t i o n mode but higher failure, b. the stresses are needed to initiate a while, cyclic magnitude test of data indicates disturbance more that for cycles the (smaller liquefaction same of stresses are needed to cause the pore pressure to r i s e to the level such failure potential) though that the d i l a t i v e response which has a s t a b i l i z i n g e f f e c t may be reduced i s not mentioned. It can be seen that while the c r i t i c a l void r a t i o method emphasizes the c l a s s i f i c a t i o n of f a i l u r e contractive dilative or behaviour at f a i l u r e , the c y c l i c stress method i s mainly concerned disturbing modes, with stresses the magnitude to cause and duration failure. In of the fact, the combination of the two methods would give the complete picture of l i q u e f a c t i o n f a i l u r e of s o i l structures when earthquake loading. subjected to Even though the method of determining l i q u e f a c t i o n p o t e n t i a l using c y c l i c test data i s able to provide some approximate solutions which are useful i n the design of actual of actual structures, there are s t i l l some questions l e f t unanswered by t h i s method, e.g.:1. pvnamlo Response - In a r r i v i n g at values f o r N <' a dynamic analysis using equivalent damping c o e f f i c i e n t s are used. The but non-linearity the change Seed and T shear moduli and and Idrlss(l969)• i n s o i l s t r e s s - s t r a i n i s approximated of shear e f f e c t i v e normal stress and modulus due to changes in compaction (hardening) i s not taken into account, 2. Time to Liquefaction - Since the time history of pore pressure response cannot be determined by the method the time to l i q u e f a c t i o n cannot be determined, 3. Maximum Strain at Liquefaction - As a result of approximating the actual s t r e s s - s t r a i n by a straight l i n e the one value of maximum s t r a i n obtained i s not a and to realistic obtain such s t r a i n value the actual s t r e s s - s t r a i n should be used, and, 4. The Effect of Consolidation during pore pressure r i s e the liquefaction potential of a soil has not on been considered. It i s the purpose of t h i s thesis to t r y to a r r i v e at a in which account. the above mentioned factors can be taken method into 45 CHAPTER 3 MODELLING OF SATURATED GRANULAR SOIL UNDER CYCLIC SIMPLE SHEAR CONDITIONS 3.1 VOLUME CHANGE CHARACTERISTICS DURING CYCLIC DRAINED SHEAR. It is a commonly observed phenomenon that a dry granular s o i l compacts or reduces i t s volume when subjected cyclic loading, earthquake e.g., machine forces. The foundation general vibration observation the amount of testing are quite scattered and experiments to amount of enclose the acceleration s o i l specimen during of stress the specimen was not known. compaction level, was or Barkan(l962) 'Rigid' containers were used i n course of v i b r a t i o n so that the state by laboratory uncorrelated and sometimes Mogaml and Kubo(1953)t and Bazant and Dvorak(1965). experienced that Apart from t h i s , the quantitative by many early investigators using contradictory, see e.g. their is the volume reduction Increases with the magnitude and duration of v i b r a t i o n . results obtained and from performance of model as well as e x i s t i n g foundations to related to the and the strain Invariably, the frequency and v e l o c i t y l e v e l i n some cases, of the input v i b r a t i o n . It i s not d i f f i c u l t to r e a l i s e that r e s u l t s obtained would be very much apparatus-dependent since the 46 actual stress and strain experienced by the s o i l specimen depends not only on the magnitude of input v i b r a t i o n but on the also dynamic response properties of the container-and-soil system. For example, i f one examines the Schaffner(l962), as shown i n Figure 3-1. data obtained by one may observe that each of the f i n a l void r a t i o curves obtained for a constant acceleration l e v e l strongly suggests the presence of resonance of the container-and-soil system at around 80 to 100 Hz. Moreover, the e f f e c t of the superimposed v e r t i c a l component of o s c i l l a t i o n , which might be applying data. vibration, present complicated due the to the interpretation Consequently, i t i s more r e a l i s t i c , i n applying laboratory method the of of the sense of test data to the analysis of actual s o i l structures, to consider the volume change of a s o i l element as a function of the stress and s t r a i n experienced by i t . In the dynamic response of a horizontal s o i l deposit to earthquake base system motion the cyclic simple shear caused by the propagation of shear waves from the base to the surface of the deposit, as shown i n Figure 2-7. most significant investigators, one. e.g., device compaction to and Silver cyclic and Seed(197D. Youd(1971#1972), used determine recently, Martin, Finn and mechanism is the With t h i s i n mind many of the recent Dmevich(1972,1972a) and shear stress the relation stress/strain Hardin the between conditions. Seed(1975) studied the and simple volume More fundamental of s o i l l i q u e f a c t i o n and a r e l a t i o n s h i p between the FIG. 3-1 Compaction of Dry (After Schaffner, Coarse 1962) Sand 48 volume change i n increment in proposed. the the drained undrained case and case the during The construction of a material pore pressure cyclic model shear to was include both the volume change and pore pressure build-up phenomena i s made possible by such a relationship. Before going into constructing the equations of the material model, the important observations concerning the various volume changes under drained c y c l i c shear are discussed f i r s t . In t h e i r investigations into the compaction subjected to S i l v e r and Seed(197l) the amount c y c l i c simple shear of and Seed and volume change loading, of sand Youd(1971,197), 2 Silver(1972) (compaction) observed that caused by c y c l i c simple shear loading on dry granular s o i l depends on, 1. the r e l a t i v e density of the s o i l , 2. the magnitude of the c y c l i c shear s t r a i n , and, 3. the t o t a l number of s t r a i n cycles the s o i l specimen has been subjected t o . Examples of experimental data obtained are as shown i n Figures 3-2 and vertical particular 3-3. It effective test, should be stress, does noted which not is that the value of the constant influence the during volume c h a r a c t e r i s t i c s s i g n i f i c a n t l y , as shown i n Figure 3-2. i s growing evidence that the compaction is caused by grains into a more 'stable volume the 1 change There change associated slight a with re-arrangement of the configuration. The amount of .550 I 10 IO2 IO3 |0 4 10 Number of Cycles FIG.3-2 Effect of Vertical Stress on Compaction (After Youd, 1972) ^ i — T "r i i i n r—i"' i i i 111 • - 1 1 1 1 1 11 1 Cycle 10 D =45% / r - - • / - ' 8 0 % / / i 0.01 i i i i i i t i 0.1 Cyclic 3-3 - 6 0 % / i t i i i i 1.0 Shear Strain - 7^ y (%) Effect of Relative Density and Shear Strain on Compaction (After Seed et al, 1972) i 10.0 51 grain re-arrangement i s governed by how much adjacent grains can move c y c l i c a l l y r e l a t i v e to one another, and, since the c y c l i c shear s t r a i n amplitude i s d i r e c t l y related to t h i s type of relative volume displacement the strong dependence of compaction change surprising. the on cyclic shear strain amplitude i s not Furthermore, mention should be made that most of non-recoverable volumetric s t r a i n (grain caused by one-dimensional compression occurs re-arrangement) i n the initial loading stage and that, 1. during drained cyclic shear, the v e r t i c a l effective the v e r t i c a l effective stress i s held constant, and, 2. during undrained c y c l i c shear, stress decreases, so that the effect of v e r t i c a l effective stress on the •compaction curves* shown i n Figures 3-2 and 3-3 which are f o r constant c y c l i c s t r a i n amplitudes i s of minor importance. the other hand, the v e r t i c a l shear modulus and so effective stress a f f e c t s the compaction which results This topic i s discussed i n section Seed and the has a s i g n i f i c a n t e f f e c t on the shear strains which result from a given ground motion and on On from therefore such an e x c i t a t i o n . 3.2. Silver(19?2) demonstrated the a p p l i c a b i l i t y of data obtained from drained c y c l i c simple shear tests to the determination of t o t a l settlement of a model dry sand subjected to harmonic base motion. deposit However, f o r non-uniform 52 c y c l i c loading, such as that Imposed by earthquakes, in t h e i r present form as shown i n Figures 3-2 directly that applicable. for a (relative Martin, soil element density) the compaction s t r a i n , Gyp» parameter so that A£ p, caused by y determined - total can this is quite on a vp = as U ( 6 v not void a ratio volumetric strain history i n volume compaction, strain similar the are cycle, f to theory , & can be the use of s t r a i n of plasticity. An c u r v e - f i t t i n g method was proposed to describe the experimental A e used additional initial accummulated a further increment an based or be hardening parameter used i n expression given and 3-3 Seed(1975) proposed Finn and with test data data:' V"> O" * 1 a P 2 "' where C^, C C^. C^ experimentally. A l * a - 2 vt> ( C € ) T- "^ + and 2 are method of (3 constants the 0.79, a values 0.45 relative density, D, r of 45$. the Figure nature process 3-4 0.80, (in of at It should be pointed out in addition the d i s c r e t e steps) of the implied t o t a l volume compaction s t r a i n function, e since in the r e s p e c t i v e l y f o r a Crystal S i l i c a Sand that the expression i s an empirical one and incremental determined of the four constants were found to be and 0.73 la) g r a p h i c a l l y presenting function given by equation (3-la) i s as shown and " counting v p the , has to be accepted number of shear s t r a i n FIG. 3 - 4 Incremental Volumetric Strain (after Martin et al, 1974) Curves V*) 5^ cycles completed i s also of For drained amplitude cyclic simple histories characteristics a are can be discrete shear the when volume the compaction four associated with the p a r t i c u l a r s o i l are known. if constants For example, i s the t o t a l volumetric compaction at the end of the 1 i-th strain cycle i and and are vp volumetric compaction and 6 vp = € ^ 1 the incremental s s t r a i n amplitude the i - t h s t r a i n cycle then, f o r equation respectively for (3-1), *4p + (3-3) Accordingly the t o t a l volumetric compaction s t r a i n completion such a of the computation Seed(l975) and a N-th was STRESS-STRAIN after cycle can be c a l c u l a t e d . carried out by Martin, the In f a c t , Finn and comparison between calculated and measured values was made as shown i n Figure 3.2 nature. tests i n which the s t r a i n monitored computed incremental RELATIONSHIP 3-5. UNDER CYCLIC SIMPLE SHEAR CONDITIONS. From a series of s t a t i c t r i a x i a l drained tests on a sand Kondner and Zelasko(1963) concluded that the r e l a t i o n s h i p between the granular axial soil stress, under axial cr , and axial compression represented by a hyperbolic curve, strain, could be £ , of closely 55 to I oI £ 0 1.2 I I 5 I 10 Number of Cycles 1 1 15 I 20 1 1 c '5 0.8 GO o 'xl •f Measured 0.4 E Computed O > 0 i 1 i 5 10 15 Number of FIG. 3 - 5 20 Cycles Comparison of Measured and Computed Volumetric Strain (after Martin et al, 1974) 56 e c - (3-4) a +b where a and b are constants determined The from experimental data. curve given by equation (3-4) i s shown i n Figure 3-6a and i t can be seen that the reciprocals of a and b give the i n i t i a l tangent modulus and maximum stress l e v e l r e s p e c t i v e l y . In terms of shear stress, T , and shear following equation can be written, see Figure T = 1 max G where G m a x T . + s t r a i n , T , the 3-6b, (3-4a) •* ^max i s the i n i t i a l tangent modulus, and, i s the maximum shear stress that can be applied to the sample, Hardin and Drnevich(1972a) showed that f o r sands G „ and r max max can be obtained from the following expressions, max = .(2-973 1 +e 1+K. -sin max where G and T max max m o v e v (3^5) 1230 n a v 4 - f f i T '.cr; (3-6) are i n p s i , ' i s the v o i d r ratio, (T i s the mean p r i n c i p a l e f f e c t i v e s t r e s s i n p s i , 0-^. i s the v e r t i c a l e f f e c t i v e s t r e s s i n p s i , K i s the c o e f f i c i e n t o f e a r t h p r e s s u r e a t r e s t , and, 0 Q 4>' i s the e f f e c t i v e angle o f s h e a r i n g r e s i s t a n c e . FIG. 3-6 Hyperbolic Stress-Strain Curve. 58 Also, equation (3-^) can be rewritten into, <fc Where G m i • = T/ /" , and f = = tabulated f o r various s o i l s . °- v T m Q /G m o . and values of f o f the sample so were In addition, they used a hollow c y l i n d r i c a l sand specimen and applied c y c l i c t o r s i o n ends 7) that to both a t y p i c a l s o i l element of the specimen was subjected to a c y c l i c simple shear stress system. From the test data on a large number of s o i l s they to were able show that the 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 can be used f o r s o i l loading ratio and as a Furthermore, elements subjected expressions function to c y c l i c simple shear f o r the shear modulus and damping of strain level were obtained. modifications to the i d e a l i z e d hyperbolic s t r e s s - s t r a i n proposed were made and are b r i e f l y l i s t e d as follows»I , A hyperbolic shear strain, ^ , i s defined instead of equation ( 3 - 7 ) the following so that expression i s used, i + G max with f. 2. Assumptions h 'r h were (3-8) r. . [ l + ae r J (3-9) made concerning the damping r a t i o D so that at any cycle and s t r a i n l e v e l T , 59 D (3-10) max with defined s i m i l a r l y to (3-9) a and b involved are d i f f e r e n t equation Numerical by Hardin from those appeared in (3-9). values and modifications though the constants f o r a's and b*s and D Drnevich(1972a). the expressions are m Q V and However, only f even are given with the applicable to the determination of average shear modulus, G, and damping ratio, D, f o r a complete s t r e s s / s t r a i n cycle, as shown i n Figure and 3-8. Neither the actual s t r e s s - s t r a i n nor the change i n shear modulus due to volume curve is 3-7 traced compaction and change i n e f f e c t i v e stress are taken into account. Martin, Finn and Seed(1975) suggested that during the l i q u e f a c t i o n process the change i n shear volume modulus due to compaction and change i n e f f e c t i v e normal stress could be taken into account by using the following expression, hv where 1* hv f a + (3-U) hf i s the horizontal shear amplitude, i s the shear s t r a i n amplitude, (Ty i s the current e f f e c t i v e normal stress, and, a and b are functions of volume compaction €vp* as follows t — strain, Fig. 3-7 Equivalent Shear Modulus for Initial Loading ON O FIG. 3 ~ 8 Definition of Equivalent Shear Damping Ratio Modulus for Symmetrical Loop and ON 62 1 A b = B, A A , A^, B 1 # 2 experimental data.. (3-13) l t +A € ^2 (3-13) W v p 1 with 2 3 vp B and 2 are constants determined Comparing equations from (3-11), (3-12) and expressions (3-4), i t can be seen that i n e f f e c t the i n (3-12) and (3-13) give the "hardening' effect of compaction with volume - i . e . , increase i n shear modulus and maximum shear stress due to volume compaction. section, a general stress-strain In relationship the next i s proposed based on the above described s o i l behavior. 3.3 MODELLING OF SATURATED GRANULAR SOIL UNDER CYCLIC SIMPLE SHEAR. 3.3.1 Hysteretic Hyperbolic S t r e s s - s t r a i n Relationship. In order to s t r e s s - s t r a i n behavior cyclic simple a of an at expressions describing the idealized shear conditions Masing(1926) type that arrive soil element under i t i s assumed here that the of hysteretic r e l a t i o n s h i p i s followed and 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 proposed by Martin, Finn and Seed(1975) i s used. These are discussed i n the following paragraphs. For the i n i t i a l loading phase of the hyperbolic stress-strain the granular s o i l relationship formulated by 63 Kondner and Zelasko(1963) and Drnevich(1972) i s used. also adopted by Hardin and Therefore, the i n i t i a l response of the granular s o i l to simple shear i s given by the following equation, T mo^ G = 1 + (3-14) mo' mo where T i s the shear stress, "t i s the shear s t r a i n , Gmo^ i s the i n i t i a l maximum tangent modulus and the maximum shear stress that can be TL« mo i s applied without failure. Following the suggestion of Hardin and Drnevich x i t i s assumed that these quantities G and TL are determined ^ mo mo n by the i n - s i t u condition of the granular s o i l and are given by the following expressions, G mo mo mo where = ,(2.973 - e ) llt76o 1 + e 2 J 1 u 1+K„ 2 1-K ( -y^in*') - ( (3-15) o n 2 > 1 * ) (3-16) " are i n r psf, mo and lmo * e i s the void r a t i o , <TQ i s the mean p r i n c i p a l e f f e c t i v e stress i n psf, i s the v e r t i c a l e f f e c t i v e stress i n psf, K Q i s the c o e f f i c i e n t of earth pressure at r e s t , and, 64 4>' i s the e f f e c t i v e angle of shearing resistance. Since we are considering the response only, i t w i l l prove more convenient conditions in Then equation terms of (3-15) i s of to horizontal layers express the stress the v e r t i c a l e f f e c t i v e stress, a~y» expressed i n the form, Therefore, f o r a given i n - s i t u condition, G mo ''•mo = c*/^ (3-18) = (3-19) pjffi For unloading and reloading the Masing(1926) type of hysteretic behavior is i n i t i a l loading curve, used. or This assumes that if the skeleton curve, which i s described by equation (3-1*0 can be represented by, X s = f (D "(3-20) 2 Then the unloading or reloading curve can be obtained from, where (f ,T_„) i s the l a s t reversal point i n the s t r e s s - s t r a i n plot. Consider a s o i l undeformed state to element a shear being strain strained level from its f, with a 81 65 .corresponding shear stress l e v e l unloaded and reloaded i n the I _. The element i s now opposite d i r e c t i o n so that the shear s t r a i n or stress undergoes a complete reversal when the shear f-^, with corresponding s t r a i n reaches a f i n a l value of shear stress, T ^ . Figure 3-9a. The s t r e s s - s t r a i n diagram i s shown i n (3-1^) while Curve OA i s given by curve AB i s given by, Tg ~ = f (~ ) f (3-22) T& 2 2 The reversal curve AB i n t e r s e c t s the negative skeleton beyond followed curve f a n i t i s assumed a would the skeleton f^'-fa. at branch when d that of the i s increased the stress-strain path be the continuation of the negative branch of curve, stress-strain BC, with history. (/, T ) Q In other erased from the when the last words, reversal point i s on the skeleton curve the absolute value the stress or strain can be used as a l i m i t i n g value f o r .switching back to the skeleton curve, thus 'erasing' (/"_, cl from our c a l c u l a t i o n . Figure 3-9b. T ) S a A general loading history i s shown i n Assuming that the current loading would bring the s t r e s s - s t r a i n point, P to P', i t can be s t r e s s - s t r a i n branches, reversal points of (f .T a AB s a and BP'A' ) . (^ .