VIBRATIONS IN SOIL WITH SPECIAL A REVIEW OF CURRENT ...

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. S c , 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 the Department

of CIVIL ENGINEERING

We accept t h i s t h e s i s as conforming to the required standard

THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1963

In presenting th i s thesis in p a r t i a l fulf i lment of

the requirements for an advanced degree at the Univers i ty of

B r i t i s h Columbia, I agree that the Library sha l l make i t free ly

avai lable for reference and study. I further agree that per

mission for extensive copying of this thesis ,for scholarly

purposes may be granted by the Head of my Department or by

his representatives,. I t i s understood that copying, or p u b l i

cat ion of this thesis for f i n a n c i a l gain sha l l not be allowed

without my written permission.

Department of C i v i l E n g i n e e r i n g

The Univers i ty of B r i t i s h Columbia, Vancouver 8, Canada.

D a 1= e A p r i l 18, 1963. ;

i i

ABSTRACT

I t i s the purpose of t h i s t h e s i s to r e v i e w , and to evaluate

to a degree, some of the current t h e o r i e s d e a l i n g w i t h the e f f e c t s

of v i b r a t i o n s on foundat ions i n contact w i t h the e a r t h .

Bas ic 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 cus sed . Cond i t ions l e a d i n g to v i b r a t i o n and

shock problems, 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 presented . The e f f e c t s of v i b r a t i o n s

on foundat ions , w i t h p a r t i c u l a r regard to se t t l ement , are c o n s i d e r

ed by means of m o d i f i c a t i o n s to the s o i l c h a r a c t e r i s t i c s as

observed by other 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 undi s turbed

and remolded c l a y samples, both 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 to secure a comparison between a c t u a l

t e s t r e s u l t s and the conc lus ions g iven by the above theory , f o r

what may be cons idered an extreme case. Cohesive s o i l i s known to

be much l e s s s e n s i t i v e to v i b r a t i o n than cohes ionles s s o i l , thus

very 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 cohes ionles s s o i l to v i b r a t i o n was i n v e s t i g a t e d

w i t h i n the l i m i t s of these t e s t s .

v i

ACKNOWLEDGEMENT

The Author wishes to express h i s thanks to h i s A d v i s o r ,

P r o f e s s o r N.D. Nathan, f o r h i s valuable suggestions and guidance.

I t was a pleasure to work under h i s s u p e r v i s i o n . The Author

a l s o wishes to express h i s indebtedness to P r o f e s s o r J.F. Muir

and the C i v i l E n g i n e e r i n g Department f o r making t h i s t h e s i s

p o s s i b l e .

A p r i l 1963.

VANCOUVER, B r i t i s h Columbia.

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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 SHEAR 9.

E. THE COEFFICIENT OF ELASTIC NON-UNIFORM SHEAR 10.

F. MODIFIED COEFFICIENTS 11.

CHAPTER I I I . MODES OF VIBRATION - SIMPLIFIED THEORIES 13.

A. VERTICAL VIBRATIONS 13.

1. One Degree of Freedom 13.

2. Two Degrees of Freedom 18.

B. ROCKING VIBRATIONS 21.

C. HORIZONTAL VIBRATIONS 23.

D. COMBINED VIBRATIONS 25.

CHAPTER IV. VALIDITY OF VIBRATION THEORIES 30.

A. GENERAL CONSIDERATIONS 30.

B. NONLINEARITY 32.

C. THE INERTIA OF THE SPRING BASE 34.

D. DIFFICULTIES ASSOCIATED WITH DAMPING 37.

CHAPTER V. DESIGN CONSIDERATIONS 40.

A. FACTORS AFFECTING DESIGN 40.

i v

B. CONTROL OF SETTLEMENT ' 44. PART TWO - EXPERIMENTAL WORK

CHAPTER VI. CONSOLIDATION TESTS. 48. A. INTRODUCTION 48. B. MATERIAL TESTED 49. C. SOURCE OF VIBRATIONS 50.

1. Ser i e s I 50. 2. Series I I and Series I I I 51.

D. TEST PROCEDURE 53. E. OBSERVATIONS 55.

CHAPTER V I I . DISCUSSION OF TEST RESULTS 57. CHAPTER V I I I . SUGGESTIONS FOR FURTHER WORK 60. SUMMARY AND CONCLUSIONS 62. BIBLIOGRAPHY 85.

LIST OP FIGURES

Page F i g u r e 1. 19.

F i g u r e 2. 19.

F i g u r e 3. 64.

F i g u r e 4. 65.

F i g u r e 5. 66.

F i g u r e 6. 67.

F i g u r e 7. 68. i

F i g u r e 8. 69.

F i g u r e 9. 70.

F i g u r e 10. 71.

F i g u r e 11. 72.

F i g u r e 12. 73.

F i g u r e 13. 74.

F i g u r e 14. 75.

F i g u r e 15. 76.

F i g u r e 16. 77.

F i g u r e 17. 78.

F i g u r e 18. 79.

F i g u r e 19. 80.

F i g u r e 20. 81.

F i g u r e 21. 82.

F i g u r e 22. 83.

F i g u r e 23. 84.

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 , pile-driving and various other sources. Oscillating machinery, such as compressors and punch presses result in unbalanced inertia forces, whereas periodic harmonic forces usually result from the unbalance of rotating parts.

If an elastically supported structure is temporarily forced out of i t s equilibrium position by an impact or sudden application or removal of a force, elastic forces are no longer in equilibrium with external forces; the structure is set in 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 7*

groups . A. Low to Medium (below 500 r.p.m.). This includes large

reciprocating machines, compressors, large blowers. Reciprocating engines operate at frequencies from 50 -250 r.p.m. but have considerable second harmonic content, so sizable dynamic forces up to frequencies of 500 r.p.m.

* Superscript numbers refer to References in the bibliography at the back.

2.

must be withstood by the foundation. B. Medium to High (300-1000 r.p.m.). Medium s i z e r e c i p r o c a t i n g

engines such a s d e i s e l and gas engines, a s w e l l a s blowers and r o t a t i n g machinery.

C. High (Above 1000 r.p.m.). High speed i n t e r n a l combustion engines, e l e c t r i c motors, steam t u r b i n e s .

The design of a foundation may require considering the e f f e c t s of v i b r a t i o n s t ransmitted through the ground from adjacent sources, p a r t i c u l a r l y i f there i s any p o s s i b i l i t y of resonance. A v i b r a t i n g foundation represents the centre of a disturbance which proceeds r a d i a l l y i n a l l d i r e c t i o n s . In t h i s way i t might produce forced v i b r a t i o n s i n s t r u c t u r e s r e s t i n g on the subgrade at a considerable distance from the centre of disturbance. The impulse t r a v e l s through the subgrade i n a wave form at 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 types 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 perpendicular to the d i r e c t i o n of wave propogation, r e s p e c t i v e l y .

Most v i b r a t i n g foundations transmit v i b r a t i o n s by means of surface waves, the most common of which are Rayleigh waves and Love waves. The former are somewhat analogous to ocean waves while the l a t t e r are a s p e c i a l type of shear wave. Since wave propogation c o n s t i t u t e s an extensive theory i n i t s e l f , i t w i l l be considered no f u r t h e r here.

3.

CHAPTER II VIBRATION PARAMETERS

A. GENERAL CONSIDERATIONS According to the theory of elasticity, any small deformation

can be resolved into the sum of the deformations accompanied only by shear and the deformations accompanied only by a change in volume. A truly solid body possesses, as a rule, only partial elasticity 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 elastic deformations in a medium, the value of residual deformation must be subtracted from the total deformation.

Two elastic constants are sufficient to describe a body whose elastic properties are identical i n a l l directions and at a l l points. The two constants commonly used for this purpose in 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 soi l on the normal stresses, and considering only elastic deformations, suggests that neither E nor V are constant. For this case E increases and V decreases with an increase in pressure. Accordingly, a tensile stress would decrease E while increasing V • This indicates that the elastic properties of soil change with variations in compressive stress.

The elastic deformations of a soi l depend on the period over which the load acts. Secondary consolidation in cohesive soils is a good example of this. For this reason, elastic deformations and hence values of the elastic 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.

of homogeneity 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 would s t i l l

be d e a l i n g w i t h a n o n - l i n e a r medium. However, due t o the complex

i t i e s i n v o l v e d , the n o n - l i n e a r a s p e c t s of the problem are commonly

i g n o r e d . The modulus of d e f o r m a t i o n , c o n s e q u e n t l y , i s t a k e n t o be

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 of f o u n d a t i o n and s o i l

mass. 3

Barkan 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 , * \ / , by

P o k r o v s k y and by Gersevanov w h i c h i n d i c a t e t h a t t h i s c o e f f i c i e n t

depends on the v e r t i c a l p r e s s u r e . He g i v e s e x p e r i m e n t a l r e s u l t s

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 showing t h a t y i s independent

of the m o i s t u r e c o n t e n t , w, but t h a t an i n c r e a s e i n sand f r a c t i o n

r e s u l t s i n a decrease i n V . T e s t s on sands show the modulus of

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 independent of w and g r a i n

s i z e , but t e s t s on c l a y cubes show E d e c r e a s i n g markedly w i t h an

i n c r e a s e i n water c o n t e n t . R e s u l t s are shown where E d e c r e a s e s

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 the v o i d r a t i o , e, and w i t h decrease 12

i n c o n f i n i n g p r e s s u r e . I t f o l l o w s t h a t , p a r t i c u l a r l y , f o r

c l a y e y s o i l s , Young's modulus i s a f u n c t i o n of the p h y s i c o -

m e c h a n i c a l p r o p e r t i e s . To determine E a c c u r a t e l y , i t i s n e c e s s a r y

t o 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 s i t e .

B. THE COEFFICIENT OF ELASTIC UNIFORM COMPRESSION

The v a l u e of Young's modulus under dynamic v e r t i c a l l o a d i n g

i s sometimes c a l l e d t h e c o e f f i c i e n t of dynamic subgrade r e a c t i o n .

I t w i l l be r e f e r r e d t o here as the c o e f f i c i e n t of e l a s t i c u n i f o r m

compression, Cu. T h i s i s e q u a l t o the magnitude of a p p l i e d

v e r t i c a l 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 s t r a i n . I t i s not

a c o n s t a n t , but an average o v e r a l l v a l u e can be t a k e n f o r a

5.

pa r t i cu la r foundation. Gu re la tes only to the e l a s t i c part of

the settlement and determines the spring charac te r i s t ic of the

s o i l base under v e r t i c a l v ib ra t ions .

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 natural c i r c u l a r frequency, f , of

a s o i l by means of a v ibra tor and ca lcu la t ing Cu from the wel l

known re l a t ionsh ip ,

However, Schleicher (1926) has shown (reproduced, i n part , i n

references 3 and 6) that settlement of a r i g i d foundation on a

per fec t ly e l a s t i c base i s inversely proport ional to the square

root of the loaded area. Settlement also i s shown to increase

s l i g h t l y wi th increase i n the length to width r a t i o of the foundat

ion , thus i t i s obvious that Cu w i l l be di f ferent for di f ferent

foundations. Schle icher ' s formulas also show a d i rec t dependence 2 3 of Cu on the factor l / ( l - V ) , where 1/ i s Poisson's r a t i o . Barkan

determines values of Cu, by means of s t a t i c loading, which increase

with decrease of bearing plate area, so adding some support to

Schle icher ' 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 ra t ios of

the square roots of the base areas, as i n Schle icher ' 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 deflect ions before and after removal of the load.

