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Vibrations in soil with special regard to foundation design : a review of current theories with experimental… Forrest, James Benjamin 1963

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VIBRATIONS IN SOIL WITH SPECIAL REGARD TO FOUNDATION DESIGN:  A REVIEW OF CURRENT THEORIES WITH EXPERIMENTAL WORK IN CLAY SETTLEMENTS by JAMES BENJAMIN FORREST B.Sc,  The U n i v e r s i t y of New Brunswick, 1959  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. i n t h e Department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1963  In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t  of  the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree a v a i l a b l e for reference  that the L i b r a r y s h a l l make i t  and study.  I f u r t h e r agree  mission for extensive copying of t h i s t h e s i s ,for  that  freely per-  scholarly  purposes may be granted by the Head of my Department or by h i s representatives,.  It  i s understood that copying, or p u b l i -  c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n .  Department of  Civil  Engineering  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Da1  =  e  A p r i l 1 8 , 1963.  ;  ii  ABSTRACT  I t i s the purpose of t h i s t h e s i s t o r e v i e w , and t o t o a degree,  evaluate  some of the c u r r e n t t h e o r i e s d e a l i n g w i t h the  effects  of v i b r a t i o n s on f o u n d a t i o n s i n c o n t a c t w i t h the e a r t h . B a s i c parameters,  the means of e v a l u a t i n g them, and t h e i r  a p p l i c a t i o n s are d i s c u s s e d .  C o n d i t i o n s l e a d i n g t o v i b r a t i o n and  shock p r o b l e m s , the s i g n i f i c a n c e of these problems and v a r i o u s c o r r e c t i o n a l methods are p r e s e n t e d .  The e f f e c t s  of v i b r a t i o n s  on f o u n d a t i o n s , w i t h p a r t i c u l a r r e g a r d t o s e t t l e m e n t , are ed by means of m o d i f i c a t i o n s t o the s o i l c h a r a c t e r i s t i c s  consideras  observed by o t h e r w r i t e r s . Long term c o n s o l i d a t i o n t e s t s were c a r r i e d out on u n d i s t u r b e d and remolded c l a y samples,  b o t h v i b r a t e d and u n v i b r a t e d .  These  t e s t s were conducted i n order t o secure a comparison between a c t u a l t e s t r e s u l t s and the c o n c l u s i o n s g i v e n by the above t h e o r y , f o r what may be c o n s i d e r e d an extreme c a s e .  Cohesive s o i l i s known t o  be much l e s s s e n s i t i v e t o v i b r a t i o n than c o h e s i o n l e s s  soil,  v e r y l i t t l e work has been done on i t i n t h i s r e g a r d .  The degree of  independence of c o h e s i o n l e s s  s o i l t o v i b r a t i o n was  w i t h i n the l i m i t s of these t e s t s .  thus  investigated  vi  ACKNOWLEDGEMENT  The A u t h o r w i s h e s t o e x p r e s s h i s t h a n k s t o h i s A d v i s o r , P r o f e s s o r N.D. N a t h a n ,  for h i svaluable  s u g g e s t i o n s and g u i d a n c e .  I t was a p l e a s u r e t o work u n d e r h i s s u p e r v i s i o n .  The A u t h o r  a l s o wishes t o express h i s indebtedness t o P r o f e s s o r J . F . Muir and t h e C i v i l  E n g i n e e r i n g Department  possible.  A p r i l 1963. VANCOUVER, B r i t i s h  Columbia.  f o r making t h i s  thesis  iii  TABLE OF CONTENTS Page ABSTRACT  i i PART I - THEORY OF VIBRATIONS  CHAPTER I .  SOURCES OF VIBRATIONS  1.  CHAPTER I I .  VIBRATION PARAMETERS  3.  A.  GENERAL CONSIDERATIONS  3.  B.  THE COEFFICIENT OF ELASTIC UNIFORM COMPRESSION  4.  C.  THE COEFFICIENT OF ELASTIC NON-UNIFORM COMPRESSION  7.  D.  THE COEFFICIENT OF ELASTIC UNIFORM  9.  E.  THE COEFFICIENT OF ELASTIC NON-UNIFORM SHEAR  10.  F.  MODIFIED COEFFICIENTS  11.  CHAPTER I I I . A.  SHEAR  MODES OF VIBRATION - SIMPLIFIED THEORIES  VERTICAL VIBRATIONS  13. 13.  1.  One Degree o f Freedom  13.  2.  Two D e g r e e s o f Freedom  18.  B.  ROCKING VIBRATIONS  21.  C.  HORIZONTAL VIBRATIONS  23.  D.  COMBINED VIBRATIONS  25.  CHAPTER I V .  VALIDITY OF VIBRATION THEORIES  30.  A.  GENERAL CONSIDERATIONS  30.  B.  NONLINEARITY  32.  C.  THE INERTIA OF THE SPRING BASE  34.  D.  D I F F I C U L T I E S ASSOCIATED WITH DAMPING  37.  CHAPTER V. A.  DESIGN CONSIDERATIONS  40.  FACTORS AFFECTING DESIGN  40.  iv  B.  CONTROL OF SETTLEMENT  '  44.  PART TWO - EXPERIMENTAL WORK CHAPTER V I .  CONSOLIDATION TESTS.  48.  A.  INTRODUCTION  48.  B.  MATERIAL TESTED  49.  C.  SOURCE OF VIBRATIONS  50.  1.  Series I  50.  2.  S e r i e s I I and S e r i e s I I I  51.  D.  TEST PROCEDURE  53.  E.  OBSERVATIONS  55.  CHAPTER V I I . CHAPTER V I I I .  DISCUSSION OF TEST RESULTS SUGGESTIONS FOR FURTHER WORK  57. 60.  SUMMARY AND CONCLUSIONS  62.  BIBLIOGRAPHY  85.  L I S T OP  FIGURES Page  Figure  1.  19.  Figure  2.  19.  Figure  3.  64.  F i g u r e 4.  65.  Figure  5.  66.  Figure  6.  67.  Figure  7.  68.  8.  69.  Figure  9.  70.  Figure  10.  71.  Figure  11.  72.  Figure  12.  73.  Figure  13.  74.  F i g u r e 14.  75.  Figure  15.  76.  Figure  16.  77.  Figure  17.  78.  Figure  18.  79.  Figure  19.  80.  Figure  20.  81.  Figure  21.  82.  Figure  22.  83.  Figure  23.  84.  i Figure  PART I  THEORY OP VIBRATIONS  1.  CHAPTER I SOURCES OF VIBRATIONS Vibrations may arise from operating machinery, earthquakes, explosions, t r a f f i c , p i l e - d r i v i n g and various other sources. O s c i l l a t i n g machinery, such as compressors and punch presses r e s u l t i n unbalanced i n e r t i a forces, whereas periodic harmonic forces usually result from the unbalance of rotating parts. If an e l a s t i c a l l y supported structure i s temporarily  forced  out of i t s equilibrium position by an impact or sudden application or removal of a force, e l a s t i c forces are no longer i n equilibrium with external forces; the structure i s set i n motion and vibrations about the equilibrium position ensue.  Therefore we may get free  vibrations produced by an i n i t i a l impulse or forced periodic vibrations from operating machinery. Vibrations from machine foundations usually f a l l into three groups A.  7* .  Low to Medium (below 500 r.p.m.).  This includes large  reciprocating machines, compressors, large blowers. Reciprocating  engines operate at frequencies  250 r.p.m. but have considerable  from 50 -  second harmonic content,  so sizable dynamic forces up to frequencies  of 500 r.p.m.  * Superscript numbers refer to References i n the bibliography at the back.  2.  must be w i t h s t o o d by the f o u n d a t i o n . B.  Medium t o H i g h (300-1000 r.p.m.).  Medium s i z e  reciprocating  engines such a s d e i s e l and gas e n g i n e s , a s w e l l a s blowers and r o t a t i n g machinery. C.  H i g h (Above 1000 r.p.m.).  H i g h speed i n t e r n a l combustion  e n g i n e s , e l e c t r i c motors, steam t u r b i n e s . The d e s i g n of a f o u n d a t i o n may r e q u i r e c o n s i d e r i n g the e f f e c t s of v i b r a t i o n s t r a n s m i t t e d through the ground from a d j a c e n t s o u r c e s , p a r t i c u l a r l y i f t h e r e i s any p o s s i b i l i t y of resonance.  A vibrating  f o u n d a t i o n r e p r e s e n t s the c e n t r e of a d i s t u r b a n c e w h i c h proceeds radially i n a l l directions. vibrations  I n t h i s way i t might produce f o r c e d  i n s t r u c t u r e s r e s t i n g on the subgrade a t a c o n s i d e r a b l e  d i s t a n c e from the c e n t r e of d i s t u r b a n c e .  The impulse t r a v e l s through  the subgrade i n a wave form a t a c e r t a i n v e l o c i t y depending l a r g e l y upon the e l a s t i c p r o p e r t i e s of the subgrade. Shock waves may c o n s i s t of two t y p e s of i n t e r i o r waves, compression waves and shear waves, i n which the v i b r a t i n g p a r t i c l e s move p a r a l l e l and p e r p e n d i c u l a r t o the d i r e c t i o n of wave p r o p o g a t i o n , respectively. Most v i b r a t i n g f o u n d a t i o n s t r a n s m i t v i b r a t i o n s by means of s u r f a c e waves, the most common of which a r e R a y l e i g h waves and Love waves.  The former are somewhat analogous t o ocean waves w h i l e the  l a t t e r are a s p e c i a l type of shear wave. constitutes  S i n c e wave p r o p o g a t i o n  an e x t e n s i v e t h e o r y i n i t s e l f , i t w i l l be c o n s i d e r e d no  further here.  3. CHAPTER II VIBRATION PARAMETERS A.  GENERAL CONSIDERATIONS  According to the theory of e l a s t i c i t y , any small deformation can be resolved into the sum of the deformations accompanied only by shear and the deformations accompanied only by a change i n volume.  A t r u l y s o l i d body possesses, as a rule, only p a r t i a l  e l a s t i c i t y because, after unloading, the body does not resume i t s i n i t i a l shape exactly.  In order to consider only e l a s t i c  deformations i n a medium, the value of residual deformation must be subtracted from the t o t a l deformation. Two e l a s t i c constants are s u f f i c i e n t to describe a body whose e l a s t i c properties are identical i n a l l directions and at a l l points. The two constants commonly used for this purpose i n engineering calculations are Young's modulus, E, and Poisson's r a t i o , V •  3 A hypothesis by Barkan  based on the dependence of the shear  modulus of a s o i l on the normal stresses, and considering only e l a s t i c deformations, suggests that neither E nor V are constant. For this case E increases and V decreases with an increase i n pressure.  Accordingly, a tensile stress would decrease E while  increasing V •  This indicates that the e l a s t i c properties of s o i l  change with variations i n compressive stress. The e l a s t i c deformations of a s o i l depend on the period over which the load acts.  Secondary consolidation i n cohesive s o i l s i s  a good example of t h i s .  For this reason, e l a s t i c deformations and  hence values of the e l a s t i c constants are greatly influenced by the rate of load application. From the above i t may be accepted that even i f the assumptions  4. o f h o m o g e n e i t y and  e l a s t i c i t y c o u l d be j u s t i f i e d , we w o u l d  be d e a l i n g w i t h a n o n - l i n e a r medium. ities  However, due  i n v o l v e d , the n o n - l i n e a r aspects  ignored.  The  modulus of d e f o r m a t i o n ,  still  to the  complex-  of the problem are  commonly  consequently,  i s taken to  a c o n s t a n t f o r any p a r t i c u l a r c o n d i t i o n s o f f o u n d a t i o n and  be  soil  mass. 3  Barkan Pokrovsky  r e f e r s t o d e t e r m i n a t i o n s of P o i s s o n ' s r a t i o , * \ /  and b y G e r s e v a n o v w h i c h i n d i c a t e t h a t t h i s  d e p e n d s on t h e v e r t i c a l p r e s s u r e . o b t a i n e d on a c l a y - s a n d m i s t u r e of the moisture  c o n t e n t , w,  r e s u l t s i n a decrease  inV  He  coefficient  gives experimental  showing t h a t y  results  i s independent  b u t t h a t an i n c r e a s e i n s a n d .  , by  fraction  T e s t s on s a n d s show t h e m o d u l u s o f  d e f o r m a t i o n , E, as b e i n g p r a c t i c a l l y i n d e p e n d e n t o f w and  grain  s i z e , b u t t e s t s on c l a y c u b e s show E d e c r e a s i n g m a r k e d l y w i t h increase i n water content.  R e s u l t s a r e shown where E  an  decreases  v e r y g r e a t l y w i t h i n c r e a s e i n t h e v o i d r a t i o , e, and w i t h  decrease  12 in  confining pressure  .  I t follows that, particularly, for  c l a y e y s o i l s , Young's m o d u l u s i s a f u n c t i o n o f t h e mechanical to  is  properties.  To d e t e r m i n e  physico-  E accurately, i t i s  c a r r y out d i r e c t t e s t i n g of u n d i s t u r b e d s o i l  from the  B.  THE  The  v a l u e o f Young's m o d u l u s u n d e r d y n a m i c v e r t i c a l  loading reaction.  be r e f e r r e d t o h e r e as t h e c o e f f i c i e n t o f e l a s t i c  compression, vertical  site.  COEFFICIENT OF ELASTIC UNIFORM COMPRESSION  sometimes c a l l e d t h e c o e f f i c i e n t of dynamic subgrade  It w i l l  necessary  Cu.  uniform  T h i s i s equal t o the magnitude of a p p l i e d  s t r e s s per u n i t of r e s u l t i n g v e r t i c a l  strain.  I t i s not  a c o n s t a n t , b u t an a v e r a g e o v e r a l l v a l u e c a n be t a k e n f o r a  5. p a r t i c u l a r foundation.  Gu r e l a t e s only to the e l a s t i c part of  the settlement and determines the spring c h a r a c t e r i s t i c of the s o i l base under v e r t i c a l v i b r a t i o n s . I t was f i r s t thought that Cu was a constant for a given s o i l . This suggested measuring the n a t u r a l c i r c u l a r frequency, f  , of  a s o i l by means of a v i b r a t o r and c a l c u l a t i n g Cu from the w e l l known r e l a t i o n s h i p , (see Chapter I I I for d e r i v a t i o n and d e f i n i t i o n s ) However, Schleicher (1926) has shown (reproduced, i n p a r t , i n references  3 and 6) that settlement of a r i g i d foundation on a  p e r f e c t l y e l a s t i c base i s i n v e r s e l y p r o p o r t i o n a l to the square root of the loaded area.  Settlement also i s shown to increase  s l i g h t l y w i t h increase i n the length to width r a t i o of the ion,  foundat-  thus i t i s obvious that Cu w i l l be d i f f e r e n t for d i f f e r e n t  foundations.  S c h l e i c h e r ' s formulas also show a d i r e c t dependence of Cu on the factor l / ( l - V 2 ) , where 1/ i s P o i s s o n ' s r a t i o . Barkan 3  determines values of Cu, by means of s t a t i c l o a d i n g , which increase with decrease of bearing plate area, so adding some support to S c h l e i c h e r ' s work.  However the experimental values were found to  vary at a s l i g h t l y lower rate than that predicted by the r a t i o s of the square roots of the base areas, as i n S c h l e i c h e r ' s equations. In Reference 5 Barkan determines Cu by the expression Cu = Pz/(Zp - Zo) where Pz = the applied v e r t i c a l pressure and Zp, Zo are the d e f l e c t i o n s before and after removal of the l o a d . Tschebotarioff points out^" that the s t a t i c rebound i s a  function  of the rate of unloading, so Cu w i l l change w i t h the frequency of  6. the e x c i t i n g force as w e l l as w i t h the magnification of amplitude near resonance.  Experimental r e s u l t s based on v i b r a t o r y tests*'  i n d i c a t e that Cu changes at a much slower rate than that based on the square root r e l a t i o n s h i p .  This i s probably due to the depend-  ence of e l a s t i c c o e f f i c i e n t s on the confining pressure, which v a r i e s w i t h depth. s o i l i s affected,  With greater contact area, a greater depth of and the influence of deeper s o i l layers on  foundation settlement  increases.  