^ ) D sb seen that are required. and ( T .X ) c s c two The have to be recorded f o r the determination of the current branch of the stress-strain curve, CPBP'A'. (/„,!) C S C branch CPB, i s required f o r the A* (a) first unloading FIG. 3-9 (b) general reloading Hysteretic Characteristics 67 (3-23) 2 2 V (fo.X^o) i s r e q u i r e d f o r the branch BP'A', a sa 2 While (^b'tgt)) i 23) t o (3-24). P passes (3-24) = s u s e l d — 2' a s c r i t e r i o n f o r s w i t c h i n g from a Note t h a t as soon as the s t r e s s s t r a i n the point B In time. this the s t r e s s o r s t r a i n v a l u e o f the l a s t but one r e v e r s a l p o i n t can be used the point the r e v e r s a l p o i n t s B and C would be e l i m i n a t e d from our c a l c u l a t i o n a t the same case, (3- last as a l i m i t i n g value f o r s w i t c h i n g back to but one u n l o a d i n g or r e l o a d i n g branch and a l s o as a c r i t e r i o n f o r ' e r a s i n g ' the l a s t and the l a s t but one r e v e r s a l p o i n t s from f u t u r e c a l c u l a t i o n . that the strain above d e s c r i p t i o n relationship under It should constitutes general a that pointed complete loading Herrera(1964) has shown sequence. be and stress- unloading combinations Coulomb f r i c t i o n and l i n e a r e l a s t i c elements e x h i b i t t h i s of behavior. Newmark and Rosenblueth(1971) out of type suggest the Masing model t o be a good r e p r e s e n t a t i o n o f s o i l b e h a v i o r and some e x p e r i m e n t a l support f o r the s u g g e s t i o n i s p r o v i d e d by Marsal(1963). 3.3.2 Volume Compaction and Pore Pressure Increments. A t y p i c a l one d i m e n s i o n a l l o a d - d e f o r m a t i o n curve f o r a laterally confined sand specimen i s shown i n F i g u r e where ab r e p r e s e n t s l o a d i n g , be u n l o a d i n g and ce 3-10a reloading. Log P (a) FIG. 3-10 One-Dimensional Soil Model ON 00 69 I t i s generally observed that while ab i s quite d i f f e r e n t from be and cd the l a t t e r two are reasonably close together to allow t h e i r representation by one curve, model as shown i n Figure 3-10b ce. A can be used to describe, to some degree of approximation, the behavior of sand dimensional compression. be assumed conceptual under one The spring element i n the model can to be non-linear so that the i d e a l i s e d unloading and reloading curve ce can be simulated. the model should be deformation curve as of a The s l i p element i n 'hardening' type i n Figure 3-11&, shown with Using load- i . e . , the force required to cause s l i p increases with the magnitude displacement. a of slip such a conceptual model i t can be seen that the volumetric s t r a i n , € v > i s given by the following equations;€ v A£ where K l r v = = < lr K + K s>'P K .A lr <3-25> P*Pmax p<p P (3-26) m a x i s the modulus (non-linear) of the spring element, K_ i s the pseudo-modulus of the s l i p element, and, s p „ i s the past maximum v e r t i c a l e f f e c t i v e s t r e s s , 'max When horizontal cyclic shear i s applied to a sand sample at equilibrium with an v e r t i c a l load, e.g., c y c l i c test, volumetric strain Since shear occurs at constant v e r t i c a l load so that a load-deformation curve obtained. simple as shown i n Figure 3-12 is the a d d i t i o n a l volume reduction caused by c y c l i c loading i s non-recoverable an a d d i t i o n a l feature has to FIG. 3-11 Force-Displacement Relationship of Slip Elements -s3 O 71 Fig. 3-12 Volume Compaction Caused by Cyclic Loading 72 be incorporated into the s l i p element of the conceptual - namely, the addition of s l i p component caused by c y c l i c shear s t r a i n as described i n previous s e c t i o n . 25) (3-26) have and *v A€ = K = y where € < lr + K s>'P <Srp + P*Pmax the (3- ^-27) p-cpJJ^ K .AP lr (3-1). i s defined as i n equation changes Equations to be replaced by the following, £J V model properties (3-28) Since compaction of the s o i l element, e.g. void r a t i o and shear modulus, more or l e s s i n the same fashion as the maximum past pressure, P ax' l ^ i " maximum past pressure, p!L , i s used i n equation (3-27) and (3-28) instead max a n e c u v a e n t m V °f P™<, max . The determination of p*~ max i s shown conceptually i n v Figure QV 3-12. To account for the undrained element, is filled with these circumstances, the same time when an (grain s l i p ) s t r a i n , AG water and drainage a soil causing pore water. a increment in volumetric portion pore has to be of i t s stress transferred to the this phenomenon l e v e l can be obtained from Martin, Finn and the compaction , i s induced by a shear s t r a i n c y c l e . Description of expression for prevented. volume reduction cannot occur at As a r e s u l t , e l a s t i c s t r a i n i n the spring element relaxed in i t may be assumed that the conceptual model shown i n Figure 3-10b Under condition water at the grain Seed(1975)» and an pressure increment, A(T » W was 73 found to be, . AC7 where n K " W ( = ' nK K x ^-29) A€ w i s the porosity of the s o i l element, w i s the bulk modulus of water. When water can be assumed to be equation (3-29) can be ACT It j » lr' vp K l r should be incompressible or K > > K w = w pointed stress needed at a l l . dimensional K l r .A (3-30) € v p out that during the settlement or This eliminates maximum (3-27) i s i s never exceeded and equation loading r s i m p l i f i e d to, l i q u e f a c t i o n process of a hroizontal deposit the past vertical i » the determination not of one- c h a r a c t e r i s t i c s of the granular material under consideration, nevertheless, the one-dimensional rebound c h a r a c t e r i s t i c s , K^ , has to be q u a n t i t a t i v e l y known i n order r that equation (3-22) can be used. In the determination of the magnitude of volumetric compaction s t r a i n due to horizontal c y c l i c s t r a i n (3-1) and (3-la) are used. experimental data pressures and for a relative Monterey confirmed. Sand densities. volumetric compaction s t r a i n conditions described Recently, on Pyke(1973) obtained at various normal The independence vertical i n the previous A comparison of the 6 ^ expressions stress under the paragraphs was function of again f o r sands of 74 different relative obtained densities Pyke(1973) by can into be one made by p l o t t i n g data graph. A 'reduced' incremental volumetric compaction s t r a i n , ^C^p, i s obtained by the product of relative density i n per cent, D, r and incremental volumetric compaction s t r a i n , A C ^ , i . e . , (3-31) D .A€ r vp vp The graphs are plotted i n Figure 3-13. the curves I t may be seen that f a l l more or less to an unique pattern suggesting the p o s s i b i l i t y of using incremental one shape function, U, f o r the s compaction s t r a i n f o r sands at d i f f e r e n t r e l a t i v e densities, namely, A € vp ^i-W^ = (3-32) (3-33) vp (3-34) where ^ , ^ , ^ 2 and ^ are constants s i m i l a r to those used i n equation (3-la). In t h i s usage i s inversely to density r the relative proportional value, D , while i(/ , ^ 2 and ij;^ are assumed to be constant f o r the same type of sand at densities. change Martin, Finn and increments completion of However, a by strain Seed(1975) assuming cycle i t i s generally that as observed different calculated the volume these shown occur i n Figure at the 3-l4a. that i n a drained c y c l i c 0.1 Shear FIG. 3-13 0.2 Strain Amplitude — 'Reduced' Incremental *f % Volumetric Strain ->3 77 shear t e s t on granular s o i l gradual volume compaction pressure build up i n the undrained stage of unloading from occurs during the maximum shear s t r a i n . In view of t h i s , i t i s not unreasonable volume compaction from test) to assume a l i n e a r value as shown assumed i n order that change increment i s added every h a l f c y c l e . involved i n t h i s Martin, Finn and 3.3.3 3 thesis the numerical (3-3*0 the volume For computations values obtained by and fy. Hardening and E f f e c t of Pore Pressure Rise cyclic suggested loading that equation the fundamental behavior of sands i n simple shear, Martin, Finn and Seed (3-11) s t r e s s - s t r a i n behavior. and to be Seed(1975) were used to determine the values In a study on under of The by an expression s i m i l a r to that of equation though the value of i f ^ i s halved 2 3-l4b. i n Figure magnitude of volume change increment i s s t i l l of q/ , y variation increment with the reduction of shear s t r a i n i t s maximum determined (or pore might be used to represent the In t h i s equation the parameters a b are constants for a given load cycle but i n general are functions of the volumetric compaction given by equations (3-12) strain, (3-13). and the shear modulus, G , i n any cycle mn ^yp. n d are a The maximum value of of loading n can be obtained i n the following manner, G mn = < hv / dT a t *= 0 (3-35) 78 From equation (3-H)t therefore, we obtain mn = JK' G For the case 3-36) a ( of i n i t i a l loading when no hardening has yet occurred we obtain mo G = ' J o T / A (3-37) l A f t e r n cycles, substitution f o r a from equation mn G i n which € cycle " / A, - — 1 2 A € + A v i n which a^ and a 3 vpJ 2 G f o r fcr^ / A m Q 1 to the nth of equation according to are constants. We can obtain an expression stress from equation (3-11) by The maximum shear stress, T t—oo (3-38) p With algebraic manipulation (3-12) and substitution of equation (3-37) we obtain gives € i s the volumetric s t r a i n accumulated of loading. (3-12) i s from equation f o r the maximum shear considering very large s t r a i n s . , i n the nth cycle of loading as (3-11), = ! ^ / o During i n i t i a l loading when (3-40) no hardening has taken place we 79 obtain (3-19) gives and a comparison with equation B l = /PlK 1 Following a procedure find that a f t e r n tan " (3 s i m i l a r to that used i n d e r i v i n g G and b vp T l 1 mo b. + 1 IZt] 2 P^3) V B -I i s the accumulated volumetric strain to define al(1975) used of the constants A. and B 1 were used to determine s t r a i n hardening. used because G mQ Herein only four hardening and T are determined m 0 the Hardin-Drnevich equations. and 1 a , 2 b^ (3-^3) to constants are independently using The four hardening constants, 2 the G ^ and define the and b , are obtained by f i t t i n g equations to six mo and the remaining four constants were used 1 b, the stress s t r a i n r e l a t i o n s h i p of sand. 1 1 a , and are constants. 2 constants X we 6. + I t should be noted that Martin et Two m n 42) cycles of loading L where again " (3-39) r e s u l t s of constant s t r a i n c y c l i c loading tests. In the case of undrained shear the increment i n pore water pressure, A<7", may w and (3-34). be computed using equations In one load cycle the t o t a l change (3-30) i n effective 80 s t r e s s w i l l be Ac^, and the e f f e c t i v e s t r e s s changes from to cr^ = y0 0 The new l e v e l o f e f f e c t i v e s t r e s s - Acr . <f <r^. w will a l s o a f f e c t the maximum shear modulus, G , and the maximum mn shear s t r e s s , '^ * a p p l i c a b l e t o the next c y c l e o f l o a d i n g . xm It i s difficult to apportion g r a i n s l i p and volume compaction. that the major c o n t r i b u t i o n the hardening between F i n n e t al(1970) to hardening comes c r e a t i o n o f more s t a b l e g r a i n t o g r a i n c o n t a c t s s l i p a t contact points rather than from Although the matter cannot be considered a t t r i b u t e most o f the hardening to conclude t h a t hardening occurs a l s o from due t o increased settled, grain slip the cyclic density. herein, we and we so i n undrained s h e a r . the case o f undrained c y c l i c shear response represented by the p o t e n t i a l v o l u m e t r i c In the g r a i n s l i p is s t r a i n , ^-yp* For undrained sands, t h e r e f o r e , moduli and maximum allowable suggests the maximum shear s t r e s s e s f o r the shear nth cycle are r e l a t e d t o the i n i t i a l v a l u e s by the f o l l o w i n g e q u a t i o n s : mn = "mn VP G. mo 'mo (3-44) 2 FvpJ [^voj aT fc 1 L + K—Z" b i + vp u Vvp v (3-45) 'vo i n which cr^ i s the i n i t i a l v e r t i c a l e f f e c t i v e s t r e s s , and Q is the current vertical effective stress. 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 then g i v e n The ^ resulting by the f o l l o w i n g : 81 r G m n 1 3.4 COMPUTATION OF + SOIL G m . f (3-46) n BEHAVIOR USING THE PROPOSED SOIL MODEL. ~ . In t h i s section the values of stress and s t r a i n and other related variables, e.g., shear modulus, damping, volume compaction and/or pore pressure build-up, etc., based on are computed the equations proposed i n the previous section and comparison i s made, whenever possible, with experimental data. A hypothetical sandy s o i l i s assumed to have the following properties:e * K Q (T y0 With 0.83 ; = 36.0° ; = 0.50 ; = 1600 psf. = these and with the help of equations (3-5) and (3-6) the i n i t i a l maximum shear modulus, G , and i n i t i a l maximum mo stress, T , can be calculated. To calculate the compaction m Q c h a r a c t e r i s t i c s values of 0.400, 0.790, 0.563 used respectively. f o r i j ^ , q> • ^3 a 2 n d and O.730 are These values were obtained based on the discussion given i n section 3.4.1 shear 3.3. Hypothetical Drained C y c l i c Simple Shear Test. In the drained cyclic simple shear test the 82 effective stress i s kept constant. (3-44) and (3-45) can be The values of a , a 1 2 > Consequently, equation reduced to the following!- b^ and b 2 take the values of 0.754, 0,406, 0.550 and 0.500 respectively as discussed i n section 3.3. By using a s t r a i n increment of 0.005$ a drained c y c l i c simple shear test calculated. section The 3.3 are with constant procedures followed. shear s t r a i n amplitude i s and equations The first is well in four s t r e s s - s t r a i n hysteresis loops were computed and the r e s u l t s Figure 3-15. described are shown i n I t can be observed that the e f f e c t of hardening represented by the increasing l e v e l of shear stress required to bring the sample to the same peak s t r a i n at each cycle. The average shear modulus or secant modulus i s also computed f o r various shear s t r a i n l e v e l s and compaction values as shown i n Figure terms of 3-l6. Similar larger 0.5% that relationships loading cycles are shown i n Figure 3-17. shown that at larger strains hardening has e f f e c t on the average 1 cycle while in The data a proportionately shear modulus. For example, at s t r a i n the average shear modulus a f t e r 5 cycles i s after strain twice at 0.05% i t i s only 10% greater. 600. Fig. 3-15 Hyperbolic Stress-Strain Loops Showing Effect of Hardening (Drained Test) .02 Fig. 3-16 .05 0.1 0.2 Cyclic Shear Strain Amplitude (%) Shear Modulus as a function of Cyclic Shear Strain and Volumetric Strain 0.5 Fig. 3-17 Shear Modulus as a function of Cyclic Shear Strain and Number of Cycles CO •Vn 86 These observations indicate the d i f f i c u l t i e s and approximation inherent i n s e l e c t i n g a single shear modulus-strain curve using in dynamic analysis of for earth structures composed of granular materials. The damping r a t i o defined by (see Figure 3-B). D = 1 (i LQ) area of loop APQ ' area of AOAB i s also calculated. v For convenience of evaluating the of the s t r e s s - s t r a i n hysteresis loop the volumetric strain i s assumed to be constant so that area compaction the following expression f o r the damping r a t i o can be obtained D = | [ l + i ] [ l - log (l+Y)A] e G where Y = - -f- (3-50) f -22- (3-51) Sin The variation amplitude, of damping ratio, D, with shear f , at d i f f e r e n t values of compaction strain strain, € , are calculated and the r e s u l t s shown i n Figure 3-18. Pyke(l973) average shear Monterey #0 apparatus. and 3-20, soil modulus Sand presented and obtained damping from data on the v a r i a t i o n of ratio with the Geonor strain simple for shear These experimental data are shown i n Figures 3-19 I t can be seen that the behavior of model, remarkable has namely shown in Figures 3-16 s i m i l a r i t y to the experimental data. the and proposed 3-18 bear FIG. 3-18 Computed Damping Ratios FIG. 3*19 Strain Shear and Modulus as a Function of Total Compaction (after Shear Pyke, 1973) Damping 68 Ratio 90 3.4,2 Hypothetical Undrained C y c l i c Simple Shear Test. In t h i s section the assumed s o i l data the previous section is used. In addition, Seed(1975) obtained by Martin, Finn and mentioned in the power law i s used f o r the one- dimensional rebound s t r e s s - s t r a i n r e l a t i o n s h i p , namely, € where vr = ^Ko^K^ 3-52) ( i s the e l a s t i c (rebound) volumetric s t r a i n , <f VQ i s the past maximum effective v e r t i c a l stress from which unloading i s taking place, <7-y i s the current e f f e c t i v e normal stress, m, n and k 0.0025 respectively. The set of unloading curves given by 3-21. The 0.430, 0.620 and are constants taken to be 2 and, one-dimensional (3-52) i s rebound obtained by d i f f e r e n t i a t i n g equation modulus shown i n Figure can then be (3-52), K vr (cr;) "" 1 The procedures i n similar to those 1 computing used for mk (<r; ) n / 2 an (3-53) m 0 undrained drained cyclic test. There additional step involved i n the c a l c u l a t i o n and calculation (3-30). of test that are i s one is the pore water pressure increments using equation Small s t r a i n increments of 0.005% a re again used. 91 However, model to illustrate during amplitude the undrained cyclic test softening cyclic shear i s chosen. behavior of the s o i l a constant stress The f i r s t f i v e s t r e s s - s t r a i n loops are shown i n Figure 3-22. The decrease i n shear modulus due to r i s e i n pore water pressure i s indicated by the increases i n s t r a i n amplitude generated by of constant amplitude shear representation of the development of successive stress. pore water pressure during c y c l i c loading are shown i n Figure 3-23. progressive increase i n the amplitude of c y c l i c shear s t r a i n and the double curvature i n the development curve, which are strain schematic and The shear A cycles pore water very c h a r a c t e r i s t i c of actual granular s o i l under undrained c y c l i c shear, are also from the model. pressure obtained Recoverable FIG. 3"2I Volumetric Strain - e One Dimensional Unloading (after Martin et al, 1974) vr Curves 93 FIG. 3 - 2 2 Hyperbolic Stress-Strain Loops (Undrained Test) Showing Softening due to Pore Pressure Rise Time FIG. 3~23 (sec) Strain and Pore Pressure during Undrained Cyclic Shear Test 95 CHAPTER 4 MECHANICAL MODEL FOR THE ANALYSIS OF LIQUEFACTION 4.1 INTRODUCTION From the discussion of s o i l l i q u e f a c t i o n i n chapters i t i s seen that the pore water earlier pressure, <r , generated during shaking controls the occurrence and extent of soil failure. analysis of r i s e of pore normal In view of t h i s , i t i s necessary to set up liquefaction pressure, stress, subjected to approximation, in a an p o t e n t i a l to calculate the maximum or maximum reduction in effective saturated horizontal sand deposit when oscillatory liquefaction base motion. As an f a i l u r e i s assumed to take place when the e f f e c t i v e normal stress, o~ = <r^. -0^, y zero, an o i s reduced to or, f o r p r a c t i c a l purposes, to l e s s than f i v e per cent of the i n i t i a l e f f e c t i v e normal stress, 0~ . yQ A complete analysis of l i q u e f a c t i o n should include the following processes, namely, 1. Dynamic response to base motion, 2. Increase i n volumetric compaction s t r a i n due to c y c l i c straining, 3. Pore water pressure generation as a r e s u l t of (2), and, 4. D i s s i p a t i o n of excess pore water pressure(consolidation). There are two main sources of n o n - l i n e a r i t y i n the 96 proposed analysis of l i q u e f a c t i o n : the n o n - l i n e a r i t y of the s o i l s t r e s s - s t r a i n r e l a t i o n s h i p and that which a r i s e s from the coupling of the above-mentioned four processes. of the complex evident coupling of the processes, As a r e s u l t which will be i n the following sections when the whole problem i s formulated, numerical procedures solution of the equations equations f o r numerical have to be obtained. computations used f o r the For convenience, will be derived immediately a f t e r each 'exact' equation i s obtained. 