Tschebotarioff points out̂ " 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 with the frequency of

(see Chapter I I I for der iva t ion and def in i t ions)

6. the exc i t ing force as we l l as wi th the magnification of amplitude

near resonance. Experimental resu l t s based on vibratory tests*'

indicate that Cu changes at a much slower rate than that based on

the square root r e l a t ionsh ip . This i s probably due to the depend

ence of e l a s t i c coeff ic ients on the confining pressure, which

varies wi th depth. With greater contact area, a greater depth of

s o i l i s affected, 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 width,

which i s the trend suggested by Schle icher ' s theore t ica l treatment.

The effects on the spring coeff ic ients of the s ize and shape

of the contact area, and the increase i n Cu with depth may be 7

considered by a procedure offered by Pauw . In th i s analogy,

the foundation-soi 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 states,

s = c + N tan 0 where

s = maximum shearing resistance

c = cohesion

N = normal pressure on the shear plane (increases with depth)

0 = angle of in te rna l f r i c t i o n

From th i s re la t ionsh ip 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 ca lcu la t ing t o t a l s t r a in by integrat ing over the in t e rva l

7.

from the foundation base down to i n f i n i t y , he calculates the

value of the spring constant for various sizes 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 ac tua l ly

occur i n cohesionless s o i l s , then dynamic t ens i l e stresses

exceeding the i n i t i a l dead load stresses w i l l inval ida te these

r e s u l t s . However i t would appear from Pauw's ca lcula t ions that

dynamic stresses even approaching the s t a t i c stresses would severe

l 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 IV. I t

should be noted here that cracks may develop i n the s o i l due to

seasonal moisture f luctuat ions and cause spring factors to vary.

Values of Cu are given by Barkan'* which vary from less than

3 kg/cm for soft clays and s i l t s to 10 kg/cm for dense grave l ly

sands. These values of spring constants are given without

reference to a pa r t i cu la r foundation, but qua l i t a t ive information

as to the effects of various foundation factors i s offered. Pauw's

coeff ic ients on the other hand are theore t ica l and are calculated

for a pa r t i cu la 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 generalizations are closer to the pa r t i cu la 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 hor izonta l e l a s t i c base w i l l depend upon

the coef f ic ien t of e l a s t i c non-uniform compression of the base,

Cp . Here, as with Cu, an average overa l l value of Cy must be

taken for a pa r t i cu la r foundation. Cy i s equal to the moment

about the axis of ro ta t ion (due to the v e r t i c a l reaction) per uni t

8.

of angular rotation, divided by the area moment of inertia 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 (rigid base), Barkan shows theoretically that the coefficient of uniform compression Cu, established under uniform loading is not equal to Cy, , established for non uniform loading. He then cites experimental evidence to confirm this. Thus the elastic properties of the soil 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 ratio C^/Cu agree to within 8$ with experimental values, Barkan suggests establishing either Cu or Cy for a site and computing the other value from this ratio. In reference 5 he determines the value of Cr from the relationship

r - M ^ - i(y>p -y>o)

where M = the applied moment (due to eccentric loading) I = the moment of inertia 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 static determinations of Cy agree well. From his experimental results i t is possible to see that Cy is even more sensitive to changes in the length-width ratio of the foundation than is Cu.

Pauw has developed expressions for the spring constants for cases of rotation about a centroidal axis in the base contact plane. In this solution, as in Barkan1s, i t is assumed that horizontal planes are not distorted, but remain plane even after

9. rotation. In a manner analogous to that used in considering vertical loading, the spring characteristic representing C v

is developed for both cohesive and cohesionless soils for different sizes of foundations. It is d i f f i c u l t to compare values of spring constants taken from Paw and Barkan because of the variations in sites, and the different parameters used. However the ratios 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 ratio of about 1.9. This compares remarkably well with Barkan1s contributions. Other calculations

2 by Converse under similar s o i l conditions show, using Pauw's coefficients, the value of this ratio 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 last two footings have a relatively large length to width ratio and a difference in foundation-soil contact area but i t is s t i l l hard to reconcile these results with Barkan's data.

D. THE COEFFICIENT OF ELASTIC UNIFORM SHEAR The elastic spring constant tending to resist the purely

elastic component of horizontal foundation movement on a horizontal base w i l l be called the coefficient of elastic 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 soil per unit of horizontal

3 displacement. Barkan gives theoretical evidence showing that just as in the case of the vertical spring characteristics, the coefficient Ct is also inversely proportional to the square root

10.

of the contact area. In this case, i t is noted that C t maintains the same relationship with Poisson's ratio,V , as was shown for Cu in Schleicher's formulas. Since V is larger in cohesive than cohesionless soils, then C c w i l l be smaller in clayey than sandy soils, under otherwise equal conditions. Experimental results are offered to corroborate.this, as well as the square root relationship; however, the latter is shown to be true only for base areas in the order of 1 or 2 square meters. Larger base areas, particularly those above 10 - 12 square meters, failed to show much connection between C t and the square root of foundation area. A dependence of this spring coefficient on the ratio of the lengths of foundation sides, as in the previous cases, is again shown. Experiments are also given which show, along with a direct relationship with vertical pressure, a dependence of C t on the duration for which the vertical pressure acts.

Pauw also offers a means of calculating this spring constant, using a procedure similar to that used for the preceding characteri s t i c s .

E. THE COEFFICIENT OF ELASTIC NON-UNIFORM SHEAR During rotation around a vertical axis, the base of a

foundation undergoes non uniform shear. The spring characteristic resisting the elastic movement in this direction w i l l , using Barkan's terms again, be called the coefficient of elastic nonuniform shear, Cyr. It is equal to the moment about the axis of rotation per unit angle of rotation, divided by the polar moment of inertia of contact base area of foundation. Experimental values show Cy as being somewhat larger than C^. 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. I n a 2

numerica l example, Pauw shows Cy as be ing approximately 1.9

times as l a rge as C t . On examination of Pauw's equations i t i s

apparent t h a t f o r other than smal l foundat ions , as cons idered i n

h i s example, Cy i s apt to be much more i n excess of C t than was

noted i n Barkan ' s experiments . Al though most of Barkan*s e x p e r i

mental r e s u l t s were d e r i v e d from smal l foundat ion s t u d i e s , one

case i s o f f e red where the r a t i o Cy / C t was only 1.16 on a

foundat ion of 15 square meter base a rea . Thi s would suggest a

weakness i n Pauw's c o e f f i c i e n t s i n l a rge founda t ions .

F . 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 re spec t

to the surface p l a n e . A c t u a l l y , foundat ions are u s u a l l y p laced

below the s u r f a c e , thus there w i l l be a r e s t r a i n i n g e f f e c t

exerted d u r i n g v i b r a t i o n by the s o i l i n contact 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 . Quant i t a t ive i n f o r m a t i o n on the values

of these s p r i n g constants i s even r a r e r than f o r the case of

h o r i z o n t a l su r f ace s . 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 plane s i m i l a r to those a l ready cons idered i n the 4

h o r i z o n t a l p l a n e . Novak cons iders 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 contact w i t h the foundat ion only as they

c o n t r i b u t e to the c o e f f i c i e n t s a s soc ia ted w i t h the h o r i z o n t a l

contact p l a n e . He o f f e r s e x p e r i m e n t a l l y obtained conc lus ions t h a t

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 increased only s l i g h t l y by t h i s e f f e c t f o r v e r t i c a l l y e x c i t e d v i b r a t i o n s . 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 increase 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 ( fo r a 10 x 5 foot foundat ion p laced 3 f ee t

12.

below ground level, 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 vertical spring characteristic.

In the case of horizontal excitation however, Novak gives evidence which shows a much greater contribution by the vertical 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 so i l surface could be reasonably reduced to investigations of a r i g i d block resting on a semi-infinite elastic solid. Since there is not yet a solution to this problem, simplifying assumptions are necessary.

The simplest solution considers a foundation placed on a weightless soil base, subject to an i n i t i a l displacement, with damping completely neglected. A linear relationship between soil reaction and foundation displacement is assumed. Taking for this case a constant value of the coefficient of uniform elastic compression, Cu, for the s o i l , then the r i g i d i t y of the base is Cu A where A is the area of foundation contact with the s o i l .

Considering z, the vertical displacement of the foundation from the equilibrium position, as positive downwards, and using d'Alembert's principle, the equation of motion for vertical vibration with one degree of freedom i s ,

W/g z M + Cu A z = 0, or z" + f 2 z = 0, 3-A-l nz 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 so i l springs are neglected. Actually there is a significant increase in vibrating mass due to the addition of the soil springs. This additional weight, V s is usually added to the foundation weight, Wf, in practice. Methods of determining V s w i l l be discussed in Chapter IV.

14.

m = W/g, the mass of the foundation and load g = the a c c e l e r a t i o n due to 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 displacement w i t h respect to time ( a c c e l e r a t i o n )

f2 _ ^ u "\ the force r e q u i r e d per u n i t of foundation nz — mass to move the foundation through a

v e r t i c a l distance of one u n i t . The general 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 f t + B cos f t 3-A-2 nz nz where A and B a r e constants determined by i n i t i a l c o n d i t i o n s .

From t h i s we see that free v i b r a t i o n s , occurring where only the i n e r t i a l forces of the foundation and the e l a s t i c forces of the base are considered, c o n s t i t u t e harmonic motion w i t h a frequency, f n z > c a l l e d the " n a t u r a l c i r c u l a r frequency of v e r t i c a l v i b r a t i o n s of the foundation". The time r e q u i r e d f o r one complete cyc l e i s 2 7T/f , which i s c a l l e d the p e r i o d , T. The frequency, which i s the r e c i p r o c a l of the p e r i o d , i s equal to ^ n z/ 2"^ and i s g e n e r a l l y given i n o s c i l l a t i o n s per second (hertz) or r e v o l u t i o n s per minute, R.P.M. I t i s seen that the n a t u r a l frequency of a foundation i s determined only by the foundation mass and the e l a s t i c i t y of the base, ( f o r t h i s , the simplest case).

In p r a c t i c e , e x c i t i n g loads imposed by machines are u s u a l l y harmonic fu n c t i o n s of time, therefore 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 equation 3-A-l to get the expression,

z" + f 2 z = p s i n OJ t 3-A-3 nz where p = P/m, the e x c i t i n g force per u n i t of foundation mass,

OJ = the frequency of the exciting force. Other symbols are as used in Eq. 3-A-l.

The complete solution of Eq. 3-A-3 is in the form z = A sin f t + B cos f t + A sin CO t 3-A-4 nz nz z Terzaghi shows^ that a beating phenomenon with a period

277"/(fnz - CO ) occurs when this complete solution is considered. Since in actual cases involving forced vibrations, the free vibrations are soon damped out, beating w i l l not be considered here. In passing, i t may be worth noting that at (x) = f__ the

HZ

period of beating is i n f i n i t e l y long, thus the amplitudes of vibration increase without limit.