Experimental work by Barkan shows Cu increasing somewhat with increasing values of the r a t i o of foundation length to w i d t h , which i s the trend suggested by S c h l e i c h e r ' s t h e o r e t i c a l treatment. The effects  on the spring c o e f f i c i e n t s of the s i z e and shape  of the contact area, and the increase i n Cu w i t h depth may be 7  considered by a procedure offered by Pauw .  In t h i s analogy,  the f o u n d a t i o n - s o i l system i s treated by considering the foundation mass to be supported by a truncated pyramid of s o i l  springs.  Pauw considers the bearing capacity of the s o i l to be determined by Coulomb's law which s t a t e s , s = c + N tan 0  where  s = maximum shearing  resistance  c = cohesion N = normal pressure on the shear plane (increases w i t h depth) 0 = angle of i n t e r n a l  friction  From t h i s r e l a t i o n s h i p he assumes that the modulus of e l a s t i c i t y i s d i r e c t l y proportional to the depth, for cohesionless s o i l s , and i s constant for cohesive s o i l s .  Considering the v e r t i c a l  stress to be spread uniformly over the area of the pyramid at any l e v e l and c a l c u l a t i n g t o t a l s t r a i n by i n t e g r a t i n g over the i n t e r v a l  7.  from the foundation base down to i n f i n i t y , he c a l c u l a t e s the value of the spring constant for various s i z e s of rectangular and c i r c u l a r foundations on both cohesionless and cohesive s o i l s . Pauw points out that as negative stresses cannot a c t u a l l y occur i n cohesionless s o i l s , then dynamic t e n s i l e stresses exceeding the i n i t i a l dead load stresses w i l l i n v a l i d a t e these results.  However i t would appear from Pauw's c a l c u l a t i o n s that  dynamic stresses even approaching the s t a t i c stresses would severel y impair the values of h i s spring constants.  This phenomenon  and i t s non-linear aspects w i l l be considered i n Chapter I V .  It  should be noted here that cracks may develop i n the s o i l due to seasonal moisture f l u c t u a t i o n s and cause spring factors to vary. Values of Cu are given by Barkan'* which vary from l e s s than 3 kg/cm sands.  for soft clays and s i l t s to 10 kg/cm  for dense g r a v e l l y  These values of spring constants are given without  reference to a p a r t i c u l a r foundation, but q u a l i t a t i v e information as to the effects of various foundation factors i s offered.  Pauw's  c o e f f i c i e n t s on the other hand are t h e o r e t i c a l and are c a l c u l a t e d for  a p a r t i c u l a r foundation.  Which of these two methods i s the  more accurate w i l l depend upon whether Pauw's assumptions - or Barkan's g e n e r a l i z a t i o n s are c l o s e r to the p a r t i c u l a r case. C.  THE COEFFICIENT OF ELASTIC NON-UNIFORM COMPRESSION  The spring constant tending to prevent overturning of a foundation l y i n g on a h o r i z o n t a l e l a s t i c base w i l l depend upon the c o e f f i c i e n t of e l a s t i c non-uniform compression of the base, Cp .  Here, as with Cu, an average o v e r a l l value of Cy must be  taken for a p a r t i c u l a r foundation.  Cy  i s equal to the moment  about the axis of r o t a t i o n (due to the v e r t i c a l reaction) per u n i t  8.  of angular rotation, divided by the area moment of i n e r t i a of the foundation base about the axis of rotation. Assuming the slope angle,  f  of the t i l t e d base of the  foundation to be constant ( r i g i d base), Barkan shows  theoretically  that the c o e f f i c i e n t of uniform compression Cu, established under uniform loading i s not equal to Cy, , established for loading.  non uniform  He then cites experimental evidence to confirm t h i s .  Thus the e l a s t i c properties of the s o i l depend not only on the size and shape of the foundation but also on the character of the load transmitted to the s o i l .  Since his theoretical values for  the r a t i o C^/Cu agree to within 8 $ with experimental values, Barkan suggests establishing either Cu or Cy for a s i t e and computing the other value from t h i s r a t i o . determines the value of C  r  r ^  In reference 5 he  from the relationship -  M i(y>p -y>o)  where M = the applied moment (due to eccentric loading) I = the moment of i n e r t i a of the base contact area of the foundation with respect to the axis of revolution. ^p,^o  = the angles of rotation measured from the i n i t i a l position following application and removal, respectively, of the moment, M.  Barkan states that dynamic and s t a t i c determinations of Cy agree well.  From h i s experimental results i t i s possible to see that  Cy i s even more sensitive to changes i n the length-width r a t i o of the foundation than i s Cu. Pauw has developed expressions for the spring constants for cases of rotation about a centroidal axis i n the base contact plane.  In t h i s solution, as i n Barkan s, i t i s assumed that 1  horizontal planes are not distorted, but remain plane even after  9. rotation.  In a manner analogous to that used i n considering  v e r t i c a l loading, the spring characteristic representing C  v  i s developed for both cohesive and cohesionless s o i l s for d i f f e r e n t sizes of foundations.  I t i s d i f f i c u l t to compare  values of spring constants taken from Paw and Barkan because of the variations i n s i t e s , and the different parameters used. However the r a t i o s of the various constants for a particular foundation should give some indication of the r e l i a b i l i t y of the two methods. Numerical  calculations by Pauw on a 3£ x 3^ foot foundation  on a dense sand show a Cy/Cu r a t i o of about 1.9. remarkably well with Barkan s 2 1  by Converse  This compares  contributions. Other calculations  under similar s o i l conditions show, using Pauw's  c o e f f i c i e n t s , the value of this r a t i o attaining about 18.  In this  case Cip was obtained from a 12 x 5 foot foundation and Cu obtained from a 12 x 7 foot foundation.  Other factors were similar.  These  l a s t two footings have a r e l a t i v e l y large length to width r a t i o and a difference i n foundation-soil contact area but i t i s s t i l l hard to reconcile these results with Barkan's data. D.  THE COEFFICIENT OF ELASTIC UNIFORM SHEAR  The e l a s t i c spring constant tending to r e s i s t the purely e l a s t i c component of horizontal foundation movement on a horizontal base w i l l be called the c o e f f i c i e n t of e l a s t i c uniform shear, C . t  This w i l l be equal to the average value of shearing stress between the base of the foundation and the s o i l per unit of horizontal 3 displacement.  Barkan gives theoretical evidence  showing that  just as i n the case of the v e r t i c a l spring characteristics, the coefficient C  t  i s also inversely proportional to the square root  10.  of the contact area.  In t h i s case, i t i s noted that C  the same relationship with Poisson's r a t i o , V , as was Cu i n Schleicher's formulas. cohesionless s o i l s , then C  c  t  maintains  shown for  Since V i s larger i n cohesive than w i l l be smaller i n clayey than sandy  s o i l s , under otherwise equal conditions. Experimental results are offered to corroborate.this, as well as the square root r e l a t i o n ship; however, the l a t t e r i s shown to be true only for base areas i n the order of 1 or 2 square meters.  Larger base areas,  p a r t i c u l a r l y those above 10 - 12 square meters, f a i l e d to show much connection between C  t  and the square root of foundation area.  A dependence of t h i s spring c o e f f i c i e n t on the r a t i o of the lengths of foundation sides, as i n the previous cases, i s again shown. Experiments  are also given which show, along with a direct  relationship with v e r t i c a l pressure, a dependence of C  t  on the  duration for which the v e r t i c a l pressure acts. Pauw also offers a means of calculating t h i s spring constant, using a procedure similar to that used for the preceding characteristics. E.  THE COEFFICIENT OF ELASTIC NON-UNIFORM SHEAR  During rotation around a v e r t i c a l axis, the base of a foundation undergoes non uniform shear.  The spring characteristic  r e s i s t i n g the e l a s t i c movement i n this d i r e c t i o n w i l l , using Barkan's terms again, be called the c o e f f i c i e n t of e l a s t i c nonuniform shear, Cyr.  I t i s equal to the moment about the axis of  rotation per unit angle of rotation, divided by the polar moment of i n e r t i a of contact base area of foundation. show Cy as being somewhat larger than C^.  Experimental values  Based upon experimental  evidence, Barkan suggests a relationship, Cy = 1.5  C. c  11. An analogous c o e f f i c i e n t i s c a l c u l a t e d by Pauw. 2 n u m e r i c a l example, Pauw shows times as l a r g e as C t .  In a  Cy as b e i n g a p p r o x i m a t e l y 1.9  On e x a m i n a t i o n of Pauw's e q u a t i o n s i t  is  apparent t h a t f o r other t h a n s m a l l f o u n d a t i o n s , as c o n s i d e r e d i n h i s example, Cy i s apt to be much more i n excess of C t t h a n was noted i n B a r k a n ' s e x p e r i m e n t s .  A l t h o u g h most of B a r k a n * s e x p e r i -  mental r e s u l t s were d e r i v e d from s m a l l f o u n d a t i o n s t u d i e s ,  one  case i s o f f e r e d where the r a t i o Cy / C t was o n l y 1.16 on a f o u n d a t i o n of 15 square meter base a r e a . weakness i n Pauw's c o e f f i c i e n t s i n l a r g e F.  T h i s would suggest a foundations.  MODIFIED COEFFICIENTS  A l l the above c o e f f i c i e n t s have been developed w i t h r e s p e c t t o the s u r f a c e p l a n e . below the s u r f a c e ,  A c t u a l l y , f o u n d a t i o n s are u s u a l l y p l a c e d  thus t h e r e w i l l be a r e s t r a i n i n g  effect  e x e r t e d d u r i n g v i b r a t i o n by t h e s o i l i n c o n t a c t w i t h the v e r t i c a l edges of the f o u n d a t i o n .  Q u a n t i t a t i v e i n f o r m a t i o n on the v a l u e s  of these s p r i n g c o n s t a n t s  i s even r a r e r t h a n f o r the case of  horizontal surfaces.  j  Pauw shows c a l c u l a t i o n s f o r s p r i n g c o e f f i c i e n t s i n the v e r t i c a l p l a n e s i m i l a r t o those a l r e a d y c o n s i d e r e d i n the 4  horizontal plane.  Novak  c o n s i d e r s the s p r i n g c o e f f i c i e n t s  of  the v e r t i c a l w a l l s i n c o n t a c t w i t h the f o u n d a t i o n o n l y as they c o n t r i b u t e t o the c o e f f i c i e n t s a s s o c i a t e d w i t h the h o r i z o n t a l contact plane.  He o f f e r s  experimentally obtained conclusions that  the v e r t i c a l s p r i n g c o e f f i c i e n t i s i n c r e a s e d o n l y s l i g h t l y by t h i s effect  for v e r t i c a l l y excited vibrations.  A c a l c u l a t i o n by  2 Converse  u s i n g Pauw's c o e f f i c i e n t s shows an i n c r e a s e i n v e r t i c a l  s p r i n g c h a r a c t e r i s t i c ( f o r a 10 x 5 f o o t f o u n d a t i o n p l a c e d 3 f e e t  12.  below ground l e v e l , on a dense sand) of less than 2 per cent. This agrees with Novak's observations; therefore a surcharge would appear to have l i t t l e effect on the v e r t i c a l spring characteristic. In the case of horizontal excitation however, Novak gives evidence which shows a much greater contribution by the v e r t i c a l s o i l walls.  This effect appears to be highly dependent upon the  type of contact maintained between the foundation and the s o i l .  13. CHAPTER III MODES OF VIBRATION - SIMPLIFIED THEORIES A.  VERTICAL VIBRATIONS  1.  ONE DEGREE OF FREEDOM. Vibrations of a foundation  placed on a s o i l surface could be reasonably reduced to investigations of a r i g i d block resting on a semi-infinite e l a s t i c solid.  Since there i s not yet a solution to t h i s problem,  simplifying assumptions are necessary. The simplest solution considers a foundation placed on a weightless s o i l base, subject to an i n i t i a l displacement, with damping completely neglected.  A linear relationship between s o i l  reaction and foundation displacement i s assumed.  Taking for this  case a constant value of the c o e f f i c i e n t of uniform e l a s t i c compression,  Cu, for the s o i l , then the r i g i d i t y of the base i s  Cu A where A i s the area of foundation contact with the s o i l . Considering z, the v e r t i c a l displacement of the foundation from the equilibrium position, as positive downwards, and using d'Alembert's p r i n c i p l e , the equation of motion for v e r t i c a l vibration with one degree of freedom i s , W/g or  z  M  + Cu A z = 0,  z" + fnz z = 0,  3-A-l  2  7  where W* = the weight of the foundation and load, *  Throughout this chapter the vibrating weight, ¥, w i l l be seen to consist of only the weight of the foundation and i t s load, i . e . the weight of the s o i l springs are neglected. Actually there i s a s i g n i f i c a n t increase i n vibrating mass due to the addition of the s o i l springs. This additional weight, V i s usually added to the foundation weight, Wf, i n practice. Methods of determining V w i l l be discussed i n Chapter IV. s  s  14. m = W/g,  the mass of the f o u n d a t i o n  and l o a d  g = the a c c e l e r a t i o n due t o g r a v i t y z" = the second d e r i v i t i v e of v e r t i c a l d i s p l a c e m e n t w i t h r e s p e c t t o time ( a c c e l e r a t i o n ) f2 _ ^ nz  u  —  The g e n e r a l  "\ the f o r c e r e q u i r e d per u n i t o f f o u n d a t i o n mass t o move t h e f o u n d a t i o n through a v e r t i c a l d i s t a n c e of one u n i t .  s o l u t i o n of the homogeneous d i f f e r e n t i a l  equation  3-A-l may be w r i t t e n as f o l l o w s : z = A s i n fnz t + B cos fnz t where A and B a r e c o n s t a n t s  3-A-2  determined by i n i t i a l  conditions.  From t h i s we see t h a t f r e e v i b r a t i o n s , o c c u r r i n g where o n l y the i n e r t i a l f o r c e s of the f o u n d a t i o n the base are c o n s i d e r e d ,  and the e l a s t i c f o r c e s of  c o n s t i t u t e harmonic motion w i t h a  f r e q u e n c y , f > c a l l e d t h e " n a t u r a l c i r c u l a r f r e q u e n c y of v e r t i c a l n z  v i b r a t i o n s of the f o u n d a t i o n " . c y c l e i s 2 7T/f  The time r e q u i r e d f o r one complete  , which i s c a l l e d the p e r i o d , T.  The f r e q u e n c y ,  which i s t h e r e c i p r o c a l of the p e r i o d , i s equal t o ^ / " ^ 2  n z  and  i s g e n e r a l l y g i v e n i n o s c i l l a t i o n s per second ( h e r t z ) or r e v o l u t i o n s per minute, R.P.M. of a f o u n d a t i o n  I t i s seen t h a t t h e n a t u r a l f r e q u e n c y  i s determined o n l y by the f o u n d a t i o n  e l a s t i c i t y of the base, ( f o r t h i s , the s i m p l e s t  mass and the  case).  