4.2 FORMULATION OF THE PROBLEM. A level sand extent so that propagation can be deposit i s assumed to be o f i n f i n i t e of shear waves through the treated as a one-dimensional problem. without any confined loss to a of generalization Consequently, considerations can be s o i l column of unit cross-section through the deposit, as shown i n figure 4-1. i s assumed deposit to be completely S o i l under the water table saturated so that when flow of water occurs as a r e s u l t of s o i l compaction the d i r e c t i o n of flow i s upward towards the ground surface. Governing equations and relevant s o i l parameters w i l l be introduced and discussed i n the following sub-sections. 4.2.1 Dynamic Response to Base Motion Consider a s o i l element of thickness dz located at height z from the base of the deposit as shown i n Figure Let 4-1. x be the horizontal displacement, 1 the horizontal shear FIG. 4-1 Idealization of the Response Problem 98 s t r e s s and y the t o t a l mass density Dynamic e q u i l i b r i u m r e q u i r e s 2 p . d z . 3t of the soil element. that, ||.dz.l 0 , or, || . For visco-elastic .o - materials in (4-x) which the s t r a i n and s t r a i n r a t e e f f e c t s are a d d i t i v e and/ non-coupled the stress-strain r e l a t i o n s h i p can be r e p r e s e n t e d by, T where In = f^iT) + (4-2) f (f) 2 f i s the shear jr i s the shear s t r a i n r a t e , and f^ and f strain, are f u n c t i o n s 2 of f and the case where the s o i l p r o p e r t i e s e q u a t i o n (4-2) Since if = can be s u b s t i t u t e d i f into f respectively. are c o n s t a n t w i t h depth (4-1) d i r e c t l y to give, , dZ 1£ = 2 * .sir _ A^ dz 9z t[Ct) f '(r) 2 2 = (4-4) and , and w r i t i n g -$r> lCf) f , and (4-5) = ^.f (/) 2 , 99 Equation (4-3) can be rewritten i n the following form by using the r e l a t i o n s given by (4-4) and (4-5) »- px 2 2 - f' l y i ) - f ' 2_(x) = 0 dz J t t f^=0 and ^2~ * I t can be seen that when the shear modulus, (4-6) 3z G the f a m i l i a r w equation n e r e G represents f o r the dynamic response of an undamped l i n e a r shear beam r e s u l t s , namely, j>x - G.^-J(X) = 0 (4-7) In general, the s o i l properties are functions of depth, (H-z), so that 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 given by equation (4-2) has to be replaced by a more general equation which contains s o i l parameters as functions of depth also. equation similar to In t h i s case (4-6) cannot be obtained i n a straight forward manner since the p a r t i a l d i f f e r e n t i a t i o n in equation (4-3) would involved advantageous to apply numerical (4-1), finite an carried more terms. approximations out I t i s more to equations I t i s shown i n Appendix I that through the use of the difference method equation (4-1) can be d i s c r e t i z e d i n the z-domain to give, m. r [M](X} + [C]{X) + [K]{X> = - .'X (4-8) b n so that a lumped mass system as shown i n Figure L m used f o r approximate analysis. matrices given i n the Appendix I 4-2 can be The components of the are rewritten here f o r 100 m, n "'I "•^j'a&5S.'^5'iS^~ m; — — — — — — — — X- hi »i k c i wmmmmmm. m-. h l-c ,k 3 3 3 2 h 2 ^2* K 2 Layer I FIG. 4 - 2 Approximation Mass System by Lumped 101 convenience :[M] = r m 0 1 0 m 0 0 0 0 2 0 " 0 0 * • • (4-9a) • • o nj [c] = ' l 2 C 0 -c + C c 0 2 3 +c 0 0 -3 3 4 c c +c 0 • • • • • . -c n • • • « • • 0 ' 0 0 • • • -k n k <* [K] = V 2 "2 0 0 "2 2 3 k k k k 0 0 +k • § 0 where fyi-* i i = n c i b = fA'ti 1 = foU* C I t should be 1-2 h + b h A ) .b,/(h. 1 1 1 1 pointed out that and width of the i - t h layer C- j,) are 1-2 1-T2 values 2 n - 1 , and . 1—2 b^ p. i , J fo(fi » (4-9c) 1-2 h^ and while (4-9b) nj i = 1 .f, i ) 1-2 1—2 c %+i i+l i+l i . z . i).b./(h..f. i) 1—2 0 0 0 • 1-2 are the thickness f , ( f - i,z- i ) 1 1-2 and 1-2 referred to the centre of the 1-th 102 layer. 4.2,2 Hyperbolic Stress-Strain Relationship, f^, and Material Damping Values, f . 1 I t was suggested element under 3 that for a soil simple shear condition, such as i n the present case of a horizontal s o i l from i n chapter deposit transmitting shear waves the base to the surface, a hyperbolic hysteretic s t r e s s - s t r a i n representation i s a close approximation to the actual stress-strain. Consequently, the, f g function described i n section 4.2.1 takes the form of equation (3-46) together the Masing behavior given i n section with 3.3.1.1. When v i b r a t i o n a l energy i s being transmitted through a material medium a portion of i t i s dissipated i n t e r n a l l y due to a number of mechanisms at the grain l e v e l , e.g. Coulomb f r i c t i o n , p l a s t i c deformation, This d i s s i p a t i o n ^ of energy viscosity, which dislocation, etc. causes a decrease i n the amplitude of v i b r a t i o n i s generally c a l l e d material damping or i n t e r n a l damping. into two main Generally, material damping i s categories - a frequency dependent type and a frequency independent type. equation (4-2) over one I n c i d e n t a l l y , i f one strain s t r e s s - s t r a i n loop i s s i m i l a r to Figure 3-3a classified and cycle, the loop assuming AB integrates that the as shown in that the i n t e g r a l s involved are integrable, one should be able to obtain the energy dissipated during s t r a i n cycle ABA, W d , as the 103 JX .df or, = yqch.df w (ir,f) = d Equation (4-10) jf (f).df + 2 w'(ir) + vq(0 (4-2) can be (4-n) used to describe a simple l i n e a r v i s c o - e l a s t i c material, e.g., a Kelvin s o l i d where, and fAf) 1 = c.'f , f (jf) = K.r . 2 (4-12) where C and K are the v i s c o - e l a s t i c constants. For harmonic v i b r a t i o n , f=Asin«/t, the energy dissipated per cycle, W, can d (4-10). be obtained from equation Since, M j f^'f) .df = Jc.f.r. dt o 2 j and f ( T ) .df 2 giving It W d can be = 0 , = C7TWA 2 of f^ ( 1 ^_ 1 3 ) seen that the energy dissipated per cycle, or the damping depends on the frequency case ^ where the w . For this particular damping stress i s l i n e a r with s t r a i n rate, the term viscous damping applies. For f r i c t i o n a l materials, constitutive equation (4-2) t^'f) = 0, the components of can be written as follows :- the 104 fM) ^ f (f) 2 where f = +f for df>0 = y -f for df<0 i s a p o s i t i v e quantity. cycle can "be obtained w d The energy dissipated per s i m i l a r l y by (4-10) to give, = f = 4.f .A This type of damping f (n.df 2 which (4-i4) is called Coulomb Damping, is frequency independent. It the can be observed from -6he above two examples that constitutive frequency relation dependent dependent stress independent (4-2) damping, component damping, or is can for be used which the responsible, structural to include strain-rate and frequency damping, which associated with the non-linear hysteretic s t r e s s - s t r a i n s t a t i c loading. • >- is under • In the equivalent l i n e a r analysis where a non-linear hysteretic material i s represented by a l i n e a r v i s c o - e l a s t i c model, damping has to be introduced a r t i f i c i a l l y through W W" vanishes. factor, C eq , In t h i s case has to be the equivalent obtained by viscous equating as damping the energy dissipated per cycle i n the non-linear material and i t s l i n e a r equivalent, so that, 105 In a t r u l y non-linear analysis, as i s i n the present analysis, where the hysteretic s t r e s s - s t r a i n relationship i s used, a r t i f i c i a l viscous damping c o e f f i c i e n t given is no (4-15) "by longer needed, instead, the function f^ can be used to describe the actual viscous material. Or, in a damping effect present in the fashion s i m i l a r to that of the method using equivalent viscous damping, i t can be used to compensate for the the i n s u f f i c i e n t damping caused by actual stress-strain W (f,T) the uncoupling of one though d this approximation by the function f^(f). into W Note that and W" i s not a necessary uncoupled determination form the of first damping type values. of damping can be obtained through v i s c o s i t y measurements while the l a t t e r can be of i t brings about convenience i n discussion as well as i n the experimental In the obtained by testing the type material i n question i n a quasi-static manner. For the present purposes, i s assumed to be of the l i n e a r viscous type, i . e . , f (f) x to = Cf (4-16) take into account of the damping e f f e c t caused by the flow of water occurring as a r e s u l t of the deformation skeleton, see e.g. Biot(1956). measure the v i s c o s i t y of saturated Yen(1967) sands near of the attempted to liquefaction. However, i t i s not clear whether the v i s c o s i t y values referred soil obtained to the actual strain-rate dependent stress component 106 or to the ' a r t i f i c i a l ' equivalent viscous damping. 4.2.3 Volume Compaction and Pore Pressure Figure 4-3b shows in chapter 3. proposed represents the s o i l container the material The model that spring-and-slip while the fluid was element inside the represents the pore water of the granular material. I t i s assumed composed skeleton Generation. that of two component, the t o t a l components, volumetric namely, and the compaction strain, the rebound (slip) £ v , is (elastic) component, ^vp» i.e., € = v Incremental volumetric € + vr € vp (4-1?) . " compaction s t r a i n , &€.y^> i s given by the following expression, e vp = Ae +1- ( f a "*2V + * r 3 a 2 +^ (4vi8) . where f shear s t r a i n l e v e l reached, i n the l a s t s t r a i n cycle (or h a l f Q i s the amplitude of shear s t r a i n , s t r a i n cycle) and the vj/s are s o i l constants cyclic simple shear or t r i a x i a l t e s t s . or the maximum determined In the case of dry granular material an a p p l i c a t i o n of a shear s t r a i n cycle amplitude with f w i l l cause the t o t a l volumetric s t r a i n to change by an amount s t r a i n , AE^, from equal to that of the volumetric compaction since there i s n_o change i n the e l a s t i c component when the e f f e c t i v e normal stays constant, i . e . , Grd. Surface MR 02 0- w + ^dZ| oz dz / / / / l i t Base Rock (a) FIG. 4 - 3 , (b) Consolidation Model 108 vr A € * ' • 0 (4-19) and A£y I f the s o i l is = vp saturated instantaneously, and relaxation w i l l take place. vertical &. The stress is drainage of result cannot take place the e l a s t i c s t r a i n component is that a portion of the transferred to the pore water so that an increment of pore water pressure, A(T , r occurs, as explained i n chapter 3 . A<T W where K< lr ir' A € vp ( 4 - 2 0 ) soil D e t a i l discussion of the volumetric s t r a i n caused b,y c y c l i c loading 4.2.4 K i s the one-dimensional'rebound modulus of the skeleton. repeated - is given in section 3.3.2 and is not here. Reconsolidation. The pressure compaction equation generation can as for a reconsolidation with pore water result of internal volumetric be derived i n the same manner as that f o r the conventional consolidation equation. The assumptions used i n the d e r i v a t i o n are i 1. Complete saturation. 2. Negligible compressibility of pore water. 3. One 4. One dimensional flow of water. dimensional compression. 109 5. V a l i d i t y of Darcy Law. Consider a section as shown typical soil i n figure 4-3a, element the unit cross- where v i s the v e l o c i t y of flow of water into the s o i l element. flow, of For the continuity of difference i n volume of flow of water into and out of the element should equal the volume change of the element i t s e l f , so that we have, v + & z - vldt £ = f^.dz.dt - U <*-> 2 L I f the flow s a t i s f i e s Darcy's Law, v where k = -k i s the c o e f f i c i e n t of permeability, f i s the unit weight of water , and w tj i s the excess pore water pressure. W If the expression f o r v i s substituted into equation the following equation i s obtained, W From equation Since, (4-17) i t can Ae^ = jp-.A<r lr v be seen that, (4-21) 110 = V - (A(T - ACT) (4-25) lr During the l i q u e f a c t i o n of a horizontal s o i l deposit the t o t a l overburden pressure i s not changed, i . e . , (4-23) so that equations (4-26) , (4-24) , (4-27) and can be combined to give, +K When the coefficient of permeability, depth and i f ( i « / ^ ) K k r -^' w i s denoted by (4 - 28) k, i s constant with c , y equation (4-28) t can be s i m p l i f i e d to, •Wvt ^vr. (4-28) Equation or (4-29) i s the desired equation governing the d i s s i p a t i o n of excess pore water pressure with pore water compaction. compaction €^ =0, pressure generation due I t can be seen that by strain function continuous to i n t e r n a l volumetric setting the volumetric i d e n t i c a l l y equal to zero, i . e . , the well known Terzaghi's Equation of Consolidation i s obtained. Inspite of the extra term i n equation (4-29), i t is s t i l l a t y p i c a l parabolic p a r t i a l d i f f e r e n t i a l equation i n one Ill dimension and i s i d e n t i c a l i n form to that f o r the of conduction heat i n a bar with heat sources d i s t r i b u t e d along the bar. For example, Sneddon(1957) gave the following equation f o r the conduction of heat i n s o l i d s , = 3t where Q (cv 2 i s the temperature k (4-30) H(r,e,t) + 0 j>c at a point with p o s i t i o n vector r, i s the thermal conductivity, j). i s the mass density of the material, c i s the s p e c i f i c heat of the material, and, K = k/(pc). The function H(f,0,t) gives the generated per unit volume per unit time. gives us the conduction. rate This of is amount Physically r i s e of temperature quite consistent of heat (H/j)c) when there i s no with the physical meaning implied i n the l a s t term of equation (4-29), namely, the rate of r i s e of pore pressure when there i s no drainage! Note that with the help of equation (4-18) the equation that (4-28) the expressed or internal as a Tf^ volumetric in compaction strain can be function of the t o t a l number of shear s t r a i n . method soil element has been The f i n i t e difference scheme f o r equation (4-29) i s derived i n Appendix integration term (4-29) can be t h e o r e t i c a l l y evaluated so cycles (or h a l f cycles) to which the subjected, last III. When i s used to determine a step by step the dynamic response of the s o i l deposit the shear strains can be computed f o r a l l 112 the layers f o r each time increment generation and d i s s i p a t i o n can he and the pore pressure incorporated. Discussion of the solution procedures i s g i v e n i n the next section, v 4.3 SOLUTION SCHEME. A numerical scheme using a step by step method i s used to solve the governing equations. since [K] i s a In (4-8) equation function of the displacement vector, {X}, a d i r e c t integration procedure i s used so that the displacement, v e l o c i t y and acceleration values at any time t can be to values at (t-At), where At i s the time step used i n the d i r e c t integration procedure. numerical integration There are several methods f o r of equations s i m i l a r to equation (4-8) and a comparison of the methods based on t h e i r accuracy present has Newmark's used i n the discussed i n Appendix I I . volumetric method(1959) of step compaction The actual Also, function, particular For the by step equations value every time the i n equation € shear s t r a i n , , f , In the as increase a in associated with the n 3-14. (4-29), i s handled same layer decreases from a maximum value, f„max „, J and computation are derived and discontinuous function, which has an incremental Figure and was chosen f o r i t s unconditional s t a b i l i t y i n the analysis of l i n e a r systems. procedures stability been given by Bathe and Wilson(1973). prupose integration related as shown in t h i s case the f i n i t e difference method of solution becomes quite s u i t a b l e . The relevant equations and 113 expressions arising from the use of the f i n i t e difference method of solution f o r equation (4-29)t or equation (4-28) f o r the more general case, are given i n Appendix I I I . A b r i e f outline of the step by step solution scheme can be given as follows :1. on the current values of f, Based and <r at time t , y the shear modulus i s calculated as described i n section 4.3.2. 2. The matrix [K] i n equation (4-8) i s then updated. 3. With the base acceleration value at (t+At) new values f o r {X} , {X} , {X} and {f} can be calculated using a direct integration method. 4. When a maximum layer has i t s shear s t r a i n decreasing from the value incremented i t s volumetric according compaction to the equations strain is described i n section 4.2.3. 5. Current value of pore water pressure, j * , w using the f i n i t e difference method are as calculated discussed i n Appendix I I I . 