Breaking down Eq. 3-A-4 we see that the f i r s t two terms constitute free vibrations as considered in Eq. 3-A-2. The remaining term, A sin CO t represents the steady state vibration

z with an amplitude A and a circular 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 n z ~ 0 j 2 ) 3-A-5

= A s t - i - T =f Ast i - or

f2 3-A-6 nz

P where A . = sr- is the foundation displacement under load P

8 m f , applied statically, nz

and n = — 2 1 * s a m a g n i ^ i c a ' k i o n factor known as the ' 1 - CU dynamic modulus.

f 2

nz

From this 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 in f i n i t y . This is the condition of resonance. Alternately, as u) gets much smaller than f i t is seen that f| decreases rapidly, therefore the amplitude of forced vibrations reaches infinitesimal 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 is always a loss of energy which tends to decrease the amplitude of a vibrating system. This phenomenon is 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 in f i n i t y . Damping is caused by departures of the mechanical properties of the soil from those of an ideal elastic body. Irreversible phenomena, occurring during the vibration cycle, give rise to losses of energy. One of the classic types of damping, which is usually applied to vibration considerations, is called viscous damping: i t constitutes a force which tends to prevent motion, and whose magnitude is directly proportional to the velocity of the motion. The assumption of such a force renders the problem amenable to mathematical solution, and is believed to be quite close to the truth in problems of foundation vibrations.

Introducing viscous damping into equation 3-A-3 we have z" + 2 cz' + f 2 z = p sin u) t 3-A-7 nz

where c = y 2— , is the damping constant and is equal to one half the damping resistance per unit of foundation mass per unit velocity of motion.

z' = the f i r s t derivative of vertical displacement with respect to time (velocity).

17. The other symbols are as defined in Eq. 3-A-2. The complete solution to Eq. 3-A-7 is z = e*"ct (A sin f n d t + B cos f n d t ) + M sin OJ t + N cos uj t 3-A-8 The f i r s t two terms represent free vibrations with a natural circular frequency, f n d» I* is seen that they w i l l be damped out by the coefficient e~c*. Considering only these terms,resubstit-ution into 3-A-7 shows the natural frequency to be:

fnd = / f 2 - c 2 3 - A ~ 9

nz

This indicates that the damping properties of a soi l decrease the natural frequency of vibration of a foundation. If c is larger than f then free vibrations are not possible. C r i t i c a l damping occurs at c = f

nz Considering now the third and fourth terms of Eq. 3-A-8, which

correspond to the steady forced vibrations of the foundation we have z = M sin OJ t + N cos co t Putting this into Eq. 3-A-7 and setting CJ t = 0 and 7f/2 we

obtain P <fnz - U 2 ) . 2 p co,

Z = 772 2 T 2 — : — 2 — 2 S I N t " 772 T 2 7 2—:—2—2 C O S 1 0 * <fnz - «^ ) + 4 c ( f ^ z - OJ ) + 4 c c j *

Adding these two components of motion vectorally, we have

z = Al2 + N 2 sin ( CJ t - )

= A* z sin (COt ),

/-2 «. where A*„ = M + N = ,—« o o

z m / ( f 2 -OJ 2 ) 2 + 4 c2uJ 2 3-A-10 nz

18.

The phase sh i f t between the exc i t ing force app l ica t ion , P,

and the displacement induced by P i s

= tan ^ 1 ̂ | = tan c —~~ f ; v - c u nz

The amplitude of forced v ib ra t ions , A # z i s equal to

J'j # A g^. where f| # i s the dynamic modulus equal to

3 - A - l l

1 1 -2 n 2

f 2

nz

2 ~ nz

3-A-12

Curves of dynamic modulus and phase angle versus cu ft nz

are shown i n F i g . 1 and F i g . 2, respec t ive ly . These were taken

from Reference 2. 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. F i g . 2

shows that when the r a t i o <*j/^ n z I s low, the phase angle i s small

for a l l damping ra t ios but as the frequency r a t i o increases, the

damping r a t i o , c / f , becomes important. When c j / f = 1 , & = 90° HZ n 9 7 nz

regardless of damping. With further increase of ^ / f ^ * ^ n e motion

becomes further out of phase with the impressed force.

2. TWO DEGREES OF FREEDOM. In cases where only very low

amplitudes of v ib ra t ion of machine foundations are permissible ,

v ib ra t ion absorbers may be used. These absorbers considerably

decrease v ibra t ions produced not only by the main ( f i r s t ) harmonics,

but also by higher harmonics of the exc i t ing force. 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. This

gives r i s e to a system having 12 degrees of freedom. In prac t ice ,

such absorbers are used almost so le ly for the v i b r o i s o l a t i o n of

PL H) H) P O P

a H 3 TO o o 3 01

P P

o cn cn

2 P

3 H . H J T O e+- C 3

fl) (5

C o ro

H j

Hs 3 P

«•-«< 3* (0 ^

P

H " O 3

•1 (D

C <D 3 o

M

P

,4

Dynamic A m p l i f i c a t i o n F a c t o r 00

/ "

\ i

/

I >

44 h—-

* s <a p o

II

p o

O ft

§ °

/ r>

< H , P »-* I-" fl> TO C *o • ro C Cfl (D

3 N o o ro

C ct - ' " i | ro •r p 3

o 3" << CL o P p CO 1 3 Hs ro P •3 O c+-H - P H -3 3 •, O TO ̂ TO

P O H,

(— ro £ O H -3 o co C ro . H r t - W »i p cn 3 3 •

ca o •

Phase-angle )f degrees §

§ 2 8 $ S S o o' o 5 11 \ \ \ \ \ \ - \ . '

c A n> N i

i p O O O p OO > U i to fc> — b b > • ft o (f ti <̂ -f*

•6T

20.

engines with v e r t i c a l cyl inders therefore the analysis can usual ly

be l im i t ed to v e r t i c a l v ibrat ions only, with two degrees of freedom.

Assuming the centres of gravi ty of a l l mass components l i e

on one s t ra ight l i n e , and neglecting damping, we can wri te 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 ibra t ions wi th spring

absorbers as fo l lows:

m. z" . + c. z. - c 0 (z0 - z, ) = 0 , / 1 1 * X 3-A-13

m 2 z 2 + c 2 ^ z 2 z l ^ = F s i n COt where m,, m 2 = the v ib ra t ing masses below and above the

absorbers, respec t ive ly ,

z, , t.^ — ^ n e v e r t i c a l displacements of the centres of gravi ty of masses m̂ and m 2 ,

c^ = CuA, the coef f ic ien t of e l a s t i c r i g i d i t y of the earth base.

c 2 = the coef f ic ien t of r i g i d i t y of the combined spring absorbers.

Other symbols are as used previously .

The so lu t ion of Eq. 3-A-13, neglecting the free v ib ra t ion

components w i l l be of the form

z^ = A^ s i n c o t and z 2 = A 2 s i n co t 3 - A - l 5

where the amplitude of forced vibrat ions of the foundation below

and above the springs are A^ and A 2 respec t ive ly . I t i s shown A

i n Reference 3 that 1 = „

where A^ i s the amplitude of forced vibrat ions without absorbers

as given by Eq. 3-A-5, and

M_ i s the magnification factor , equal to s :2 , v <c 2

M s = 1 - U + / i ) « I + 6 £ , - S « . > 3-A-16

Here = f^/co a n d £ l z = ^ n l z ^

f , =1 , the frequency of natural vibration of the niz Jm, + m~ i i • - . 1 — assuming no absorbers,

21.

FT vhere f ^ = j — ' the frequency of natural vibra t i o n of the

n « 2 foundation above the springs assuming no movement of the lower foundation,

K = , the frequency of Jm, + m~ , . , 1 1 2 complete system,

and JJ = n̂ /m̂

From Eq. 3-A-16 i t i s seen that a s £ ^ approaches zero, Mg and

hence the amplitude of forced vibrations approaches zero. Since

£ ^ = f n ^ / C c > t i t follows that i f the natural frequency of foundation

vibrations above the spring absorbers i s small i n comparison with

the frequency of engine rotation, the amplitude, A-̂ of the lower

foundation with absorbers i s small i n comparison with the amplitude

of vibrations of the same foundation without absorbers. On the

other hand, as fc,^ approaches i n f i n i t y , or becomes very large,

the absorbers w i l l have less e f f e c t , i n fact even become harmful

beyond the r e l a t i v e l y short range of |MSJ<1.

In summary, the natural frequency of the foundation above the

absorbers must be kept low r e l a t i v e to engine speed, or the

absorbers may be p o s i t i v e l y harmful. Since f ^ = __2 , the required m2

effect can be achieved by reducing the s t i f f n e s s of the absorber

springs; when the l i m i t to t h i s reduction (imposed by strength

requirements) i s reached, the foundation above the absorbers can

be increased i n weight.

B. ROCKING VIBRATIONS

Rocking vibrations usually occur i n high foundations under

machines having unbalanced horizontal components of exciting forces

and exciting moments. Let us assume that the e l a s t i c resistance

22. of the soi l to sliding of the foundation is in f i n i t e l y great and that the centre of inertia of the foundation mass and the centroid of i t s horizontal base area l i e on a vertical line in the vertical plane in which rocking occurs. In this 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 is 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)

- ¥ Q y>" + £ M + M sin UJ t = 0 3-B-l or - W-y?" + WUp - Cv I<P + M sin uJ t = 0

where ¥ = the moment of inertia of the foundation and machine mass with respect to the axis of rotation,

LWY*1 = the moment of the foundation-machine weight about the axis of rotation,

L = the vertical distance between the axis of rotation and the centre of gravity of the mass,

CÎ^P = the moment of soi l reaction about the axis of rotation, I = the moment of inertia of the foundation contact area

with respect to the axis of rotation and Cy, = the coefficient of elastic non-uniform soil compression.

The complete solution of this 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, fn^f is determined as

f nv»

CÎ - WL ¥o

C«, I — ^ — , since CÎ » ¥L 3-B-2

o

23.

The terms representing free vibra t ions may be put i n the form

<P= C s i n ( f n ( p t +</>o) 3-B-3

where C, ^PQ are constants, representing the amplitude and phase

angle, and are determined from i n i t i a l condit ions.

The so lu t ion for the amplitude, A^ , of forced vibra t ions for

th i s case w i l l give

A „ = *

which has a form s imi la r to Eq. 3-A-5.

Prom Eq. 3-B-3 i t i s seen that the natural frequency of

v ib ra t ion i s a function of I , the moment of i n e r t i a of the found

at ion contact area about the axis of ro t a t ion . Since I i s d i r e c t l y

proport ional to the t h i r d power of the foundation length,

(perpendicular to the axis of ro t a t ion ) , any change i n th i s length

w i l l have an immense effect O n ̂ nif) and consequently on A ^ .

Considering the above foundation motion but including damping

we have the natural frequency

f ^ = It" - c 2 3-B-5

The amplitude A ^ s imi la r to Eq. 3-A-10, i s

M A Vd

¥ Q J(f 2 - c 2 ) 2 + 4 C2LO: 3-B-6

C. HORIZONTAL VIBRATIONS

When a foundation mass res t ing on the ground surface i s

subjected to a hor izontal force, shearing deformations develop

wi th in 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 act ion of the hor izonta l forces w i l l

24.

occur mainly i n the d i r ec t i on of these forces.

Assuming a hor izonta l force P^ s inoj t act ing on the centre

of mass of a foundation, then, analogous to Eq. 3-A-3, ve have ,2 nx x" + f_„ = p t s i n cot 3 - C - l

where x = the hor izontal displacement of the centre of gravi ty of the foundation,

2 C c A

f = —jjj— , the square of the natural frequency i n shear,

3-C-2

C c = the coeff ic ient of e l a s t i c uniform shear i n the s o i l .