I n p r a c t i c e , e x c i t i n g l o a d s imposed by machines are u s u a l l y harmonic f u n c t i o n s of t i m e , t h e r e f o r e we w i l l s u b s t i t u t e an e x c i t i n g f o r c e , p s i n oJ t , i n t o e q u a t i o n  3-A-l t o get the  expression, z" + fnz z = p s i n OJ t where p = P/m, the e x c i t i n g f o r c e per u n i t of f o u n d a t i o n 2  3-A-3 mass,  OJ = the frequency of the exciting force. Other symbols are as used i n Eq. 3-A-l. The complete solution of Eq. 3-A-3 i s i n the form z = A sin fnz t + B cos fnz t + Az s i n CO t  3-A-4  Terzaghi shows^ that a beating phenomenon with a period 277"/(f  nz  - CO ) occurs when t h i s complete solution i s considered.  Since i n actual cases involving forced vibrations, the free vibrations are soon damped out, beating w i l l not be considered In passing, i t may be worth noting that at (x) = f__ the  here.  HZ  period of beating i s i n f i n i t e l y long, thus the amplitudes of vibration increase without l i m i t . Breaking down Eq. 3-A-4 we see that the f i r s t two terms constitute free vibrations as considered i n Eq. 3-A-2. The remaining term, A sin CO t represents the steady state vibration z with an amplitude A and a c i r c u l a r frequency, to, equal to that of z the impressed force.  Substituting this into Eq. 3-A-3 we determine  the amplitude of forced vibrations as, A_ = m ( f  =  A  nz ~  0  s t - i -  j  2  3-A-5  )  =f st A  T  i - or  2 nz  f  where A . = 8  and  n '  =  P  3-A-6  sr- i s the foundation displacement under load P m fnz , applied s t a t i c a l l y ,  —2 * 1  1 - CU f nz 2  s  a  m a  g i^i 'ki n  c a  o n  dynamic modulus.  factor known as the  From t h i s i t may be seen that when the natural frequency of the foundation coincides with the forced frequency, the dynamic modulus has a value of i n f i n i t y . resonance.  This i s the condition of  Alternately, as u) gets much smaller than f  i t is  seen that f| decreases rapidly, therefore the amplitude of forced vibrations reaches i n f i n i t e s i m a l values. For u) much smaller than f  n z  , f| approaches one and the dynamic nature of the loading need  no longer be considered. Under f i e l d conditions there i s always a loss of energy which tends to decrease the amplitude of a vibrating system.  This  phenomenon i s called damping, and because of i t , even under conditions of resonance, the amplitude of forced vibrations w i l l never actually go to i n f i n i t y .  Damping i s caused by departures  of the mechanical properties of the s o i l from those of an ideal e l a s t i c body.  Irreversible phenomena, occurring during the  vibration cycle, give r i s e to losses of energy.  One of the classic  types of damping, which i s usually applied to vibration considerations, i s called viscous damping:  i t constitutes a force which  tends to prevent motion, and whose magnitude i s d i r e c t l y proportional to the v e l o c i t y of the motion.  The assumption of such a  force renders the problem amenable to mathematical solution, and i s believed to be quite close to the truth i n problems of foundation vibrations. Introducing viscous damping into equation 3-A-3 we have z" + 2 cz' + fnz z = p s i n u) t 2  3-A-7  where c = y — , i s the damping constant and i s equal to one half the damping resistance per unit of foundation mass per unit v e l o c i t y of motion. 2  z' = the f i r s t derivative of v e r t i c a l displacement with respect to time (velocity).  17. The other symbols are as defined i n Eq. 3-A-2. solution to Eq. 3-A-7 z = e*"  (A s i n f  ct  n d  The complete  is  t + B cos f  The f i r s t two terms represent  n d  t ) + M s i n OJ t + N cos uj t  3-A-8  free vibrations with a natural  c i r c u l a r frequency, f »  I* i s seen that they w i l l be damped out  by the c o e f f i c i e n t e~ *.  Considering  nd  c  ution into 3-A-7 f  only these  terms,resubstit-  shows the natural frequency to be:  nd = / f - c nz 2  2  3  - ~ A  9  This indicates that the damping properties of a s o i l decrease the natural frequency of vibration of a foundation. than f  then free vibrations are not possible.  If c i s larger  C r i t i c a l damping  occurs at c = f nz Considering now the t h i r d and fourth terms of Eq. 3-A-8, which correspond to the steady forced vibrations of the foundation we have z = M sin  t + N cos co t  OJ  Putting this into Eq. 3-A-7  7f/2 we  and setting CJ t = 0 and  obtain P < nz - U ) . S I N 2T2—:—2—2 f  Z  =  772  < nz - «^ f  2  ) +  4  t  " 772  2 p co, T272—:—2—2COS  ( f ^ - OJ  c  + 4 ccj*  )  z  Adding these two components of motion vectorally, we have z =  Al  2  + N  2  s i n ( CJ t -  = A* s i n (COt z  /-2 «. where A*„ = M + N = z  )  ),  ,—« m /(f -OJ nz 2  o 2  )  2  + 4  c uJ 2  o 2  3-A-10  1  0  *  18. The phase s h i f t between the e x c i t i n g force a p p l i c a t i o n , P, and the displacement induced by P i s = tan ^ ^| = tan 1  c  f;  v  —~~  3-A-ll  -cu  nz The amplitude of forced v i b r a t i o n s , A J'j  #  A ^. where f| g  #  i s equal to  i s the dynamic modulus equal to  2 2 n  1  # z  1 -  2  f  2  ~ nz  3-A-12  nz Curves of dynamic modulus and phase angle versus cu ft are shown i n F i g . 1 and F i g . 2, r e s p e c t i v e l y . from Reference 2.  nz These were taken  I t may be seen from F i g . 1 that the effects of  damping are very c r i t i c a l i n the v i c i n i t y of resonance. shows that when the r a t i o <*j/^  nz  I  s  Fig. 2  low, the phase angle i s small  for a l l damping r a t i o s but as the frequency r a t i o increases, damping r a t i o , c / f  , becomes important. HZ  regardless of damping.  the  When c j / f  = 1 , & = 90° nz With further increase of ^ / f ^ * ^ motion n 97  n e  becomes further out of phase with the impressed f o r c e . 2.  TWO DEGREES OF FREEDOM.  In cases where only very low  amplitudes of v i b r a t i o n of machine foundations are p e r m i s s i b l e , v i b r a t i o n absorbers may be used.  These absorbers considerably  decrease v i b r a t i o n s produced not only by the main ( f i r s t ) harmonics, but also by higher harmonics of the e x c i t i n g f o r c e .  Absorbers are  commonly placed between the supporting frame or base of the machine and the sub-base, or foundation i n contact with the ground. gives r i s e to a system having 12 degrees of freedom.  This  In p r a c t i c e ,  such absorbers are used almost s o l e l y for the v i b r o i s o l a t i o n of  Dynamic A m p l i f i c a t i o n  Factor 00  PL H) H) P O P  a H  3 TO  •1  (D  o o o 3 cn cn 01  3  H . H J T O  e+- C  fl)  (5  3  Hj  C o ro Hs 3 P «•-«<  3* (0  ^  i  <D  2 P  P P  \  C  H" O  3  3 o  M  / "  * s  /  P  p  <a  I  >  / r>h — ,4 44  o  II  O  §  °  P  P h a s e - a n g l e )f <  ft  p o  H,  P »-* I-" fl>  C *o ro C  §2 8 $ So  TO  o' o  §  S 5  11 \ \ \ \ \ \  •  degrees  -  \  .  '  Cfl (D  N 3  o o ct- '"i  •r p  |  ro C ro 3 o  3" << CL o P CO 1 p 3 H s ro P c+•3 O HHP 3 3 •, O  TO ^ TO P (— ro O H, O  H-  3 co Co ro . r t - W »i p cn 3 3 • ca o  •  •6T  £ H  c A n>  i N  >  p  iU  O O  i  to  O  fc>  p  —  OO b b  > •fto (f ti <^ -f*  20. engines w i t h v e r t i c a l c y l i n d e r s therefore the analysis can u s u a l l y be l i m i t e d to v e r t i c a l v i b r a t i o n s only, w i t h two degrees of freedom. Assuming the centres of g r a v i t y of a l l mass components l i e on one s t r a i g h t l i n e , and neglecting damping, we can w r i t e the d i f f e r e n t i a l equations of forced v e r t i c a l v i b r a t i o n s w i t h spring absorbers as f o l l o w s : m. z " . + c. z. - c (z - z, ) = 0 ,/ * 2 2 2^ 2 l^ s i n COt where m,, m = the v i b r a t i n g masses below and above the absorbers, r e s p e c t i v e l y , 0  0  1  m  z  +  c  1  X  z  z  =  3-A-13  F  2  z, , t.^ — ^ v e r t i c a l displacements of the centres of g r a v i t y of masses m^ and m , n e  2  c^ = CuA, the c o e f f i c i e n t of e l a s t i c r i g i d i t y of the earth base. c  2  = the c o e f f i c i e n t of r i g i d i t y of the combined spring absorbers.  Other symbols are as used p r e v i o u s l y . The s o l u t i o n of E q . 3-A-13, neglecting the free v i b r a t i o n components w i l l be of the form z^ = A^ s i n c o t  and  z  2  = A s i n co t 2  3-A-l 5  where the amplitude of forced v i b r a t i o n s of the foundation below and above the springs are A^ and A r e s p e c t i v e l y . 2  A  i n Reference 3 that  I t i s shown  1 = „  where A^ i s the amplitude of forced v i b r a t i o n s without absorbers as given by E q . 3-A-5, and M_s i s the magnification f a c t o r , equal to :2 , v <c 2 M  Here  s = =  1-  U+/i)  f^/co  «I+6£,-S  and£  l z  =  ^nlz^  «. >  3-A-16  21.  FT  vhere f ^ = j — ' «2 n  K  the frequency of n a t u r a l v i b r a t i o n of the f o u n d a t i o n above the s p r i n g s assuming no movement of the lower f o u n d a t i o n ,  f , ==1 , the frequency of of n a t u r a l v i b r a t i o n of the niz Jm, + m~ , . , 1 2 complete system, — assuming no absorbers, i  1  and JJ =  i  -  •  1  n^/m^  From Eq. 3-A-16 i t i s seen t h a t a s £ ^ approaches hence the amplitude £^ =  .  f ^/Cc> t n  zero, M  of f o r c e d v i b r a t i o n s approaches  g  and  zero.  Since  i t f o l l o w s t h a t i f the n a t u r a l frequency of f o u n d a t i o n  v i b r a t i o n s above the s p r i n g absorbers i s small i n comparison  with  the frequency of engine r o t a t i o n , the amplitude, A-^ of the lower foundation w i t h absorbers i s small i n comparison w i t h the amplitude of v i b r a t i o n s of the same foundation without absorbers. other hand, asfc,^approaches  i n f i n i t y , or  On the  becomes very l a r g e ,  the absorbers w i l l have l e s s e f f e c t , i n f a c t even become harmful beyond the r e l a t i v e l y short range  of |M J<1. S  In summary, the n a t u r a l frequency of the foundation above the absorbers must be kept low r e l a t i v e t o engine speed, or the absorbers may be p o s i t i v e l y harmful.  Since f ^ = __2 , the r e q u i r e d m  2  e f f e c t can be achieved by reducing the s t i f f n e s s of the absorber s p r i n g s ; when the l i m i t to t h i s r e d u c t i o n (imposed by s t r e n g t h requirements)  i s reached, the f o u n d a t i o n above the absorbers can  be i n c r e a s e d i n weight. B.  ROCKING VIBRATIONS  Rocking v i b r a t i o n s u s u a l l y occur i n h i g h foundations under machines having unbalanced h o r i z o n t a l components of e x c i t i n g f o r c e s and e x c i t i n g moments.  L e t us assume t h a t the e l a s t i c  resistance  22. of the s o i l to s l i d i n g of the foundation i s i n f i n i t e l y great and that the centre of i n e r t i a of the foundation mass and the centroid of i t s horizontal base area l i e on a v e r t i c a l l i n e i n the v e r t i c a l plane i n which rocking occurs.  In t h i s case the position of the  foundation w i l l be determined by one independent variable, the angle of rotation  of the foundation around the axis passing  through the centroid of foundation-soil contact area and perpendicular to the plane of vibrations. At any time t , at which the foundation i s rotated by a small angle ^ , the equation of motion of the foundation vibration under the time-dependent exciting moment M sinajt w i l l be (neglecting damping) - ¥ y>" + £ M + M s i n Q  or  - W-y?" +  where ¥  WUp  -  C  v  UJ  t = 0  I<P + M s i n  3-B-l uJ  t = 0  = the moment of i n e r t i a of the foundation and machine mass with respect to the axis of rotation,  LWY* = the moment of the foundation-machine weight about the axis of rotation, 1  L = the v e r t i c a l distance between the axis of rotation and the centre of gravity of the mass, C^I^P  = the moment of s o i l reaction about the axis of rotation, I = the moment of i n e r t i a of the foundation contact area with respect to the axis of rotation and  Cy,  = the c o e f f i c i e n t of e l a s t i c non-uniform s o i l compression.  The complete solution of t h i s equation w i l l have a form similar to that of Eq. 3-A-4 containing terms of free and forced vibrations. Solving for the condition of free vibrations, the natural frequency, f ^ n  f  i s determined as C^I - WL  f  nv»  ¥  o  C«, I —^— o  , since C^I » ¥L  3-B-2  23. The terms representing free v i b r a t i o n s may be put i n the form <P= C s i n ( f  n ( p  t +</> )  3-B-3  o  where C, ^P are constants, representing the amplitude and phase Q  angle, and are determined from i n i t i a l c o n d i t i o n s . The s o l u t i o n for the amplitude, A ^ , of forced v i b r a t i o n s for t h i s case w i l l give A „  =  *  which has a form s i m i l a r to E q . 3-A-5. Prom Eq. 3-B-3 i t i s seen that the natural frequency of v i b r a t i o n i s a function of I , the moment of i n e r t i a of the founda t i o n contact area about the axis of r o t a t i o n .  Since I i s d i r e c t l y  p r o p o r t i o n a l to the t h i r d power of the foundation length, (perpendicular to the axis of r o t a t i o n ) , any change i n t h i s length w i l l have an immense effect O n ^  and consequently on A ^ .  nif)  Considering the above foundation motion but i n c l u d i n g damping we have the n a t u r a l frequency f  ^ =  It"  -  c  The amplitude A ^ A  s i m i l a r to Eq. 3-A-10, i s M  Vd ¥  C.  3-B-5  2  Q  J(f  2  -  c ) 2  2  +  4  C LO 2  :  3-B-6  HORIZONTAL VIBRATIONS  When a foundation mass r e s t i n g on the ground surface  is  subjected to a h o r i z o n t a l force, shearing deformations develop w i t h i n the s o i l .  I f the resistance of the s o i l to compression i s  large i n comparison to i t s resistance to shear, then displacement of the foundation under the a c t i o n of the h o r i z o n t a l forces w i l l  24. occur mainly i n the d i r e c t i o n of these f o r c e s . Assuming a h o r i z o n t a l force P^ s i n o j t a c t i n g on the  centre  of mass of a foundation, then, analogous to E q . 3-A-3, ve have ,2 x" + f_„ 3-C-l nx = p s i n cot where x = the h o r i z o n t a l displacement of the centre of g r a v i t y of the foundation, t  2  C  f  c  A  = —jjj— , the square of the natural frequency i n shear, 3-C-2 C  c  = the c o e f f i c i e n t of e l a s t i c uniform shear i n the  soil.  From t h i s equation i t i s seen that t x = Trs i n co t . P  3-C-3  Consider next the case of r o t a t i o n a l v i b r a t i o n s with respect to a v e r t i c a l axis passing through the centre of g r a v i t y of foundation mass and the centroid of the base area.  We have, under  a h o r i z o n t a l e x c i t i n g moment M s i n to t , the equation z  z^ " f z V = z ^ where W = the moment of i n e r t i a of the v i b r a t i n g mass w i t h respect to the v e r t i c a l a x i s , ¥  +  C  J  M  s  i  n  +  J z = the polar moment of the foundation base area, = the angle of r o t a t i o n of the foundation with respect to the v e r t i c a l a x i s , Cy, = the c o e f f i c i e n t of e l a s t i c non-uniform shear. The amplitude for t h i s case w i l l be (f  2  -CO ) 2  and the n a t u r a l frequency of v i b r a t i o n w i l l be  3-C-4  The f o r e g o i n g e q u a t i o n s and s o l u t i o n s were o b t a i n e d i n a manner analogous t o t h a t used f o r v e r t i c a l v i b r a t i o n s . m o d i f i c a t i o n s due t o v i s c o u s damping, noted f o r the cases w i l l be s i m i l a r D.  The  vertical  here.  