6. Current e f f e c t i v e normal stresses, g-^, are then calculated "by 0~ - (7"-(7^, f o r use i n the next time step. v v A computer program has been written solution 4-5. based on this scheme and the o v e r a l l flow chart i s shown i n figure A l i s t i n g of the computer program i s given i n Appendix 114 READ IN DATA NUMERICAL INTEGRATION FOR X, X, X & f ETC. PORE PRESSURE, VOLUME COMPACTION, IP REQD. CHANGE SHEAR MODULUS & OTHER PARAMETERS. RECONSCNIDATION CALCUL/LTION PRINT OUTPUT &/0R TAPE OUTPUT FOR PLOTTING. MORE [NCREMENTS ?. YES NO FIG. 4 - 4 Flow Chart 115 CHAPTER 5 APPLICATION OF THE MECHANICAL MODEL - AN ANALYSIS OF LIQUEFACTION OF A SOIL DEPOSIT 5.1 A HYPOTHETICAL SOIL DEPOSIT The method of analysis developed i n e a r l i e r chapters was applied to a hypothetical s o i l deposit described section. The equivalent Assumed same deposit was also analysed by using the l i n e a r analysis so that values f o r the in this comparison can be deposit are as shown i n Figure The v a r i a t i o n of r e l a t i v e density with depth was made to that of the Niigata deposit analysed by Seed et that the problem this thesis 5-1. similar al(l96l) so i s as close to r e a l s i t u a t i o n as possible. However, mention should be made that i t i s not of made. the intention to project the r e s u l t of the analysis to the actual f i e l d condition at N i i g a t a . The maximum and minimum void r a t i o s were assumed be 1.00 and 0.50 so that the void r a t i o s at various r e l a t i v e density can be calculated. For convenience, the bulk density of the s o i l above water table, ground surface which was 5*0 feet beneath i n t h i s case, was assumed to be 122.0 pcf while the buoyant unit weight was assumed to be 60.0 pcf. deposit i n order was to sub-divided into a p p l i c a t i o n of the previously 14 layers described method of The s o i l that analysis Ground Surface o Water Table ===== "T^5J0 ft or 50 SAND 200.0ft T T = 1 2 buoy. 2 s PCf 0 6 0 P 0 100 H a. c f UJ a 150 ' ^ Bedrock ^ ^ ^ ' ^ ^ 200 ^ 1 (a) FIG. 5-1 Properties of a Soil Deposit could be made. Table 5-1. up. The properties of each layer are as shown i n Notice the numbering of the layers i s from bottom E f f e c t i v e shearing a n g l e w a s assumed to the c o e f f i c i e n t of earth pressure at r e s t , K , be was Q 30 and calculated from an equation given by Blshop(1958). K 1. = Q The i n i t i a l mean e f f e c t i v e stress e f f e c t i v e v e r t i c a l stress, <r 0 the use of obtained. system , as (5-2) and shown Column 9 gives calculated based the and (3-6) initial the initial maximum shear i n columns 7 and 8 of Table 5-1 the masses of the were lumped mass on a unit cross-section s o i l column while column 10 gives the i n i t i a l spring s t i f f n e s s e s . of damping values i s discussed i n section 5.2 from , by, v equations (3-5) mo m Q calculated 0 maximum shear modulus, G , stress,t <r£ was ^.(l.+2.K ) .<4 <T = Through (5-1) - s i n d)' Choice 5*2. RESPONSE CHARACTERISTICS OF THE SOIL DEPOSIT. In order to investigate the steady-state response of the s o i l deposit to harmonic base motion through the ij> this proposed obvious step-by-step use of non-linear analysis the volume change parameter used i n equation is the (4-29) was set to zero. The reason for since transient response i s included i n the integration technique, steady-state vibration TABLE 5-1 • VALUES FOR LATER APPROXIMATION TO SOIL DEPOSIT LATER NO. THICKNESS ft. DEPTH TO CTR ft. EFF VERT STRESS psf D r 14 5.0 2.5 50. 7-5 15.0 305.0 760.0 13 12 5.0 10.0 1210.0 50. 11 25.0 1810.0 10 10.0 10.0 35.0 2410.0 9 8 10.0 10.0 45.0 55.0 7 6 10.0 20.0 5 G mo kip/ft T _ m. K mo psf i slug 1 kip/ft 0 594. 938. 85. 212. 9.5 19.0 118.8 338. 506. 28.4 50. 1184. 1448. 1733. 2010. 674. 3010.0 55. 60. 37.9 37.9 118.4 144.8 841. 37.9 173.3 201.0 65. 70. 2283. 1009. 37.9 228.3 65.0 3610.0 4210.0 2557. 37.9 80.0 5110.0 2923. 20.0 100.0 6310.0 75. 80. 1177. 1428. 3369. 1764. 4 20.0 120.0 7510.0 85. 2099. 3 2 20.0 20.0 30.0 HO.O 160.0 185.0 8710.0 9910.0 11410.0 85. 3813. 4106. 56.9 75.8 75.8 255.7 146.1 190.6 2434. 75.8 85. 4380. 2770. 85. 4700. 3189. 75.8 94.8 1 % 50. 187.6 190.6 205.3 219.0 156.7 119 occurs only a f t e r a few cycles of v i b r a t i o n and i f hardening or softening, i . e . change i n maximum shear modulus, maximum shear stress, T^, f a r as m n Moreover, to minimize possible the duration of the transient v i b r a t i o n , the amplitude of base acceleration input was made to gradually increase i n the f i r s t few cycles to the required l e v e l and kept constant, thereafter. that and i s allowed f o r every cycle the steady state would never be achieved. as G » After a few t r i a l s i t was found i f the amplitude of base acceleration i s made to r i s e as a sine function within the f i r s t two and h a l f cycles transient v i b r a t i o n has i t s least linear system influence. The response of non- depends not only on the frequency of the input loading but also on the magnitude. Consequently, complete response c h a r a c t e r i s t i c s requires a l o t of computer runs. view of t h i s , an a r b i t r a r i l y fixed base acceleration l e v e l of 0.065g was used. A time acceleration plot f o r the base input i s as shown i n Figure 5-2. Even though the hysteretic quite cent, soil model (3-30), at addition, due inside the s o i l to the stepwise-llnear structure. nature integration procedure a small amount of viscous points. 0.10 per s t r a i n l e v e l s around viscous damping i s s t i l l needed to take into account of the e f f e c t of the flow of water needed contributes large damping, e.g. 20 to 30% i n terms of damping r a t i o defined by equation In In of the damping is to eliminate high frequency v i b r a t i o n at the reversal Small amount of viscous damping was added through f n Time FIG. 5 - 2 Input Base (sees) Acceleration 121 defined i n equation viscous (4-2). s t i f f n e s s values, k c a i ft are i t proportional a value of vibrations given 0.0008 was used surface acceleration was obtained 0.002 by equation gives while p =0.002 It should be noted that changed but the high at r e v e r s a l points are eliminated. 2 %. for the By viscous (4-24) i t can be shown that a /$ value of 0.0008 gives approximately while initial (5-3) c a l c u l a t i n g the energy dissipated per cycle as the k the amplitude of response i s s l i g h t l y force to jS- i,t=0 88 response as i n Figure 5-3b. frequency linear t=0* namely, ± response as shown i n Figure 5-3& gave that damping can be used and that the value of the viscous damping c o e f f i c i e n t s , c When Also i t was assumed a damping ratio of 0.6% For a l l analyses described i n t h i s chapter a ^5 value of 0.002 was used. Step-by-step i n t e g r a t i o n procedures were c a r r i e d out for at least 10 cycles of v i b r a t i o n and i t was found most cases steady-state response could be reached. 3b shows a typical acceleration. time of the acceleration Figure 5-5. Figure 5- of the surface Twelve frequencies were used and amplitude to that of computed for each frequency. in response in A t y p i c a l s t r e s s - s t r a i n response of a layer i s as shown i n Figure 5-4. ratio history that of the steady-state the base acceleration The r e s u l t obtained i s the surface Input was plotted For comparison purposes, the "SHAKE" program 122 cu u o> in c o Q> U U < Time u c o ^ o 2. f\ /I /I /I /I /I 0 v. 0) O u (sees) -Z VJ VI -4. _L 2. Time FIG. 5 - 3 VI VI VI VJ 3. 4. (sees) Surface Acceleration Response 123 .RESPONSE OF HORIZONTAL SOIL DEPOSIT TO HARMONIC BASE ACC. STRESS-STRAIN FOR LAYER 12 0.04 0 0.02 Shear FIG. 5-4 0.02 0.04 Strain (%) Stress - Strain Response Steody State Surface Acceleration Base Acceleration 125 written by Schnabel et al(1972) was also used and i s plotted i n the same f i g u r e . the result It can be seen that while the equivalent l i n e a r analysis used i n the "SHAKE" program gives a pronounced peak at the resonant frequency, the non-linear analysis gives a much less pronounced peak i n the response curve. A frequency- plausible explanation i s that when both the equivalent l i n e a r and the non-linear models are to the same s t r a i n l e v e l , the former has a larger amount of s t r a i n energy which i s responsible curve. subjected Apart from this, f o r the peaked the two methods give close a c c e l e r a t i o n r a t i o within the frequency response reasonably range of 2 to 10 hertz. A t y p i c a l surface acceleration by the "SHAKE" program response calculated i s shown i n Figure 5-6. Comparing with Figure 5-3, i t can be observed that the equivalent linear analysis gives the usual s i n u s o i d a l surface a c c e l e r a t i o n while in the non-linear case the acceleration response i s no sinusoidal. Jennings(1964) method to obtain the response system with non-linear used of a a numerical a obtained sinusoidal one. by interesting integration single-degree-of-freedom hysteretic force-deflection r e l a t i o n s h i p and observed that the a c c e l e r a t i o n not longer response was This i s i n agreement with the r e s u l t the proposed non-linear analysis. It is to note from Figure 5-3 and Figure 5-6 that both the response computed by the 'SHAKE' program and that computed by the proposed non-linear analysis have more or less the same 127 phase s h i f t with respect to the input acceleration wave form. 5.3 MAGNITUDE OF PORE PRESSURE RISE AND ITS EFFECT ON THE DYNAMIC RESPONSE OF THE DEPOSIT. In this used as the previous section base input section. a simulated earthquake record to The the deposit described in was the acceleration values were calculated from a random acceleration function suggested by Bogdanoff, Goldberg and Bernard(I96I), namely, J x (t) = • .1. jjt.exp(-5jt).cos(wjt+ej) g where j^y ^ l w < S^i 2 w t>0 (5-4) are r e a l p o s i t i v e numbers, < 3 w < 4 w ' • • » J » W 62t Ojt • • » t 0j » a n d are real random variables uniformly d i s t r i b u t e d over the i n t e r v a l 0 to 27T. A large number of terms can be used when of an accelerogram is needed. best representation In a study of the seismic response of structure-foundation systems Parmelee et followed the and al(l968) suggestion of Bogdanoff et al(196l) and assumed to accelerogram. function used was be constant Ten terms when were generating used and a simulated the acceleration given by, 10 x ( t ) = O.50.t.exp(-0.333t).£ cos(wjt+elj) g 1 in which the frequencies, Wj, and phase angles, in Table 5-2. t>0 8-j, are (5-5) shown In t h i s section, the same accelerogram scaled 128 to various values of maximum acceleration Figure 5-7 analyses. was used shows a plot of the accelerogram. can be seen that a f t e r 15 seconds the amplitude of relatively for - a l l It motion small so that the analyses can be stopped at is this time without losing any s i g n i f i c a n t data. For the I n i t i a l parameter, was comparison could be 5-8a Figure still made shows computer the kept run, Identically with results using the computer volume change zero so that a obtained otherwise. surface acceleration obtained by the non-linear analysis while Figure 5-8b by the shows the same program "SHAKE". obtained In both cases the acceleration values of the base motion were scaled so that the maximum acceleration was point 0.065g. Though direct both i s no responses. material softening the be present Thus, so f a r , i t can be seen when there hardening due to compaction nor material due to r i s e i n pore water pressure, both the method of d i r e c t integration of the and to comparison of the surface acceleration responses i s not possible the same general c h a r a c t e r i s t i c s seem to in point method of wave non-linear equations propagation with strain-compatible material constants give more or less the harmonic as well as to random e x c i t a t i o n . show the typical strain dynamical and stress same response to Figure 5-9a and b response of a layer obtained by the non-linear a n a l y s i s . In order that the pore pressure generated during shaking might be accounted f o r the respective «J/. values for FIG. 5 ~ 7 Simulated Earthquake Accelerogram TABLE 5-2 FREQUENCIES AND PHASE ANGLES FOR THE SIMULATED EARTHQUAKE ACCELEROGRAM 3 UK (rad/sec) 8. (radians) 1 6.00 3.7663 2 8.00 1.3422 «J 3 10.00 4.8253 4 0.2528 5 11.15 12.30 6> 13.25 1.8834 7 14.15 1.3320 8 16.20 1.7852 9 17.35 0.1517 10 19.15 2.4881 4.5204 TABLE 5-3 %-VALUES FOR DIFFERENT LAYERS Layer No. Relative Density it) +1 14 50. 0.0000 13 50. 0.5200 12 50. 0.5200 11 50. 0.5200 10 55. 0.4727 9 60. 0.4333 8 65. 0,4000 7 70. 0.3714 6 75. 0.3467 5 80. 0.3250 4 85. 0.3059 3 85. 0.3059 2 85. 0.3059 1 85. 0.3059 131 Time FIG. 5-8 (sec.) Surface Acceleration Response (no Pore Pressure ) 132 each layer were calculated based on the Inverse proportional r e l a t i o n s h i p between 4^ i n section 3.3.2 soil deposit Figure 3-11, Since a n d r e l a t i v e density, D , as discussed r of chapter 3. has Assuming that the present the volume change c h a r a c t e r i s t i c s given by the vj/^ values l i s t e d i n Table 5-3 are obtained. the top layer (layer 14) i s above the water table i t s volume change parameter \\>^ was set to zero so that excess pore water pressure due to compaction before, the acceleration maximum of were 0.065g was could not be generated. values obtained. As were scaled down so that a In t h i s analysis the layers assumed to be separated by impervious boundaries and r e - d i s t r i b u t i o n of pore pressure was not allowed. As previously discussed the integration procedures were carried out up to 15 seconds. and Figure 5-10a and b show "the surface acceleration displacement responses and Figure 5-Ha, b and c show the s t r a i n , stress and pore water pressure of the 11th layer which i s situated between depths of 20.0 to 30.0 feet. shows a s t r e s s - s t r a i n plot f o r the 11th l a y e r . which liquefaction occurred marked i n a l l the response pressure The time 5-12 at was around 8.5 second and i t i s plots. From Figure 5-8a and 5-10a the pore Figure r i s e s during i t can be the seen liquefaction surface acceleration tends to be attenuated. The that as process relatively large acceleration response that occurs at a time of 9 seconds when pore pressure i s not generated i s not seen i n Figure 5- 10a where pore water pressure i s taken into account. As 133 FIG. 5 - 9 Strain and Stress Responses for Layer II (no Pore Pressure) FIG. 5-10 Surface Acceleration and Displacement Response (Pore Pressure Generated) 0.6 — . — [ T ..., , I , , , 1 — i 1 0.2 r i i J\\ *A A ~ — — - \ A/ \/ v~ •A 0.4 1 0 c to — -0.2 (a) -0.4 -0.6 400.h 1 T 1 1 I I 1 1 1 1 1 1 1 1 r 1 1 1 r I 1 1 1 : 1 1 1 T — | — i — i — i — r T 1 1 1 I I L 1 T r ^2000. a. CO CO £ 1000. Q_ <P i— O (c) 0. 5. Time (sec) FIG. 5-11 Stress, Strain and Pore Response 15. 10. for Layer 11 Pressure FIG. 5-12 Stress - Strain Response for Layer II (Pore Pressure Generated ) 137 expected the acceleration response within the f i r s t three seconds was not much affected by the Increase i n pore pressure which as shown i n Figure 5-13 did not r i s e to a high value within t h i s time i n t e r v a l . 5-9a and 5-Ha shows the dramatic e f f e c t of Figures the pore pressure r i s e on the layer. s t r a i n response the 11th In the case where no pore pressure was generated the peak s t r a i n response was +0.05$ same of peak at around 2 seconds. was also obtained i n the case where pore pressure was generated. In the l a t t e r case, however, the +0.05$ s t r a i n response at around 2 seconds was dwarfed by the s t r a i n at 8 seconds, a shown short i n Figure 5 - l l a . the input base maximum strain some doubt on the analysis before increasing acceleration response are decreasing. the time +0.25$ liquefaction, as The softening behaviour of the s o i l l s well i l l u s t r a t e d by the when The strain as well response even as the stress Moreover, the large difference i n levels developed i n the two cases casts validity of using an equivalent f o r the study linear of s o i l l i q u e f a c t i o n as was done by Seed and I d r l s s ( I 9 6 7 ) . The offset of the s t r a i n response of the 11th layer, and hence the surface displacement response, the to one side of zero axis a f t e r l i q u e f a c t i o n i s probably due to the s l i g h t non-symmetry of the input base motion. stress-strain response Figure 5-12 shows the of the 11th layer and i t can be seen that even a f t e r l i q u e f a c t i o n the stress and s t r a i n were still 138 describing the describe. reveals kind Close of loops which examination of they were supposed to the stress-strain plot the s l i g h t one-sidedness of the stress response a f t e r 9 seconds. The reason i s that at such low l e v e l of confining stress the f a i l u r e stress i s very small so that a small stress increment near t h e . f a i l u r e stress can induce a very large strain. 5-13 Figure shows the d i s t r i b u t i o n of pore pressure in the 200-foot deep s o i l pore water pressure was deposit. Initially, an increasing the excess function of depth. However, as time went on the s o i l layers situated and. .50 feet pressure. strains depth interval Consequently, at later water pressures. liquefaction between 10 began to develop higher excess these layers developed larger times which led to even higher excess pore The interaction occurred in the seemed 11th to go on until layer which i s situated between 20 and 30 feet. 5.4 EFFECT OF PORE PRESSURE DISSIPATION. To dissipation investigate on the effect of pore pressure the l i q u e f a c t i o n process the complete coupled analysis, i . e . including dyanmic response, pore water pressure generation and d i s s i p a t i o n , was performed on the s o i l described earlier. deposit The simulated earthquake input described i n section 5*3 of t h i s chapter was again used. Analyses were carried out using three d i f f e r e n t values of the permeability, Pore Pressure (psf) (Note •• values for I. sec. curve should be divided by I0<) FIG. 5-13 Pore Pressure Distribution at Various Times (no Dissipation) 140 k. In each analysis, the value of the permeability was assumed to be constant f o r the entire depth Even though the permeability capability computer values was not program that vary used since from a matter of the set layer deposit. up to accept to layer the the purpose was to obtain a general idea on the influence of As was of pore pressure dissipation. f a c t , such investigation i s very l i k e l y the f i r s t of i t s kind carried out to date(1975). Terzaghi permeability and values encountered i n analyse three Peck(l96?) for engineering different approximately types practice. cases 0.000003, 0.0003 and 0.03 these various listed feet with per correspond a of soil It was to of that are decided permeability second range values of respectively and impervious fine nearly sands, medium permeable sands and very permeable clean From here onwards these these cases are sands. three cases w i l l be referred to as case D l , D2 and D3 r e s p e c t i v e l y . for to shown Pore pressure in Figure distribution 5-l4a, b and c respectively. Without pore pressure liquefied at around t h i s chapter. When was soil layer dissipation of pore water pressure allowed but with a very low value of permeability, as of liquefaction agrees with the 11 8 second as described i n section 5«3 of case Dl, l i q u e f a c t i o n occurred at zone dissipation, was boundary the 12th s h i f t e d upwards. conditions of the layer, in i . e . the This behaviour problem since Pore 0 Pressure IOOO. (psf) 2000. Pore Pressure 0 1000. \ \• I 50. • / / • 100 100. I50.h I50.h (psf) 2000. • I / • Pore Pressure .0 1000. (psf) 2000. • • . / / / / 1/ • • / // • • • 3 CD 01 CO CD O CO 200 200. 