From t h i s equation i t i s seen that P t

x = Tr- s i n co t . 3-C-3

Consider next the case of ro ta t iona l v ibra t ions with respect

to a v e r t i c a l axis passing through the centre of gravi ty of

foundation mass and the centroid of the base area. We have, under

a hor izonta l exc i t ing moment M s i n to t , the equation z

¥ z ^ " + Cf J z V = M z s i n ^ +

where W = the moment of i n e r t i a of the v ib ra t ing mass with respect to the v e r t i c a l ax i s ,

J = the polar moment of the foundation base area, z = the angle of ro ta t ion of the foundation with respect

to the v e r t i c a l ax i s ,

Cy, = the coef f ic ien t of e l a s t i c non-uniform shear.

The amplitude for th i s case w i l l be

( f 2 -CO 2 ) 3-C-4

and the natural frequency of v ib ra t ion w i l l be

The foregoing equations and s o l u t i o n s were obtained i n a manner analogous to 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 . The m o d i f i c a t i o n s due to viscous damping, noted f o r the v e r t i c a l cases w i l l be s i m i l a r here.

D. COMBINED VIBRATIONS A foundation 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 frequencies. A s o l u t i o n f o r t h i s case i s given i n Reference 7. In p r a c t i c e , however, only-one to three types of motion u s u a l l y need to be considered.

V i b r a t i o n s causing combined compression and shear to the s o i l w i l l now be d e a l t w i t h . Take f i r s t the case where the centre 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 contact area 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 which v i b r a t i o n s are assumed to occur. Consider the e x c i t i n g forces ( 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 at 0, and a couple, M s i n c o t . Under these forces the foundation w i l l undergo motion determined by the three parameters x, z and ^ . The parameters x and z are the h o r i z o n t a l and v e r t i c a l displacements r e s p e c t i v e l y , of the centre 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 ^ represents the r o t a t i o n of the foundation w i t h respect to an a x i s through 0 perpendicular to the plane of v i b r a t i o n s .

By equating the sum of the v e r t i c a l , h o r i z o n t a l and r o t a t i o n a l f orces together w i t h the i n e r t i a forces (d'Alembert), to zero we a r r i v e at the three equations (see References 3 or 5) of motion,

mz" + Cu A z = P s i n to t 3-D-l z

mx" + C Ax - C, AL f = P v s i n to t 3-D-2

26. M m ip" - C t A L x + (Cp I - ¥ L + C E A L 2 ) = M s i n UJ t where M = the moment of i n e r t i a of the mass w i t h respect to the m a x i s through 0, normal to the plane of v i b r a t i o n s .

P , P x = 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 force P,

M = the e x c i t i n g moment, and other terms are as defined p r e v i o u s l y .

I t may be noted that since Eq. 3-D-l contains n e i t h e r x nor y3 i t i s independent and can be considered separately as was done i n Eq. 3-A-l. Eqs. 3-D-2 on the other hand are interdependent and must be considered together.

Considering the case of free v i b r a t i o n s , i . e . Px = M = 0, Eqs. 3-D-2 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 s i n (f t +0< ) ; W = B s i n ( f t + tx ) 3-D-3 a n r a n

The two p r i n c i p a l n a t u r a l frequencies of v i b r a t i o n corresponding to these two degrees of freedom are

f 2 f 2 = 1

n l » rn2 2 *

where # = M m/M m o, 3-D-4

2 M = M + m L , the moment of i n e r t i a of the v i b r a t i n g

m o m mass wi t h respect to the a x i s passing through the c e n t r o i d of the base contact area perpendicular to the plane of v i b r a t i o n s ,

f. „ , f are as given i n Eqs. 3-B-2 and 3-C-2 r e s p e c t i v e l y . n"P ' nx s u r J

(The frequency w i t h respect to 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 obtained independently as before).

3 According to Barkan , fn2» ^he smaller of the two n a t u r a l

frequencies i s smaller than e i t h e r of the two l i m i t i n g frequencies f or f n x > while i s l a r g e r . I t may be seen by s u b s t i t u t i n g Eqs. 3-D-3 i n t o the f i r s t of Eqs. 3-D-2 that the r a t i o

27. of amplitudes of the' two vibration components,

nx 2 2 nx n 3-D-5

where ± n is the natural frequency of vibration of the foundation. If f Q = fn2> then P is positive and i t is seen that x and *p

tend to be in the same direction. 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. Gn the other hand, i f f = fn2» the higher frequency, then w i l l be negative, and A and B w i l l be 180° out of phase. Then vibration takes place about a point above the centre of gravity of the vibrating mass. Thus, from Eqs. 3-D-4, 3-D-5, either of two forms of vibration are possible, depending on the foundation size and soi 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 tut j <f = A^ sin uJ t. 3

Under an impressed horizontal force P sin cc) t we obtain , by the usual technique of resubstitution in the i n i t i a l differential equations, values of amplitudes: A = x

C f I - U + C, A L M CU m 4 ( c o 2 )

p; A ^ = Ct A L P

3-D-6 Under an exciting moment, M sin to>t, the amplitudes are

2̂ C . A L A =

A ( u ? ) M A

C t A - m CO

A (co)' M,

3-D-7

28.

where A(oj2) = m Mffl ( f 2 ^ - c o ) ( f 22 -CO2). 3-D-8

Considering either of the two resonance conditions, either CU = f n j or OJ = fn2» i t may be seen from Eq. 3-D-8 that

\̂ (cu) = 0* 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 is seen that when the exciting frequency CU is very small with respect to f , then P = L and rocking takes place about the centroid of base contact area. As O) increases, P increases (the centre of rotation moves down) until at CO = f„„> ft = o°

(The centre of rotation is at an infinite depth) and sliding motion alone takes place. When CU exceeds f ^ , the motions become out of phase. Further increase in 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, unt i l , when UJ is much greater than f , ft approaches zero and purely rotational oscillations 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 vertical line. Taking the case where there is 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-l, 3-D-2 (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 - Cc A L<p = 0 m z" + Cu A z - Cu A£ ^ = 0

29. MmyJ' - Cz A L x + (Cv I - ¥ L + Cu Ae 2 + A L2)</>- Cu Ae z = 0

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 is a decrease in the two smaller natural frequencies and an increase in the highest natural frequency from those values found in Eqs. 3-D-l and 3-D-2 for zero eccentricity. He states that for foundations having eccentricities up to 5 per cent of the length of a foundation dimension, this effect may be neglected, and computations may be performed using Eqs. 3-D-l and 3-D-2, where eccentricity £ = 0.

It is 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 specific plane of action - that in which the vibratory impulse is applied. Furthermore, since the designer usually has some control over eccentricity, and because of the complications involved, this factor has generally been ignored, particularly when i t occurs

7

outside the major plane of vibrations. According to Pauw , the error in determining the fundamental frequency when damping is neglected is usually less than 2 per cent. It is the purpose of most foundation designs to avoid a frequency ratio, 6c>/fnz> in the vicinity of one, while outside this vicinity of frequency ratio values, the dynamic amplitude factor, as seen by Fig. 1, is quite small. This, combined with the d i f f i c u l t i e s involved in evaluating damping has resulted in i t s being neglected in most vibration calculations. The result is 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 vertical vibration with one degree of freedom. This mode of vibration should be representative of a l l conditions of motion and is obviously the simplest to analyse.

The earliest quantitative results on the dynamic character-i s t i c s of foundations were acquired, according to Terzaghi and Tschebotarioff 1, by DEGEBO (German Research Society for Soil Mechanics). Tests are described using a machine called a two mass oscillator (similar in principle to the machine described in this thesis, see CHAP. VI, Part B) to determine experimentally the natural frequencies of the sites. It was originally hoped that a useful relationship between measured dynamic properties and resistance to static loads could be established. Although tables of allowable bearing pressure versus natural frequency of soils have been given**, i t is seen that the variations in bearing pressure are extremely high as compared with the useful variations in frequency. This indicates that correllation by this means must be poor.

Using the elementary equation for natural frequency, f = / k/m , where k is the spring constant equal to Cu A in Eq. 3-A-l and m = f s , the mass of the foundation and soil

g undergoing vibration, experimental values of k and ¥ were

s determined. In this method, different weights of vibrators were

31. used while the area, which was thought to control the equivalent weight of s o i l , was held constant. It was found that ¥ varied

s

between very wide limits. ¥ith increasing values of exciting force, the resonant frequency was found to decrease. This was explained as being due to an increase in ¥ with increasing load. It was shown conclusively by these tests that the natural frequency was dependent not only upon the foundation under consideration, but on adjoining structures as well. Thus we are dealing with the natural frequency of a particular site, 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 linear.

(Even in the cases where they are considered to increase with increasing depth below the surface, as in Pauw's work.)

2. Consideration of the foundation-soil as a lumped parameter system, in which the weight of the springs is added to the weight of the foundation load and the damping forces, i f considered, are lumped together as one source of viscous damping.

3. The d i f f i c u l t y in choosing proper values of damping, and of the equivalent mass of the soil springs. Foundation soil is an imperfectly elastic 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 in theoretical considerations.

32. B. NONLINEARITY Consider again the coefficient of uniform compression, Cu.

We have already considered in CHAP. II the r e l i a b i l i t y of the average values of Cu. 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 is found to coincide with the frequency of the exciting forces, there actually exists a relatively linear relationship between the

9 3 foundation displacement and the soil reaction . Barkan states that the linear relationship between magnitude of exciting force and amplitude of vibrations depends upon this. This seems reasonable and f a i r l y easy to measure. His graph on page 90 shows some observations.

8 4 According to Lorenz , and Novak , the assumption that the

soil acts as a linear spring without mass does not lead to satisfactory results as shown by large scale tests. Increases in exciter force lead to decreases in natural frequency, observations f i r s t made by DEGEBO in 19346. Lorenz points out that i f the decrease in natural frequency is attributed to the increase in vibrating s o i l mass, then the spring characteristic could be considered constant. However he states that as the exciter area is increased, with constant contact pressure, the natural frequency increases. As this must cause an increase in the vibrating s o i l mass, the natural frequency should instead decrease. According to Lorenz this indicates non-harmonic vibrations, therefore he would replace Cu A in the equations of CHAP. I l l with a reactive force that does not change linearly with deflection.

Q Den Hartog states that there is no exact solution for

33.

undamped vibrations with a curved spring characteristic. He gives an approximate solution of a nonlinear characteristic based on the assumption that the motion is sinusoidal. This method is adapted by Lorenz to determine nonlinear spring characteristics for a s o i l . Lorenz also provides three other methods of determining the characteristic, 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 soils (using method one) i t appears that nonlinearity is more significant in cohesive soils. Lorenz suggests that this is due to the fact that in clayey soils pore pressures reduce the increase in effective stress quite extensively during application of small loads whereas at higher loads pore pressures f a l l off quite rapidly. He states that these results apply equally well to torsion and shear. Thus i t seems that nonlinearity is less a problem in cohesionless soils, where most vibration problems occur.

Novak (as well as Lorenz) justifies his approximate solutions 4

for nonlinear vibration by the fact that vibration records picked up by an oscilliscope usually indicate that the motion of a foundation under a harmonically applied force is i t s e l f almost purely harmonic. This also would indicate that the f i r s t harmonic of the solution is 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 in s o i l parameters due to foundation peculiarities. These were vibratory tests on a loess loam with variations in static pressure from

34. 0.194 to 0.482 kg/cm and with square base plate areas between 0.5 and 1.5 square meters. Measurements were taken only after a satisfactory elastic state of the soil had been reached, i.e. residual deformations became negligible. It was noted that the numerical value of the spring characteristic of the soil varied only very slightly within the range of pressures used in the test. Novak infers from this that the subsoil behaves as a stratum of limited size, rather than as the theoretical semi-infinite elastic base, and so justifies the simplified handling of vibration problems.