COMBINED VIBRATIONS  A f o u n d a t i o n r e s t i n g on a s o i l base has 6 degrees of freedom and can e x h i b i t 6 i n d i v i d u a l fundamental f r e q u e n c i e s . f o r t h i s case i s g i v e n i n Reference  7.  I n p r a c t i c e , however, only-  one t o t h r e e types of motion u s u a l l y need t o be V i b r a t i o n s c a u s i n g combined compression w i l l now  be d e a l t w i t h .  A solution  considered.  and shear t o the  soil  Take f i r s t the case where the c e n t r e of  g r a v i t y , 0, of the v i b r a t i n g mass and the c e n t r o i d of  the base  c o n t a c t a r e a l i e on a v e r t i c a l l i n e on the one major p r i n c i p a l plane i n w h i c h v i b r a t i o n s are assumed t o occur.  C o n s i d e r the e x c i t i n g  f o r c e s ( i n the plane of v i b r a t i o n s ) broken down i n t o a f o r c e , P s i n tut,  a p p l i e d a t 0, and a c o u p l e , M s i n c o t .  Under these f o r c e s the  f o u n d a t i o n w i l l undergo motion determined  by the t h r e e parameters  x, z and ^ .  The parameters x and z are the h o r i z o n t a l and  vertical  d i s p l a c e m e n t s r e s p e c t i v e l y , of the c e n t r e of v i b r a t i n g mass from i t s p o s i t i o n of r e s t .  The angle  ^  r e p r e s e n t s the r o t a t i o n of  the f o u n d a t i o n w i t h r e s p e c t t o an a x i s through 0 p e r p e n d i c u l a r t o the plane of v i b r a t i o n s . By e q u a t i n g the sum  of the v e r t i c a l , h o r i z o n t a l and  rotational  f o r c e s t o g e t h e r w i t h the i n e r t i a f o r c e s ( d ' A l e m b e r t ) , t o zero a r r i v e a t the t h r e e e q u a t i o n s mz"  (see References  s i n to t  + Cu A z = P  3 or 5) of  we  motion, 3-D-l  z mx"  + C  Ax - C, AL  f  = P  v  sin  to  t 3-D-2  26. M  C  -  ip"  m  A L x + (Cp I - ¥ L + C  t  A L ) = M s i n UJ t 2  E  where M m = the moment of i n e r t i a of the mass w i t h r e s p e c t t o the a x i s through 0, normal t o the plane of v i b r a t i o n s . P , P  = the v e r t i c a l and h o r i z o n t a l components of the e x c i t i n g f o r c e P,  x  M = the e x c i t i n g moment, and other terms are as d e f i n e d p r e v i o u s l y . I t may  be noted t h a t s i n c e Eq. 3-D-l  c o n t a i n s n e i t h e r x nor  y3 i t i s independent and can be c o n s i d e r e d s e p a r a t e l y as was i n Eq. 3 - A - l .  Eqs. 3-D-2  and must be c o n s i d e r e d  on the other hand are  interdependent  together.  C o n s i d e r i n g the case of f r e e v i b r a t i o n s , i . e . Px = M = Eqs. 3-D-2  done  0,  have p a r t i c u l a r s o l u t i o n s of the form  x = A a s i n ( fn t +0<  )  Wr = B a s i n ( f nt + tx )  ;  3-D-3  The two p r i n c i p a l n a t u r a l f r e q u e n c i e s of v i b r a t i o n  correspond-  i n g t o these two degrees of freedom are f n l » nf2 2  =  2  r  where  2*  # M  1  = =  m o  f. „ , f n"P ' nx  3-D-4  M /M , m  mo  2 + m L , the moment of i n e r t i a of the v i b r a t i n g mass w i t h r e s p e c t t o the a x i s p a s s i n g through the c e n t r o i d of the base c o n t a c t area p e r p e n d i c u l a r t o the plane of v i b r a t i o n s , are as g i v e n i n Eqs. 3-B-2 and 3-C-2 r e s p e c t i v e l y . M  m  s  u  r  J  (The frequency w i t h r e s p e c t t o the t h i r d degree of freedom, d e s c r i b i n g motion i n the z d i r e c t i o n , i s o b t a i n e d i n d e p e n d e n t l y  as  before). 3 A c c o r d i n g t o Barkan , f 2» ^he n  s m a l l e r of the two n a t u r a l  f r e q u e n c i e s i s s m a l l e r than e i t h e r of the two l i m i t i n g f  or f > w h i l e n x  Eqs. 3-D-3  i s larger.  I t may  i n t o the f i r s t of Eqs. 3-D-2  be seen by  frequencies  substituting  t h a t the r a t i o  27. of amplitudes of the' two vibration components, nx 2  2  nx where ± If f  Q  n  n  3-D-5  i s the natural frequency of vibration of the foundation.  = f 2> then P  i s positive and i t i s seen that x and *p  n  tend to be i n the same d i r e c t i o n .  This means that the foundation  w i l l undergo rocking vibrations with respect to a point situated at a distance  below the centre of mass of the foundation.  the other hand, i f f be negative, and A  = f 2» the higher frequency, then  will  n  and B  w i l l be 180° out of phase.  vibration takes place about a point of the vibrating mass.  Gn  Then  above the centre of gravity  Thus, from Eqs. 3-D-4, 3-D-5, either of  two forms of vibration are possible, depending on the foundation size and s o i l properties but independent of the i n i t i a l conditions of foundation motion. Considering Eqs. 3-D-2 for the case of forced vibration we arrive at solutions of the form x = A  x  sin t u t j  <f = A^ s i n uJ t .  3 Under an impressed horizontal force P s i n cc) t we obtain , by the usual technique of resubstitution i n the i n i t i a l equations, values of amplitudes: M m CU C I - U + C, A L A = x 4(co ) f  p;  A^ =  C  t  differential A L P 3-D-6  2  Under an exciting moment, M s i n to>t, the amplitudes are 2^ C A - m CO C.AL M, A M A = t  A(u?)  A  (co)'  3-D-7  28. where A(oj )  = m M  2  ( f ^ -co)  (f  2  ffl  -CO ).  2  3-D-8  2  2  Considering either of the two resonance conditions, either CU = f j or OJ = f 2» i t may be seen from Eq. 3-D-8 n  ^\ (cu)  n  =  0*  that  Therefore the amplitudes as given by Eqs.  3-D-6  approach i n f i n i t y or, more r e a l i s t i c a l l y , w i l l increase without limit. By considerations similar to those associated with Eq.  3-D-5  i t i s seen that when the exciting frequency CU i s very small with , then P  respect to f  = L and rocking takes place about the  centroid of base contact area.  As O) increases,  P increases (the  centre of rotation moves down) u n t i l at CO = f„„> ft =  o°  (The centre of rotation i s at an i n f i n i t e depth) and s l i d i n g motion alone takes place. of phase.  When CU exceeds f ^ , the motions become out  Further increase i n CO decreases  P  and lowers the axis  of rotation from a position i n f i n i t e l y high above the foundation (which i t attained as P  passed from positive to negative) towards  the centre of mass, u n t i l , when UJ i s much greater than f  , ft  approaches zero and purely rotational o s c i l l a t i o n s occur. In the above case of combined vibrations, the centre of gravity of the foundation and machine mass and the centroid of foundation contact area were considered to l i e on a v e r t i c a l  line.  Taking the case where there i s an eccentricity of the mass, £  ,  (in the plane of vibration) with respect to the centroid of base area we have the equations, corresponding to equations 3-D-2  3-D-l,  (but considering only free vibrations resulting from some  i n i t i a l disturbance such as an impact) m x" + C  c  A x - C  c  A L<p  m z" + Cu A z - Cu A £ ^  =  = 0  0  29. M yJ' - C m  z  A L x + (C  v  I - ¥ L + Cu Ae  2  +  A  L )</>- Cu Ae z = 0 2  3-D-9 It may be noted that a l l these equations are interrelated  3 and must be considered together.  Barkan  shows that the effect of  this eccentricity i s a decrease i n the two smaller natural frequencies and an increase i n the highest natural frequency from those values found i n Eqs. 3-D-l and 3-D-2  for zero eccentricity.  He states that for foundations having e c c e n t r i c i t i e s up to 5 per cent of the length of a foundation dimension, t h i s effect may neglected, and computations  be  may be performed using Eqs. 3-D-l and  3-D-2, where eccentricity £ = 0. It i s obvious that the above treatment of vibrations deals only with very special cases.  In practice, however, i t i s usually  considered possible to narrow down vibratory motion determinations to a s p e c i f i c plane of action - that i n which the vibratory impulse i s applied. Furthermore,  since the designer usually has some  control over eccentricity, and because of the complications involved, t h i s factor has generally been ignored, p a r t i c u l a r l y when i t occurs 7 outside the major plane of vibrations.  According to Pauw , the  error i n determining the fundamental frequency when damping i s neglected i s usually less than 2 per cent.  I t i s the purpose of  most foundation designs to avoid a frequency r a t i o , 6c>/f > i n the nz  v i c i n i t y of one, while outside this v i c i n i t y of frequency r a t i o values, the dynamic amplitude factor, as seen by F i g . 1, i s quite small.  This, combined with the d i f f i c u l t i e s involved i n  evaluating damping has resulted i n i t s being neglected i n most vibration calculations.  The result i s that solutions for cases  other than those presented above, are not available at present.  30. CHAPTER IV VALIDITY OF VIBRATION THEORIES A.  GENERAL CONSIDERATIONS  In order to evaluate the preceding vibration theory we shall consider more extensively the condition of v e r t i c a l vibration with one degree of freedom.  This mode of vibration should be represent-  ative of a l l conditions of motion and i s obviously the simplest to analyse. The e a r l i e s t quantitative results on the dynamic characteri s t i c s of foundations were acquired, according to Terzaghi  and  Tschebotarioff , by DEGEBO (German Research Society for S o i l 1  Mechanics).  Tests are described using a machine called a two mass  o s c i l l a t o r (similar i n p r i n c i p l e to the machine described i n this thesis, see CHAP. VI, Part B) to determine experimentally the natural frequencies of the s i t e s .  I t was o r i g i n a l l y hoped that a  useful relationship between measured dynamic properties and resistance to s t a t i c loads could be established. Although tables of allowable bearing pressure versus natural frequency of s o i l s have been given**, i t i s seen that the variations i n bearing pressure are extremely high as compared with the useful variations i n frequency.  This indicates that c o r r e l l a t i o n by t h i s means must  be poor. Using the elementary equation for natural frequency, f  = / k/m , where k i s the spring constant equal to Cu A i n  Eq. 3-A-l and  m =  f  s , the mass of the foundation and s o i l g  undergoing vibration, experimental values of k and ¥ determined.  s  were  In this method, different weights of vibrators were  31. used while the area, which was  thought to control the  weight of s o i l , was held constant.  I t was  equivalent  found that ¥  varied  s between very wide l i m i t s .  ¥ith increasing values of exciting  force, the resonant frequency was  found to decrease.  explained as being due to an increase i n ¥ It was was  This  was  with increasing load.  shown conclusively by these tests that the natural frequency  dependent not only upon the foundation under consideration,  but on adjoining structures as well.  Thus we are dealing with  the natural frequency of a p a r t i c u l a r s i t e , not merely of a particular s o i l . The most apparent weaknesses of the present methods of handling vibration calculations are: 1.  The usual assumption that the spring constants are l i n e a r . (Even i n the cases where they are considered to increase with increasing depth below the surface, as i n Pauw's work.)  2.  Consideration of the foundation-soil as a lumped parameter system, i n which the weight of the springs i s added to the weight of the foundation considered,  load and the damping forces, i f  are lumped together as one source of viscous  damping. 3.  The d i f f i c u l t y i n choosing proper values of damping, and the equivalent mass of the s o i l  of  springs.  Foundation s o i l i s an imperfectly e l a s t i c material whose 12 14 4 many properties  '  *  vary with distance from the surface,  moisture content, manner of loading, state of stress, presence of subterranean water, etc.  These cannot a l l be taken into account  i n theoretical considerations.  32. B.  NONLINEARITY  Consider again the c o e f f i c i e n t of uniform compression, Cu. We have already considered average values of Cu.  i n CHAP. II the r e l i a b i l i t y of the  Here we shall discuss primarily the effects  of i t s nonlinearity on the vibration theories.  Since the  frequency of forced vibrations of machine foundations i s found to coincide with the frequency of the exciting forces, there actually exists a r e l a t i v e l y linear relationship between the 9  3  foundation displacement and the s o i l reaction .  Barkan  states  that the linear relationship between magnitude of exciting force and amplitude of vibrations depends upon t h i s . able and f a i r l y easy to measure.  This seems reason-  His graph on page 90 shows some  observations. 8  4  According to Lorenz , and Novak , the assumption that the s o i l acts as a linear spring without mass does not lead to satisfactory results as shown by large scale t e s t s . exciter force lead to decreases i n natural frequency, f i r s t made by DEGEBO i n 1934 . 6  Increases i n observations  Lorenz points out that i f the  decrease i n natural frequency i s attributed to the increase i n vibrating s o i l mass, then the spring characteristic could be considered  constant.  However he states that as the exciter area  i s increased, with constant contact pressure, increases.  the natural frequency  As this must cause an increase i n the vibrating s o i l  mass, the natural frequency should instead decrease.  According to  Lorenz t h i s indicates non-harmonic vibrations, therefore he would replace Cu A i n the equations of CHAP. I l l with a reactive force that does not change l i n e a r l y with deflection.  Q Den Hartog  states that there i s no exact solution for  33. undamped vibrations with a curved spring c h a r a c t e r i s t i c .  He  gives an approximate solution of a nonlinear characteristic based on the assumption that the motion i s sinusoidal.  This method i s  adapted by Lorenz to determine nonlinear spring characteristics for a s o i l .  Lorenz also provides three other methods of determin-  ing the c h a r a c t e r i s t i c , a l l , probably, of a lower degree of accuracy than the f i r s t .  Since the Den Hartog method does not  consider damping, i t w i l l give an improper interpretation near conditions  of resonance.  From plotted characteristics for  several s o i l s (using method one)  i t appears that nonlinearity i s  more s i g n i f i c a n t i n cohesive s o i l s .  Lorenz suggests that t h i s  i s due to the fact that i n clayey s o i l s pore pressures reduce the increase  i n effective stress quite extensively during application  of small loads whereas at higher loads pore pressures f a l l off quite rapidly. to torsion and  He states that these results apply equally well shear.  Thus i t seems that nonlinearity i s less a  problem i n cohesionless s o i l s , where most vibration problems occur. Novak (as well as Lorenz) j u s t i f i e s his approximate solutions  4 for nonlinear vibration  by the fact that vibration records picked  up by an o s c i l l i s c o p e usually indicate that the motion of a foundation under a harmonically applied force i s i t s e l f almost purely harmonic.  