6 (b) (a) FIG. 5-14 200. Pore Pressure Distribution (c) for Different k Values 142 drainage can only take place at the top surface. When the permeability value of the s o i l was made larger, as i n case D2, liquefaction d i d not occur distributions second respectively. similar pore pressure occurred at much l a t e r time, as shown by the 815-second and though curves in Figures 5-l4a and When the permeability value was made 0.03 feet per second the pore pressure f o r s o i l s t r a t a situated depths 0 deposit between and 100 feet a c t u a l l y decreased during the i n t e r v a l of time 8 to 15 seconds. statements b could be In general terms the following made i n r e l a t i o n to the p a r t i c u l a r s o i l analysed. 1. Relatively low permeability values per second) tends to s h i f t (around 0.000003 feet the zone of l i q u e f a c t i o n upwards and d i s s i p a t i o n of pore pressure does not improve the resistance to l i q u e f a c t i o n failure of the soil deposit to any s i g n i f i c a n t extent. 2. Medium to large permeability value (0.0003 to 0.03 feet per second) Improves the resistance to l i q u e f a c t i o n f a i l u r e by d i s s i p a t i n g the high pore pressure pressure may decrease instead generated. Pore increase a f t e r the peak disturbance i s passed even though there are t r a i l i n g weak disturbances fed to the deposit. that I t may be visualized i f the earthquake motion consists of strong pulses separated by weak ones the maximum pore pressure near the surface would be even smaller. 3. Within the time i n t e r v a l of 15 seconds the magnitude and 143 d i s t r i b u t i o n of pore pressure below the 100-foot depth i s l i t t l e affected, i f any, by the d i s s i p a t i o n process. the earthquake and uniform duration, waves consist of a r e l a t i v e l y low-valued amplitude say If 60 acceleration waves with long to 80 seconds, l i q u e f a c t i o n of deeper zones i s possible. Quantitative strictly applicable section 5*1. terms of the r e s u l t s obtained i n t h i s to the chapter are p a r t i c u l a r deposit described i n Qualitative extrapolation to other deposits i n effects of some of the s o i l parameters on the l i q u e f a c t i o n process can be made. In addition, the numerical solution of the coupled problem Is shown to be practicable well as feasible. Computer time f o r numerical integration for any one case described i n t h i s for a as section, i . e . integrating time history of 15 seconds at 0.01 second i n t e r v a l , i s approximately 15 seconds CPU time. 144 CHAPTER 6 SUMMARY AND CONCLUSIONS / 6.1 SUMMARY The problem of s o i l l i q u e f a c t i o n has concern been a main i n the design of foundations f o r earthquake-resistant structures on saturated granular soils. Previous investigations indicated that f o r horizontal s o i l deposits the important factor accumulated i s the, amount of pore during the period of shaking. water pressure In t h i s thesis i t i s assumed that f o r the saturated granular s o i l comprising the deposit, l i q u e f a c t i o n occurs when the e f f e c t i v e normal stress reaches a value of 0.05 of i t s model initial value. A material i s proposed whereby such changes i n pore water pressure can be calculated by following the s t r a i n history element subjected to shear deformation. assumed to be of i n f i n i t e on of a soil The s o i l deposit i s extent and f i n i t e thichness resting a firm bedrock i n order that s i m p l i f i c a t i o n of the problem to one of the shear-beam type derived i s possible. Equations are to describe the coupled processes involved during the progress of l i q u e f a c t i o n , namely, 1. Dynamic response to base motion, 2. Increase i n volumetric straining, compaction strain due to cyclic 145 3. Pore water pressure generation as a r e s u l t of (2), and, 4. Dissipation of excess pore water pressure(consolidation). Solution of the numerical i n t e g r a t i o n . problem was achieved a lumped-mass-system integration method dynamical using The f i n i t e difference method was used to a r r i v e at an approximation of the continuous by by was equation. soil i n a r a t i o n a l way. then used to deposit A step-by-step solve the resulting A separate f i n i t e difference scheme was used to solve the consolidation equation. Several example problems were solved f o r an assumed deposit by varying some of the s o i l parameters so that factors a f f e c t i n g the l i q u e f a c t i o n process can be studied. 6.2 CONCLUSIONS Conclusions reached in this research are as follows:1. The hyperbolic h y s t e r e t i c s t r e s s - s t r a i n volumetric compaction parameter, can be 'strain used serving effectively relationship with as a 'hardening' to describe the l i q u e f a c t i o n behavior of s o i l under dynamic loading. 2. Using the proposed s o i l model f o r a deposit a r e l a t i v e l y • f l a t ' frequency response i s obtained. that the resonant capacity This indicates of the non-linear model i s lower than i t s l i n e a r equivalent. 3. When pore water pressure accumulated during shaking reaches 146 A value of about 90.% of stress, very large the strains initial account doubt on of equivalent but without pore pressure r i s e . the v a l i d i t y linear of analysis concerning l i q u e f a c t i o n normal develop In comparison with those computed f o r the same deposit into effective using This throws some strain to taking compatible obtain information potential. For r e l a t i v e l y low permeability values (0.000003 to feet per second) i n the deposit a l o c a l i z e d type of l i q u e f a c t i o n i s produced so that s t i f f n e s s deterioration and strength due to pore pressure r i s e f o r a p a r t i c u l a r layer occur at a much faster rate than others. usual case strata zone when In the the r e l a t i v e densities of the surface are lower than of 0.0003 those l i q u e f a c t i o n occurs of near the deeper strata the the surface, i.e. at depth between 20.0 to 50.0 f e e t . R e l a t i v e l y large permeability values (around 0.03 feet second) reduce the tendency strength deterioration near the surface per to l o c a l i z e s t i f f n e s s and to a single layer. the maximum pore For water strata pressure accumulated during the time of earthquake distrubance, i s reduced and larger magnitude base motion i s require to In general, the r a t i o of the pore pressure accumulated to cause liquefaction. the i n i t i a l e f f e c t i v e normal stress decreases with after the 50.0 feet mark. depth For s o l i deposits without e s p e c i a l l y weak layers l i q u e f a c t i o n at depth greater than 14? 100.0 feet i s very u n l i k e l y . For s o i l s t r a t r a located 50.0 feet or more beneath the ground surface, the pore pressure accumulated i s not much affected by the dissipation process even permeability value i s as high as 0.03 Therefore, when a permeable feet when the per second. s o i l deposit i s shaken by earthquake motions consisting of r e l a t i v e l y low, uniform amplitude and long duration acceleration waves, l i q u e f a c t i o n at larger depths ( i . e . 50.0 and 100.0 is 3 In feet) possible. SUGGESTIONS FOR FURTHER RESEARCH view of the p a r t i a l s t a b i l i z a t i o n e f f e c t produced by sandy material confining due to i t s dllatant pressures, incorporation behavior at low of dilatancy into 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 i s needed so that the analysis may be continued near and at l i q u e f a c t i o n (zero or near zero e f f e c t i v e s t r e s s ) . A thorough investigation of the volume c h a r a c t e r i s t i c s i s necessary i n order compaction function 1. a continuous can be generalised that compaction the volume to Include, v a r i a t i o n of volumetric compaction as a function of s t r a i n or stress, and, 2. e f f e c t of a general state of s t r e s s . A correlation study with f i e l d data should be c a r r i e d out. 148 LIST OF REFERENCES Ambraseys, N., and Sarma, S., I969. "Liquefaction of S o i l s Induced by Earthquakes." B u l l , of the Seism. Soc. of Amer., v o l . 59» no. 2, A p r i l 1969, PP. 651-664. Barkan, D.D., I 9 6 2 , "Dynamics of Bases and Foundation." McGraw H i l l Book Co., Inc., I962. Bathe, K.J., and Wilson, E.L., I973t " S t a b i l i t y and Accuracy Analysis of Direct Integration Methods." I n t e r n l . J . Earthquake Engg. and Struct. Dyn., v o l . 1, 1973. PP. 283-291. Bazant, Z., and Dvorak, A., 1965. "Effects of Vibrations on Sand and the Measurement of Dynamic Properties." Proc. 6th I n t e r n l . Conf. S o i l Mech. & Found. En*rg., v o l 1, 1965. PP. 161-164. Blot, M.A., 1956, "Theory of Propagation of E l a s t i c Waves i n Fluid-Saturated Porous S o l i d . I - Low-Frequency Range." J . Acoustical Soc. of Amer., v o l . 28, no. 2, March 1956, pp. 168-178. - Bogdanoff, J.L., Goldberg, J.E., and Bernard, M.L., I 9 6 I , "Response of a Simple Structure to a Random Earthquake Type Disturbance." B u l l . Seism. Soc. Amer., v o l . 511 no. 2, A p r i l 1961, PP. 293-310. Casagrande, A., 1936, "Characteristics of Cohesionless S o i l s A f f e c t i n g the S t a b i l i t y of Slopes and Earthf i l l s . J . Boston Soc. C i v i l Engrs., Jan. 1936. 1 1 149 Castro, G., 19^9. "Liquefaction of Sands." Ph.D. Thesis, Harvard Univ., Cam., Mass., I969. E d i t o r i a l Committee of the "General Report.", I968, "General Report on the Niigata Earthquake of 1964." Finn, W.D.L., 1972, " S o i l Dynamics - Liquefaction of Sands." Proc. I n t e r n l . Conf. Microzonation f o r Safer Construction Research arid Application, Seattle, Washington, USA, Oct., 1972, pp. 87-111. Finn, W.D.L., Emery, J . J . , and Gupta, Y.P., 1970, "A Shake Table Study of the Liquefaction of Saturated Sands during Earthquakes.", Proc. 3rd European Sym. Earthquake Engg., Sept. 1970, PP. 253-262. Finn, W.D.L., Emery, J . J . , and Gupta, Y.P., 1971, "Liquefaction of Large Samples of Saturated Sand on a Shake Table.", Proc. 1st Can. Conf. Earthquake Engg., Van., UBC, May 1971, PP. 97-110. Finn, W.D.L., Emery, J . J . , and Gupta, Y.P., 1971a, " S o i l Liquefaction Studies Using a Shake Table." Closed Loop, System Corporation, Fall/Winter, 1971. Finn, W.D.L., Pickering, D.J., and Bransby, P.L., 1971. "Sand Liquefaction i n T r i a x i a l and Simple Shear Tests." J . S o i l Mech. & Found. Div. ASCE, v o l . 97, SM 4, A p r i l 1971. PP. 639-659. t F l o r i n , V.A., and Ivanov, P.L., 1961, "Liquefaction of Saturated Sandy S o i l s . " Proc. 5th I n t e r n l . 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Engg., v o l . 1, 1957, PP. 367-372. 152 Maslng, G., 1926, "Eigenspannlngen und Verfestigung beim Messing." Proc. 2nd Internl Cong. Applied Mech., 1926, PP. 332-335. Mogami, R., and Kubo, 1953, "The Behaviour of S o i l during Vibration." Proc. 5th I n t e r n l . Conf. S o i l Mech. and Found. Engg., v o l . 1, Zurich, 1953, PP. 182-185. Newmark, N.M., 1959 "A Method of Computation for S t r u c t u r a l Dynamics." J . Engg. Mech. Div., ASCE, v o l . 85, no. EM3, July 1959. Newmark,.N.M., and Rosenblueth, E., 1971, "Fundamentals of Earthquake Engineering." Prentic H a l l , Inc., Eaglewood C l i f f s , N.J., USA, 1971. PP. 162-163. Osaki, Y., 1968, "Building Damage and S o i l Condition." General Report on the Niigata Earthquake of 1964. Committe of "General Report", I968, pp. 355-383. Parmelee, R.A., Perelman, D.S., Lee, S.L., and Keer, L.M., 1968, "Seismic Response of Structure Foundation System." J . Engg. Mech. Div., ASCE, v o l . 94, EM 6, Dec. 1968, pp. 1295-1315. Peacock, W.H., and Seed, H.B., I968, "Sand Liquefaction under C y c l i c Loading Shear Conditions." J . S o i l Mech. & Found. Div., ASCE, v o l . 94, SM 3, May 1968, pp. 689-708. Pyke, R.M., 1973. "Settlement and Liquefaction of Sands under M u l t i D i r e c t i o n a l Loading." Ph.D. Thesis, Univ. of C a l i f . , Berkeley, C a l i f . , 1973, P. 280. 153 Schnabel, P.B., Lysmer, J . , and Seed, H.B., 1972, "A Computer Program for Earthquake Response Analysis of Horizontally Layered S i t e s . " Earthquake Engineering Research Centre, Report Number EERC 72-12, Dec. 1972. Seed, H.B., I968, "Landslides during Earthquakes due to Liquefaction." J . S o i l Mech. & Found. Div., ASCE, v o l . 94, SM 5, Sept. 1968, pp. 1053-1122. Seed, H.B., 1972, "Dams and S o i l s . " The San Fernando Earthquake of February 9, 1971, and Public P o l i c y . Special Subcommittee of the Joint Committee on Seismic Safety, C a l i f o r n i a Legislature, July 1972. Seed, H.B., and I d r i s s , I.M., I 9 6 7 , "Analysis of S o i l Liquefaction : Niigata Earthquake." J . S o i l Mech. & Found. Div., ASCE, v o l . 93, SM 3, May 1967, PP. 83-108. Seed, H.B., and I d r i s s , I.M., 19^9, "Influence of S o i l Conditions on Ground Motions during Earthquakes." J . S o i l Mech. & Found. Div., ASCE, v o l . 95, SM 1, Jan. I969, PP. 99-137. Seed, H.B., and I d r i s s , I.M., 1971, "Simplified Procedures f o r Evaluating S o i l Liquefaction Potential." J . S o i l Mech. & Found. Div., ASCE, v o l . 97, SM 9, Sept. 1971, pp. 1249-1273. Seed, H.B., and Lee, K.L., I 9 6 6 , "Liquefaction of Saturated Sands during Cyclic Loading." J . S o i l Mech. & Found. Div., ASCE, v o l . 92, SM 6, Nov. 1966, pp. 105-134. Seed, H.B., and S i l v e r , M.L., 1972, "Settlement of Dry Sands during Earthquakes." J . S o i l Mech. & Found. Div., ASCE, v o l . 98, SM 4, A p r i l 1972, pp. 381-397. 154 S i l v e r , M.L., and "Volume Changes J . S o i l Mech. & Sept. 1971, PP. Seed, H.B., 1971. i n Sands during C y c l i c Loading." Found. Div., ASCE, v o l . 97. SM 9. 1171-1182. Sneddon, I.N., 1957. "Elements of P a r t i a l D i f f e r e n t i a l Equations." McGraw H i l l Book Company, Inc., 1957. State Earthquake Investigation Commission, 1908, "The C a l i f o r n i a Earthquake of A p r i l 18, 1906." Report of the State Earthquake Investigation Commission, Published by the Carnegie Institute of Washington, 1908. Terzaghi, K., and Peck, R.B., I968, " S o i l Mechanics i n Engineering P r a c t i c e s . " John Wiley Publishing Company, F i r s t Corrected P r i n t i n g , 1968. Yen, B.C., I967, "Viscosity of Saturated Sand near Liquefaction." Proc. I n t e r n l . Sym. Wave Propag. & Dyn. Properties of Earth Materials, New Mexico, I967. PP. 677-888. Yoshimi, Y., 1967. "An Experimental Study of Liquefaction of Saturated Sands." S o l i and Foundation, v o l . 7. no. 2, June 1967, PP. 20-32. 155 APPENDIX. I LUMPED MASS SYSTEM APPROXIMATION FOR NON-LINEAR SHEAR BEAMS The equilibrium equation, equation i n chapter 4 i s re-written here f o r - s$ (4-1), obtained convenience, 8 =0 When the s o i l parameters are functions of depth, (H-z), where H i s the t o t a l depth of the s o i l deposit and z i s the height from the surface of the bedrock, equation (4-2) has to be replaced by a more general equation, namely, and also, Equations T = f^JT.z) f = j>(z) ( 1 - 2 ) and (1-3) + (1-2) f (f,z) 2 (1-3) can be substituted into (1-1) to give, Consider, intervals, f o r example, so that the z-domain O^^z^z^ i s sub-divided into . , . <z =H , n and length of each i n t e r v a l be denoted by h^, i . e . , l e t the 156 Also, j> f l e t x., 1 at f. and / p. represent 1 the values of x, i , e t and c ,. and denote the p a r t i c u l a r 2,i' function at z^ by ' f The 1 2,i central = f 2 i' i> (f ^- ) z difference 6 quotient for Tz^2^' ^ a z ^ z i^ s given by, A2 H f (ir (f 2.1 i " + ,i-i>/< f 2 i h i l * i> + h + and a s i m i l a r expression can be written f o r f^(f,z) when into these expressions are following equation can be i(h Now, i+1 +h ).^ .x i 1 i substituted so that (1-3)• the obtained, + (f^..^ - f + <2,i-i " 2,i+*> f 1 § i + i ) f = 0 tt- ) 8 write, t (f,z) z tM,z) -L- = .t (1-9) 3x dz Since f i-Jr ( Consequently, equations x i * i-l x ) / h i (1-9) and (I-10) l) fo(f? 1 1-2 I f a new symbol i s defined as follows :- ( I give us, - 1 0 ) 157 i (1-12) I h. . f. ( I - 1 1 ) can be w r i t t e n i n a simple equation f 2,i-i *' i ' k ( i x " i - i x form, (1-13) } S i m i l a r l y , i t can be shown t h a t , f 2.1+4 F ' k (1-14) i+l*( i+l " i ^ x x (1-15) ^i'( i ~ i-l^ x f i.i+i c ' ,= x i+l*( i+l x i) A.Z. where, c Equations equation (1-17) i (1-13) (1-16) i ^ x 1 (I-l6) through ( 1 - 8 ) and 1-2 the can following be substituted 'discretized• into dynamic equation i s obtained, X V x i + c +c (-^ i -c i + 1 i + 1 ).^ i-1 X.. X > i 1 + *i-l + (" i i k k + i l k - i l) k + X + x Note that, i(h i + 1 +h ).j> i i i i+l * = 0 (1-18) 158 and since the average density within the i n t e r v a l i s usually known instead of the density value at z^, m^ i s "better calculated by, V *\fi-i- i = h ^/i+i^i+l + I t should be pointed out that the f i n i t e difference (1-18) holds i n general f o r i=2,3,4,... , n - l . observing that when i = l the variables x For i = l and and x Q i n equation (1-18) are i n fact the v e l o c i t y and of the base Q be used appearing displacement respectively so that the symbols x^ and x^ are used instead. subscript 1 9 ) equation i=n, i . e . at both end points, the same equation can by " ( I And, of n+l when i=n the variables are set i d e n t i c a l l y equal with the to zero. system of n equations thus obtained can be represented by A a simple matrix equation![M].{'x} where x i' x {x}, i a n d + [C].{x} + [K].{x> = {P > b {xj- and {x} are column vectors whose components x i Sive us the acceleration, velocity displacement respectively of the point at z^, {x} = (x x £ x^ . . . x ) T x {x} = (x x 2 x^ . . . x ) T x {x} = (x x 2 x . . . x ) T x {P } = b (1-20) (k x +c x 1 b 1 b 3 0 0 n n n . . . 0) T and 159 where the s u p e r s c r i p t T r e f e r s t o the transpose o f the row v e c t o r i n t o a column v e c t o r and x^ and x^ r e f e r t o v e l o c i t y and displacement The matrices the base respectively. [M], [C] and [K] as o b t a i n e d from e q u a t i o n (1-20) are as f o l l o w s / [M] = m 0 1 0 m 0 • 0 ' • 0 0 2 0 0 m^ • • • 0 0 0 , 0 • (I -21) • * • m n j [C] = c c 1 + " 2 0 C [K] = ~2 C 0 2 3 " 3 2 C + C - 3 c c c 0 • • 0 0 • • 0 • • 0 • • • • - n 3 4 + c • * • • 0 0 0 0 0 0 • • 0 -k 0 t • 0 • • 0 • • • " l + k k " 2 0 k 2 k 2 + k -k 3 3 3 k +k 3 4 • • • • 0 0 0 0 C c (I -22) n - k vector displacement vector, i s preferred. {X) and {X} L e t {x} be the -23) n F o r base a c c e l e r a t i o n type o f l o a d i n g the displacement (I relative be the r e l a t i v e corresponding 160 velocity and acceleration vectors and l e t x^ be the variable describing the displacement of the base as a function of time, i t can be seen that {x} Similar expressions can be acceleration. equation (1-24) When written equation f o r the v e l o c i t y and i s substituted into (1-24) (1-20), i t can be shown that the following equation i s obtained, [Ml{X) + [ C ] . { X ) kx [K].