Soil becomes elastic due to repeated stressing and i t s stress-12 4

strain relationship becomes linear ' . If the characteristic for any site 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 this, then nonlinearity of the spring characteristic can probably be ignored. Therefore the above theories of vibrations w i l l probably give relatively adequate results. However i f the frequency of vibration could result in amplitudes that are markedly different from those involved during measurement, then nonlinearity should be considered by one of the above methods.

C. THE INERTIA OF THE SPRING BASE A factor of very definite importance in the previous

vibratory considerations is the effective soil weight, ¥ , or the weight of vibrating s o i l . Tschebotarioff states that there is no clearly defined limit for the s o i l mass, which has a very complex motion, and so an "equivalent weight" of soil must be considered. As Pauw puts i t , "Everyone agrees that the mass of vibrating s o i l

35.

must be added, d i f f e r e n c e s l i e only i n the assumptions used to determine i t " .

Pauw's method of estimating the apparent mass of ¥ i s to s

compute a concentrated mass that 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 that k i n e t i c energy i s equal to 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 equivalent mass, undergoing v i b r a t i o n s at the surface of the ground, and solves f o r the equivalent mass. The e f f e c t e d s o i l volume i s again assumed to be that enclosed by the pyramid r e f e r r e d to on page 6, between the plane of the foundation base and a plane at an i n f i n i t e distance below the foundation. Apparent mass f a c t o r s f o r various modes of v i b r a t i o n i n cohesionless s o i l are derived i n t h i s way. In a s i m i l a r manner, he estimates the apparent mass moments of i n e r t i a of the s o i l to 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 cohesionl e s s s o i l s . The i n t e g r a l used f o r the determination of the apparent mass f o r a cohesive s o i l does not converge, however, so f o r t h i s case Pauw's method i s not a p p l i c a b l e . Pauw states that 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 cohesionless s o i l s . For cohesive s o i l s there i s so l i t t l e experimental information that c o r r e l a t i o n s could not be made.

Experimental methods using 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, instead of a constant P may be

4 8 used ' to determine ¥ . For t h i s case the e x c i t e r force increases

s w i t h the square of the frequency as shown by Eq. 6-C-l. P l o t t i n g

36. the amplitude of v i b r a t i o n s versus the frequency w i l l give a curve which w i l l show, according to the theory 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 asymptotic to a s t r a i g h t l i n e at a constant amplitude, o , as the frequency increases without bound. Here

m r 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, mQ, and m i s the 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 . Since mQ and r

m r are known and the value of o may be measured,, m may be

m -determined. Subtracting the mass of the v i b r a t o r from t h i s value of m w i l l give the apparent mass. These methods seem to o f f e r reasonably good values f o r the p a r t i c u l a r foundation c o n d i t i o n s represented by the v i b r a t o r , but i t i s not a simple matter to p r o j e c t them to other foundations on the same s i t e . Novak shows, experimentally, an increase i n apparent mass of 40 - 60 per cent when foundations are placed below the surface, even when a i r spaces are l e f t around the foundation.

The idea of t r e a t i n g a foundation as the t h e o r e t i c a l weightl e s s s p r i n g by adding i t s apparent weight to the foundation load i s i t s e l f a very good approximation, assuming of course t h a t the foundation s p r i n g analogy i s otherwise r e l i a b l e . Various t e x t s show that by considering a v i b r a t i n g spring as w e i g h t l e s s , no appreciable 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 considered as being concentrated at i t s free end.

Novak r e f e r s to a s i m p l i f i e d theory (by Sechter) of v e r t i c a l forced v i b r a t i o n s of a r i g i d foundation 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. This work i s c i t e d to prove t h e o r e t i c a l l y that 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 considering the body to v i b r a t e on a massless base, provided the mass of

3 7 .

the foundation is suitably increased, and the damping coefficient is appropriately selected.

Prom the above i t would appear that the increase in vibrating mass must be considered in 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 is s t i l l rather uncertain.

D. DIFFICULTIES ASSOCIATED WITH DAMPING The effects of damping on foundation vibrations can be of

major importance in cases of resonance of forced vibrations or during shock loading. According to Tschebotarioff^, no reliable or practical methods have been developed to permit numerical determination, and for this reason damping is often neglected. With regard to forced vibrations, Eq. 3-A-9 shows that i f the damping constant c is small relative to the natural frequency without damping f , then damping w i l l have l i t t l e effect on the

H Z

resonant frequency and may be neglected. However i f damping is 3

large, i t w i l l have a large effect. According to Barkan the effe of the damping reaction of the s o i l on the amplitudes of free vibration of a foundation is rather considerable, even in cases of small c. Since foundations under machines with a steady regime of work are usually designed to avoid any possibility of resonance then damping effects may be neglected in 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 coefficient for the particular foundation conditions represented by the vibrator. Terzaghi^

3 8 .

shows a method of c a l c u l a t i n g the damping c o n s t a n t by e q u a t i n g

the excess energy output of t h e v i b r a t o r t o t h a t s u p p l i e d t o the

f o r c e of v i s c o u s damping. L i t t l e e vidence of c o r r e l a t i o n s w i t h

e x p e r i m e n t a l o b s e r v a t i o n s are o f f e r e d .

Novak no t e s from p r e v i o u s l y d e s c r i b e d experiments t h a t the

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 w i t h i n c r e a s e i n

f o u n d a t i o n a r e a . T h i s was a l s o noted by L o r e n z . Novak s t a t e s

t h a t a mere decrease i n m o i s t u r e of a l o e s s loam caused by two

weeks of drought r e s u l t e d i n a 50 per cent i n c r e a s e i n the damping

c o e f f i c i e n t . R e s u l t s by Pauw show t h a t v i s c o u s damping i s l e s s

f o r l a r g e a m p l i t u d e s . Thus we see some of the problems 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 p r e s e n t s a s i m p l i f i e d s o l u t i o n f o r v i b r a t i o n s on a

s e m i - i n f i n i t e e l a s t i c s o l i d . T h i s does not c o n s i d e r damping, but

graphs p l o t t e d u s i n g e q u a t i o n s from t h i s t h e o r y show a s i m i l a r i t y

w i t h resonance cu r v e s f o r a system w i t h one degree of freedom

s u b j e c t e d t o damping. T h i s i n d i c a t e s t h a t even an i d e a l l y e l a s t i c

s o i l has a damping e f f e c t on t h e a m p l i t u d e of f o u n d a t i o n v i b r a t i o n ,

due t o i n e r t i a e f f e c t s . From B a r k a n 1 s c u r v e s i t may be seen t h a t

f o r t h i s case damping depends on the v a l u e s of P o i s s o n ' s r a t i o as

w e l l as upon the s i z e and weight of the f o u n d a t i o n . A decrease

of damping w i t h i n c r e a s e of s t a t i c c o n t a c t p r e s s u r e i s a l s o shown.

P r o p e r v a l u e s of damping are as y e t d i f f i c u l t t o a s c e r t a i n . 3

Some e x p e r i m e n t a l v a l u e s f o r t i m b e r 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. Damping c o e f f i c i e n t s may

be d e t e r m i n e d , i n a d d i t i o n t o methods mentioned above, 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 , from the phase s h i f t

between 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 d e f o r m a t i o n * ^ and

39. 3 from h y s t e r e s i s l o o p s » There have been few comparisons between

any of t h e s e methods. I t appears t h a t the extreme i n s t a b i l i t y of

the damping f a c t o r s i s the 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 i n v o l v e d

i n t h e o r e t i c a l t r e a t m e n t s of v i b r a t i o n s where damping must be

c o n s i d e r e d .

CHAPTER V DESIGN CONSIDERATIONS

40

A. FACTORS AFFECTING DESIGN A foundation resting on a so i l may undergo intolerable

movement due to shear failure resulting from compressive, or undue settlement. For this reason, most of the problems involved in 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. It would appear that most of these have l i t t l e value, since correlation between different methods is usually poor. In many cases these foundations are extremely uneconomical, and in a few they are apt to be inadequate.

It has been long realized that the total pressure on the s o i l , both static and dynamic, could be larger than the bearing capacity of the s o i l . In order to insure against this possibility occurring, the dynamic load was often translated into an equivalent static load and added to the permanent static load, or dead load. Terzaghi^ refers to a design formula used by Rausch which states that the equivalent static load should be arrived at by adding to the ordinary static load 3 times the dynamic load.

Barkan, referring to evidence that dynamic pressure may induce settlements up to many times those of static loads, says that this invalidates the type of relationship used by Rausch. His criterion of foundation design is to limit the static pressure and the

4 1 .

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, in 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 is some component of force acting in this direction. Of greatest practical interest is usually the case of purely vertical vibrations accompanied simultaneously by a gyratory movement of the foundation as was considered in CHAP. III.

Any general limit for the permissible amplitude of vibrations would be d i f f i c u l t to suggest. This is partly due to the fact, already mentioned, that foundations undergoing relatively small vibrations, especially under low frequency loads, may induce strong conditions of resonance in adjoining structures. Barkan notes cases where amplitudes of 0.4 to 0.5 mm. did not have harmful effects; but based on a l l his experience he suggests a maximum permissible value of 0.20 mm. An amplitude that would be acceptable in one area may be entirely unsatisfactory in another due to residential or precise manufacturing use of the land.

2 Converse refers to experiments by Mutual Life Insurance

Company that indicate vibrations are noticed by people when the amplitude exceeds 0.36/f inches. He states that vibrations which result in accelerations of 5 per cent that of gravity are sometimes used as a criterion for nuisance limit.

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 foundation mass (Tschebotarioff suggests l e a v i n g c a v i t i e s near the four corners of a foundation 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 foundations under r e c i p r o c a t i n g machinery, the n a t u r a l frequency, f n , i s g e n e r a l l y greater than the impressed frequency. For t h i s case, minimum height of foundation means minimum mass and maximum f ^ , so s a f e t y w i t h regard to resonance i s increased. Smaller areas, which decrease the e f f e c t i v e mass of s o i l , and cork i s o l a t i o n pads may also be h e l p f u l .

For high frequency machines, large block foundations w i t h large masses are g e n e r a l l y employed. Large areas which increase the equivalent earth mass may prove s a t i s f a c t o r y .

For machines whose frequencies are i n danger of producing resonance, spring supports are commonly used t o lower the n a t u r a l frequency, t^. These may be laminated, h e l i c a l or even compressed a i r . Cork provides a good s p r i n g pad i f i t i s kept dry and i n good c o n d i t i o n . Asbestos i s commonly used because of i t s d u r a b i l i t y . As i s seen i n Section A - 2 of CHAP. I l l , i n order f o r absorbers t o have a favourable e f f e c t on the amplitude of foundation v i b r a t i o n s , the frequency of n a t u r a l v i b r a t i o n s of the mass above the springs should be as small as p o s s i b l e i n comparison w i t h the frequency of engine r o t a t i o n . In high 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 frequency 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 decreasing the r i g i d i t y of the absorbers because strength requirements may l i m i t t h i s . Here a decrease i n n a t u r a l frequency can u s u a l l y be

43. achieved by p r o v i d i n g instead an increase i n the foundation mass above the springs.