This also would indicate that the f i r s t harmonic  of the solution i s satisfactory.  Experimental values are plotted  which show extremely good correlation with a characteristic calculated by one  of Novak's approximate formulae.  Novak cites  experimental work carried out to determine variations i n s o i l parameters due to foundation p e c u l i a r i t i e s .  These were vibratory  tests on a loess loam with variations i n s t a t i c pressure from  34. 0.194 0.5  to 0.482 kg/cm and 1.5  and with square base plate areas between  square meters.  Measurements were taken only after  a satisfactory e l a s t i c state of the s o i l had been reached, i . e . residual deformations became negligible.  I t was  noted that  numerical value of the spring characteristic of the s o i l  the  varied  only very s l i g h t l y within the range of pressures used i n the t e s t . Novak infers from t h i s that the subsoil behaves as a stratum of limited size, rather than as the theoretical semi-infinite e l a s t i c base, and so j u s t i f i e s the simplified handling of vibration problems. S o i l becomes e l a s t i c due  to repeated stressing and i t s stress12 4  s t r a i n relationship becomes linear  ' .  If the characteristic for  any s i t e can be measured for the amplitude of motion that w i l l occur, and i f proper interpretation of foundation effects can be combined with t h i s , then nonlinearity of the spring characteristic can probably be ignored.  Therefore the above theories of vibrations  w i l l probably give r e l a t i v e l y adequate r e s u l t s .  However i f the  frequency of vibration could r e s u l t i n amplitudes that are markedly different from those involved during measurement, then nonlinearity should be considered by one C.  THE  INERTIA OF THE  of the above methods. SPRING BASE  A factor of very definite importance i n the previous vibratory considerations i s the effective s o i l weight, ¥ , or the weight of vibrating s o i l .  Tschebotarioff  states that there i s no  c l e a r l y defined l i m i t for the s o i l mass, which has a very complex motion, and so an "equivalent weight" of s o i l must be considered. As Pauw puts i t , "Everyone agrees that the mass of vibrating  soil  35.  must be added, d i f f e r e n c e s l i e o n l y i n the assumptions used t o determine i t " . Pauw's method of e s t i m a t i n g the apparent mass of ¥  i s to s  compute a c o n c e n t r a t e d mass t h a t would have the same k i n e t i c energy as the r e a l s o i l mass.  He uses the elementary r e l a t i o n s h i p which  s t a t e s t h a t k i n e t i c energy i s equal t o one h a l f the mass m u l t i p l i e d by the square of the v e l o c i t y .  I n t e g r a t i n g f o r the k i n e t i c energy  of each p a r t i c l e w i t h i n the e f f e c t e d s o i l volume, he equates t h i s to the k i n e t i c energy of an e q u i v a l e n t mass, undergoing v i b r a t i o n s at the s u r f a c e of the ground, and s o l v e s f o r the e q u i v a l e n t mass. The  e f f e c t e d s o i l volume i s a g a i n assumed t o be t h a t e n c l o s e d  the pyramid r e f e r r e d t o on page 6,  by  between the plane of the  f o u n d a t i o n base and a plane a t an i n f i n i t e d i s t a n c e below the foundation.  Apparent mass f a c t o r s f o r v a r i o u s modes of v i b r a t i o n  i n c o h e s i o n l e s s s o i l are d e r i v e d i n t h i s way.  I n a s i m i l a r manner,  he e s t i m a t e s the apparent mass moments of i n e r t i a of the s o i l t o be used i n r o t a t i o n problems d e a l i n g w i t h both cohesive' and less s o i l s .  The  cohesion-  i n t e g r a l used f o r the d e t e r m i n a t i o n of the  apparent mass f o r a cohesive s o i l does not converge, however, so for  t h i s case Pauw's method i s not a p p l i c a b l e .  Pauw s t a t e s t h a t  apparent mass terms found by t h i s procedure were i n e x c e l l e n t agreement w i t h f i e l d measurements f o r c o h e s i o n l e s s s o i l s . c o h e s i v e s o i l s t h e r e i s so l i t t l e e x p e r i m e n t a l  For  information that  c o r r e l a t i o n s c o u l d not be made. Experimental  methods u s i n g a v i b r a t o r w i t h a  constant  e c c e n t r i c i t y and r o t a t i n g mass, i n s t e a d of a c o n s t a n t P may 4  used '  be  8  t o determine ¥ . For t h i s case the e x c i t e r f o r c e i n c r e a s e s s w i t h the square of the frequency as shown by Eq. 6 - C - l . Plotting  36. the amplitude of v i b r a t i o n s v e r s u s the frequency w i l l g i v e a curve which w i l l show, a c c o r d i n g t o the t h e o r y of damped v i b r a t i o n s of a l i n e a r s p r i n g , the amplitude of v i b r a t i o n s u l t i m a t e l y becoming a s y m p t o t i c t o a s t r a i g h t l i n e a t a c o n s t a n t amplitude, r  o m  , as the frequency i n c r e a s e s w i t h o u t bound.  i s the e c c e n t r i c i t y of the r o t a t i n g mass, m , Q  t o t a l v i b r a t i n g mass of the v i b r a t o r and s o i l . m are known and the v a l u e of determined.  Here  and m i s the Since m  and r  Q  r o m -  may  be measured,, m may  be  S u b t r a c t i n g the mass of the v i b r a t o r from t h i s v a l u e  of m w i l l g i v e the apparent mass.  These methods seem t o o f f e r  r e a s o n a b l y good v a l u e s f o r the p a r t i c u l a r f o u n d a t i o n c o n d i t i o n s r e p r e s e n t e d by the v i b r a t o r , but i t i s not a simple matter p r o j e c t them t o other f o u n d a t i o n s on the same s i t e .  to  Novak shows,  e x p e r i m e n t a l l y , an i n c r e a s e i n apparent mass of 40 - 60 per  cent  when f o u n d a t i o n s are p l a c e d below the s u r f a c e , even when a i r spaces are l e f t around the f o u n d a t i o n . The i d e a of t r e a t i n g a f o u n d a t i o n as the t h e o r e t i c a l  weight-  l e s s s p r i n g by adding i t s apparent weight t o the f o u n d a t i o n l o a d i s i t s e l f a v e r y good a p p r o x i m a t i o n , assuming of course t h a t the f o u n d a t i o n s p r i n g analogy i s o t h e r w i s e r e l i a b l e .  Various t e x t s  show t h a t by c o n s i d e r i n g a v i b r a t i n g s p r i n g as w e i g h t l e s s , no a p p r e c i a b l e e r r o r w i l l r e s u l t i f one t h i r d of the a c t u a l s p r i n g mass i s c o n s i d e r e d as b e i n g c o n c e n t r a t e d a t i t s f r e e end. Novak r e f e r s t o a s i m p l i f i e d t h e o r y (by S e c h t e r ) of v e r t i c a l f o r c e d v i b r a t i o n s of a r i g i d f o u n d a t i o n r e s t i n g on a s e m i - i n f i n i t e e l a s t i c base.  T h i s work i s c i t e d t o prove t h e o r e t i c a l l y t h a t t h i s  c o n d i t i o n can be i n v e s t i g a t e d w i t h s u f f i c i e n t accuracy by c o n s i d e r ing  the body t o v i b r a t e on a massless base, p r o v i d e d the mass of  37.  the foundation i s suitably increased, and the damping c o e f f i c i e n t i s appropriately selected. Prom the above i t would appear that the increase i n vibrating mass must be considered i n order to avoid errors when applying current vibration theories. Whether this mass can be calculated accurately by Pauw's methods or adequately predicted from experimental observations i s s t i l l rather uncertain. D.  DIFFICULTIES ASSOCIATED WITH DAMPING  The effects of damping on foundation vibrations can be of major importance i n cases of resonance of forced vibrations or during shock loading.  According to Tschebotarioff^, no r e l i a b l e  or p r a c t i c a l methods have been developed to permit numerical determination, and for t h i s reason damping i s often neglected. With regard to forced vibrations, Eq. 3-A-9 shows that i f the damping constant c i s small r e l a t i v e to the natural frequency without damping f  , then damping w i l l have l i t t l e effect on the HZ  resonant frequency and may be neglected.  However i f damping i s 3  large, i t w i l l have a large e f f e c t .  According to Barkan  the effe  of the damping reaction of the s o i l on the amplitudes of free vibration of a foundation i s rather considerable, even i n cases of small c.  Since foundations under machines with a steady regime  of work are usually designed to avoid any p o s s i b i l i t y of resonance then damping effects may be neglected i n these  computations.  The same experimental methods (described by both Novak and Lorenz) which are used to determine the apparent mass may also be used to determine the damping c o e f f i c i e n t for the particular foundation conditions represented by the vibrator.  Terzaghi^  38.  shows a method o f c a l c u l a t i n g t h e d a m p i n g c o n s t a n t b y the excess  energy output  of the v i b r a t o r t o t h a t s u p p l i e d t o  f o r c e of v i s c o u s damping. experimental  equating  Little  evidence  the  of c o r r e l a t i o n s w i t h  o b s e r v a t i o n s are o f f e r e d .  Novak n o t e s from p r e v i o u s l y d e s c r i b e d experiments damping c o e f f i c i e n t i n c r e a s e s r e m a r k a b l y foundation area.  T h i s was  t h a t a mere d e c r e a s e  with increase i n  a l s o n o t e d by L o r e n z .  i n moisture  t h a t the  Novak s t a t e s  of a l o e s s loam caused by  two  weeks o f d r o u g h t r e s u l t e d i n a 50 p e r c e n t i n c r e a s e i n t h e d a m p i n g coefficient. for  R e s u l t s by Pauw show t h a t v i s c o u s d a m p i n g i s l e s s  large amplitudes.  Thus we  see some o f t h e p r o b l e m s i n v o l v e d  i n a t t e m p t i n g t o c o n s i d e r damping.  5 Barkan presents a s i m p l i f i e d s o l u t i o n semi-infinite elastic  solid.  graphs p l o t t e d u s i n g equations  T h i s does not c o n s i d e r damping, f r o m t h i s t h e o r y show a  w i t h r e s o n a n c e c u r v e s f o r a s y s t e m w i t h one s u b j e c t e d t o damping. s o i l has  f o r v i b r a t i o n s on  degree of freedom  of f o u n d a t i o n  From B a r k a n s c u r v e s  for  t h i s c a s e d a m p i n g d e p e n d s on t h e v a l u e s o f P o i s s o n ' s r a t i o of the f o u n d a t i o n .  of damping w i t h i n c r e a s e of s t a t i c Proper  be  vibration,  to i n e r t i a effects.  w e l l a s u p o n t h e s i z e and w e i g h t  i t may  elastic  due  1  but  similarity  T h i s i n d i c a t e s t h a t e v e n an i d e a l l y  a d a m p i n g e f f e c t on t h e a m p l i t u d e  a  contact pressure  v a l u e s o f d a m p i n g a r e as y e t d i f f i c u l t  A  seen t h a t as  decrease  i s a l s o shown. to  ascertain.  3 Some e x p e r i m e n t a l v a l u e s  f o r timber mats, used under c o n d i t i o n s  of shock l o a d i n g , are g i v e n by Barkan. be d e t e r m i n e d ,  Damping c o e f f i c i e n t s  i n a d d i t i o n t o methods m e n t i o n e d a b o v e ,  by  o b s e r v a t i o n s of damping of f r e e v i b r a t i o n s , f r o m the phase b e t w e e n t h e e x i s t i n g p e r i o d i c f o r c e and  the s o i l  may  shift  deformation*^  and  39. from h y s t e r e s i s  any o f t h e s e m e t h o d s . the  3  loops »  There have been few c o m p a r i s o n s between I t appears t h a t t h e extreme i n s t a b i l i t y of  damping f a c t o r s i s t h e c h i e f r e a s o n f o r u n c e r t a i n t i e s  involved  i n t h e o r e t i c a l t r e a t m e n t s o f v i b r a t i o n s w h e r e damping must be considered.  40  CHAPTER V DESIGN CONSIDERATIONS A.  FACTORS AFFECTING DESIGN  A foundation resting on a s o i l may undergo intolerable movement due to shear f a i l u r e resulting from compressive, or undue settlement.  For t h i s reason, most of the problems involved  i n designing against vibrations are related to control of these two phenomena. In the past, various minimum foundation weights were recommended for machines by different persons.  These weights  were based on operating speeds, horsepower, type of engine, number of cylinders, etc.  I t would appear that most of these have l i t t l e  value, since correlation between different methods i s usually poor. In many cases these foundations are extremely uneconomical, and i n a few they are apt to be inadequate. It has been long r e a l i z e d that the total pressure on the s o i l , both s t a t i c and dynamic, could be larger than the bearing capacity of the s o i l .  In order to insure against t h i s p o s s i b i l i t y occurring,  the dynamic load was often translated into an equivalent s t a t i c load and added to the permanent s t a t i c load, or dead load. Terzaghi^ refers to a design formula used by Rausch which states that the equivalent s t a t i c load should be arrived at by adding to the ordinary s t a t i c load 3 times the dynamic load. Barkan, r e f e r r i n g to evidence that dynamic pressure may induce settlements up to many times those of s t a t i c loads, says that this invalidates the type of relationship used by Rausch.  His c r i t e r i o n  of foundation design i s to l i m i t the s t a t i c pressure and the  41.  amplitude of vibrations caused by dynamic load.  This would  appear to be a more rational method. The c r i t i c a l factors to consider then, i n the design of foundations subjected to vibrations, are. amplitude and therefore resonance.  Resonance w i l l occur i f the operating frequency of  the machine or one of i t s harmonics coincides with the natural frequency of one of the six degrees of foundation freedom, assuming there i s some component of force acting i n t h i s direction. Of greatest p r a c t i c a l interest i s usually the case of purely v e r t i c a l vibrations accompanied simultaneously by a gyratory movement of the foundation as was considered i n CHAP. I I I . Any general l i m i t for the permissible amplitude of vibrations would be d i f f i c u l t to suggest.  This i s partly due to the fact,  already mentioned, that foundations undergoing r e l a t i v e l y small vibrations, especially under low frequency loads, may induce strong conditions of resonance i n adjoining structures.  Barkan  notes cases where amplitudes of 0.4 to 0.5 mm. d i d not have harmful effects; but based on a l l h i s experience he suggests a maximum permissible value of 0.20 mm.  An amplitude that would be  acceptable i n one area may be entirely unsatisfactory i n another due to r e s i d e n t i a l or precise manufacturing use of the land. 2 Converse  refers to experiments by Mutual L i f e Insurance  Company that indicate vibrations are noticed by people when the amplitude exceeds 0.36/f inches. He states that vibrations which result i n accelerations of 5 per cent that of gravity are sometimes used as a c r i t e r i o n for nuisance l i m i t . Frequency and amplitude are most easily regulated by control of the spring characteristic by means of foundation area or by  42. such methods as p i l e s , e t c . , by v a r i a t i o n i n f o u n d a t i o n mass ( T s c h e b o t a r i o f f suggests l e a v i n g c a v i t i e s near the f o u r c o r n e r s of a f o u n d a t i o n f o r t h i s purpose) or by a d d i t i o n a l components such as s p r i n g dampers. For f o u n d a t i o n s under r e c i p r o c a t i n g machinery, the n a t u r a l f r e q u e n c y , f , i s g e n e r a l l y g r e a t e r than the impressed n  frequency.  