{X} + m I t i s quite apparent obtained that the f i n i t e (1-25) b n difference equation i s the same as that f o r the dynamic response system of n dashpots. discrete Thus, a masses lumped connected mass by model springs can be used of a and to approximate the continuous shear beam model f o r the dynamic response of the s o i l column discussed i n chapter 4, The value of the masses are obtained by lumping h a l f the mass of each layer to i t s corresponding nodes, z^ ^ and z^. s t i f f n e s s e s of the springs and the damping the dashpots are obtained from f relationship, and (I-l?), equation 1 and f 2 The coefficients for of the s t r e s s - s t r a i n (4-2), and through the use of (1-12) Also, the width of the s o i l column can be made 161 d i f f e r e n t from u n i t y and i t s e f f e c t can be taken i n t o account by m u l t i p l y i n g the c o r r e s p o n d i n g width o f l a y e r , b^, i n t o the appropriate c^ and terras. So t h a t , i n g e n e r a l , the components m^,. k^ o f the m a t r i c e s [M], [C] and [K] are g i v e n by the following i - m c k i *«/>i-*'V i = h £-j>i+£' i+l- i+l + b i = tilh-i'H-tf'*! i = f 2 ^ i - i ' Z i - ^ - h / ih-h-tf b i '/ ( I - 2 6 ) <Vi-*> where i r e f e r s t o the l a y e r number o f the lumped mass system. 162 APPENDIX II NUMERICAL INTEGRATION OF THE DYNAMIC RESPONSE EQUATION The dynamic response equation for. the lumped mass system i s re-written here f o r convenience, [M].{X} + [C].{X} + [KJ.{X} = {P} (II-l) where {p} i s the i n e r t i a force vector, •m. {p} II.1 - (II-2) Incremental S t r e s s - s t r a i n . Let where t T t + At, ( H - 3 ) i s an instant of time at which a l l the quantities i n equation ( I I - l ) increment. are known and equation (II-l) Since At is a small time has to he s a t i s f i e d at every instant, we have, [M] {X} + [CJ {X} + [K] {X} T T T T T T [M] {x} + [C] {x} + [K] { } t t t t t X t = (P} = {P} where the subscript r e f e r s to the instant of the (II-5) T p a r t i c u l a r quantity takes on i t s value. (II-6) t time at which Since the mass 163 matrix i s a constant matrix, [M] so that = T [M] {X} T T [M] t = [ M], [MJ {x} - t = t (X} [M] - {X} T t However, the [Cj and [Kj matrices depend on the shear of the s o i l , as a relation, result of ways non-linear strain stress-strain and thus on the displacement of the masses so that (II-7) cannot equations s i m i l a r to that of these the (II-7) be obtained matrices i n a s t r a i g h t forward manner. to arrive at expressions similar There are two (II-7) using to A crude method i s to replace [K]^ approximation methods. for by [K]^, so that, [K] {X} T - T [K] {X} t = [K] t t {x} T - {x} 1 t (II-8) A better method i s as follows »Define, {F} so that, [K] {X} [K].{X} T T - (H-9) [K] {X} t t Wt 3F; 5X = where [K], - {X}, W t jj [ K] .|{X} t 3F^ ax. J- T - ^ >t} X (11-10) (11-11) 164 and 9). i s the component of the column vector defined i n ( I I Therefore, i n general, [K] {X} T - T [K] {X> t = t LK] .|{X} t (11-12) - {X} | T t where [K]^ can take the value of [K]^ f o r crude approximation or i t can be evaluated approximation. according Similar to procedures (11-11) f o r better can be used to obtain IF equation (II-6) i s subtracted from equation ( I I 5) and taking note of (II-7) and (11-12), an incremental form for the dynamic response equation can be obtained as, [M]|{X} + T - + {X} j t [Kj [{x> t T [C] |{x} t -{x> t J T - {X} J t = {P} - {P} T T (11-13) I t can be seen that the only unknown i n t h i s equation are the acceleration, v e l o c i t y and displacement vectors As such, at time T. the numerical procedures proposed by Newmark(1959) can be applied to the equation so that {X}^, {x} T and { x } T can be expressed i n term of {*X} , {X}^ and { x } . t II.2 t Step by Step Integration In Newmark's method of step by step integration two parameters, o< and displacement at time |3 are used T so that can be expressed the v e l o c i t y and i n terms of the acceleration, v e l o c i t y and displacement at time t, and of the 165 unknown acceleration at time T. The r e l a t i o n s can be written as follows :{X} T = {X} + (l-cx).At.{X} + *.At.{X} {X} T = {X} + At.{X} t t t + (i-/5).(At)?{X} t '+ Newmark( 1959) j8.(At)?{x} and /3=i proposed that (11-14) T t (11-15) T f o r unconditionally- stable integration procedure, which i n c i d e n t a l l y to a constant average acceleration method of i n t e g r a t i o n . I f «=(l/2) and /6=(l/6) method as proposed are used interval and /3=^ °<=i the acceleration At=T-t {Q) acceleration are used. increment during the time i s represented by {Q}^,. i . e . , = T the l i n e a r by Wilson and Clough(1952) i s obtained. For the present purpose If corresponds (x} T - {x} (11-16) t and i f the following simplifying symbols are used, then {a} t = At.{x} {b} t = At.{X} equations (II-14) t (H-17) t + 2-.(At)?{X} and (11-15) (11-18) t can be written into a simple form, namely, {X} -{X} T {X} T t {X> t = { a ) + c*.At.{Q} = {b} + y5.(At)?{Q} t (II-19) T t T (11-20) 166 Equations (II-16), (11-13) into (11-19) and to give (11-20) can be substituted a matrix equation involving only one unknown, {Q} , namely, [D] {Q} t where [D] and {P} t T = T {P} (11-21) T + p. (At) .[K] = [M] + ot.At. [C]t = { P } - {P} - [ C ] { a } T Consequently, t t {Q} T t = t - [K] {b) t t calculated T (11-23) (11-24) [Dj^P}^, and {x}^, { x } and { X ) can be T (11-22) 2 using equations (11-16), (11-19) and (11-20). In the numerical step by step integration, a t y p i c a l sequence of calculations that have to be performed i n one time step, At, i s as follows »1. Based on { x } t > {X} and { X } at time t , (11-17) and t t (II- 18) are evaluated, 2. {a} t = At.{x} {b} t = At,{x} t Based on {x}^ and { x } , {f} t m t {i} t where + t t i.(At) .{x} 2 t and { f } are calculated by, t = CaJ{x} (11-25) = [^]{x} (11-26) t t 167 l/h = [a] -l/h 0 1 l/h 2 2 0 0 0 0 - l / h ^ l/h^ 0 0 -lA 3. With the {i\ {t} and t evaluated according to values, t the [C] appropriate n [ K ] and t l/h n t are expressions i n appendix I . 4. [D]^. i s then calculated using equation (11-22). 5. Using the base acceleration value at time T, x b T , i t is possible to evaluate, r -\ T bT m n and 6. {Q} T (P} = T {P} - {P} - [CJ {a} T i s solved velocity and by t (11-24) displacement t so that t - [K] {b} t the vectors, {X) , T t acceleration, {x) T and fX} T can be found by, {X} n {X) = {X} + { a } +o<.At.{Q} (X} = {X} + {b} + |3.(At) .{Q} . T T 7. t t t The steps 1 through 6 are increment. T 2 t repeated T f o r the next time 168 An option i s b u i l t into the analysis to bring the s t r e s s strain point I f the strains integration rates, closer to the actual s t r e s s - s t r a i n curve. and strain determined are accepted as true s t r a i n s and the stress-strain relationships can be T t C ] . {X} . T However, T and used r e s t o r i n g force, [ K ] . { X ) , T rates to and true each strain stress-strain-rate determined the in the 'true' 'true' damping force, these r e s t o r i n g and damping forces do not necessarily s a t i s f y the equilibrium equation 5). An artificial 'external' force, { P . ™ by the following can be applied to the }, system (II- defined so that equilibrium i s restored, {P e r r The {P_„_ for the } = {P} - [M].{X} T T - [C] .{X} } can be added to {P}m next time increment. T i n step T - [K] .{X} T number CX . 5 for I t should be pointed out that t h i s method of correction forces i s only good the magnitude of {P-„_ T where } i s small as i n the present case X of step-by-step integration with small time increment. 169 APPENDIX I I I FINITE DIFFERENCE EQUATIONS FOR PORE PRESSURE DISSIPATION In the general case where the permeability and compressibility of the s o i l are functions of depth equation (4-19) i s used, *cr w 3 IF = K ae^ d<r„ k l r - ^ ^ ' 1 7 ) + K lr-"iT Following the same approach as i n appendix I, i t i s assumed that the z-domain i s divided into n i n t e r v a l s so that, 0<z,<z <z l c. } o . Q . ,<z =H, n where H i s the t o t a l depth of the deposit. For convenience the following notations w i l l be used :hj to denote the length of the i-th i n t e r v a l , i . e . , h W 1 ,T i = Z i" i-1 Z (IH-2) to denote the f i n i t e difference approximation to W i,T at z=z^ and t=T, or, = °w< f > z T (IH-3) 170 Moreover, attention should be given to the following m v k^ 1 # and ^ which are used to denote the values at the centre of their values €vp^ i^ at are z the used. the i-th interval. symbols respective To indicate 'nodes* the symbols m (z ), k(z,) and v 1 i m i s the coefficient of volume v compressibility so that, 1/K m (III-4) lr For the f i n i t e difference approximation of equation (III-l) consider f i r s t the term on the l e f t hand side of the equation and use a forward difference quotient for approximation so that, w St t=T z=z, acr ( W A central i.T AT- W + J difference i.T quotient (III-5) ) / A T will approximation of the f i r s t term on the be used for the right hand side of equation ( I I I - l ) , namely, r .A( * l r 3z 2f 2£) t=T dz w W L i+1*' z=z - W 1+1,T l J i+l V * l+1 ( And for the 1,T W 111 h + * h 1 " 1-1,T h. W (III-6) )^ (z ) v 1 second term on the right hand side of equation ( I I I - l ) a backward f i n i t e difference quotient w i l l be used, 17.1 l 3t r m ( v - T Z l ) .AT T v * (III-7) It can be shown that by combining equations ( I I I - l ) , (III-5), ( I I I - 6 ) and (III-7) of the a finite difference equation following form can be obtained, W " MVl+l.I i.T AT + + where, R, W 1.T = 1 N a n It d W can + i+l/ h f,T= be 1.T seen U that 1 I - » 8 (III-9) 2 (h ^ I h *w' i+l - ( / — i — — = 1 A M "l-l.T * <Vl>-Wi.T] + l + AT r . ~ hi)»n (z ) h v 1 (111-10) 1 f ( 1 1 1 - U ) equation (III-8) resembles the usual f i n i t e difference s o l u t i o n of a one dimensional consolidation problem except •generation f o r the of 1 pore values of m and £ instead the v of v p last water term which pressure. represents the In order that the at the centre of the layer can respective values at the nodal be points used the following average values are used, ffi / . v< l> = z m v,i+l' H-l * h + h 1 + 1 h l m v,i' l h (111-12) 172 vp 1T m (z, )_ v IT c A£ vp,l+1,T mv.i+l A G vp,i,' mv . i (111-13) Expressions (III-10) and ( I I I - l l ) can be modified into the following, 2. AT r AW g l.T The w- v,i+l- i+l = 4. final form (m h vp.i+l.T mv,i+l *i + m (111-14) v,i'V' i h Srp,i,T A + (111-15) m of the equation v.l is still (III-8) expressions (III-9), (111-14) and (111-15) w i l l the analysis. be but used i n LISTING OF LIQUEFACTION PROGRAM - A U G , 75 LIQPG 1 173 APPENDIX IV C**************************************** C* INPUT CARDS FOR LIQUEFACTION PROGRAM * C**************************************** C C (20AU) (1 1- TITLE CARD) 2. NMA T , N P T Y P E , N V O L , N L A Y E R , N L I N E L , N D A M P , N B T , N B B , N C R V WHAT = NUMBER OF MATERIAL TYPES. C C C NPTYPE C C C C C C C C C C C C C C C C C NVOL NLAYER NLINEL NDAMP NBT NBB NCRV . . (20IU) (1 C = PRO B K E M T Y P E NUMBER, 1 FOR DYNAMIC RESPONSE ONLY, 2 FOR DYN. RSSP. PLUS PORE PRESS. R I S E OR V O L . CHANG 3' F O R D Y N . R E S P . PLUS PORE PRESS. PLUS DISSIPATION. = 0 FOR PORE PRESSURE RISE. = 1 FOR VOLUME CHANGE (SETTLEMENT). = NUMBER OF L A Y E R S . = 0 FOR NON-LINEAR MATERIAL (CHANGE MODULI) = 1 FOR LINEAR MATERIAL (CONSTANT MODULI) = 1 FOR DAMPING C O E F F I C I E N T S INPUT. = 2 FOR DAMPING PROPPORTIONAL TO MASS AND S T I F F N E S S . = 0 FOR ZERO PORE PRESSURE AT T O P BOUNDARY. =1 F O R ZERO G R A D I E N T AT T O P BOUNDARY. = 0 FOR ZERO PORE PRESSURE AT BOTTOM BOUNDARY, ='1 F O R ZERO G R A D I E N T AT BOTTOM BOUNDARY. = 0 F O R N O T C A L L I N G S U B R O U T I N E TO C H E C K S T R A I N REVERSAL = 1 F O R C A L L I N G S U B R O U T I N E TO CHECK FOR S T R A I N REVERSAL C C 3. C MATYP(I), NSUBD(I), H(I) , ( 2 1 4 , 2 F 1 0 . U) , WIDTH (I) (NLAYER C C C C C U. C MATYP(I) NSUBD(I) H(I) WIDTH(I) A1 ( I ) , A 2 ( I ) D1 ( I ) , D 2 ( I ) = M A T E R I A L T Y P E NUMBER O F I = NUMBER O F S U B D I V I S I O N OF = THICKNESS OF I - T H LAYER. = WIDTH OF I - T H L A Y E R . , A 3 (I) , B 1 (I) , B 2(I) , B 3(I) , C 1 ,DEN-H (I) , DEN-V (I) ,PERM (I) , - T H LAYER. I - T H LAYER FOR PP CARDS) DISSIPAT (I) , C 2 (I) ,C3 (I) ,C4 (I) , ALFA (I) ,BETA (I) . C (6F12.4) C C C 5 . NEQ,INTYP,NC,NCPR,NCPRM,NPCON,NCONT R NPLD,DT — ( 3 * N M AT (5I4,F10.3) CARDS (1 C A NEQ = 0 = 1 = 2 FOR CONSTANT AMPLITUDE HARMONIC INPUT. F O R EARTHQUAKE RECORD INPUT. F O R G R A D U A L L Y I N C R E A S I N G A M P L I T U D E TO T Y P E C C INTYP = = 0 1 FOR WILSON AND CLOUGH'S METHOD OF I N T E G R A T I O N . FOR NEWMARK'S METHOD O F I N T E G R A T I O N . C C NC NCPP = = NUMBER NUMBER C C NCPRM NPCON = = NUMBER OF INTEGRATIONS F O R P R I N T I N G MAXIMUM V AL. 1 F O R P R I N T I N G OUT F I N A L V A L U E S FOR C O N T I N U I N G NEXT C C C " = 0 C C C C C C OF INTEGRATIONS OF INTEGRATIONS FOR EACH PRINT OUT FOR NO P R I N T I N G OUT INITIAL 1. NCONT = 0 F O R ZERO NPLD = = = 1 0 1 FOR NON-ZERO I N I T I A L CONDITIONS. F O R NO O U T P U T ON U N I T 7 F O R P L O T . F O R OUTPUT ON U N I T 7 FOR P L O T . CONDITIONS. T L I S T I N G OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 2 174 C 6. FOR NEQ=1, NCARD,NREC,NFTS,RDFR C (2014) (1 CARD) C NCARD = NUMBER OF EQ REC CARDS. C NREC = NUMBER OF EQ REC PER CARD. C NFTS = 0 GOR NUMBER OF GEE'S INPUT. C . RDFR = FACTOR TO BE MULTIPLIED TO ACC RECORDS. C ' =1 FOR FPS UNIT INPUT. C FMT C (20A4) C FORMAT FOR EQ REC INPUT. C NEQ=0 C NEQ=0, NC, NFTS,AMP,OMEGA C (2014) C NC NUMBER OF SINE CYCLES. C NCFTS C NFTS = 0 FOR NUMBER OF GEE'S EQ INPUT C =1 FOR FPS UNIT INPUT C AMP = AMPLITUDE OF SINUSOIDAL INPUT C OMEGA - CRICULAR FREQ. OF S I N . INPUT. C 7. T I T L E C (20A4) (1 CARD) C DESCRIPTION FOR BASE MOTION INPUT. C C 8. NEQ=0 S NCONT=0 THAT IS ALL NO MORE CARDS REQD. C NEQ=0 & NCONT=1 I N I T I A L VALUES INPUT ACCORDING TO TM1VAL. C NEQ=1 & NCONT=0 INPUT EARTHQUAKE RECORDS. C NEQ= 1 5 NCONT=1 INPUT INITIAL. VALUES AND EQ. REC. C** DYNAMIC ANALYSIS FOR LIQUEFACTION OF A HORIZONTAL SOIL DEPOSIT C SIX TYPES OF PROBLEMS CAN BE SOLVED, NAMELY: C 1. DYNAMIC RESPONSE OF SOIL DEPOSIT ONLY C 2. DYNAMIC RESPONSE WITH PORE PRESSURE GENERATION C 3 . DYNAMIC RESPONSE WITH PORE PRESSURE GENERATION AND DISSIPATION C AND FOR EACH OF THE ABOVE EITHER C I) LINEAR CASE (NLINEL= 1) , I . E . STIFFNESS ETC I S CONSTANT C I I ) NON-LINEAR CASE (NLINEL=0) C SO NPTYPE=1 CORRESPONDS TO TYPE ONE PROBLEM, ETC. C LCPPG IS USED TO CONTROL CALLING OF PORE PRESS GEN SUB-PROG. C LPPDIS I S USED TO CONTROL CALLING OF DISSIPATION SUB-PROG. C LPLOT I S USED TO CONTROL OUTPUT FOR PLOTTING PROGRAM. C INTYP IS USED TO INDICATE TYPE OF INTEGRATION SCHEME USED: C INTYP=0 WILSON AND CLOITGH'S METHOD IS USED C INTYP=1 NEWMARK'S METHOD OF INTEGRATION IS USED. C LDOF1 IS USED TO INDICATE ONE-DEGREE OF FREEDOM C LOGICAL* 1 LVDIR ( 2 0 ) ,LREV ( 2 0 ) ,LRG ( 2 0) ,DODEVP ( 2 0 ) , L L I Q ( 2 0 ) , LREAD,LCPPG,LPPDIS,LPLOT,LDOF1,LLIN,LSTOP,LMCLG, LJRV,LCKF3,LIT,LPMAX COMMON /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB . /LCONTL/LDOF1,LLIN,LSTOP,LMCLG,LJRV,LCKFB,LIT,LPMAX . /LOGCOM/LVDIR,LR EV,LRG,DODEVP,LLIQ . / P R O P T Y / T I T L E ( 2 0 ) , P R O P ( 2 0 , 1 7 ) , H ( 2 0 ) ,DEPTH ( 2 0 ) ,WIDTH ( 2 0 ) ,WT ( 2 0 ) , M A T Y P ( 2 0 ) ,SIGZ ( 2 0 ) ,CC ( 2 0 ) ,GG ( 2 0 ) , NSUBD ( 2 0 ) , GAMR ( 2 0 ) , L I S T I N G OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 3 175 TAUR(20) ,CM(40) ,SK(40) ,GMN(20) ,TMN(20) /DYNEQT/DYM (40) /PEP (20) ,FORER(20) /CALV AL/DIRN (20) ,XD2 (20) , ABXD2 (20) ,XD1 (20) ,XD0 (20) ,BD(20) , BA (20) ,BB (20) , GD2 (20) , GD 1 (2 0) , G A MM A (20) , G A AO (20) , GAMMAX (2 0) ,TAXY(20) ,EVP (20) ,SK1R (20) , DPP (20) ,PP(20) /CPVALU/BAP(20) ,BBP(20) ,XD2P (20),XD1P (20) ,XD0P (20) ,GD2P (20) , GD1P (20) ,TAXYP (20) /REV PA R/NRPT (20) ,GGAMR(20,8) ,TTAUR(20,8) ,DIRNR(20,8) DIMENSION FMT (20) , AE (10) , DTJMIN (17) DATA LREAD/.TRUE./,LCPPG/.FALSE./, LPPDIS/. FALSE./,LPLOT/. F A L S E . / DATA TNOT/0.0/,UG1/0.0/,NCPRC/0/,NRECC/0/,NPRMC/0/ MM c 1 2 9001 1003 1004 7 C** C C c** 1001 8 READ(5,1) T I T L E FORMAT(20 AU) WRITE (6,2) T I T L E FORMAT(1H1,10X,20A4) READ (5,3) NMAT,NPTYPE,NVOL,NLAYER,NLINEL,NDAMP,NBT,NBB,ITERRV, ICKFB FORMAT (2014) I F (NLINEL.EQ. 1) LLIN=.TRUE. I F (NPTYPE.GE.3) LPPDIS=.TRUE. I F (NLAYER. EQ. 1) LDOF1=. TRUE. LJRV = . F A L S E . LCKFB = .FALSE. I F (ITERRV. EQ. 0) LJRV=.TRUE, I F (ICKFB. GE.1) LCKFB=. TRUE. I F (LLIN) WRITE (6, 9001) FORMAT ('0** LINEAR MATERIAL IS USED **•) N = NLAYER N2=N*2 N21=N2-1 IF(NLAYER.GE.1) GO TO 1003 WRITE (6,6) FORMAT(1H0,•** ERROR 2, ERROR IN INPUT **•) STOP DO 1004 1=1,N READ(5,7) MATYP (I) ,NSUBD (I) ,H (I) , WIDTH (I) FORMAT(2I4,2F10.4) I F (NMAT.GE.1) GO TO 1001 WRITE (6,4) FORMAT (1 HO,'** ERROR 1,ERROR IN INPUT * * « ) STOP ORDER OF INPUT CONSTANTS ARE: A 1,A2,A3,B1,B2,B3,C1,C2,C3,C4,D1,D2, DEN-H,DEN-V,PERM.,ALFA,BETA WRITE INPUT DATA WRITE (6,8) NMAT,NPTYPE NVOL NLAYER,NLINEL,NDAMP,NBT, NBB, ITERRV, ICKFB FORMAT ('0 NMAT =',13,' ; NPTYPE =',13,' ; NVOL =',13, ' ; NLAYER = « , I 3 , ' ; NLINEL =',I3,' ;'/' NDAMP =',13, « ; NBT =',I3,« ; NBB =',I3, ; ITERRV = ' , 1 3 , ' ; ICKFB =',13) WRITE(6,9) f r 1 L I S T I N G OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 4 176 9 FORMAT(1H0,'***********************'/* i C C * * * * * * * * * * * * * * * * * * * * * * * * MATERIAL PROPERTIES * ' / 1 I F NDAMP=1, CC INPUT AS FORCE/UNIT VELOCITY 2, ALFA=PROP(1, 16) AND BETA=PROP (1,17) DO 1002 I=1,NMAT READ(5,5) (DUMIN (IJ) , I J = 1 , 17) FORMAT (6F12.U) WRITE(6,10) I , (DUMIN (I J) , IJ= 1,17) FORMAT (1X, 'MATERIAL TYPE', 13,' : A " S AND B«'S : ' , 1 P6 E1 4. 3/20 X , • C ^ S AND D " S :',6E14. 3/20 X, • DEN. - H = , 0 P F 5 . 1 , « P C F ; ' , 3 X , •DEN.-V = ' , F 5 . 1 , « P C F ; PERM. =',1PE10.3,' ; ', * ALFA =',E11.3,3X,'BETA = ',E11.3) DO 1012 IJ=1,N IF (MATYP (IJ) . NE. I) GO TO 1013 DO 1014 IK=1,17 PROP ( I J , I K ) =DUMIN (IK) CONTINUE CONTINUE CONTINUE WRITE(6,11) FORMAT(1H0,'*********************/1X,'* LAYER PROPERTIES *'/1X, 5 10 , 1014 1013 1012 1002 11 m 12 13 C C C C C 1009 1016 1007 C C t********************i/j WRITE (6, 12) FORMAT (2X 'LAY.NO. MATYP NSUBD WRITE (6,13) ( I , MATYP (I) , NSUBD (I) ,H(I) FORMAT(1X,I4,3X,2I8,2F10.2) f THICK , WIDTH (I) ASSIGN LOWER MOST BOUNDARY AS PROBLEM' BOUNDARY FORM DIAGONAL MASS MATRIX FIND DEPTH AND SIGZ AT CENTRE OF EACH LAYER DO 1007 1=1,N I J = N-I+1 IK = IJ+1 DT2 = 0. 5*H (IJ) UGI = DT2*PROP (I J , 13) CA = DT2*PROP ( I J , 14) IF (I.GT. 1) GO TO 1009 WT(N) = UGI DEPTH (N) = DT2 SIGZ(N) = CA GO TO 1016 WT(IJ) = UGI+UG1 DEPTH(IJ) = DEPTH (IK) +DT1 + DT2 S I G Z ( I J ) = SIGZ (IJ+1) +CA+BK DT1 = DT2 UG1 = UGI BK = CA CONTINUE CHANGE MASS UNIT TO LB-SEC.SQ./FT. UNIT CA = 1./32. 