The e f f e c t s of earthquakes must be taken i n t o account i n designing s t r u c t u r e s i n many areas. When analyzing the damage caused by earthquakes we are at a great disadvantage i n that we do not know what component of the v i b r a t i o n caused the damage. I f the n a t u r a l frequency of a s t r u c t u r e i s low compared w i t h the frequency of the earthquake, we may assume that 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 that 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 w a l l s ; however, i n some st r u c t u r e s such as high smoke stacks, i t i s p o s s i b l e f o r resonance to occur. I f t h i s happens, f a i l u r e may r e s u l t .

Terzaghi says^ that the best measure of the d e s t r u c t i v e impulse of an earthquake i s the amount of energy i t supplies to the foundation per u n i t time. Since t h i s energy i s d i r e c t l y r e l a t e d to 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 to the a c c e l e r a t i o n of g r a v i t y . This r a t i o , as measured f o r any earthquake, i s u s u a l l y much lower, on rock outcrops than on loose alluvium. S i m i l a r remarks apply to underground explosions. T s c h e b o t a r i o f f * 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 the square of the a c c e l e r a t i o n i n f e e t per second, per second,as measured by an accelerograph, d i v i d e d by the square of the frequency i n cycles per second, measured by a seismograph.

This d i s c u s s i o n has d e a l t w i t h foundation design w i t h regard to general a c c e p t a b i l i t y . Settlement i n p a r t i c u l a r w i l l be d e a l t w i t h i n the f o l l o w i n g s e c t i o n .

44.

B. CONTROL OF SETTLEMENT It is well known that foundations subjected to shock and

vibrations may undergo settlements many times larger than those caused by static loads. The harmful results of such settlements are obvious. These settlements are usually considered to be characteristic of cohesionless soils, but they may also be exhibited to some degree by cohesive ones. The physical processes which cause changes in so i l properties are not clar i f i e d yet, but experiments show that vibrations cause changes in the fr i c t i o n , cohesion and hydro dynamic properties of a s o i l , and also on the elastic properties such as Young's modulus, the shear modulus, and the limits of elasticity and plasticity.

In the usual settlement analysis of foundations under static loads, elastic deformations, which are usually small in comparison with residual deformations, are ignored. Elastic constants seem to bear no relationship to the coefficient of compressibility, a , calculated in static consolidation tests**. Because of this i t v 7

is very hard to predict settlements merely from the elastic properties of the s o i l .

3

Barkan shows experimental results which indicate a decrease in the coefficient 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 static and dynamic conditions. He shows that dynamic penetrations, while being much greater than static penetrations in 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 is 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. Penetration here was 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 particles, i t would follow that these observations could be applied directly towards predicting values of settlement. The above information indicates an increasing effect of vibrations on the mechanical properties of a soil with increase in grain size. The principal vibration parameter seems to be acceleration, and so the inertial force, and hence the density of the involved particles may be of some importance.

3

Plotted graphs of the minimum void ratio, 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 test. A relationship between the slope of this 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. Fairly 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. Barkan states that this is true for a l l other types of soil as well. He also notes what he terms the "threshold of vibratory compaction" which is the limit of vibratory acceleration following which settlements occur. This might correspond to

46. Terzaghi's " c r i t i c a l range of frequencies" w i t h i n which excessive settlement occurs. Presumably t h i s i s because, between these frequencies, amplitudes are very large and f o r a given frequency, large amplitude means large a c c e l e r a t i o n . Terzaghi s p e c i f i e s the c r i t i c a l range of frequencies as being between one h a l f and one and one h a l f the fundamental frequency of the f o u n d a t i o n - s o i l system, although he says i t i s f a i r l y independent of the s i z e of

3

the v i b r a t o r . Experimental work by Barkan shows the angle of f r i c t i o n decreasing very l i t t l e at frequencies below 140 c.p.s. f o r a c o n s o l i d a t i o n t e s t on a sand but decreasing very r a p i d l y i n the range from about 140 to 240 c.p.s. This again could i n d i c a t e the c r i t i c a l range, since confined l a b o r a t o r y samples w i l l show resonant frequencies much higher''"* than those f o r s i m i l a r s o i l s i n the f i e l d . The threshold of v i b r a t o r y compaction i s known to r i s e w i t h an increase i n c o n f i n i n g pressure. Under higher c o n f i n i n g pressures, sand i s l e s s s u s c e p t i b l e to compaction. Barkan explains t h i s i n terms of an increase i n apparent cohesion under these c o n d i t i o n s . I f t h r e s h o l d of compaction i s defined i n terms of a c c e l e r a t i o n , then increase i n a c c e l e r a t i o n i m p l i e s an increase 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 , according to the r e l a t i o n s h i p , force = ma where m i s the mass of the s o i l p a r t i c l e s undergoing forced 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 r e l a t i v e e f f e c t of t h i s a p p l i e d force could be expected to decrease w i t h increased c o n f i n i n g pressure a c t i n g on the s o i l . An increase i n t h i s c o n f i n i n g pressure r e s u l t s i n a decrease i n the r e l a t i v e magnitude of the i n e r t i a f o r c e s .

Tests by Barkan on sands show the p e r m e a b i l i t y , under conditions of v i b r a t i o n , i n c r e a s i n g w i t h higher a c c e l e r a t i o n s .

4 7 .

The r a t e of increase 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 buildup of pore pressure during v i b r a t i o n s , followed by a t a p e r i n g o f f . Therefore v i b r a t i o n s must cause a decrease i n s o i l volume at a r a t e r a p i d enough to o f f s e t any increase i n p e r m e a b i l i t y . This i n d i c a t e s a decrease i n e f f e c t i v e pressure w i l l occur during 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 occurring under conditions of shock. These observations would seem to 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 VI

CONSOLIDATION TESTS

A. INTRODUCTION

P l a s t i c c l a y s are commonly thought 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 l i t t l e

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 c l a y c o n s o l i d a t i o n . S i n c e machine bases are not apt t o be

p l a c e d 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 r a t h e r t h a n shear

f a i l u r e w i l l be the main c o n s i d e r a t i o n . F o l l o w i n g the i n i t i a l

i n s t a n e o u s e l a s t i c compression, t h e r e are two t y p e s of s e t t l e m e n t

e x h i b i t e d by c l a y m a t e r i a l s d u r i n g l o a d i n g . I t was the purpose

of t h i s work t o c o n s i d e r these t y p e s of s e t t l e m e n t w i t h r e g a r d t o

the f o l l o w i n g q u e s t i o n s ;

(a) P r i m a r y c o n s o l i d a t i o n , depending on the 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 the 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 r a t e of s e t t l e m e n t ?

(b) Secondary c o n s o l i d a t i o n or p l a s t i c f l o w , depending 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 the amount and/or the 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 here was an attempt t o e x p l o r e the

above p o i n t s but was not i n t e n d e d t o be an e x h a u s t i v e t r e a t m e n t

of the problem. A p o s s i b l e t e s t procedure was checked and some

of t h e d i f f i c u l t i e s t o be expected i n f u r t h e r work a l o n g these

l i n e s were determined.

Two p o s s i b l e cases of f o u n d a t i o n response were c o n s i d e r e d :

(1) The e f f e c t of v i b r a t i o n s from o t h e r s o u r c e s on a s t a t i c a l l y l o a d e d s t r a t a ( S e r i e s I T e s t s ) ,

(2) The e f f e c t of h a r m o n i c a l l y v a r y i n g l o a d superposed on s t a t i c l o a d ( S e r i e s I and I I T e s t s ) .

I n p r a c t i c e , the dynamic component of l o a d i s l i k e l y t o be

49.

a very small f r a c t i o n of the s t a t i c load ( a p p l i e d load plus overburden). For t h i s reason, t e s t s (see (2) above) were conducted at small dynamic loads.

I t should be noted that the choice of frequencies here was not too good. Although the two frequencies used were w i t h i n the range of resonant frequencies to be expected f o r foundations i n the f i e l d , i t i s known th a t the n a t u r a l frequency of a confined l a b o r a t o r y sample i s many times higher than t h i s . For t h i s reason the impressed frequencies were w e l l below the resonant range f o r the samples and would not be expected to represent the c r i t i c a l 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 blue marine c l a y of approximately the f o l l o w i n g makeup:

CHEMICAL ANALYSIS 1 5

s i o 2 - 58.5#

A10 3 - 21.1$ F e 2 0 3 - 8.696

CaO e.5fo

MgO 0.5$

l o s s - 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 period of 18 days during which time about 83 per cent of the suspended m a t e r i a l as measured by the hydrometer, s e t t l e d out.

50. Where exposed, t h i s c l a y weathers to form a brown, cracked, sandy-like surface.

Undisturbed samples were obtained 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 mile north and 50 f e e t above the Fraser R i v e r at Port Haney, B.C. Samples f o r t e s t S e r i e s I I and I I I were taken from the base of a 30 f o o t c l i f f which faces west and undergoes a slow fl o w i n g movement. The exact l o c a t i o n from which the samples of Series I were taken i s unknown.

This c l a y appears to be very homogeneous, and has the f o l l o w i n g p r o p e r t i e s :

s p e c i f i c g r a v i t y - 2.79 l i q u i d l i m i t - 45 per cent p l a s t i c l i m i t - 28 per cent shrinkage l i m i t - 26 per cent

C. SOURCE OF VIBRATIONS 1. S e r i e s I . Two methods of v i b r a t i o n were used i n these

t e s t s . In S e r i e s I , a l / l 5 horsepower e l e c t r i c motor was merely b o l t e d to one of the h o r i z o n t a l braces of a standard 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 to by screwing a \ i n c h long \ inch diameter b o l t i n t o the r o t a t i n g shaft of the motor. In t h i s way, v i b r a t i o n s quite p e r c e p t i b l e to the touch were generated throughout the frame and t r a v e l l e d up i n t o the standard c o n s o l i d a t i o n samples which were set on the c o n s o l i d a t i o n frame. Thin rubber pads were placed under the legs of the c o n s o l i d a t i o n frame i n order to prevent v i b r a t i o n s from t r a v e l l i n g along the f l o o r and a f f e c t i n g adjacent non-vibrated t e s t s .

The speed of t h i s motor was taken as 1740 r e v o l u t i o n s per

51.

minute, from the name p l a t e , or 29 cy c l e s per second. No values of i n t e n s i t y of dynamic force or of amplitude were obtained, but v i b r a t i o n s of the frame appeared to be near that 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 to persons, without being p a r t i c u l a r l y obnoxious.

2. S e r i e s I I and Seri e s I I I . V i b r a t i o n s were ap p l i e d to the samples of Series 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. This consisted of two gears which r o t a t e d i n opposite d i r e c t i o n s and on which were mounted e c c e n t r i c masses. These masses were symmetrically placed w i t h respect to the v e r t i c a l plane passing through the meshing teeth of the two gears and perpendicular to the plane of the gears. In t h i s way, the r e s u l t ant of the combined h o r i z o n t a l components of c e n t r i f u g a l force a p p l i e d to the axles was equal to zero. The f i n a l r e s u l t was that the v i b r a t i n g u n i t acted under an a l t e r n a t i n g v e r t i c a l force of a purely harmonic nature.