For t h i s c a s e , minimum h e i g h t of f o u n d a t i o n means minimum mass and maximum f ^ , so s a f e t y w i t h r e g a r d t o resonance  i s increased.  S m a l l e r a r e a s , which decrease the e f f e c t i v e mass of s o i l , and c o r k i s o l a t i o n pads may  a l s o be h e l p f u l .  F o r h i g h frequency machines, l a r g e b l o c k f o u n d a t i o n s w i t h l a r g e masses are g e n e r a l l y employed. the e q u i v a l e n t e a r t h mass may prove  Large areas which i n c r e a s e satisfactory.  For machines whose f r e q u e n c i e s are i n danger of p r o d u c i n g resonance, s p r i n g supports are commonly used t o lower the n a t u r a l f r e q u e n c y , t^. air.  These may be l a m i n a t e d , h e l i c a l or even compressed  Cork p r o v i d e s a good s p r i n g pad i f i t i s kept d r y and i n  good c o n d i t i o n . durability.  Asbestos i s commonly used because of i t s  As i s seen i n S e c t i o n A - 2 of CHAP. I l l ,  i n order  f o r a b s o r b e r s t o have a f a v o u r a b l e e f f e c t on the amplitude of f o u n d a t i o n v i b r a t i o n s , the f r e q u e n c y of n a t u r a l v i b r a t i o n s of the mass above the s p r i n g s s h o u l d be as s m a l l as p o s s i b l e i n comparison w i t h the f r e q u e n c y of engine r o t a t i o n .  I n h i g h frequency machines  a s u i t a b l e r e l a t i o n s h i p i s e a s i l y achieved. I f dampers are d e s i r e d f o r low f r e q u e n c y machines, the proper r e l a t i o n s h i p i s g e n e r a l l y d i f f i c u l t to e s t a b l i s h j u s t by d e c r e a s i n g the r i g i d i t y of the absorbers because s t r e n g t h requirements limit this.  may  Here a decrease i n n a t u r a l f r e q u e n c y can u s u a l l y be  43. a c h i e v e d by p r o v i d i n g i n s t e a d an i n c r e a s e i n the f o u n d a t i o n above the The  mass  springs. e f f e c t s of earthquakes must be t a k e n i n t o account i n  designing  s t r u c t u r e s i n many a r e a s .  caused by earthquakes we  When a n a l y z i n g the damage  are a t a g r e a t disadvantage i n t h a t we  not know what component of the v i b r a t i o n caused the damage. the n a t u r a l f r e q u e n c y of a s t r u c t u r e i s low compared w i t h f r e q u e n c y of the earthquake, we may  do  If  the  assume t h a t the amplitude of  v i b r a t i o n of the s t r u c t u r e w i l l be equal to t h a t of the  earthquake.  L u c k i l y , t h i s i s s a t i s f a c t o r y f o r most low b u i l d i n g s and r e t a i n i n g walls;  however, i n some s t r u c t u r e s such as h i g h smoke s t a c k s , i t  i s p o s s i b l e f o r resonance t o o c c u r .  I f t h i s happens, f a i l u r e  may  result. T e r z a g h i s a y s ^ t h a t the b e s t measure of the  destructive  impulse of an earthquake i s the amount of energy i t s u p p l i e s the f o u n d a t i o n  per u n i t t i m e .  to  S i n c e t h i s energy i s d i r e c t l y r e l a t e d  t o the impressed a c c e l e r a t i o n , earthquakes are u s u a l l y c l a s s i f i e d by the r a t i o of t h e i r a c c e l e r a t i o n s t o the a c c e l e r a t i o n of g r a v i t y . T h i s r a t i o , as measured f o r any  earthquake, i s u s u a l l y much lower,  on r o c k o u t c r o p s than on l o o s e a l l u v i u m . t o underground e x p l o s i o n s .  S i m i l a r remarks a p p l y  Tschebotarioff*  c l a s s i f i e s these by  means of what he c a l l s the energy r a t i o ; t h i s i s equal to square of the a c c e l e r a t i o n i n f e e t per measured by an a c c e l e r o g r a p h ,  second, per  the  second,as  d i v i d e d by the square of  the  f r e q u e n c y i n c y c l e s per second, measured by a seismograph. T h i s d i s c u s s i o n has d e a l t w i t h f o u n d a t i o n to general  acceptability.  design with  S e t t l e m e n t i n p a r t i c u l a r w i l l be  w i t h i n the f o l l o w i n g s e c t i o n .  regard dealt  44. B.  CONTROL OF SETTLEMENT  It i s well known that foundations subjected to shock and vibrations may undergo settlements many times larger than those caused by s t a t i c loads. are obvious.  The harmful results of such settlements  These settlements are usually considered to be  characteristic of cohesionless s o i l s , but they may also be exhibited to some degree by cohesive ones.  The physical processes  which cause changes i n s o i l properties are not c l a r i f i e d yet, but experiments  show that vibrations cause changes i n the f r i c t i o n ,  cohesion and hydro dynamic properties of a s o i l , and also on the e l a s t i c properties such as Young's modulus, the shear modulus, and the l i m i t s of e l a s t i c i t y and p l a s t i c i t y . In the usual settlement analysis of foundations under s t a t i c loads, e l a s t i c deformations, which are usually small i n comparison with residual deformations, are ignored.  E l a s t i c constants seem  to bear no relationship to the c o e f f i c i e n t of compressibility, a , calculated i n s t a t i c consolidation tests**. v 7  Because of this i t  i s very hard to predict settlements merely from the e l a s t i c properties of the s o i l .  3 Barkan shows experimental results  which indicate a decrease  in the c o e f f i c i e n t of internal f r i c t i o n of sands during vibration. He shows that this effect increases with increasing energy supplied to the s o i l , with increasing grain size, and with decreasing cohesion. Tschebotarioff* describes penetration tests on samples under s t a t i c and dynamic conditions. He shows that dynamic penetrations, while being much greater than s t a t i c penetrations i n sands, seem to have no similar result on clays.  He explains this as being due  45. to the fact that the shearing strength of sands i s dependent on external pressures which are varying due to vibration, whereas the cohesive bonds of clays are not broken, being independent of external pressure.  Progressive  slippage of grains of sand occur  during instants of decreased contact pressures. was  Penetration here  found to be governed by the number of repetitions and intensity  of dynamic force, and not by frequency.  Since residual settlement  entails primarily a rearrangement of the s o i l p a r t i c l e s , i t would follow that these observations  could be applied d i r e c t l y towards  predicting values of settlement.  The above information  indicates  an increasing effect of vibrations on the mechanical properties of a s o i l with increase i n grain size.  The p r i n c i p a l vibration  parameter seems to be acceleration, and so the i n e r t i a l force, and hence the density of the involved p a r t i c l e s may  be of some  importance. 3 Plotted graphs  of the minimum void r a t i o , e, which can be  reached under given vibration versus the acceleration of the applied vibrations have an appearance not unlike the e versus pressure curves obtained from a consolidation t e s t .  A relationship  between the slope of t h i s curve and the water content appears to exist.  These curves show, for a sand, that maximum vibratory  compaction occurs at about 80 per cent saturation.  F a i r l y close  results are obtained for a perfectly dry sand, but between these values of water content compaction decreases by as much as 75 per cent of the maximum. types of s o i l as well.  Barkan states that t h i s i s true for a l l other He also notes what he terms the  "threshold  of vibratory compaction" which i s the l i m i t of vibratory acceleration following which settlements occur.  This might correspond to  46. T e r z a g h i ' s " c r i t i c a l range of f r e q u e n c i e s " settlement  occurs.  frequencies,  w i t h i n which  Presumably t h i s i s because, between these  a m p l i t u d e s are v e r y l a r g e and f o r a g i v e n  l a r g e amplitude means l a r g e a c c e l e r a t i o n . the c r i t i c a l range of f r e q u e n c i e s one  and  excessive  frequency,  Terzaghi s p e c i f i e s  as b e i n g between one h a l f  one h a l f the fundamental frequency of the  system, a l t h o u g h he says i t i s f a i r l y  and  foundation-soil  independent of the s i z e of  3 the v i b r a t o r .  E x p e r i m e n t a l work by Barkan  f r i c t i o n decreasing for  shows the angle of  v e r y l i t t l e at f r e q u e n c i e s  below 140  a c o n s o l i d a t i o n t e s t on a sand but d e c r e a s i n g  the range from about 140  t o 240  c.p.s.  the c r i t i c a l range, s i n c e c o n f i n e d resonant frequencies the f i e l d .  The  very r a p i d l y i n  This again could  l a b o r a t o r y samples w i l l show  much higher''"* than those f o r s i m i l a r s o i l s i n  i n confining pressure.  Under h i g h e r  sand i s l e s s s u s c e p t i b l e t o compaction.  t h i s i n terms of an i n c r e a s e conditions.  indicate  t h r e s h o l d of v i b r a t o r y compaction i s known t o r i s e  w i t h an i n c r e a s e pressures,  c.p.s.  confining  Barkan e x p l a i n s  i n apparent c o h e s i o n under these  I f t h r e s h o l d of compaction i s d e f i n e d i n terms of  a c c e l e r a t i o n , then i n c r e a s e  i n a c c e l e r a t i o n i m p l i e s an  i n the a l t e r n a t i n g v i b r a t o r y f o r c e , a c c o r d i n g  increase  to the r e l a t i o n s h i p ,  f o r c e = ma where m i s the mass of the s o i l p a r t i c l e s undergoing f o r c e d motion and  a i s the a p p l i e d a c c e l e r a t i o n .  The  relative  e f f e c t of t h i s a p p l i e d f o r c e c o u l d be expected t o decrease w i t h increased  c o n f i n i n g p r e s s u r e a c t i n g on the s o i l .  An i n c r e a s e  t h i s c o n f i n i n g p r e s s u r e r e s u l t s i n a decrease i n the  relative  magnitude of the i n e r t i a f o r c e s . T e s t s by Barkan on sands show the p e r m e a b i l i t y , under c o n d i t i o n s of v i b r a t i o n , i n c r e a s i n g w i t h h i g h e r  accelerations.  in  47.  The  r a t e of i n c r e a s e goes up w i t h a decrease i n g r a i n s i z e .  These t e s t s a l s o showed a steady b u i l d u p of pore p r e s s u r e v i b r a t i o n s , f o l l o w e d by a t a p e r i n g o f f .  during  Therefore v i b r a t i o n s  must cause a decrease i n s o i l volume a t a r a t e r a p i d enough t o o f f s e t any i n c r e a s e i n p e r m e a b i l i t y .  T h i s i n d i c a t e s a decrease  i n e f f e c t i v e p r e s s u r e w i l l occur d u r i n g v i b r a t i o n , and would agr w i t h the phenomena of spontaneous l i q u e f a c t i o n o c c u r r i n g under c o n d i t i o n s of shock.  These o b s e r v a t i o n s  would seem t o i n d i c a t e  an i n c r e a s i n g e f f e c t of v i b r a t i o n s on the hydro dynamic p r o p e r t i of a s o i l , w i t h decrease i n g r a i n s i z e .  PART I I  EXPERIMENTAL WORK  48. CHAPTER V I CONSOLIDATION TESTS A.  INTRODUCTION  Plastic  c l a y s a r e commonly t h o u g h t t o be r e l a t i v e l y  independent of v i b r a t i o n s .  As a r e s u l t of t h i s , v e r y  little  work has been c a r r i e d out t o determine t h e e f f e c t s of v i b r a t i o n s on  clay consolidation.  placed  S i n c e m a c h i n e b a s e s a r e n o t a p t t o be  d i r e c t l y on a s o f t c l a y , s e t t l e m e n t  failure will  be t h e m a i n c o n s i d e r a t i o n .  instaneous e l a s t i c  compression, there  e x h i b i t e d by c l a y m a t e r i a l s of t h i s work t o c o n s i d e r the  following  during  rather  than shear  Following  the i n i t i a l  a r e two t y p e s o f  loading.  settlement  I t was t h e p u r p o s e  these types of settlement  with regard t o  questions;  (a)  P r i m a r y c o n s o l i d a t i o n , d e p e n d i n g on t h e p e r m e a b i l i t y - does v i b r a t i o n a f f e c t t h e p e r m e a b i l i t y s u f f i c i e n t l y t o be r e f l e c t e d i n the rate of settlement?  (b)  S e c o n d a r y c o n s o l i d a t i o n o r p l a s t i c f l o w , d e p e n d i n g on s t r u c t u r a l adjustment accompanied by y i e l d i n g of g r a i n bonds - w i l l v i b r a t i o n a f f e c t t h e amount a n d / o r t h e r a t e of y i e l d i n g ? The  t e s t program d e s c r i b e d  above p o i n t s  h e r e was a n a t t e m p t t o e x p l o r e t h e  b u t was n o t i n t e n d e d t o be a n e x h a u s t i v e  of t h e problem.  A possible  treatment  t e s t p r o c e d u r e was c h e c k e d and some  o f t h e d i f f i c u l t i e s t o be e x p e c t e d i n f u r t h e r w o r k a l o n g  these  l i n e s were d e t e r m i n e d . Two p o s s i b l e  cases of foundation  r e s p o n s e were  (1)  The e f f e c t o f v i b r a t i o n s f r o m o t h e r s o u r c e s on a loaded s t r a t a (Series I Tests),  (2)  The e f f e c t o f h a r m o n i c a l l y v a r y i n g l o a d ( S e r i e s I and I I T e s t s ) .  load  considered: statically  s u p e r p o s e d on s t a t i c  I n p r a c t i c e , t h e d y n a m i c component o f l o a d i s l i k e l y t o be  49.  a v e r y s m a l l f r a c t i o n of the s t a t i c l o a d ( a p p l i e d l o a d p l u s overburden).  For t h i s r e a s o n , t e s t s (see (2) above) were  conducted a t s m a l l dynamic l o a d s . I t s h o u l d be noted t h a t the c h o i c e of f r e q u e n c i e s here not t o o good.  was  A l t h o u g h the two f r e q u e n c i e s used were w i t h i n the  range of r e s o n a n t f r e q u e n c i e s t o be expected f o r f o u n d a t i o n s i n the f i e l d ,  i t i s known t h a t the n a t u r a l f r e q u e n c y of a c o n f i n e d  l a b o r a t o r y sample i s many times h i g h e r than t h i s .  F o r t h i s reason  the impressed f r e q u e n c i e s were w e l l below the resonant range f o r the samples and would not be expected t o r e p r e s e n t the  critical  case. B.  MATERIAL TESTED  Long term c o n s o l i d a t i o n t e s t s were c a r r i e d out on Haney c l a y , a b l u e marine c l a y of a p p r o x i m a t e l y the f o l l o w i n g makeup: CHEMICAL A N A L Y S I S  15  sio  A10  2  3  Fe 0 2  3  -  58.5#  -  21.1$  -  8.696  CaO  e.5fo  MgO  0.5$  loss  -  4.8#  A g r a i n s i z e d i s t r i b u t i o n curve p l o t t e d from a hydrometer t e s t , i s shown i n F i g . 23.  Readings were taken over a  p e r i o d of 18 days d u r i n g which time about 83 per cent of the suspended m a t e r i a l as measured by the hydrometer, out.  settled  50. Where exposed, t h i s c l a y weathers t o form a brown, c r a c k e d , sandy-like  surface.  