172 DO 1033 1=1,N WIDTH') ,I=1,N) L I S T I N G OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 177 1033 C C C** C WT(I) = WT(I) *CA*WTDTH(I) FIND DAMPING VALUE AND STIFFNESS VALUE FOP. EACH LAYER FORM STIFFNESS AND DAMPING MATRIX. NOTE HALF BAND WIDTH = 2 CALL SYSCKM (NCONT) C C 15 16 24 25 26 27 WRITE I N I T I A L PROPERTIES OF THE LUMPED MASS SYSTEM WRITE(6,15) FORMAT(1 H1,• * * I N I T I A L PROPERTIES OF THE LUMPED MASS SYSTEM** /1H0,' NO. ALFA BETA»,10X,»G•,11X,•C»,11X,•M' 7X,'DEPTH',8X,•SIGZ'/) WRITE (6, 16) (I,PROP (1,16) ,PROP (1,17) ,-GG (I) ,CC(I) , WT (I) ,DEPTH (I) S I G Z ( I ) ,1=1 ,N) FORMAT(1X,14,1P7E12.3) WRITE (6,24) FORMAT('0***INITIAL STIFFNESS MATRIX***') J = 2*N-1 WRITE (6, 25) (SK (I) ,1=1, J , 2) FORM AT(' DIAGONAL TERMS :•/(10X,1P10E11.3)) WRITE(6,26) (SK (I) ,1=2, J , 2) FORMAT(' OFF-DIAGONAL TERMS :•/(10X,1P10E11.3) ) WRITE(6,27) FORMAT ('0***INITIAL DAMPING MATRIX***') WRITE(6,25) (CM (I) ,1=1, J , 2) WRITE(6,26) (CM (I) ,1=2, J,2) C C** . START DYNAMIC ANALYSIS C READ (5,18) NEQ,INTYP,NC,NCPR,NCPRM,NPCON,NCONT,NPLD,DT 18 FORMAT(8I4,3F10.0) WRITE (6,28) NEQ,INTYP,NC,NCPR,NCPRM,NPCON,NCONT,NPLD 28 FORMAT ('6*** CONTROL NUMBERS FOR DYNAMIC RESPONSE ANALYSIS ***• 10X,'NEQ =',I4,» ; INTYP =',I4,« ; NC =',14 ' ; NCPR =',I4,» ;•/10 X,'NCPRM =',14,' ; NPCON = 14,' ; NCONT =',14,' ; NPLD =',I4,' ;•) LPLOT = .FALSE. RDFR = 1. I F (NPLD. EQ. 1) LPLOT=. TRUE. I F (NCPRM.EQ.0) NCPRM=NCPR I F (NEQ. NE. 1) GO TO 1056 READ(5,30) NCARD,NREC,NFTS,RDFR,FMT 30 FORMAT(3I4,F10. 0/20A4) WRITE (6,29) NCARD,NREC,NFTS,RDFR,FMT 29 FORMAT('0*** EARTHQUAKE RECORD USED AS INPUT ***'/10X,'NCARD =• 14,' ; NREC =',I4,» ; NFTS =',I2,' ; MULTIPLYING 'FACTOR =',F5.2/10X,'FORMAT OF INPUT : ',20A4) NCINP = NCARD*NREC GO TO 1057 1056 READ(5,22) NCINP,NFTS,AMP,OMEGA 22 FORMAT (214,2F10.4) L I S T I N G OF LIQUEFACTION PROGRAM - AUG, LIQPG 75 1?8 31 , 1057 3333 19 1050 23 20 WRITE(6,31) NCINP,AMP,OMEGA,NFTS FORMAT(* 0*** SINUSOIDAL BASE INPUT I S USED * * * » / 1 0 X , . * NUMBER OF BASE ACC. INPUT =',15,' ; AMPLITUDE = », 1PE11.3,' ; OMEGA =\,E11.3,' ; NFTS = ' , 0 P I 2 , « .») NREC = 1 0 READ (5,3) NS.UBDT WRITE (6,3333) NSUBDT I F (NSUBDT.EQ.0) NSUBDT=1 F O R M A T ( » 0 * * * * * SUB-DIVISION OF TIME INTERVAL FOR », •CONSOLIDATION =',I3,» ****••) ACSC = RDFR I F (NFTS.EQ.0) ACSC=32.17*RDFR I F (DT.LE.0.1) GO TO 1050 WRITE(6,19) DT F O R M A T ( » 0 * * * TIME INTERVAL TOO LARGE, D T = « , F 6 . 3 , » ! » * * * ' ) STOP READ (5,1) T I T L E WRITE(6,23) T I T L E FORMAT(«0*****»,20A4,•*****»/) WRITE (6,20) DT FORMAT (* 0** TIME INTERVAL USED, D T = , F 6 . 3 , » SEC. ***») IF(NCONT.EQ.0) GO TO 1052 CALL TM1VAL(UG1,DT,TNOT) , C C I N I T I A L CYCLE C 1052 21 WRITE(6,21) FORMAT(1H1,* OUTPUT VALUES ARE IN THE FOLLOWING ORDER : * * ' / » ** LAY.NO.,ACC., VEL., DISPL., GAMMA., TAU, EVP, PP. . • **«//) C C START R E P E T I T I V E INTEGRATION, (PP GEN. AND PP DIS. I F C 100 1051 1034 UG1 = 0.0 I F (. NOT. LREAD) GO TO 1034 CALL GETACC(AE,FMT,DT,AMP,OMEGA,NEQ,NREC,NCINP,TNOT) DO 1051 1=1,NREC A E ( I ) = ACSC*AE(I) LREAD = .FALSE. NRECC = NRECC+1 NCPRC = NCPRC+1 NPRMC = NPRMC+1 INCR = INCR+1 ITERNO = 0 CA = INCR TIME = CA*DT+TNOT DDT = DT I F (INCR.GT.NC) GO TO 9999 DGI = AE (NRECC) -UG 1 UGII = UGI UG1 = AE (NRECC) I F (NRECC. NE. NREC) GO TO 1023 NRECC = 0 REQUIRED) LISTING OF LIQUEFACTION PROGRAM -• AUG, 75 LIQPG 179 LREAD = .TRUE. 1023 CONTINUE C C************** 1054 CALL SYSCKM(NCONT) r j * * * * *********** C 1029 DO 1063 1=1,N BAP (I) = BA(I) BBP(I) = BB(I) XD2P(I) = XD2(I) XD1P (I) = XD1 (I) XDOP (I) = XDO (I) GD2P(I) = GD2(I) GD1P (I) = GD1 (I) GAMMAO (I) = GAMMA (I) 1063 TAXYP(I) = TAXY(I) C CALL NUMERICAL INTEGRATION SCHEME C c**************** LIT = .FALSE. 1064 CALL INTSCH(DDT,UG1,UGII) C**************** C I F (LJRV) GO TO 1062 C ( 3 * * * * * * * * * * * * * * * * 1065 CALL CHEKRV (DT,DDT,UG1,UGT,UGII,ITERNO,NRETN) C**************** C GO TO (1062,1054,1064,1065),NRETN WRITE(6,93) NRETN 93 FORMAT (' 0 * * SOMETHING WRONG, NRETN = ,I4,» ***•) STOP C 1062 CONTINUE IF (NPTYPE. LT. 2) GO TO 1035 LSTOP = .FALSE. C , c*************** CALL PPGEN(TIME,LCPPG,LSTOP) IF(LSTOP) GOTO 9 9 9 9 C*************** c 1061 C I F (.NOT.LPPDIS) GO TO 1058 c**************** CALL PPDISN(DT,NSUBDT,LCPPG) C**************** C GO TO 1035 1058 I F (.NOT.LCPPG) GO TO 1035 DO 1059 1=1,N 7 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 8 180 1059 C 1035 1666 C C IF (.NOT.DODEVP(I)) GO TO 1059 PP (I) = PP (I) +DPP (I) CONTINUE , DO 1666 1=1,N IF (PP(I) .GT,SIGZ(I)) PP (I) =SIGZ (I) CONTINUE • . LPMAX = .FALSE. CALL MAXVAL (TIME,UG1) IF(LPLOT) WRITE (7) TIME, UG1 , (I, ABXD2 (I) , XD1 (I) , XDO (I) , GAMMA (I) , TAXY (I) , EVP (I) ,PP(I) ,I=1,N) 1055 I F (NCPRC.LT.NCPR) GO TO 1667 NCPRC = 0 C DISPLACEMENT IS REFERRED TO THE TOP OF THE SOIL LAYER WRITE(6,90) TIME,UG1 , GG (1) 90 FORMAT (1H0, •TIME = ,F6.3,' S E C ; ACC. = ,F8.t»,' FPSS', 1P2E13.4) IF (NPTYPE.LE. 1) WRITE (6,91) (I, ABXD2 (I) ,XD1 (I) ,XD0 (I) , GAMMA (I) , TAXY (I) ,1=1, N) 91 FORMAT (1X,IU, 1P5E1 2. 3) IF (NPTYPE. GT. 1) WRITE(6,92) (I,ABXD2(I) ,XD1 (I) ,XD0 (I) ,GAMMA (I) , TAXY (I) ,EVP (I) ,PP (I) ,1=1 ,N) 92 F0RMAT(1X,IU,1P7E12.3) 1667 I F (NPRMC.LT.NCPRM) GO TO 100. NPRMC = 0 LPMAX = .TRUE. CALL MAXVAL (TIME,UG1) . GO TO 100 9999 IF (NPCON.NE.1) GO TO 9998 WRITE(8) TIME,UG 1 , DT DO 1060 1=1,N 1060 WRITE (8) I,XD2 (I) ,XD1 (I) ,XD0 (I) ,TAXY (I) , EVP (I) , PP (I) ,GAMMA (I) , GAMMAO (I) , GAMR (I) ,TAUR (I) , DIRN (I) 9998 I F (LPLOT) END FILE 7 C LPMAX = .TRUE. CALL MAXVAL (TIME,UG1) C WRITE (6,150) INCR 150 FORMAT(* 0******************************************* */ • * NORMAL TERMINATION OF PROGRAM *'/ » * NUMBER OF INCREMENTAL CALCULATION =',14,' *•/ , • , * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i ) STOP C *********************** C * END OF MAIN PROGRAM * C *********************** END BLOCK DATA LOGICAL*1 LVDIR (20) ,LREV (20) ,LRG (20) , DODEVP (20) ,LLIQ (20) , LDOF1,LLIN,LSTOP,LMCLG L I S T I N G OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 9 181 COMMON /NCONTL/N,NDAM P,INCR,INTYP,NVOL,NBT,NBB /LCONTL/LDOF1,LLIN,LSTOP,LMCLG /LOGCOM/LVDIR,LREV,LRG DODEVP,LLIQ /PROPTY/TITLE (20) ,PROP(20, 17) ,H (20) ,DEPTH(20) ,WIDTH(20) ,WT (20) , MATYP(20),SIGZ(20) ,CC(20) ,GG (20) ,NSUBD(20) ,GAMR(20) , TAUR (20) ,CM (40) , SK (40) ,GMN (20) ,TMN (20) . /DYNEQT/DYM (40) , PEP (20) ,FOREP (20) . /CALVAL/DIRN(20) ,XD2(20) ,ABXD2(20) ,XD1 (20) ,XD0(20) , B D ( 2 0 ) , BA (20) ,BB(20) ,GD2 (20) ,GD1 (20) ,GAMMA (20) , GAM MAO (20) , . GAMMAX (20) ,TAXY (20) , EVP (20) ,SK1R (20) ,DPP (20) ,PP (20) . /CPVALU/BAP (20) ,BBP(20) ,XD2P (20) ,XD1P(20) ,XD0P(20) ,GD2P(20) , GD1P(20) ,TAXYP(20) . , /REVPAR/NRPT (20) ,GGAMR (20, 8) ,TTAUR (2 0,8) , DIRNR (20,8) DIMENSION BLOC(360) ,BLOC2 (160) EQUIVALENCE (BLOC (1) , DIRN (1) ) , (BLOC2 (1) ,BAP (1) ) DATA INCP/0/, LDOF1/.FALSE./, L L I N / . F A L S E . / , LSTOP/.FALSE./, LMCLG/.FALSE./, L V D I R / 2 0 * . F A L S E . / , LEEV/2 0*.FALSE./, LRG/20*.FALSE./, DODEVP/20*.FALSE./, LLIQ/20*.FALSE./, DEPTH/20*0.0/, WT/20*0.0/, S I G Z / 2 0 * 0 . 0 / , CC/20*0.0/, GG/20*0.0/, BLOC/360*0.0/, BLOC2/16O*0.0/, FORER/20*0.0/, NRPT/20*0/, GGAMR/20*0.0/, TTAUR/20*0.0/ END SUBROUTINE TM1VAL(TNOT,UG1,DT) COMMON " . /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB . /PROPTY/TITLE(20) ,PROP(20,17) ,H(20) ,DEPTH (20) ,WIDTH (20) ,WT(20) , HATYP(20) ,SIGZ (20) ,CC (2 0) ,GG (20) ,NSUBD (20) ,GAMP. (20) , .. TAUR (20) ,CM (40) , SK (40) , G M N (20) ,TMN (20) . /CALVAL/DIRN (20) ,XD2 (20) ,ABXD2 (20) ,XD1 (20) ,XD0 (20) ,BD(20) , BA(20) ,BB(20) ,GD2(20) ,GD1 (20) ,GAMMA(20) ,GAMMAO(20) , GAMMAX (20) , TAXY (20) ,EVP (20) , SK1R (20) , DPP (2 0) ,PP (20) . /REVPAR/NRPT (20) ,GGAMR (20,8) ,TTAUR (20,8) , DIRNR (20,8) WRITE (6,21) FORMAT ('0*** SUBROUTINE TM1VAL I S CALLED ***•/) READ (8) TN0T,UG1,DT1 WRTTE(6,22) TN0T,UG1,DT1 FORMAT(* TIME =',F7.3,' SEC.; A C C = « , F 7 . 3 , ' FPSS; TIME', » INTERVAL = , F 5 . 3 , ' SEC.'/) TEM = ABS (DT-DT1) I F (TEM. GT. 0. 0005) GO TO 888 WRITE (6,23) FORMAT('ONO. ACC',9X,'VEL•,8X,'DISP',7X,'TAUXY•,9X,'EVP•, 10X,*PP',7X,'GAMMA',6X,'GAMMAO',8X,'GAMR »,8X,'TAUR', 7X,'DIRN'/) . . . „ . 21 22 f , 23 C DO 101 1=1,N READ (8) a,XD2 (I) , XD1 (I) ,XD0 (I) , TAXY (I) ,EVP(I) ,PP(I) ,GAMMA (I) , GAMMAO (I) , GAMR (I) , TAUR (I) , DIRN (I) I F ( J . NE.I) GO TO 887 BA (I) = XD2 (I) *DT BB(I) = (XD1 ( I ) + 0 . 5*BA (I) ) *DT LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 10 182 101 24 888 25 887 26 999 14 15 11 17 12 18 102 WRITE(6,24) J,XD2(I) ,XD1 (I) ,XD0(I) ,TAXY(I) ,EVP(I) ,PP(I) , GAMMA (I) , GAMMAO(T) , GAMR (I) ,TAUR(I) ,DIRN(I) FORMAT (1X,12,1X,1P10E12.4,0PF6.1) RETURN' WRITE (6, 25) FORMAT (»0**** SOMETHING WRONG, D T " S ARE NOT EQUAL !!! ****«) GO TO 999 WRITE(6,26) FORMAT(* 0**** SOMETHING WRONG, I NOT EQUAL TO J I ! ! ****») STOP END SUBROUTINE GETACC{AE,FMT,DT,AMP,OMEGA,NEQ,NREC,NCINP,TNOT) LOGICAL*1 LDOF1,LLIN,LSTOP,LMCLG,LO,L1,L2,L3 COMMON . /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB . /LCONTL/LDOF1,LLIN,LSTOP,LMCLG DIMENSION AE (10) ,FMT(20) DATA LO/.FALSE./,L1/.FALSE./,L2/.FALSE./,L3/.FALSE./ , IF (L3) GO TO 999 IF (INCR. GT. NCINP) GO TO 99 > I F (LO) GO TO 11 IF (L1) GO TO 12 IF (L2) GO TO 13 IF (NEQ. NE. 0) GO TO 14 LO = .TRUE. GO TO 11 • I F (NEQ.NE.1) GO TO 15 L1 •= .TRUE. GO TO 12 . FNEQ = FLOAT (NEQ) RNPI = FNEQ*3.1415926 RNPI = 1./RNPI L2 = .TRUE. GO TO 18 FI = INCR T = FI*DT DO 17 1=1,10 T = T+DT TEM = T+TNOT TEM = TEM*OKEGA AE(I) = AMP*SIN (TEM) CONTINUE RETURN READ (5, FMT, END=99) (AE (IJ) ,IJ= 1 , NREC) RETURN I F (TNOT.LT.DT) GO TO 13 WRITE(6,102) TNOT FORMAT(•0**** GRADUALLY INCREASING AMPLITUDE REQUIRED, BUT TNOT *, • IS NON-ZERO, TNOT = ,F6.2,« SECS. ****') GO TO 99 F I = INCR T = FI*DT DO 19 1=1,10 , 13 L I S T I N G OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 11 183 C 21 20 19 99 22 999 T = T+DT TEM = T*OMEGA ACCT =. AMP*SIN (TEM) TEM = TEM*RNPI TEM IS DIVIDED BY N PI I F (TEM. GE. 1.0) GO TO 21 TEM = TEM*1.570796 TEM = SIN (TEM) ACCT = ACCT*TEM GO TO 20 LO = .TRUE. L2 = .FALSE. A E ( I ) = ACCT CONTINUE RETURN DO 22 1=1,10 A E ( I ) = 0.0 L3 = .TRUE. RETURN END SUBROUTINE SYSCKM(NCONT) LOGICAL*1 LVDIR (20) ,LREV (20) ,LRG (20) ,DODEVP (20) ,LLIQ (20) , LDOF1,LLIN,LSTOP,LMCLG,LJUMP1,LJUMP2,L3,L4 COMMON . /NCONTL/N,NDA MP,INCR,INTYP,NVOL,NBT,NBB . /LCONTL/LDOF1,LLIN,LSTOP,LMCLG . /LOGCOM/LVDIR ,LR EV,LRG,DODEVP, LLTQ. . /PROPTY/TITLE(20) ,PROP(20,17) ,H(20) ,DEPTH(20),WIDTH(20) ,WT(20) , MATYP{20) ,SIGZ(20) ,CC(20) ,GG(20) ,NSUBD(20) ,GAMR(20) , TAUR (20) ,CM (40) ,SK (40) ,GMN (20) ,TMN (20) . /CALVAL/DIRN(20) ,XD2(20) , ABXD2(20) , XD1 (20) ,XDO (20) , BD (20) , BA(20) , BB ( 20) ,GD2(20) ,GD1 (20) ,GAMMA (20) ,GAM M AO (20) , GAMMAX (2 0)",TAXY (20) ,EVP (20) , SK1R (20) , DPP (20) ,PP (20) . /REVPAR/NRPT (20) ,GGAMR (20,8) , TT AUR (20,8) , DIR NR (20,8) DIMENSION FK (20) DATA LJUMP1/.FALSE./,LJUMP2/.FALSE./,KPRT/0/ DATA RLIM1/0.0500/, RLIM2/0.05/ C C THIS PORTION FINDS SHEAR MODULUS FOR A LAYER c IF C C 27 15 C C (LJUMP1) GO TO 14 HERE IS FOR LINEAR CASE OR FIRST INCREMENT DO 15 1=1,N GMN (I) = PROP (1,1) TMN(I) = PROP(I,4) GG (I) = GMN (I) I F (.NOT. LLIN) LJUMP1 = .TRUE. GO TO 50 HERE IS NON-LINEAR CASE, AND INCR > 1 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 12 184 14 17 C C 19 2005 29 28 31 30 DO 16 1=1,N I F (.NOT. LVDIR (I) ) GO TO 16 IF (.NOT.DODEVP(I)) GO TO 17 EVPI = EVP (I) CEP 1 = 1. +EVPI/ (PROP (1,2) + PROP (1,3) *EVPI) CEP2 = 1.+EVPI/(PROP (1,5)+PROP (1,6) *EVPI) RATS = 1.-PP (I)/SIGZ (I) SATT = RATS IF (RATT.LT. RLIM1) RATT=RLIM 1 TMN(I) = P R O P (1,4) *CEP2*RATT RATT = SQRT (RATS) IF (RATT. LT.RLIM2) RATT=RLIM2 GMN (I) = PROP (I, 1) *CEP1*RATT CONTINUE J = NRPT(I) GMNT = GMN (I) TMNT = TMN (I) DIRNI = DIRN (I) GAMI = GAMMA (I) TAUI = TAXY (I) GAMAB = ABS (GAMI) TAUAB = ABS (TAUI) IF (. NOT. LRG (I) ) GO TO 20 LRG(I) = -FALSE. HERE REVERSAL OCCURS, INCREASE NRPT BY 1 I F (J.LT.R) GO TO 29 . IF (J.EQ.8) GO TO 29 WRITE (6,2005) INCR,I,J FOR MAT (' **** AT INCR',14," FOR LAY.»,I3,« » ****!) STOP J = 7 NRPT (I) = J GO TO 40 TEM1 = GAMR(I)*GMNT/TMNT TEM2 = TAUR (I) /TMNT IF (J.EQ.O) GO TO 30 TDIF = TEM2-TTAUR (I, J) TPRD = DIRNR (I, J) *TDIF IF (TPRD.GT.0.0) GO TO 31 J = J+1 NRPT (I) = J DIRNR (I, J) = SIGN (1. ,TDIF) GGAMR (I, J) = TEM 1 TTAUR (I, J) = TEM2 GO TO 40 J = 1 NRPT (I) = 1 DIRNR (1,1) = SIGN (1. ,TEM2) GGAMR (1,1) = TEM 1 TTAUR(I,1) = TEM 2 GO TO 40 J > 8 AND =',19, LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 13 185 C 20 C C C C 21 C I F (J.EQ.O) GO TO 18 CHECK FOR REDUCTION IN NRPT, I E . CHECK FOR ENVELOP CURVES :NOTE THAT THE LAST BUT ONE REVDRSAL POINT IS ALSO THE LIMIT FOR SWITCHING TO ENVELOP CURVE I F (J.GT. 1) GO TO 21 TEMP = ABS (TTAUR (I, 1) ) *TMNT IF (TAUAB.LT.TEMP) GO TO 40 J = 0 NRPT (I) = 0 GO TO 18 TEMP = TTAUR (I,J-1) *TMNT CHECK FOR CONSISTENCY BETWEEN DIRN AND TTAUR'S TDIF = TEMP-TTAUR (I, J) *TMNT TPRD = DIRNI*TDIF IF (TPRD) 24,25,25 24 KPRT = KPRT+1 J = J-1 NRPT (I) = J IF (KPRT. GT. 50) GO TO 20 IF (J.NE.O) GO TO 26 WRITE(6,2004) INCR,I,J GO TO 20 26 J1 = J+1 WRITE (6,2004) TNCR,I,J, (TTAUR(I,K) ,K=1,J1) ,TAUI 2004 FORMAT (1X,314,1P9E13.3) GO TO 20 25 I F (TDIF) 22,23, 23 23 . IF (TAUI.LT.TEMP) GO TO 40 J = J-2 NRPT (I) = J GO TO 20 22 IF (TAUI.GT.TEMP) GO TO 40 J = J-2 NRPT (I) = J GO TO 20 40 TEMP = GGAMR(I,J)*TMNT/GMNT GGT = 1. 0/(1.0 + 0.005*GMNT*ABS (GAMI-TEMP)/TMNT) GO TO 116 18 GGT = 1.0/(1.0+0.01*GMNT*GAMAB/TMNT) 116 GGT = 0.98*GGT*GGT*GMNT RLG = 1./PROP (1,1) IF (GGT*RLG.LT.0.005) GGT=0.005/RLG GG(I) = GGT C IF (TAUAB. LT. TMNT) GO TO 16 KPRT = KPRT+1 IF (KPRT.GT.50) GO TO 16 IF (KPRT. EQ. 50) GO TO 45 IF (TAUI) 42,43,43 42 I F (DIRNI.GT.0.0) GO TO 16 WRITE(6,2001) INCR,I LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 186 2001 43 2002 45 2003 C 16 C FORMAT(' ** AT I N C R ' , 1 4 , ' LAY.• , 1 3 , • TAXY < -TMN *) GO TO 16 I F (DIRNI.LT.0.0) GO TO 16 WRITE(6,2002) INCR,I . , FORMAT (' ** AT I N C R ' , 1 4 , ' L A Y . ' , 1 3 , • TAXY > TMN') GO TO 16 WRITE (6,2003) FORMAT(* *** NO MORE WRITING FOR |TAXY| > |TMN| ***') CONTINUE 50 CONTINUE C THIS PORTION FORMS SYSTEM DAMPING C C** C C CC AND G VECTORS CONTAIN LAYER DAMPING AND STIFFNESS VALUES C AND K ARE BANDED DAMPING AND STIFFNESS MATRICES C 57 56 59 C 58 52 C AND STIFFNESS MATRICES IF (LJUMP2) GO TO 58 LJUMP2 = .TRUE. AA1 = PROP(1,16) BB1 = PROP (1,17) DO 56 1=1,N IF (NDAMP.EQ.1) GO TO 57 CC(I) = BB1*GG(I) GO TO 56 CC(I) = PROP (I, 16) CONTINUE DO 59 1=1,N 12 = 2*1 I2M1 = 12-1 CCI = CC (I) *WIDTH (I)/H (I) CCI1 = CC (1 + 1) *WIDTH (1+1)/H (1+1) CM(I2M1) = CCI IF (NDAMP.EQ. 2) CM (I2M1) =CM (I2M1) +AA1*WT (I) IF (I.GE.N) GO TO 59 CM(I2M1) = CM (I2M1)+CCI1 CM (12) = -ecu CONTINUE DO 52 1=1,N FK(I) = GG (I) *WIDTH (I)/H (I) DO 51 1=1,N 12 = 2*1 I2M1 = 12-1 FKI = FK(I) FKI1 = FK (1+1) SK (I2H1) = FKI IF (I.GE.N) GO TO 51 SK(I2M1) = SK (12M1) + FK11 . LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 15 187 SK(I2) = -FKI1 CONTINUE IF (INCR. LT. 113) GO TO 9999 IF (INCR.GT. 120) GO TO 9999 ; WRITE(6,2999) INCR, (GAMMA (K) , K= 1 , N) WRITE (6,2999) INCR, (TAXY(K) ,K=1,N) FORMAT (1X,I4,1P10E12.3/(5X,10E12. 3) ) RETURN END SUBROUTINE SOLB2 COMMON /NCONTL/N,NDAMP,INCR,INTYP,N70L,NBT,NBB . /DYNEQT/DYM (40) ,PEP(20) ,FORER(20) DIMENSION T(20) C DYM BECOMES THE LOWER TRIANGULAR MATRIX C T IS THE UPPER TRIANGULAR MATRIX C NOTE L33 IS DYM (5) , ETC, T12 IS T ( 1 ) , T23 IS T (2) ETC. IF (ABS(DYM(1)).GT.1.0E-75) GO TO 103 WRITE (6, 10) DYM (1) 10 FORMAT («0*** ZERO ENCOUNTERED IN ROW 1 !!! *** DYM (1)=•,E12.4) STOP 103 CONTINUE PEP(1) = PEP (1)/DYM (1) IF(N.EQ.1) RETURN DO 101 1=2,N IM1 = 1-1 MN = 2*IM1 NN = MN+1 MM = MN-1 AMN = DYM (MN) • T (IM1) = AMN/DYM (MM) DYM (NN) = DYM (NN)-T (IM1) *AMN IF (ABS (DYM (NN) ) .GT. 1. OE-75) GO TO 104 WRITE(6,11) I , NN , DYM (NN) 11 FORMAT (* 0*** ZERO ENCOUNTERED IN ROW',13,' !!! *** DYM(',I2, •)=',E12.4) STOP 104 CONTINUE PEP(I) = (PEP(I) -AMN*PEP(IM1) )/DYM(NN) 101 CONTINUE C BACK SUBSTITUTION DO 102 1=2,N MN = N-I + 1 NN = MN+1 102 PEP,(MN) = PEP(MN) -T(MN) *PEP(NN) RETURN END SUBROUTINE CHEKSV(DT,DDT,UG1,UGI,UGII,ITER NO,NRETN) LOGICAL*1 LDOF1,LLIN,LSTOP,LMCLG,LJUMP,LITE,LCPPG,LPRT, LVDIR (20) , LREV (20) , LRG (20) ,DODSVP (20) , LLIQ (20) , LRGL(20),LJRV,LCKF3,LIT,LPMAX COMMON /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB /LCONTL/LDOF1,LLIN,LSTOP,LMCLG,LJRV,LCKFB,LIT,LPMAX 51 C C C C C2999 9999 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 16 188 C C 102 101 103 104 100 111 9 124 C /LOGCOM/LVDIR,LREV,LRG,DODEVP,LLIQ /PROPTT/TITLS (20) , PROP (20, 17) , H (20) , DEPTH (20) , WIDTH (20) ,WT (20) MATYP (2 0) ,SIGZ (20) ,CC (20) ,GG (20) , NSUBD (20) , G AMR (20) , TAUR (20) ,CM (40) ,SK{40) ,GMN (20) ,TMN (20) /DYNEQT/DYM (40) ,PEP (20) ,FORER (20) /CALVAL/DIRN (20) ,XD2(20) ,ABXD2(20) ,XD1 (20) , XDO (20) ,BD(20) , BA (20) , BB (20) ,GD2 (20) ,GD1 (20) ,GAMMA (20) ,GAMMAO (20) , GAMWAX(20) ,TAXY(20) ,EVP(20) ,SK1R(20) ,DPP(20) ,PP(?0) . /CPVALU/BAP (20) ,BBP (20) ,XD2P (20) ,XD1P (20) ,XD0P (20) ,GD2P (20) , GD1P (20) ,TAXYP (20) DIMENSION NREV (20) ,GAMRO (20) ,TAURO (20) DATA JCALL/0/,LJUMP/.FALSE./,LPRT/.TRUE./,JPRINT/0/, LRGL/20*. FALSE. / IF (INCR. EQ. 50) LPRT=. FALSE. I F (LJUMP) GO TO 100 IF (JCALL.GE.1) GO TO 101 DO 102 1=1,N LVDIR (I) = . FALSE. DO 103 1=1,N IF (GAMMA (I) .EQ.0.0) GO TO 103 IF (LVDIR (I) ) GO TO 103 DIRN(I) = SIGN (1. , GAMMA (I) ) LVDIR (I) = .TRUE. CONTINUE JCALL = JCALL+1 I F (JCALL. EQ. 1) GO TO 131 DO 104 1=1,N • I F (. NOT. LVDIR (I) ) GO TO 100 CONTINUE LJUMP = .TRUE. I F (ITERNO.NE.O) GO TO 111 NDT = 0 NDDT = • 10 NTDT = 0 ITERNO = ITERNO+1 IF (ITERNO.LT.11) GO TO 124 WRITE (6,9) ITERNO FORMAT(' ***ITERNO =',14,' ***») STOP CONTINUE CHECK GAMMA FOR REVERSAL LITE = .FALSE. DO 105 1=1,N LREV (I) = .FALSE. IF (LRGL (I) ) GO TO 105 IF (.NOT.LVDIR(I) ) GO TO 105 TEMY = GAMMA (I)-GAMMAO (I) IF (TEMY. EQ. 0.0) GO TO 105 TEMY = DIRN (I) *TEMY IF (TEMY. GT. 0.0) GO TO 105 LREV (I) = .TRUE. LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 17 189 LRG (I) = .TRUE. GAMRO (I) = GAMR (I) TAURO (I) = TAUR (I) GAMR (I) = GAMMAO (I) TAUR (I) = TAXYP(I) GAMMAX (I) = GAMMAO (I) DIRN (I) = -DIRN (I) IF (ABS(GAMMAO(I)) .LT.1.E-8) GO TO 105 LITE = .TRUE. 105 CONTINUE IF ( (.NOT.LITE) .OR.NTDT. EQ. 10) GO TO 131 C SINCE RE-CALCULATION IS REQUIRED, CHANGE GAMR'S AND DIRN'S BACK C TO ITS VALUES BEFORE CHECKING FOR REVERSAL. DO 123 1=1,N IF (. NOT. LREV (I) ) GO TO 123 GAMR (I) = GAMRO (I) TAUR (I) = TAURO (I) DIRN(I) =-DIRN(I) 123 CONTINUE C NREV IS A NO. USED FOR INDICATING NEAREST TENTH OF (DT) FOR C (GAMMA-DOT)=ZERO. DO 106 1=1,N I F (. NOT. LREV (I) ) GO TO 106 IF (ABS (GAMMAO (I) ) .GT. 1. E-8) GO TO 120 NREVI = 1 0 GO TO 121 120 ' TEMY = GD1P (I) TEMP = TEMY-GD1 (I) I F (ABS (TEMP) .GT. 1. E-20) GO TO 133 IF (LPRT) WRITE (6,3) I NCR , ITERNO, I 3 FORMAT(12X,'INCR =»,I4,», ITERNO =',13,', LAYER',13, ', (GD1-GD1P)=0«) NREVI = 1 0 GO TO 121 133 TEMY = 10.0*TEMY/TEMP NREVI = IFIX (TEMY) IF (NREVI.LE.O) NREVI=0 IF (NREVI.GE.10) NREVI=10 121 CONTINUE NREV (I) = NREVI 106 CONTINUE C DO 107 KNT=1,11 K = KNT-1 DO 108 1=1,N IF (.NOT.LREV(I)) GO TO 103 IF (NREV (I) .EQ*. K) GO TO 109 108 CONTINUE 107 CONTINUE 109 NDT = K IF (NDT.GT.NDDT) NDT=NDDT C IF (NDT. NE. 0) GO TO 112 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 18 190 4 113 112 C 134 117 5 116 DO 113 1=1,N LRG (I) = - FALSE. GAMMA (I) = GAMMAO(I) IF (LRGL (I) ) GO TO 113 IF (. NOT. LREV (I) ) GO TO 11 3 I F (NREV (I) . NE. 0) GO TO 113 LRG (I) = .TRUE. LRGL (I) = .TRUE. DIRN(I) = -DIRN(I) IF (LPRT) WRITE(6,4) INCR,ITERNO,I,NDT FORMAT(12X,«INCR =«,I4,', ITERNO =',13,»: ,» CHANGES MOD. , NDT =«,I2) GAMR (I) = GAMMAO (I) TAUR (I) = TAXYP (I) CONTINUE CALL SYSCKM(O) NRETN = 3 LIT = .TRUE. RETURN IF(ITERNO.EQ.1.AND.NDT.EQ.10) GO TO 134 FN = FLOAT (NDT) UG1 = UG1-UGII DDT = 0.1*FN*DT UGII = 0.1*FN*UGI UG1 = UG1+UGII ',13, LIT = .TRUE. CALL INTSCH (DDT,UG1,UGII) DO 116 1=1,N LRG (I) = . FALSE. I F (LRGL (I) ) GO TO 1 16 IF (. NOT. LREV (I) ) GO TO 116 I F (NPEV(I) .EQ.NDT) GO TO 117 TEMY = GAMMA (T)-GAMMAO (I) IF (TEMY. EQ. 0.0) GO TO 116 TEMY = DIRN (I) *TEMY IF (TEMY. GT. 0.0) GO TO 116 LRG (I) = .TRUE. LRGL (I) = . TRUE. I F (LPRT) WRITE (6,4) INCR,ITERNO,I,NDT IF (LPRT) WRITE (6,5) I, ABXD2 (I) , XD1 (I) ,XDO (I) , GAMMA (I) , TAXY (I) ,EVP (I) ,PP (I) FORMAT(20X,I4,1P7E12.3) DIRN (I) = -DIRN (I) GAMR(I) = GAMMA (I) TAUR (I) = TAXY (I) CONTINUE NTDT = NTDT+NDT NDDT = 10-NTDT FN = FLOAT(NDDT) DDT = 0.1*FN*DT UGII = 0.1*FN*UGI LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 19 191 131 DG1 = UG1+UGII NRETN = 2 I F (NTDT.GE.10) NRETN=4 RETURN NRETN = 1 DO 132 100 C IK 103 102 121 132 1 = 1 , N . LRGL (I) = .FALSE. RETURN END SUBROUTINE INTSCH(DT,UG1,UGI) LOGICAL*1 LJUMP,L2,L3,L4,LDOF1,LLIN,LSTOP,LMCLG, LJRV,LCKFB,LIT,LPMAX COMMON . /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB . /LCONTL/LDOF1,LLIN,LSTOP,LMCLG,LJRV,LCKFB,LIT,LPMAX . /PROPTY/TITLE(20) , P R O P ( 2 0 , 1 7 ) ,H(20) ,DEPTH(20) ,WIDTH(20) , W T ( 2 0 ) , MATYP (20) ,SIGZ (20) ,CC (20) ,GG (20) ,NSUBD (20) ,GAMR ( 2 0 ) , TAUR(2 0) ,CM (40) , SK (40) , GMN (2 0) , TMN (20) . /DYNEQT/DYM (40) ,PEP (20) ,FORER(20) . /CALVAL/DIRN(20) , X D 2 ( 2 0 ) , A B X D 2 ( 2 0 ) , X D 1 (20) , X D 0 (20) , B D ( 2 0 ) , BA (20) , B B (20) , G D 2 (20) , G D 1 (20) , G AMMA (20) , GAMMAO ( 2 0) , GAMMAX(20),TAXY(20) , EVP ( 2 0 ) , S K 1 R ( 2 0 ) , D P P (20) , P P ( 2 0) . /CPVALU/BAP (20) ,BBP(20) , X D 2 P ( 2 0 ) , X D 1 P ( 2 0 ) , X D 0 P ( 2 0 ) , G D 2 P ( 2 0 ) , G D 1 P (20) ,TAXYP (20) . /REVPAR/NRPT(2 0 ) ,GGAMR (20,3) , T T A U R ( 2 0,8) , DIRNR (20,8) DIMENSION FCC E M (20) , FOCEC (20) , FOCEK (20) ,FORERO(20) DATA LJUMP/.FALSE./ IF (LJUMP) GO TO 100 LJUMP = .TRUE. DFINT = 0 . 