The magnitude of t h i s force i s given by the formula F = mr (JJ" s i n cu t 6-C-l

where m/2 i s the qua n t i t y of the e c c e n t r i c mass added to each gear, r 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 w i t h respect to

the centre of the re s p e c t i v e gear a x l e . C o /is the angular v e l o c i t y of r o t a t i o n of the gears i n radians

per second, and i s equal to 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 the dynamic force was governed by varying the s i z e of the e c c e n t r i c masses, which were measured to one l/lOO of a gram, and the e c c e n t r i c i t y , which was measured to one l/lOOO of an i n c h .

The gears of the o s c i l l a t o r were dri v e n by means of a

52. f l e x i b l e d r i v e coupled to a 4-step p u l l e y . This p u l l e y was i n tur n connected by means of a v - b e l t , see F i g . 5, to 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. The frequency of o s c i l l a t i o n was taken ast*-'/2 7r> from Eq. 6-C-l. The speed of the motor was taken to be a constant value. A c t u a l 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 per cent. - By ad j u s t i n g the d r i v i n g r a t i o of the two p u l l e y s , the angular v e l o c i t y , (jj , could be adjusted 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 diametric r a t i o s of the p u l l e y s , and the speed of the motor. This c o n s t i t u t e d the frequency c o n t r o l .

Since two v i b r a t e d t e s t s were c a r r i e d out, i n both s e r i e s I I and I I I , using d i f f e r e n t frequencies, i t was necessary to completely i s o l a t e one t e s t from the other. In a d d i t i o n , i t was des i r e d to apply v i b r a t i o n s to the t e s t samples through the loading head only, hence i t was necessary to prevent v i b r a t i o n s from being transmitted through the frame or through the wires supporting the crossbeam through which the load i s transmitted to the loading head.

The f i r s t of these conditions was met by b u i l d i n g a second ta b l e over the o r i g i n a l c o n s o l i d a t i o n frame i n order to support one of the samples. This t a b l e d i d not come i n t o contact w i t h the c o n s o l i d a t i o n frame at any p o i n t , see F i g . 6.

The second c o n d i t i o n mentioned above, that of c o n t r o l l i n g the means of access of the v i b r a t i o n s to the samples, i s also i l l u s t r a t e d by F i g . 6. The loading cross beam and i t s associated counter-balance i s supported i n each case not by means of a post mounted on the s o n s o l i d a t i o n frame, as i s the usual 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 to note that i n s p i t e of the v e r t i c a l nature of the v i b r a t i o n s , h o r i z o n t a l p i n

53.

c o n n e c t e d 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 t h e two

wooden f rames t o each o t h e r had t o be removed because of t h e i r

a b i l i t y t o t r a n s m i t v i b r a t i o n s . The t e n s i o n r o d s (see a l s o

P i g . 3) t h a t t r a n s m i t t h e end r e a c t i o n s of t h e l o a d i n g beams down

t o t h e h o r i z o n t a l b r a c e s o f t h e c o n s o l i d a t i o n frame were removed

f r o m t h e frame and c o n n e c t e d b y means o f a n c h o r s t o t h e c o n c r e t e

f l o o r .

I n t h i s way t h e r e was no p h y s i c a l c o n n e c t i o n between t h e

p a r t s o f t h e a p p a r a t u s a s s o c i a t e d w i t h t h e two d i f f e r e n t

c o n s o l i d a t i o n s a m p l e s . The v i b r a t i n g o s c i l l a t o r s were s u p p o r t e d

by means o f l o a d i n g frames w h i c h f i t t e d over t h e s t a n d a r d l o a d i n g

f rames o f t h e c o n s o l i d a t i o n d e v i c e , see P i g . 6. These added frames

were c o m p l e t e l y s u p p o r t e d by a b a l l b e a r i n g w h i c h r e s t e d over t h e

c e n t r e o f t h e l o a d i n g head and t r a n s m i t t e d v i b r a t i o n s t h r o u g h t h e

s t a n d a r d l o a d i n g c r o s s beam i n t o t h e l o a d i n g head and hence i n t o

t h e s a m p l e . The maximum m a g n i t u d e o f t h e dynamic l o a d was l i m i t e d

by t h e s t a t i c w e i g h t o f t h e o s c i l l a t o r , t o p r e v e n t b o u n c i n g of t h e

o s c i l l a t o r on t h e s a m p l e .

D i r e c t r e a d i n g s o f the a m p l i t u d e o f a p p l i e d dynamic l o a d were

a t t e m p t e d by means o f f o u r SR -4 s t r a i n gauges bonded on t h e

l o a d i n g c r o s s b a r ( t h e w i r e s l o a d i n g t o t h e s e s t r a i n gauges c a n

be seen i n F i g . 6-b) of t h e c o n s o l i d a t i o n frame and c o n n e c t e d

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 . However

t h e r e q u i r e d degree o f m a g n i f i c a t i o n o f t r a n s d u c e r o u t p u t c o u l d

n o t be a c h i e v e d so t h i s p r o c e d u r e was a b a n d o n e d , and t h e computed

a m p l i t u d e was a c c e p t e d as c o r r e c t .

D. 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 a p p a r a t u s

54. on 2\ i n c h diameter, one inch 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 cases, w i t h a loading range of \ to 8 kilograms per square centimeter (kg/cm ). Settlement was measured to one 1/10,000 of an in c h .

Three s e r i e s of t e s t s were c a r r i e d out. Serie 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

f o l l o w s : Test 1 - I - Undisturbed, unvibrated. Test 2 - I - Undisturbed, v i b r a t e d . Test 3 - I - Remolded, unvibrated. Test 4 - I - Remolded, v i b r a t e d .

Each increment was allowed to s e t t l e f o r approximately one week. V i b r a t i o n s were appl i e d as stated i n s e c t i o n C - l . Observations and r e s u l t s are shown i n Part E.

Series 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 - Undisturbed, unvibrated. Test 4 - I I - Remolded, unvibrated. Test 6 - I I - Undisturbed, v i b r a t e d at 16 cycles per second. Test 7 - I I - Undisturbed, v i b r a t e d at 24.6 cycles per second. Dynamic loads app l i e d were as f o l l o w s :

S t a t i c Load Test 6-II Test 7-II Total Dynamic $ of Dynamic $ of

Pressure Force Load S t a t i c Load S t a t i c ( k g / c m 2 ) (kg) (kg) (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 vere a p p l i e d as described i n Par t C-2. Observations are given i n Par t E.

Se 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 - Undisturbed, unvibrated. Test 4 - I I I - Undisturbed, unvibrated. Test 6 - I I I - Undisturbed, v i b r a t e d at 24.6 cycles per second. Test 7 - I I I - Undisturbed, v i b r a t e d at 16.0 cycles per second. Dynamic loads were appl i e d as f o l l o w s :

S t a t i c Load Test 6-III Test 7-III To t a l Dynamic fo of Dynamic fo of

Pressure Force Load S t a t i c Load S t a t i c (kg/cm 2) (kg) (kg) <kS>

i 7.5 0 0 0 0 \ 15.0 0.75 5.0 .575 3.82 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

Observations are shown i n p a r t E.

E. OBSERVATIONS

Water contents were determined by drying samples at approximately 105° C u n t i l constant weight was reached.

Water contents 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. Series II Tests

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

Series III Tests 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

Void r a t i o versus l o g of pressure curves f o r t e s t s of s e r i e s I, II, and III are shown i n F i g . 7, 8 and 9 r e s p e c t i v e l y .

F i g . ' s 10 to 13 i n c l u s i v e show settlement versus 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 to 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 II. F i g . ' s 17 to 19 show the settlement - 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 III. F i g . ' s 20 through 22 show settlement - 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 II.

57. CHAPTER VII

DISCUSSION OF TEST RESULTS

I t may be concluded from F i g . ' s 10 to 22 which show settlement 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 to these c o n s o l i d a t i o n samples there 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.

In P a r t B of Chapter V i t was suggested that the red 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 g r a i n s i z e . I t was al s o suggested that 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 pe r m e a b i l i t y increased 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 realm of cl a y s one might expect the e f f e c t of v i b r a t i o n s to be very minor, but from the second observation, a c l a y sample might be expected to show a more r a p i d r a t e of settlement at l e a s t during the stage of primary c o n s o l i d a t i o n . In the observations made here, however, there seems to 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 unvibrated samples, i n e i t h e r the primary or the secondary c o n s o l i d a t i o n range.

In 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 of v i b r a t i o n were confined to 16 and 24.1 cycles per second. From the d i s c u s s i o n i n Part B of Chapter V i t i s l i k e l y that the apparent independence of settlement to v i b r a t i o n s i n t h i s case i s due to the range of frequencies being w e l l below the c r i t i c a l range re q u i r e d to induce settlement. Observations, already r e f e r r e d t o , by Tschebotarioff, to the e f f e c t that frequency has no e f f e c t on penetration values of a sand could also be due to t h i s same phenomena.

58. F i g . ' s 7, 8 and 9 show curves of voids r a t i o versus l o g

of pressure. I t may be seen i n F i g . ' s 7 and 9 that the p l o t s representing v i b r a t e d undisturbed samples show a trace of concavity upwards i n t h e i r s teeply s l o p i n g p o r t i o n s . According to Terzaghi and Peck**' t h i s i n d i c a t e s e x t r a s e n s i t i v i t y .

F i g . ' s 7, 8 and 9 i n d i c a t e , w i t h respect to the undisturbed samples, that 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 unvibrated samples. The samples of s e r i e s I , shown i n F i g . 6, were procured by persons other than the author and there i s reason to b e l i e v e that they suffered a measure of disturbance, hence the e f f e c t of s t r u c t u r e may have been diminished previous to t e s t i n g .

In F i g . ' s 7 and 9, the maximum past pressures as i n d i c a t e d by Casagrande's method are only about 75 per cent as high f o r v i b r a t e d as f o r unvibrated samples. In F i g . 8 the average past pressure of the two v i b r a t e d samples i s about 90 per cent of that i n d i c a t e d by the unvibrated sample. Since t h i s trend was noted i n a l l three s e r i e s of t e s t s , i t appears that 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 that are governed extensivel y by past h i s t o r y . The e f f e c t on the past pressure p r e d i c t i o n seems to be s i m i l a r f o r both the more subtle 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 loading of s e r i e s I I and I I I . The undisturbed, v i b r a t e d sample tes 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 to 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 applying 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. In a d d i t i o n , samples test e d i n s e r i e s I t e s t s remained i n storage f o r more than a year

59. before being used. What e f f e c t t h i s prolonged p e r i o d , (during which the e f f e c t i v e c o n f i n i n g pressure may have been reduced) could 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 only be guessed at. I t would conceivably contribute to the more gradual change of slope 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 conclusion i s supported by the f a c t that the past pressures 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 undisturbed 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 corresponding values 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 that v i b r a t i o n s r a p i d l y destroy 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 past loading 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 to a narrow range below the p r e - c o n s o l i d -a t i o n l o a d . This gives the c l a y a s e n s i t i v i t y greater than that which could be expected from r e s u l t s on unvibrated samples. Below 75 per cent of the apparent value of the past pressure, v i b r a t e d samples behave s i m i l a r l y to unvibrated 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 again show a very s i m i l a r pressure-voids r a t i o r e l a t i o n s h i p . This e f f e c t might be extremely dependent upon the load increment r a t i o . With a p a r t i c u l a r value 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 to show up i n one of the settlement-time p l o t s , as shown i n F i g . ' s 10 to 22. Again i t must be remembered that the above represents 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

Although there 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 settlement of s o i l s and on the eval u a t i o n of various dynamic s o i l parameters, most of t h i s has been confined to cohesionless s o i l s . This i s probably due to the much greater dependence of the behaviour of cohesionless s o i l s on a p p l i e d v i b r a t i o n s . Nevertheless cohesive s o i l i s a l s o known to be somewhat a f f e c t e d by dynamic f a c t o r s , the extent of which i s s t i l l l a r g e l y unknown.