U n d i s t u r b e d samples were o b t a i n e d  from the p i t of the Haney  B r i c k and T i l e Company which i s about one h a l f m i l e n o r t h and f e e t above the F r a s e r R i v e r a t P o r t Haney, B.C. S e r i e s I I and  50  Samples f o r t e s t  I I I were t a k e n from the base of a 30 f o o t  cliff  w h i c h f a c e s west and undergoes a slow f l o w i n g movement.  The  exact  l o c a t i o n from which the samples of S e r i e s I were t a k e n i s unknown. T h i s c l a y appears t o be v e r y homogeneous, and has  the  following properties: specific gravity  -  liquid limit  - 45 per  cent  plastic limit  - 28 per  cent  shrinkage l i m i t  - 26 per  cent  C.  SOURCE OF VIBRATIONS  1.  Series I.  tests.  Two  2.79  methods of v i b r a t i o n were used i n these  I n S e r i e s I , a l / l 5 horsepower e l e c t r i c motor was  b o l t e d t o one  merely  of the h o r i z o n t a l braces of a s t a n d a r d c o n s o l i d a t i o n  frame, see F i g . 3.  The  n a t u r a l e c c e n t r i c i t y of t h i s motor  was  added t o by screwing a \ i n c h l o n g \ i n c h diameter b o l t i n t o the r o t a t i n g s h a f t of the motor.  I n t h i s way,  vibrations quite  p e r c e p t i b l e t o the touch were generated throughout the frame  and  t r a v e l l e d up i n t o the s t a n d a r d c o n s o l i d a t i o n samples which were s e t on the c o n s o l i d a t i o n frame.  T h i n rubber pads were p l a c e d  under the l e g s of the c o n s o l i d a t i o n frame i n order t o p r e v e n t v i b r a t i o n s from t r a v e l l i n g a l o n g the f l o o r and a f f e c t i n g a d j a c e n t non-vibrated t e s t s . The  speed of t h i s motor was  t a k e n as 1740  revolutions  per  51.  minute, from t h e name p l a t e , or 29 c y c l e s p e r second.  No v a l u e s  of i n t e n s i t y of dynamic f o r c e or of amplitude were o b t a i n e d , b u t v i b r a t i o n s of t h e frame appeared t o be near t h a t range where v i b r a t i o n s f i r s t become p e r c e p t i b l e t o p e r s o n s , w i t h o u t  being  p a r t i c u l a r l y obnoxious. 2.  S e r i e s I I and S e r i e s I I I . V i b r a t i o n s were a p p l i e d t o  the samples of S e r i e s I I and I I I by means of an o s c i l l a t o r shown i n F i g . 4.  T h i s c o n s i s t e d of two gears which r o t a t e d i n o p p o s i t e  d i r e c t i o n s and on w h i c h were mounted e c c e n t r i c masses.  These  masses were s y m m e t r i c a l l y p l a c e d w i t h r e s p e c t t o the v e r t i c a l p l a n e p a s s i n g through t h e meshing t e e t h of t h e two gears and perpendicular  t o the p l a n e of t h e g e a r s .  I n t h i s way, the r e s u l t -  ant of t h e combined h o r i z o n t a l components of c e n t r i f u g a l f o r c e a p p l i e d t o t h e a x l e s was equal t o z e r o .  The f i n a l r e s u l t was t h a t  the v i b r a t i n g u n i t a c t e d under an a l t e r n a t i n g v e r t i c a l f o r c e of a p u r e l y harmonic  nature.  The magnitude of t h i s f o r c e i s g i v e n by the f o r m u l a F = mr  (JJ"  s i n cu t  6-C-l  where m/2 i s t h e q u a n t i t y o f t h e e c c e n t r i c mass added t o each gear, r i s t h e e c c e n t r i c i t y of the r o t a t i n g mass w i t h r e s p e c t t o the c e n t r e o f t h e r e s p e c t i v e gear a x l e . C o / i s the a n g u l a r v e l o c i t y o f r o t a t i o n of t h e gears i n r a d i a n s per second, and i s equal t o 2 77/60 times the r e v o l u t i o n s per minute. The  i n t e n s i t y of t h e dynamic f o r c e was governed by v a r y i n g  the s i z e of t h e e c c e n t r i c masses, which were measured t o one l/lOO of a gram, and the e c c e n t r i c i t y , which was measured t o one l/lOOO of an i n c h . The  gears of t h e o s c i l l a t o r were d r i v e n by means o f a  52. f l e x i b l e d r i v e coupled t o a 4-step p u l l e y .  T h i s p u l l e y was i n  t u r n connected by means of a v - b e l t , see F i g . 5, t o a s i m i l a r p u l l e y mounted on a ^ horsepower e l e c t r i c motor. of o s c i l l a t i o n was t a k e n as *-'/2 7r>  The f r e q u e n c y  from Eq. 6 - C - l .  t  the motor was t a k e n t o be a c o n s t a n t v a l u e .  Actual  The speed of measurements  showed a v a r i a t i o n i n angular v e l o c i t y of about one p e r c e n t . - By a d j u s t i n g t h e d r i v i n g r a t i o of the two p u l l e y s , t h e a n g u l a r v e l o c i t y , (jj , c o u l d be a d j u s t e d w i t h i n the l i m i t s c o n t r o l l e d by the d i a m e t r i c r a t i o s o f the p u l l e y s , and the speed of t h e motor.  This  constituted  the f r e q u e n c y c o n t r o l . S i n c e two v i b r a t e d t e s t s were c a r r i e d out, and  I I I , using d i f f e r e n t frequencies,  i s o l a t e one t e s t from t h e o t h e r .  i n both s e r i e s I I  i t was n e c e s s a r y t o c o m p l e t e l y  I n a d d i t i o n , i t was d e s i r e d t o  a p p l y v i b r a t i o n s t o t h e t e s t samples through t h e l o a d i n g head hence i t was n e c e s s a r y t o p r e v e n t v i b r a t i o n s from b e i n g  only,  transmitted  through t h e frame or through the w i r e s s u p p o r t i n g t h e crossbeam through which t h e l o a d i s t r a n s m i t t e d The  t o the loading  f i r s t of these c o n d i t i o n s was met by b u i l d i n g a second  t a b l e over t h e o r i g i n a l c o n s o l i d a t i o n one  o f t h e samples.  consolidation The  head.  frame i n order t o support  T h i s t a b l e d i d n o t come i n t o c o n t a c t w i t h t h e  frame a t any p o i n t , see F i g . 6.  second c o n d i t i o n mentioned above, t h a t o f c o n t r o l l i n g the  means o f access of t h e v i b r a t i o n s t o t h e samples, i s a l s o i l l u s t r a t e d by F i g . 6.  The l o a d i n g  c r o s s beam and i t s  associated  c o u n t e r - b a l a n c e i s supported i n each case not by means of a p o s t mounted on the s o n s o l i d a t i o n frame, as i s the u s u a l case ( F i g . 3) but by a t r i a n g u l a r wooden frame.  I t i s i n t e r e s t i n g t o note  that  i n s p i t e of t h e v e r t i c a l n a t u r e o f t h e v i b r a t i o n s , h o r i z o n t a l p i n  53. connected  b r a c e s w h i c h o r i g i n a l l y were u s e d t o t i e  wooden f r a m e s  t o each o t h e r  3)  that  t r a n s m i t the  The t e n s i o n r o d s  end r e a c t i o n s  t o the h o r i z o n t a l braces of the f r o m t h e frame  two  had t o be removed b e c a u s e o f  a b i l i t y to transmit v i b r a t i o n s . Pig.  the  and c o n n e c t e d  of the  (see  their  also  l o a d i n g beams down  c o n s o l i d a t i o n frame were  b y means o f a n c h o r s  removed  to the  concrete  floor. I n t h i s way t h e r e was no p h y s i c a l parts  of the  apparatus  associated  c o n s o l i d a t i o n samples.  c o n n e c t i o n between  w i t h t h e two  frames  of the  consolidation device,  were c o m p l e t e l y centre  of the  sample.  by the  oscillator  weight  on t h e  bar  These added  to prevent  the  through  the  into limited  b o u n c i n g of  the  of the  of a p p l i e d dynamic l o a d were  SR-4 s t r a i n gauges bonded on t h e l o a d i n g t o t h e s e s t r a i n gauges c o n s o l i d a t i o n frame  and  the r e q u i r e d degree of m a g n i f i c a t i o n so t h i s p r o c e d u r e  a m p l i t u d e was a c c e p t e d  as  of t r a n s d u c e r  was a b a n d o n e d ,  can  connected  t h r o u g h a B a l d w i n s t r a i n i n d i c a t o r t o an o s c i l l i s c o p e .  D.  frames  over  o f t h e d y n a m i c l o a d was  amplitude  (the w i r e s  be s e e n i n F i g . 6-b)  n o t be a c h i e v e d  loading  l o a d i n g head and hence  oscillator,  of the  b y means o f f o u r  loading cross  standard  bearing which rested  beam i n t o t h e  of the  supported  sample.  Direct readings attempted  the  were  l o a d i n g h e a d and t r a n s m i t t e d v i b r a t i o n s  The maximum m a g n i t u d e  static  over  see P i g . 6.  s u p p o r t e d by a b a l l  standard loading cross the  different  The v i b r a t i n g o s c i l l a t o r s  by means o f l o a d i n g f r a m e s w h i c h f i t t e d  the  However  output  and t h e  could computed  correct.  TEST PROCEDURE  T e s t s were c a r r i e d out i n s t a n d a r d c o n s o l i d a t i o n  apparatus  54. on 2\  i n c h d i a m e t e r , one i n c h t h i c k samples.  A s t a t i c load  increment r a t i o of 2 was used i n a l l c a s e s , w i t h a l o a d i n g range of \ t o 8 k i l o g r a m s per square c e n t i m e t e r (kg/cm ).  Settlement  was measured t o one 1/10,000 of an i n c h . Three s e r i e s of t e s t s were c a r r i e d o u t . S e r i e s I - Four c o n s o l i d a t i o n t e s t s were c a r r i e d out as follows: Test 1 - I - U n d i s t u r b e d , u n v i b r a t e d . Test 2 - I - U n d i s t u r b e d , v i b r a t e d . Test 3 - I - Remolded, u n v i b r a t e d . Test 4 - I - Remolded, v i b r a t e d . Each increment was a l l o w e d t o s e t t l e f o r a p p r o x i m a t e l y one week.  V i b r a t i o n s were a p p l i e d as s t a t e d i n s e c t i o n C - l .  O b s e r v a t i o n s and r e s u l t s are shown i n P a r t E. S e r i e s I I - Four c o n s o l i d a t i o n were c a r r i e d out as f o l l o w s : Test 3 - I I - U n d i s t u r b e d , u n v i b r a t e d . Test 4 - I I - Remolded, u n v i b r a t e d . Test 6 - I I - U n d i s t u r b e d , v i b r a t e d a t 16 c y c l e s per second. Test 7 - I I - U n d i s t u r b e d , v i b r a t e d a t 24.6 c y c l e s per second. Dynamic l o a d s a p p l i e d were as f o l l o w s : S t a t i c Load Test 6 - I I Total Dynamic $ of Pressure Force Load Static (kg 2) (kg) (kg) / c m  Test 7 - I I Dynamic $ of Load Static (kg)  i  7.5  0.11  1.47  0.11  1.33  \  15.0  0.22  1.47  0.19  1.27  1  30.0  0.44  1.47  0.38  1.27  2  60.0  0.87  1.45  0.77  1.29  4  120.0 ,  1.74  1.45  1.40  1.17  8  240.0  2.60  1.07  3.10  1.29  V i b r a t i o n s v e r e a p p l i e d as d e s c r i b e d i n P a r t C-2. O b s e r v a t i o n s are g i v e n i n P a r t E. S e r i e s I I I - Four c o n s o l i d a t i o n t e s t s as f o l l o w s : Test 3 - I I I - U n d i s t u r b e d , u n v i b r a t e d . Test 4 - I I I - U n d i s t u r b e d , u n v i b r a t e d . Test 6 - I I I - U n d i s t u r b e d , v i b r a t e d a t 24.6 c y c l e s per second. Test 7 - I I I - U n d i s t u r b e d , v i b r a t e d a t 16.0 c y c l e s per second. Dynamic l o a d s were a p p l i e d as f o l l o w s : S t a t i c Load  Test 6 - I I I Total Force (kg)  Pressure (kg/cm ) 2  Test 7 - I I I  Dynamic Load (kg)  fo of Static  Dynamic Load < S>  fo of Static  0  0  k  i  7.5  0  0  \  15.0  0.75  5.0  1  30.0  1.46  4.85  1.50  5.0  2  60.0  3.00  5  3.00  5.0  4  120.0  3.00  2.5  3.00  2.5  8  240.0  3.00  1.25  3.00  1.25  .575  3.82  O b s e r v a t i o n s a r e shown i n p a r t E. E.  OBSERVATIONS  Water c o n t e n t s were determined by d r y i n g samples a t a p p r o x i m a t e l y 105° C u n t i l constant weight was  reached.  Water c o n t e n t s were as f o l l o w s : S e r i e s I Tests Test 1-1  Test 2-1  Test 3-1  Test 4-1  S t a r t of t e s t  42.5  44.7  42.5  41.5  End of t e s t  25.2  30.0  25.8  27.3  56. S e r i e s II T e s t s Test 3-II  Test 4-II  Test 6-II  Test 7-II  S t a r t of t e s t  42.2  37.9  39.8  41.8  End of t e s t  31.7  23.5  30.2  29.8  S e r i e s III T e s t s Test 3-III  Test 4-III  Test 6-III  Test 7-III  S t a r t of t e s t  40.6  44.9  46.2  46.1  End of t e s t  26.8  26.3  27.2  29.5  V o i d r a t i o v e r s u s l o g of p r e s s u r e curves f o r t e s t s of s e r i e s I, I I , and III a r e shown i n F i g . 7, 8 and 9 r e s p e c t i v e l y . Fig.'s  10 t o 13 i n c l u s i v e show s e t t l e m e n t v e r s u s l o g of time  curves f o r t e s t s 1 and 2 of s e r i e s I.  F i g . ' s 14 t o 16 show  s i m i l a r curves f o r t e s t s 3, 6 and 7 of s e r i e s I I . F i g . ' s  17 t o 19  show the s e t t l e m e n t - l o g time r e l a t i o n s h i p f o r t e s t s 3, 4, 6 and 7 of s e r i e s I I I . F i g . ' s  20 through 22 show s e t t l e m e n t - l o g time  curves f o r t e s t s 3 and 4 of s e r i e s I and t e s t 4 of s e r i e s I I .  57. CHAPTER V I I DISCUSSION OF TEST RESULTS I t may be concluded from F i g . ' s 10 t o 22 which show s e t t l e m e n t p l o t t e d versus the l o g of time t h a t f o r the range of v i b r a t i o n s a p p l i e d t o these c o n s o l i d a t i o n samples t h e r e i s no p e r c e p t i b l e e f f e c t on e i t h e r the r a t e or the magnitude of settlement. I n P a r t B of Chapter V i t was suggested t h a t t h e r e d u c t i o n of i n t e r p a r t i c l e f r i c t i o n caused by v i b r a t i o n s decreased w i t h decreasing  grain size.  I t was a l s o suggested t h a t the e f f e c t of  v i b r a t i o n s towards i n c r e a s i n g the p e r m e a b i l i t y i n c r e a s e d w i t h decreasing g r a i n s i z e .  P r o j e c t i n g the f i r s t of these  considerations  i n t o the r e a l m o f c l a y s one might expect the e f f e c t of v i b r a t i o n s t o be v e r y minor, but from the second o b s e r v a t i o n , a c l a y sample might be expected t o show a more r a p i d r a t e of s e t t l e m e n t d u r i n g the stage of p r i m a r y c o n s o l i d a t i o n .  at least  I n the o b s e r v a t i o n s  made h e r e , however, t h e r e seems t o be no d i f f e r e n c e i n the c o n s o l i d a t i o n behaviour of v i b r a t e d and u n v i b r a t e d  samples, i n  e i t h e r the p r i m a r y or the secondary c o n s o l i d a t i o n range. I n any i n t e r p r e t a t i o n of these r e s u l t s i t must be noted that the frequencies c y c l e s p e r second.  of v i b r a t i o n were c o n f i n e d t o 16 and 24.1 From t h e d i s c u s s i o n i n P a r t B of Chapter V  i t i s l i k e l y t h a t t h e apparent independence of s e t t l e m e n t t o v i b r a t i o n s i n t h i s case i s due t o the range of f r e q u e n c i e s w e l l below the c r i t i c a l range r e q u i r e d t o induce Observations,  being  settlement.  a l r e a d y r e f e r r e d t o , by T s c h e b o t a r i o f f , t o the  e f f e c t t h a t f r e q u e n c y has no e f f e c t on p e n e t r a t i o n v a l u e s of a sand c o u l d a l s o be due t o t h i s same phenomena.  58. F i g . ' s 7, 8 and 9 show curves of v o i d s r a t i o v e r s u s l o g of p r e s s u r e . representing  I t may  be seen i n F i g . ' s 7 and  v i b r a t e d undisturbed  samples show a t r a c e of  upwards i n t h e i r s t e e p l y s l o p i n g p o r t i o n s . and Peck**' t h i s i n d i c a t e s e x t r a  9 t h a t the p l o t s concavity  According to Terzaghi  sensitivity.  F i g . ' s 7, 8 and 9 i n d i c a t e , w i t h r e s p e c t t o the  undisturbed  samples, t h a t the v i b r a t e d samples s u f f e r a much more abrupt break i n t h e i r c o n s o l i d a t i o n curves than do the u n v i b r a t e d The  samples.  