5 AFINT = 0.16666667 IF (INTYP. EQ. 1) AFINT=0.25 CONST 1 = DFINT*DT C O N S T 2 = AFINT*DT*DT FORM RIGHT HAND SIDE OF DYNAMICAL EQUATION. DO 1 0 1 1 = 1 , N TEM1 = 0.0 TEM2 = 0.0 DO 102 IJ=1,3 I I = I+IJ-2 = 2* (.1-1) +IJ-1 IF (IK.LE.O) GO TO 103 IF (IK.GE.2*N) GO TO 103 TEM 1 = T E M 1 + C M (IK) *BAP (II) T E M 2 = T E M 2 + SK(IK) *BBP(II) CONTINUE CONTINUE PEP(I) = -WT (I) * U G I - T E M 1 - T E M 2 IF (.NOT. LCKFB) GO TO 1 0 1 IF (.NOT.LIT) GO TO 1 2 1 PEP (I) = PEP (I)-FORERO (I) GO TO 1 0 1 PEP (I) = PEP(I)-FORER (I) LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 20 192 101 C 105 C C C C C 107 106 115 116 114 C CONTINUE FORM LEFT HANDSIDE OF DYNAMICAL EQUATION DO 10S 1=1,N I J = 2*1-1 DYH (IJ) = WT (I) +CONST1 *CM (IJ) +CONST2*SK (IJ) I J = IJ+1 DYM(IJ) = CONST1*CM(IJ)+CONST2*SK (IJ) CONTINUE CALL SOLB2 CALCULATE ACC., VEL.,DISPL., GAMMA AND TAXY ETC. DO 106 1=1,N TEH 1 = PEP (I) XD2 (I) = XD2P (I) + TEM1 ABXD2 (I) = XD2 (I) +UG1 XD1 (I) = XD1P (I) + BAP (I) +TEM1*C0NST1 XDO (I) = XDOP (I) + BBP (I) +TEM1 *CONST2 BA (I) = XD2 (I) *DT BB(I) = XD1 (I) *DT+BA (I) *CONST1 IF (I.NE. 1) GO TO 107 TEM 1 = 1./H(1) GD2 (1) = XD2 (1) *TEM 1 GD1 (1) = XD1 (1) *TEM 1 GAMMA (1) = 100. 0*XD0 (1) *TEM1 GO TO 106 TEM 1 = 1./H(I) GD2(I) = (XD2 (I) -XD2 (1-1) ) *TEM1 GD1 (I) = (XD1 (I)-KD1 (1-1) ) *TEM1 GAMMA (I) = 100. 0* (XDO (I)-XDO (1-1) ) *TEM1 CONTINUE DO 114 1=1,N GMNT = GMN (I) TMNT = TMN (I) IF (LLIN) GO TO 116 J = NRPT (I) I F (J.EQ.O) GO TO 115 TEM 1 = 0. 01* (GAMMA (I) -GGAMR (I, J) *TMNT/GMNT) TAXY (I) = TTAUR (I, J) *TMNT +GMNT*TEM1/(1.0+0.5*GMNT*ABS(TEM1)/TMNT) GO TO 114 TEM 1 = 0.01*GAMMA(I) TAXY (I) = GMNT*TEM1/(1.0 + GMNT*ABS (TEM 1) /TMNT) GO TO 114 TAXY (I) = 0.01 *GMNT*GAMMA (I) CONTINUE DO 111 1=1,N FOCEM (I) = WT (I) *ABXD2 (I) FOCEC(I) = CC (I) *WIDTH (I) *GD1 (I) FOCEK(I) = TAXY (I) *WIDTH (I) LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 21 193 IF (I.EQ.N) GO TO 113 11 = 1+1 FOCEC(I) = FOCEC (I) -CC (11) *WIDTH (11) *GD1 (11) FOCEK (I) = FOCEK (I) -TAXY (11) *WIDTH (11) 113 CONTINUE TEM 1 = FOCEH(I) TEM2 = FOCEC(I) TEM3 = FOCEK (I) TEM4 = TEM1+TEM2+TEM3 FORERO(T) = FORER (I) FORER (I) = TEM4 TEM4 = ABS(TEM4) TEM5 = ABS (TEM1) +ABS (TEM2) +ABS (TEM3) IF (TEM4.LE.TEM5) GO TO 111 LSTOP = .TRUE. WRITE (6,2001) INCR,.I,GDI (I) ,GD2 (I) ,TEM 1, TEM2, TEM 3 , FORER (I) 2001 F0RMAT(1X,2I4,2X,1P6E14.4) 111 CONTINUE IF (LSTOP) STOP RETURN END SUBROUTINE PPGEN(TIME,LCPPG,LSTOP) LOGICAL* 1 LVDIR (20) ,LREV (20) , LRG (2 0) , DODEVP (20) , LLIQ (20) , LLPR(2 0) , LUPMP (20) ,LUPMC (20) ,LNUPK (20) ,LXZY (20) , LPRINT,LCALL,LCPPG,LSTOP COMMON . /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB /LOGCOM/LVOIR,LREV,LRG,DODEVP,LLIQ . /PROPTY/TITLE(20) ,PROP (20,17),H (20),DEPTH (20),WIDTH (20) ,WT (20) , MATYP (20) ,SIGZ (20) ,CC (20) ,GG (20) ,NSUBD (20) ,GAMR (20) , TAUR (20) ,CM (40) , SK (40) ,GMN (20) ,TMN (20) . /CALVAL/DIRN (20) ,XD2 (20) ,ABXD2 (20) ,XD1 (20) ,XD0 (20) ,BD(20) , BA (20) ,BB (20) ,GD2 (20) ,GD1 (20) ,GAMMA (20) ,GAMMAO (20) , GAMMAX (20) ,TAXY (20) ,EVP(20) ,SK1R(20) ,DPP (20) ,PP (20) . /CPVALU/BAP (20) ,BBP(20) ,XD2P(20) ,XD1P(20) ,XD0P(2O) ,GD2P(20) , GD1P (20) ,TAXYP (20) DIMENSION YMXP (20) ,DEVP (20) ,DEVPP (20) ,DPPP (20) ,TFYMP (20) , DEVPT(20) ,YMXT(20) ,VCC1 (20) ,RDENK (20) DATA LCALL/.FALSE./ C , c ********** C C C INITIALIZATION FOR FIRST CALL NOTE PP(I) AND EVP (I) HAS TO BE INITIALIZED IN THE MAIN PROGRAM ********** IF (LCALL) GO TO 105 LCALL = .TRUE. NPRINT = 0 LPRINT = .TRUE. NWLIQ = 0 VCC2 = 0.790 VCC3 = 0.5625 VCC4 = 0.730 POWTM = 0.43 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 22 194 106 C Q C C C C C C 105 C C 101 131 C 102 C C 103 C POWTN = 0.620 SK2T = 0.0025 POWTMC = 1.-POWTM DO 106 1=1,N YMXT(I) = GAMMA (I) BD(I) = GAMMA (I) YMXP (I) = O.'O TFYMP(I) = 0 . 0 DPPP(I) = 0 . 0 LUPMP(I) = .TRUE. VCC1 (I) = 0. 5*PROP (1,7) TEMP = POWTN-POWTM TEMP = POWTM*SK2T*SIGZ (I)**TEMP RDENK(I) = 100.0/TEMP SK1R(I) = RDENK (I) *SIGZ (I) **POWTMC LLPR(I) = • FALSE. RETURN ********** CHECK GAMMA FOR CROSSING ZERO VALUE AND DETERMINE GAMMAX LXZY = .TRUE. FOR GAMMA CROSSING ZERO STRAIN AXIS LUPMP = .TRUE. FOR GAMMA INCREASING IN MAG. DURING LAST INTERVAL LUPMC = .TRUE. FOR GAMMA INCREASING IN MAG. DURING CUR. INTERVAL LNUPK = .TRUE. FOR NEW PEAK OCCURING DURING CURRENT INTERVAL ********** CONTINUE LCPPG = .FALSE. DO 122 1=1,N DPP (I) = 0 . 0 LUPMC (I) = .FALSE. DODEVP (I) = .FALSE. LXZY (I) = .FALSE. TEMP = BD (I) TEMC = GAMMA (I) TPRD = TEMP*TEMC IF (TPRD) 102,103, 101 GAMMAO AND GAMMA ARE ON THE SIDE, CHECK FOR GAMMAX TEMPI = ABS(YMXT(I)) IF (ABS (TEMC) .LE.TEMPI) GO TO 131 YMXT(I) = TEMC IF (ABS (GAMMAX (I)) . LT. ABS (YMXT (I) ) ) GAMMAX (I) =YMXT (I) I F (ABS (TEMC) .GT. ABS (TEMP) ) LUPMC (I) =. TRUE. GO TO 104 CONTINUE GAMMA CHANGES SIGN HERE - I.E. CROSSES ZERO, AXIS, STOP PPGEN LXZY (I) = .TRUE. GO TO 104 CONTINUE EITHER GAMMA OR GAMMAO OR BOTH EQUAL 0 IF (TEMP. NE. 0.0) GO TO 102 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 23 195 104 121 122 C c C C LOPHC(I) = .TRUE. GO TO 121 LNUPK(I) = .FALSE. I F (LUPMP (I) . AND. . NOT. LUPMC (I) ) LNUPK (I) = . TRUE. IF (LUPMC (I)) GO TO 121 LCPPG = .TRUE. DODEVP (I) = '.TRUE. LUPMP(I) = LUPMC (I) CONTINUE ********** CHECK FOR PRINTING GAMMAX ********** IF {.NOT. LCPPG) GO TO 888 DO 107 1=1,N I F (LLPR (I) . AND.LXZY (I) ) GO TO 109 107 CONTINUE GO TO 108 109 I F (. NOT- LPRINT) GO TO 108 NPRINT = NPRINT*1 IF (NPRINT.GT.20) LPRINT=.FALSE. WRITE (6,5) INCR 5 FORMAT('0 INCR =',I4,» GAM-MAX VOL CHANG " TIME •) DO 110 1=1, N IF (. NOT. LLPR (I) ) GO TO 110 WRITE (6,6) I,YHXP(I) ,DEVPP(I) ,DPPP(I) ,TFYMP(I) 6 FORMAT (1 OX, 14 ,1P3E 12. 3, 0PF8. 2) LLPR (I) = -FALSE. 110 . CONTINUE 108 CONTINUE C c ********** C CALCULATE VOLUME CHANGE OR PORE PRESSURE INCREMENT Q 114 115 116 PRES CHAN', ********** DO 130 1=1,N DEVP(I) = 0 . 0 DPP (I) = 0 . 0 IF (.NOT.DODEVP(I)) GO TO 130 IF (. NOT. LNUPK (I) ) GO TO 116 TEMC = EVP (I) TEMP = ABS (YMXT (I) ) IF (TEMP.GT. 1.E-10) GO TO 115 DODEVP(I) = .FALSE. YMXT (I) =0.0 DEVPT(I) = 0 . 0 GO TO 130 TPRD = VCC1 (I)*(TEMP-0.79*TEMC+0.5625*TEMC*TEMC/(TEMP+0.73*TEMC) ) IF (TPRD.LT.0.0) GO TO 114 DEVPT (I) = TPRD I F (LUPMC(I)) GO TO 130 TEMP = ABS (YMXT (I) ) IF (TEMP.LT. 1.E-10) GO TO 130 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 196 C C 117 TEMC = ABS (BD (I) -GAMMA (I) ) IF (LXZY (I)) TEMC=ABS (BD (I) ) DEVP(I) = TEMC*DEVPT (I)/TEMP EVP (I) = EVP (I) +DEVP (I) IF (NVOL.EQ.O) GO TO 117 DPP (I) = DEVP(I) PP (I) = EVP (I) GO TO 118 '. • , CALCULATE PORE PRESSURE INCREMENT SGVP = SIGZ (I)-PP (I) I F (SGVP.GT.0.05*SIGZ (I) ) GO TO 112 SGVP = 0.05*SIGZ (I) I F (LLIQ (I) ) GO TO 112 LLIQ(I) = -TRUE. WRITE ( 6 , 3 ) INCR, I 3 FORMAT(' **** AT INCR =',I4,» ; LAYER',13,' LIQUEFIED ! ****') NWLIQ = NWLIQ+1 N02 = N/2 IF (NWLIQ.LE.N02) GO TO 112 WRITE(6,200U) 2004 FORMAT('0** HALF OF DEPOSIT HAS LIQUEFIED ! **') LSTOP = .TRUE. GO TO 888 112 DEVPS = 0.01*DEVP(I) TEMC = RDENK(I)*SGVP**POWTHC DPP(I) = TEMC*DEVPS SK1R(I) = TEMC 118 CONTINUE IF (. NOT. LXZY (I) ) GO TO 130 LLPR(I) = .TRUE. YMXP(I) = GAMMAX (I) GAMMAX (I) = 0.0 YMXT(I) = GAMMA(I) DEVPP (I) = DEVPT (I) DPPP(I) = DPP (I) 130 CONTINUE 888 DO 889 1=1,N 889 BD(I) = GAMMA (I) RETURN END SUBROUTINE PPDISN(DT,NSUBDT,LCPPG) C SOLUTION OF ONE-DIMENSIONAL HEAT OR CONSOLIDATION PROBLEMS. C MAXIMUM 20 LAYERS C FINITE DIFFERENCE SOLUTION C PROP (I,J) = PROPERTIES OF SOIL LAYER FROM MAIN PROGRAM C SK1R(I) = ONE-D REBOUND MODULUS FOR THE LAYER C H(I) = THICKNESS OF SOIL LAYER C NSUBD (I) = NUMBER OF SUBDIVISIONS IN THE LAYER C PPO (I) = PORE PRESSURES AT THE START OF TIME INTERVAL C DPP (I) = INCREMENTS OF PORE PRESSURE FOR THE TIME INTERVAL C REFERRED TO MID-LAYER C DT = TIME INTERVAL FOR ITERATION LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 197 C C C C C C C C 101 130 131 132 133 C 15 102 103 49 C C N NTB,NBB DZ(I) CV(I) AM (I) NUMBER OF SOIL LAYERS 0 FOR W=0 AT ALL TIMES AT TOP OR BOTTOM BOUNDARY, = 1 FOR (DW/DN) =0 AT ALL TIMES. = DEPTH INTERVAL = COEFFICIENTS OF CONSOLIDATION C.V*DT/(DZ (I) **2) — LOGICAL*1 LVDIR (20) ,LREV(20) ,LRG (20) ,DODEVP(20) ,LLIQ(20) , LDOF1,LLIN,LSTOP,LMCLG,LCPPG,LJUMP,L3,L4 COMMON /LCONTL/LDOF1,LLIN,LSTOP,LMCLG /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB /LOGCOM/LVDIR,LREV,LRG,DODEVP,LLIQ /PROPTY/TITLE(20) ,PROP (20,17) ,H (20) ,DEPTH (20) ,WIDTH (20) ,WT (20) HATYP (20) ,SIGZ (20) ,CC (20) ,GG (20) ,NSUBD (20) ,GAMR (20) , TAUR (2 0) ,CM (40) ,SK (40) ,GMN (2 0) ,TMN (20) /CALVAL/DIRN (20) ,XD2(20) , ABXD2(2 0) ,XD1 (20) ,XD0 (20) ,BD(20) , BA(20) ,BB(20) ,GD2 (20) ,GD1 (20) , GAMMA (20) , GAMMAO (20) , GAMMAX (2 0) ,TAXY (20) ,EVP (20) , SK1R (20) , DPP (20) ,PP (20) DIMENSION W (100) ,TW (100) ,CV (20) , AM (20) ,DZ (20) DATA LJUMP/.FALSE./, W/100*0.0/, TW/100*0.0/ IF (LJUMP) GO TO 25 LJUMP = .TRUE. WRITE(6,101) FORMAT (•0*** DESCRIPTION OF CONSOLIDATION PROBLEM ***«/) I F (NBT. EQ. 0) WRITE(6,130) IF (NBT. EQ. 1) WRITE(6,131) IF (NBB. EQ. 0) WRITE(6,132) I F (NBB. EQ. 1) WRITE(6,133) FORMAT (' AT THE TOP BOUNDARY THE CONDITION IS ( W=0 )') FORMAT (' AT THE TOP BOUNDARY THE CONDITION IS ( DW/DN = 0 )') FORMAT (' AT THE BOTTOM BOUNDARY THE CONDITION IS ( W=0 )•) FORMAT (' AT THE BOTTOM 30UNDARY THE CONDITION IS ( DW/DN = 0 ) ' NTOT IS THE TOTAL NUMBER OF SOIL POINTS FOR PP CALCULATION NTOT = 0 DO 15 1=1,N IF (NSUBD (I) . EQ. 0) NSUBD(I)=2 FN = FLOAT(NSUBD(I)) DZ (I) = H (I) /FN NTOT = NTOT+NSUBD(I) CONTINUE NTOT = NTOT+1 WRITE (6, 102) NTOT FORMAT (» TOTAL NUMBER OF SOIL POINTS FOR PP CALCULATION =',I4) IF (NTOT.LE.100) GO TO 49 WRITE(6,103) FORMAT ('0*** NTOT IS GREATER THAN 100, PROCESSING STOPS.***') STOP CONTINUE INITIALIZING THE VARIABLES DO 16 IJ=1,NTOT LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 26 198 16 105 C C 135 201 50 109 14 107 108 C C 25 17 C 19 W (IJ) = 0. 0 TW(IJ) = 0.0 CONTIN.UE WRITE (6,105) FORMAT('0*** INITIAL PROPERTIES OF SOIL LAYERS ***•/) CHECK THAT ALL AM'S ARE LESS 0.5 LSTOP = .FALSE. IF (NSUBDT.LE.10) GO TO 201 WRITE (6,1 35) NSUBDT FORMAT(•0***** NSUBDT IS TOO LARGE !! =•,14,' ***•*•) STOP RFNDT = FLOAT (NSUBDT) DDT = 0.01605*DT/RFNDT DO 14 1=1,N TEMP = DZ (I) TEMP = TEMP*TEMP AH (I) = DDT*PROP ( I , 15) *SK1R (I)/TEMP IF (AM (I).LT.0.5) GO TO 14 WRITE(6,109) I FORMAT (•0*** AM VALUE IS TOO HIGH FOR LAYER',13,' ***•) LSTOP = .TRUE. CONTINUE WRITE(6,107) FORMAT ('0 LAY.NO. K-S K1R-S THICK. •AM-S NSUBD-S'/) WRITE (6, 108) (I, PROP (I, 15) ,SK1R(I) ,H(I) ,AM(I) , NSUBD (I) ,I=1,N) FORMAT (1X,I6,4X,1P2E12.3,0PF12.2,1PE12.3,3X,0PI6) I F (LSTOP) STOP RETURN PORE PRESSURE REDISTRIBUTION STARTS HERE CONTINUE DO 37 K=1,NSUBDT NP = 0 DO 26 1=1,N NN = NSUBD(I) TEMP = DZ (I) TEMP = TEMP*TEMP AMI = DDT*PROP (1,15) *SK1R (I) /TEMP I F (AMI.LT.0.5) GO TO 17 LSTOP = .TRUE. GO TO 50 CONTINUE DO 26 IJ=1,NN NP = NP+1 IF (I.NE. 1) GO TO 19 IF (IJ.NE. 1) GO TO 21 CALCULATION FOR BOTTOM BOUNDARY IF (NBB. EQ. 0) GO TO 18 TW(1) = 2. *AMI* (W (2)-W (1) )+W (1) GO TO 18 IF ( I J . NE. 1) GO TO 21 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 27 199 C C 21 18 26 C 27 C C CALCULATION FOR INTERFACE OF 2 LAYERS TEM 1' = DZ (I) /SK1F. (I) TEMP = DZ (1-1)/SK1R (1-1) TEMP = TEMP+TEM1 CONST = 2.*DDT/TEMP A = PROP(I-1.,15)/DZ(I-1) B = PROP (I, 15) /DZ (I) TW(NP) = CONST* (A*W (NP-1) + B*W (NP+1) - (A + B) *W (NP) )+W (NP) GO TO 18 CALCULATION FOR POINTS IN A LAYER TW(NP) = AMI*(W(NP-1) + W (NP+1)-2. *W (NP) ) + W (NP) CONTINUE CONTINUE CALCULATION FOR TOP BOUNDARY IF (NBT. EQ. 0) GO TO 27 NP = NP+1 TW(NP) = 2. *AMI* (W (NP-1)-W (NP) )+W (NP) CONTINUE ADD DPP TO HT(I) TO GIVE W (I) NP = 0 DO 30 1=1,N DPI = DPP (I) *RFNDT NN = NSUBD (I) .IF (T.EQ. N) NN=NN+1 DO 30 IJ=1,NN NP = NP+1 IF (NP.EQ.LAND.NBB.EQ.O) GO TO 33 IF (NP.EQ.NTOT.AND. NBT. EQ.O) GO TO 33 IF (I.NE. 1. AND.IJ. EQ. 1) GO TO 32 C NORMAL SOIL POINTS, BOTTOM BOUND. PT. AND TOP BOUND. PT. W(NP) = TW(NP)+DPI GO TO 33 C INTERFACE 32 W (NP) = TW (NP) +0 . 5 * (DPI + DPP (1-1) ) 33 CONTINUE 30 CONTINUE 37 CONTINUE C FIND PP(I) WHICH IS AVERAGE OF W (I) FOR A LAYER NP = 1 DO 35 1=1,N NP = NP-1 NN = NSUBD (I) +1 FN = FLOAT (NN) A = 0.0 DO 36 IJ=1,NN NP = NP+1 36 A = A+TW (NP) 35 PP(I) = A/FN C WRITE (6,9999) (TW (K) , K= 1 , NP) C999 FORMAT (' OWT (HP) : , 1P5E14. 3, (/9X, 1P5E1 4. 3) ) C WRITE (6,9998) (DPP (K) , K= 1 , N) C998 FORMAT (' DPP (I) :»,1P5E14.3, (/9X,1P5E14.3)) 1 LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 200 C C997 C C996 120 C 101 102 103 104 150 C C 200 2001 WRITE (6, 9997) (W (K) , K=1 , NP) FOR MAT(' W(NP) :»,1P5E14.3, (/9X,1P5E14.3)) WRITE (.6,9996) (PP (I) ,T= 1, N) FORMAT (' PP (I) :», 1P5E14.3, (/9X,1P5E14.3)) RETURN END SUBROUTINE MAXVAL(TIME,UG1) LOGICAL*1 LDOF1,LLIN,LSTOP,LMCLG,LJRV,LCKFB,LIT,LPMAX COMMON . /LCONTL/LDOF1,LLIN,LSTOP,LMCLG,LJRV,LCKFB,LIT,LPMAX . /NCONTL/N,NDAMP,INCR,INTYP,NVOL,NBT,NBB . /CALVAL/DIRN (20) ,XD2 (20) ,ABXD2 (20) ,XD1 (2 0) ,XD0 (20) ,BD (20) , . BA(20) ,BB(20) ,GD2(20) ,GD1 (20) , GAMMA (20) ,GAMMAO(20) , GAMMAX (20) ,TAXY (20) , EVP (20) ,SK1R (20) ,DPP (20) ,PP (20) DIMENSION ACCX(20) ,GAMX(20) ,TMAX(20) ,EVPX(20) ,PPMX(20) , TFM (20,5) DATA ACCX/20*0.0/, GAMX/20*0.0/, TMAX/20*0.0/, EVPX/20*0.0/, PPMX/20*0.0/, TFM/100*0.0/, ACCINX/0.0/, T1/0.0/, TP/0.0/ IF (LPMAX) GO TO 200 TEM = ABS (UG1) IF (ACCINX.GE.TEM) GO TO 120 ACCINX = TEM T1 = TIME CONTINUE DO 150 1=1,N TEM = ABS (ABXD2(I)) IF (ACCX (I) .GE.TEM) ACCX(I) = TEM TFM (1,1) = TIME TEM = ABS (GAMMA (I) ) IF (GAMX (I) .GE.TEM) GAMX (I) = TEM TFM (1,2) = TIME TEM = ABS (TAXY (I) ) IF (TMAX (I) .GE.TEM) TMAX(I) = TEM TFM (1,3) = TIME TEM = ABS (EVP (I) ) IF (EVPX (I) .GE.TEM) EVPX(I) = TEM TFM (1,4) = TIME TEM = ABS (PP (I) ) I F (PPMX(I) .GE.TEM) PPMX(T) = TEM TFM (1,5) = TIME CONTINUE RETURN GO TO 101 GO TO 102 GO TO 103 GO TO 104 GO TO 150 PRINT OUT MAXIMUM VALUES BEFORE CONTINUE WRITE (6,2001) TP,TIME FORMAT ( TERMINATION LISTING OF LIQUEFACTION PROGRAM - AUG, 75 LIQPG 29 201 . »0********************************************* • * MAXIMUM VALUES OCCURRED -',F8.3,' TO',F8.3,' / SEC *'/ • ******************************************* ' LAY. TIME ABS ACC -TIME STRAIN TIME ', •STRESS TIME VOL STRAIN TIME PORE PRESS'/) DO 2000 1=1, N WRITE (6,2002) I, TFM ( 1 , 1) , ACCX (I) ,TFM (I , 2) , GAMX (I) ,TFM (I, 3) , TM AX (I) , TFM (1,4) ,EVPX(I) ,TFM (1,5) ,PPMX (I) FORMAT (14,1X,5 (0PF7. 2, 1PE11. 3) ) WRITE (6,2003) ACCINX,T1 FORMAT('0** MAX BASE ACC =',1PE11.3,' : TIME =',0PF5.2, ' SECS **') ; 2000 2002 2003 C C RESET VARIABLES TO ZERO ACCINX = 0,0 DO 160 1=1,N ACCX(I) =0.0 GAMX(I) = 0.0 TMAX(I) = 0.0 EVPX(I) =0.0 PPMX(I) =0.0 160 CONTINUE TP = TIME RETURN END EXECUTION TERMINATED $SIG
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Mechanical model for the analysis of liquefaction of horizontal soil deposits Lee, Kwok Wing 1975
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Title | Mechanical model for the analysis of liquefaction of horizontal soil deposits |
Creator |
Lee, Kwok Wing |
Date Issued | 1975 |
Description | During the development of liquefaction in a soil deposit subjected to vibration there are two processes which work in opposite directions. The volume compaction tendency under cyclic loading causes the pore water pressure to rise, and the dissipation of excess pore water pressure (consolidation) decreases it. Recently, Martin, Finn and Seed (l975) studied the mechanics of pore water pressure generation of a soil sample subjected to cyclic loading and a relation between shear strain cycles, volume compaction and pore water pressure increment was established. A material model based on this relationship is developed in this thesis for saturated granular soil under cyclic simple shear conditions. The model includes a hysteretic stress-strain relationship, volume compaction, pore water pressure rise and dissipation. Using this proposed comprehensive material model, a global mechanical model is constructed to simulate the liquefaction (including consolidation) behavior of a thick horizontal deposit when subjected to horizontal base motion. In this way the coupled problems of dynamic response, pore water pressure rise and consolidation of the deposit under seismic loading can be analysed. The numerical techniques used to solve such problems are discussed in detail. The response of a typical saturated sand deposit under earthquake loading is determined using the proposed model and the results show that the model can predict the various phenomena that saturated sand deposits exhibit during earthquakes. The global model also makes clear the influence of permeability on the liquefaction potential of the soil deposit. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0063017 |
URI | http://hdl.handle.net/2429/19649 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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