From the above experiments, the Author concludes that 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 settlement, p a r t i c u l a r l y w i t h regard to cohesive s o i l s , s t r i c t c o n t r o l of frequency and magnitude of impressed v i b r a t i o n s i s mandatory. I t seems qu i t e p o s s i b l e that under proper conditions of frequency, amplitude, a c c e l e r a t i o n , e t c . , a c e r t a i n maximum dens i t y may be a t t a i n e d by cohesive s o i l , as i s the case w i t h cohesionless s o i l s .

I t would be i n t e r e s t i n g to discover the v a r i a t i o n i n resonant frequency of a confined sample w i t h changes i n s t a t i c pressure, degree of c o n s o l i d a t i o n and magnitude of dynamic load. This 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 c o n d i t i o n s .

Although s t a t i c and dynamic e l a s t i c constants are not the same**', perhaps a r e l a t i o n s h i p between them could be discovered.

12

¥ork along these l i n e s t h a s been of f e r e d by Wilson and D i e t r i c h who show some connection between modulus of e l a s t i c i t y , E, determined by both s t a t i c and dynamic methods, and compressive strength. They also 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

6 1 .

c h a r a c t e r i s t i c s . A g r e a t d e a l m o r e w o r k s h o u l d be d o n e a l o n g t h e s e

l i n e s . The v a l u e o f b e i n g a b l e t o p r e d i c t d y n a m i c p r o p e r t i e s f r o m

t h e s t a n d a r d c l a s s i f i c a t i o n s y s t e m s i s v e r y o b v i o u s , s h o u l d t h i s

be p o s s i b l e .

Some w o r k h a s b e e n d o n e o n c o m p a c t e d c l a y , i n t h i s r e g a r d ,

b u t t h i s m a t e r i a l r e q u i r e s a g r e a t d e a l m o r e i n v e s t i g a t i o n ,

p a r t i c u l a r l y a s f a r a s d i f f e r e n t w a t e r c o n t e n t s a r e c o n c e r n e d . I t

w o u l d seem t h a t a t t e m p t s t o c l a s s i f y s o i l s a c c o r d i n g t o t h e i r

d y n a m i c c h a r a c t e r i s t i c s s h o u l d i d e a l l y be c a r r i e d o u t i n c o n j u n c t i o n

w i t h s t u d i e s t o d e t e r m i n e t h e a c t u a l b e h a v i o u r o f t h e s e s o i l s

u n d e r a l l c o n d i t i o n s o f d y n a m i c l o a d i n g . U n t i l t h e r e i s a b e t t e r

a p p r e c i a t i o n o f t h e l a t t e r c o n s i d e r a t i o n , t h e f o r m e r l a c k s v e r y

m u c h o f i t s p o t e n t i a l v a l u e .

SUMMARY AND CONCLUSIONS

62.

Current v i b r a t i o n t h e o r i e s were evaluated and t h e i r

a p p l i c a t i o n s to foundations were considered. I t was noted t h a t

these t h e o r i e s are a l l based upon very simple assumptions (a mass-

l e s s v i b r a t i n g spring) and t h e r e f o r e do not allow any q u a n t i t a t i v e

c o n s i d e r a t i o n of such v a r i a b l e s as water content, r a t e of lo a d

a p p l i c a t i o n s , s t a t e s of s t r e s s or other a l l i e d f a c t o r s .

N o n l i n e a r i t y of s o i l s p r i n g c h a r a c t e r i s t i c , damping and

eq u i v a l e n t weight of v i b r a t i n g s o i l were d i s c u s s e d . Their

i n f l u e n c e and l i m i t s of a p p l i c a t i o n i n the above t h e o r i e s were

covered b r i e f l y .

The more rec e n t trend i n d e s i g n i n g foundations a g a i n s t

dynamic l o a d i n g , towards the c o n t r o l l i n g of amplitude and

frequency* was compared to some of the former p r i m i t i v e methods,

where horsepower, number of c y l i n d e r s and the l i k e were used'in

e m p i r i c a l r e l a t i o n s h i p s . I t was concluded t h a t the e a r l y methods

were of l i t t l e v a l u e .

The e f f e c t s of v i b r a t i o n s on the p r o p e r t i e s of s o i l s were

di s c u s s e d 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 and extents of

settlement. Long term v i b r a t e d and un v i b r a t e d 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 to check the a c t u a l settlement of a p l a s t i c c l a y

as compared wi t h current v i b r a t i o n t h e o r i e s and knowledge. Three

s e r i e s of t e s t s were conducted. ' V i b r a t i o n s were a p p l i e d i n S e r i e s

I i n a manner analogous to waves t r a n s m i t t e d through a s o i l

stratum. S e r i e s I I and I I I samples were v i b r a t e d by a ha r m o n i c a l l y

v a r y i n g l o a d which was a p p l i e d d i r e c t l y through the l o a d i n g heads

of the c o n s o l i d a t i o n samples and v a r i e d from about 1 to 5 per cent

63.

of the s t a t i c l o a d .

Settlement behaviour under a l l types of v i b r a t i o n was found

to be very s i m i l a r . In a d d i t i o n , no d i f f e r e n c e i n the time-

settlement r e l a t i o n s h i p c o u l d be n o t i c e d between v i b r a t e d and

u n v i b r a t e d samples f o r e i t h e r undisturbed or remolded samples i n

e i t h e r the primary or secondary stage.

V o i d r a t i o versus l o g pressure curves a l s o showed no d i f f e r

ence between v i b r a t e d and u n v i b r a t e d t e s t s f o r remolded samples,

but f o r undisturbed samples there was a n o t i c e a b l e departure of

v i b r a t e d samples from the usual t r e n d . The v i b r a t e d e-log p

curves f o r undisturbed samples commonly showed a f a i n t c o n c a v i t y

upward i n t h e i r steeper p o r t i o n s , s i m i l a r to t h a t c h a r a c t e r i s t i c

of extremely s e n s i t i v e s o i l s . These curves broke very a b r u p t l y

and i n d i c a t e d past pressures s l i g h t l y lower than those f o r un

v i b r a t e d samples. However, outside the r e l a t i v e l y narrow range

of the break i n the v i b r a t e d e - l o g p curve, p l o t s f o r both

v i b r a t e d and u n v i b r a t e d samples were again i d e n t i c a l .

T h i s suggests t h a t v i b r a t i o n s may a f f e c t those p r o p e r t i e s

which are e x t e n s i v e l y governed by the past h i s t o r y of the c l a y

stratum.

I t would appear from these experiments t h a t v i b r a t i o n s w i l l

have an e f f e c t on the settlement of a c l a y only i n the v i c i n i t y

of the maximum past pressure. However the v i b r a t i o n s a p p l i e d i n

these t e s t s were w i t h i n the frequency range of 15 - 30 c y c l e s per

second. These are known to be w e l l below the resonant frequency

of c o n f i n e d samples and are t h e r e f o r e probably not the c r i t i c a l

ones with regard to settlement.

64.

FIG. 3 Standard C o n s o l i d a t i o n Frame

(b)

FIG. 4

O s c i l l a t o r U n i t

66,

(b)

FIG. 5

O s c i l l a t o r Drive U n i t , c o n s i s t i n g of an e l e c t r i c motor, 3 f o u r - s t e p p u l l e y s and 2 f l e x i b l e d r i v e s .

(c) FIG. 6

M o d i f i e d C o n s o l i d a t i o n Frame

©I I o •H -P oJ U a>

13 •H O

F i g . 7 : - • C o n s o l i d a t i o n curves showing Voids R a t i o Versus The Log Of Pressure.

Ser i e s I Test 2— I — Un.diisT., Test 3 - I—/?em., Unyife Test 4 - I — • • L/ib. Test 1 - I-CVndist. Uni/ib.

0.3 0.5 1.0 2.0 3.0 5.0 , 10

V e r t i c a l Pressure - kg. per sq. cm. 00

F i g .

C o n s o l i d a t i o n c u r v e s s h o v i n g V o i d s R a t i o V e r s u s The Log Of P r e s s u r e .

S e r i e s TL i Test 7 - I I Undut . wib. T e s t 6 - I I ' ' • ' T e s t 3 - I I * U w i b . Te s t 4 - I I Remol. ••

1 0

0.3 0.5 1.0 2.0 3.0 5.0 10 V e r t i c a l Pressure - kg per sq. c m .

I i ; L ;—; 1 _ 0.1 1.0 10 100 1000 10,000

Time i n minutes

Time i n minutes

10,000

\

1000 10,000

Time i n minutes

Fi£ , 22 Consolidat ion curves showing settlement v . s . log of time. Series I

Test 3 Test 4

Series II Test 4

- 8 kg per sq. cm.

'em (jjno/b. Uib .

0.1 1.0 10 100 Time i n minutes

1000 10,000

FIG. 23

Grain Size D i s t r i b u t i o n

85. BIBLIOGRAPHY

1. Ts c h e b o t a r i o f f , G.P.: " S o i l Mechanics, Foundations and Earth S t r u c t u r e s " , McGraw-Hill, New York, 1951.

2. Leonards, G.A.: "Foundation Engineering", McGraw-Hill, New York, 1962, Chap. 18, "Foundations Subjected to Dynamic Forces", by F.J. Converse.

3. Barkan, D.D.: "Dynamics of Bases and Foundations", McGraw-Hill, New York, 1962.

4. Novak, M.: "The V i b r a t i o n s of Massive Foundations on S o i l " , Publ. of the In t e r n . Assoc. f o r Bridge and S t r u c t u r a l Engineeri 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 Foundation Engineering, Cambridge, 1936.

6. Terzaghi, K.: " T h e o r e t i c a l S o i l Mechanics", Wiley, New York, 1943.

7. Pauw, A.: "Dynamic Analogy f o r Foundation-Soil Systems", A.S.T.M. Spec. Tech. Publ. No. 156, 1953.

8. Lorenz, 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. Publ. No. 156, 1953.

9. Den Hartog: "Mechanical V i b r a t i o n s " , McGraw-Hill, New York, 1934. 10. Bernhard, R.K. and F i n e l l i , J . : " P i l o t Studies on S o i l Dynamics",

A.S.T.M. Spec. Tech. Publ. No. 156, 1953. 11. Shannon and Wilson, S e a t t l e , Wash.: "Test R e s u l t s , June 1962,

unpublished." 12. Wilson, 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

pressure on E l a s t i c and Strength P r o p e r t i e s of Clay", A.S.C.E. Shear Conference, Boulder, Colorado, June I960.

13. Taylor, D.W.: "Fundamentals of S o i l Mechanics", Wiley, New York, 1948.

14. Eastwood, W.: "The Factors which A f f e c t Natural 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 Bearing Power of Foundations i n Sand", Proceedings of the 3rd I n t e r n . Conference of S o i l Mechanics and Foundation Engineering, Switzerland, 1953.

15. Armstrong, J.E.: " S u r f i c i a l Geology of the New Westminster Area", Geol. Surv. of Canada, Paper 57-5, 1957.

16. Terzaghi, K. and Peck, R.s " S o i l Mechanics i n Engineering P r a c t i c e " , Wiley, New York, 1948.

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