samples of s e r i e s I , shown i n F i g . 6, were p r o c u r e d by persons  other than the author and there i s r e a s o n to b e l i e v e t h a t t h e y s u f f e r e d a measure of d i s t u r b a n c e , may  have been d i m i n i s h e d I n F i g . ' s 7 and  hence the e f f e c t of s t r u c t u r e  p r e v i o u s to t e s t i n g .  9, the maximum p a s t p r e s s u r e s as  indicated  by Casagrande's method are o n l y about 75 per cent as h i g h f o r v i b r a t e d as f o r u n v i b r a t e d  samples.  I n F i g . 8 the average p a s t  p r e s s u r e of the two v i b r a t e d samples i s about 90 per cent of t h a t i n d i c a t e d by the u n v i b r a t e d  sample.  S i n c e t h i s t r e n d was  noted  i n a l l t h r e e s e r i e s of t e s t s , i t appears t h a t v i b r a t i o n s can have an e f f e c t on those p r o p e r t i e s of a c l a y t h a t are governed l y by p a s t h i s t o r y .  The  extensive-  e f f e c t on the p a s t p r e s s u r e p r e d i c t i o n  seems t o be s i m i l a r f o r both the more s u b t l e v i b r a t i o n s of s e r i e s I and the d i r e c t l y a p p l i e d dynamic l o a d i n g of s e r i e s I I and I I I . The  undisturbed,  v i b r a t e d sample t e s t e d i n s e r i e s I l a c k s some of  the sharpness of break i n c o n s o l i d a t i o n curve noted i n s e r i e s I I and  I I I , but i t i s not p o s s i b l e t o say whether t h i s i s due  to  the  d i f f e r e n c e i n the method of a p p l y i n g the v i b r a t i o n s or due  to  the  p o s s i b l e d i f f e r e n c e i n sampling procedure.  I n a d d i t i o n , samples  t e s t e d i n s e r i e s I t e s t s remained i n s t o r a g e f o r more t h a n a year  59. b e f o r e b e i n g used.  What e f f e c t t h i s prolonged p e r i o d , ( d u r i n g  which the e f f e c t i v e c o n f i n i n g p r e s s u r e may  have been reduced)  c o u l d have on the c o n s o l i d a t i o n c h a r a c t e r i s t i c s can o n l y be guessed at.  I t would c o n c e i v a b l y c o n t r i b u t e to the more g r a d u a l change of  s l o p e i n the c o n s o l i d a t i o n curves noted i n F i g . V I I .  This  c o n c l u s i o n i s supported by the f a c t t h a t the p a s t p r e s s u r e s i n s e r i e s I are about 1.6  and 2.1 kg/cm  f o r the v i b r a t e d and  u n d i s t u r b e d samples, r e s p e c t i v e l y . These are l e s s than 90 per cent of the c o r r e s p o n d i n g v a l u e s found f o r s e r i e s I I and I I I , These t e s t s i n d i c a t e t h a t v i b r a t i o n s r a p i d l y d e s t r o y the s t r u c t u r e of the c l a y r e s u l t i n g from i t s p a s t l o a d i n g h i s t o r y , but t h i s e f f e c t i s r e s t r i c t e d t o a narrow range below the p r e - c o n s o l i d ation load.  T h i s g i v e s the c l a y a s e n s i t i v i t y g r e a t e r than t h a t  which c o u l d be expected from r e s u l t s on u n v i b r a t e d samples.  Below  75 per cent of the apparent v a l u e of the p a s t p r e s s u r e , v i b r a t e d samples behave s i m i l a r l y to u n v i b r a t e d ones i n the same range. Beyond the p r e c o n s o l i d a t i o n l o a d , both types of c o n s o l i d a t i o n t e s t s a g a i n show a v e r y s i m i l a r p r e s s u r e - v o i d s r a t i o r e l a t i o n s h i p . e f f e c t might be extremely dependent upon the l o a d increment  This ratio.  W i t h a p a r t i c u l a r v a l u e of t h i s r a t i o , the e f f e c t s of s t r u c t u r a l breakdown would be expected  t o show up i n one of the  time p l o t s , as shown i n F i g . ' s 10 t o 22.  settlement-  A g a i n i t must be  remembered t h a t the above r e p r e s e n t s the non-resonance c o n d i t i o n of v i b r a t i o n .  60.  CHAPTER V I I I SUGGESTIONS FOR FURTHER WORK A l t h o u g h t h e r e has been a measure of work done on the e f f e c t s of v i b r a t i o n s on the s e t t l e m e n t  of s o i l s and on the e v a l u a t i o n of  v a r i o u s dynamic s o i l parameters, most of t h i s has been c o n f i n e d t o cohesionless  soils.  T h i s i s p r o b a b l y due t o the much g r e a t e r  dependence of the behaviour of c o h e s i o n l e s s vibrations.  Nevertheless  s o i l s on a p p l i e d  c o h e s i v e s o i l i s a l s o known t o be some-  what a f f e c t e d by dynamic f a c t o r s , the e x t e n t of which i s s t i l l l a r g e l y unknown. From the above experiments, the Author concludes t h a t i n any work d e a l i n g w i t h v i b r a t o r y induced s e t t l e m e n t ,  p a r t i c u l a r l y with  r e g a r d t o c o h e s i v e s o i l s , s t r i c t c o n t r o l of f r e q u e n c y and magnitude of impressed v i b r a t i o n s i s mandatory.  I t seems q u i t e p o s s i b l e t h a t  under proper c o n d i t i o n s of f r e q u e n c y , a m p l i t u d e , a c c e l e r a t i o n , e t c . , a c e r t a i n maximum d e n s i t y may be a t t a i n e d by c o h e s i v e s o i l , as i s the case w i t h c o h e s i o n l e s s  soils.  I t would be i n t e r e s t i n g to d i s c o v e r the v a r i a t i o n i n r e s o n a n t f r e q u e n c y of a c o n f i n e d  sample w i t h changes i n s t a t i c  p r e s s u r e , degree of c o n s o l i d a t i o n and magnitude of dynamic l o a d . T h i s i n t u r n might show some c o r r e l a t i o n w i t h f i e l d  conditions.  A l t h o u g h s t a t i c and dynamic e l a s t i c c o n s t a n t s  are not the  same**', perhaps a r e l a t i o n s h i p between them c o u l d be  discovered. 12  ¥ork a l o n g these l i n e s h a s been o f f e r e d by W i l s o n and D i e t r i c h t  who show some c o n n e c t i o n  between modulus of e l a s t i c i t y , E, d e t e r m i n -  ed by both s t a t i c and dynamic methods, and compressive s t r e n g t h . They a l s o i n d i c a t e a q u a l i t a t i v e comparison based on p l a s t i c i t y  61.  characteristics.  A great  lines.  of  the be  The v a l u e  standard  deal  being  more w o r k  able  classification  to  s h o u l d be  predict  systems  is  done  dynamic  very  along  properties  obvious,  should  these from  this  possible. Some w o r k h a s  but  this  material  particularly would  seem t h a t  dynamic with under  as  all  to  far  as  its  of  water  classify  the  actual  dynamic  latter  be  are  carried of  Until the  regard,  concerned.  according  behaviour  loading.  in this  investigation,  contents  consideration,  value.  clay,  more  soils  should i d e a l l y  of  potential  a great deal  to  determine  the  on compacted  different  attempts  conditions  appreciation  done  requires  characteristics  studies  much o f  been  out  to  there  former  their  in  these is  It  conjunction soils a  lacks  better very  62. SUMMARY AND  CONCLUSIONS  C u r r e n t v i b r a t i o n t h e o r i e s were e v a l u a t e d  and t h e i r  a p p l i c a t i o n s t o f o u n d a t i o n s were c o n s i d e r e d .  I t was n o t e d  these t h e o r i e s  assumptions  a r e a l l b a s e d upon v e r y s i m p l e  less v i b r a t i n g spring)  and t h e r e f o r e  consideration  of such v a r i a b l e s  applications,  states  Nonlinearity equivalent influence covered  do n o t a l l o w  of s o i l  spring  of load  factors.  c h a r a c t e r i s t i c , damping and  w e i g h t o f v i b r a t i n g s o i l were d i s c u s s e d . and l i m i t s  Their  o f a p p l i c a t i o n i n t h e above t h e o r i e s  more r e c e n t  dynamic l o a d i n g , f r e q u e n c y * was  trend  i n designing  foundations  against  t o w a r d s t h e c o n t r o l l i n g o f a m p l i t u d e and  compared t o some o f t h e f o r m e r p r i m i t i v e methods,  where h o r s e p o w e r , number o f c y l i n d e r s and t h e l i k e were empirical  relationships.  were o f l i t t l e The  I t was  concluded that  on t h e p r o p e r t i e s  Long term v i b r a t e d  and u n v i b r a t e d  were c a r r i e d o u t t o c h e c k t h e a c t u a l  I  compared w i t h c u r r e n t  of s o i l s  settlement  vibration theories  stratum.  a n a l o g o u s t o waves t r a n s m i t t e d  Series  consolidation of a p l a s t i c  and k n o w l e d g e .  through a  I I and I I I samples were v i b r a t e d  l o a d w h i c h was  of t h e c o n s o l i d a t i o n  applied  were  and e x t e n t s o f  o f t e s t s were c o n d u c t e d . ' V i b r a t i o n s were a p p l i e d  i n a manner  varying  t h e e a r l y methods  w i t h the i n t e n t i o n of p r e d i c t i n g r a t e s  settlement.  used'in  value.  e f f e c t s of v i b r a t i o n s  discussed  series  were  briefly.  The  as  ( a mass-  any q u a n t i t a t i v e  as w a t e r c o n t e n t , r a t e  of s t r e s s or other a l l i e d  that  by a  clay Three  i n Series  soil harmonically  d i r e c t l y through the l o a d i n g  samples and v a r i e d  tests  heads  from about 1 t o 5 p e r c e n t  63. of the  static  load.  Settlement t o be  very  behaviour  similar.  under a l l t y p e s  I n a d d i t i o n , no  settlement  relationship  unvibrated  samples f o r e i t h e r  either  the p r i m a r y  Void r a t i o  c o u l d be  versus  but  for undisturbed  vibrated curves  for undisturbed  upward i n t h e i r of extremely and  curves  usual trend.  soils.  The  lower  This  and  concavity  characteristic  than those  for  un-  However, o u t s i d e t h e r e l a t i v e l y n a r r o w r a n g e  unvibrated  suggests  of  broke v e r y a b r u p t l y  of the break i n the v i b r a t e d e - l o g p c u r v e , p l o t s vibrated  samples,  vibrated e-log p  to t h a t  These c u r v e s slightly  differ-  a noticeable departure  steeper p o r t i o n s , s i m i l a r  samples.  a l s o showed no  u n v i b r a t e d t e s t s f o r remolded  indicated past pressures  vibrated  or r e m o l d e d samples i n  samples commonly showed a f a i n t  sensitive  time-  stage.  samples t h e r e was  samples f r o m t h e  found  n o t i c e d between v i b r a t e d and  log pressure  ence between v i b r a t e d and  d i f f e r e n c e i n the  undisturbed  or s e c o n d a r y  of v i b r a t i o n was  samples were a g a i n  t h a t v i b r a t i o n s may  for  both  identical.  affect  those p r o p e r t i e s  w h i c h a r e e x t e n s i v e l y g o v e r n e d by t h e p a s t h i s t o r y of t h e  clay  stratum. I t w o u l d appear f r o m t h e s e have an e f f e c t  on t h e  settlement  o f t h e maximum p a s t p r e s s u r e . these  t e s t s were w i t h i n t h e  second.  of a c l a y  o n l y i n the  regard to  vicinity  However t h e v i b r a t i o n s a p p l i e d i n  frequency  r a n g e o f 15  These a r e known t o be w e l l b e l o w t h e  o f c o n f i n e d samples and ones w i t h  experiments t h a t v i b r a t i o n s w i l l  are t h e r e f o r e p r o b a b l y  settlement.  - 30  resonant not  the  c y c l e s per frequency critical  64.  FIG. 3 S t a n d a r d C o n s o l i d a t i o n Frame  (b) FIG. 4 Oscillator  Unit  66,  (b) FIG. Oscillator electric  5  Drive Unit,  c o n s i s t i n g of  motor, 3 f o u r - s t e p  2 flexible  drives.  pulleys  an and  (c) FIG. Modified  6  Consolidation  Frame  Fig.  7 :  - •  C o n s o l i d a t i o n c u r v e s showing V o i d s R a t i o Versus The Log Of P r e s s u r e .  ©I  Series Test Test Test Test  I  o  •H -P oJ  U  a> •H O  I 2— 34 1-  I — Un.diisT., I—/?em., U n y i f e I— • • L/ib. I-CVndist. Uni/ib.  13  0.3  0.5  1.0 Vertical  2.0  3.0  P r e s s u r e - kg.  5.0 per  sq.  , cm.  10  00  Fig. Consolidation curves shoving V o i d s R a t i o V e r s u s The L o g Of P r e s s u r e . Series Test Test Test Test  10  TL 76 3 4 -  II II II II  i Undut. wib. '' •' * Uwib. Remol. ••  0.3  0.5  1.0 Vertical  2.0 Pressure  3.0 - kg per  5.0 sq.  10 cm.  I  0.1  i  1.0  ;  L  ;—;  10  1  100 Time  in  minutes  _  1000  10,000  Time  in  minutes  Time i n  minutes  10,000  \  1000 Time  in  minutes  10,000  - 8 kg per s q . cm.  F i £ , 22 Consolidation curves showing settlement v . s . l o g of time. Series I Test 3 Test 4 Series II Test 4 0.1  'em (jjno/b. Uib .  1.0  10  100  Time i n minutes  1000  10,000  FIG.  Grain Size  23  Distribution  85. BIBLIOGRAPHY 1.  T s c h e b o t a r i o f f , G.P.: " S o i l Mechanics, F o u n d a t i o n s and E a r t h S t r u c t u r e s " , M c G r a w - H i l l , New York, 1951.  2.  Leonards, G.A.: "Foundation E n g i n e e r i n g " , M c G r a w - H i l l , New York, 1962, Chap. 18, "Foundations S u b j e c t e d t o Dynamic F o r c e s " , by F . J . Converse.  3.  Barkan, D.D.: "Dynamics of Bases and F o u n d a t i o n s " , M c G r a w - H i l l , New York, 1962.  4.  Novak, M.: "The V i b r a t i o n s of Massive F o u n d a t i o n s on S o i l " , P u b l . of the I n t e r n . Assoc. f o r B r i d g e and S t r u c t u r a l E n g i n e e r i n g , Z u r i c h , 1960.  5.  Barkan, D.D.: " F i e l d I n v e s t i g a t i o n s of the Theory of V i b r a t i o n of Massive Foundations under Machines", I n t e r n . Conference of S o i l Mechanics and F o u n d a t i o n E n g i n e e r i n g , Cambridge, 1936.  6.  T e r z a g h i , K.: 1943.  7.  Pauw, A.: "Dynamic Analogy f o r F o u n d a t i o n - S o i l Systems", A.S.T.M. Spec. Tech. P u b l . No. 156, 1953.  8.  L o r e n z , H.: " E l a s t i c i t y and Damping E f f e c t s of O s c i l l a t i n g Bodies on S o i l " , A.S.T.M. Spec. Tech. P u b l . No. 156, 1953.  9.  Den H a r t o g : "Mechanical V i b r a t i o n s " , M c G r a w - H i l l , New York,  " T h e o r e t i c a l S o i l Mechanics", W i l e y , New  York,  1934.  10.  Bernhard, R.K. and F i n e l l i , J . : " P i l o t S t u d i e s on S o i l Dynamics", A.S.T.M. Spec. Tech. P u b l . No. 156, 1953.  11.  Shannon and W i l s o n , S e a t t l e , Wash.: "Test R e s u l t s , June unpublished."  12.  W i l s o n , S.D. and D i e t r i c h , R . J . : " E f f e c t of C o n s o l i d a t i o n p r e s s u r e on E l a s t i c and S t r e n g t h P r o p e r t i e s of C l a y " , A.S.C.E. Shear Conference, B o u l d e r , C o l o r a d o , June I960.  13.  T a y l o r , D.W.: York, 1948.  14.  Eastwood, W.: "The F a c t o r s which A f f e c t N a t u r a l Frequency of V i b r a t i o n of. Foundations and the E f f e c t of V i b r a t i o n s on the B e a r i n g Power of F o u n d a t i o n s i n Sand", P r o c e e d i n g s of the 3rd I n t e r n . Conference of S o i l Mechanics and F o u n d a t i o n E n g i n e e r i n g , S w i t z e r l a n d , 1953.  15.  Armstrong, J.E.: " S u r f i c i a l Geology of the New Westminster A r e a " , G e o l . Surv. of Canada, Paper 57-5, 1957.  16.  T e r z a g h i , K. and Peck, R.s " S o i l Mechanics i n E n g i n e e r i n g P r a c t i c e " , W i l e y , New York, 1948.  "Fundamentals  1962,  of S o i l Mechanics", W i l e y , New  

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