Soil Dynamics and Machine Foundations Swami Saran_2

.

óóóóóó '0~"

SOIL DYNAMICS . AND"

MACHINE

'..

FOUNDATIONS

By Dr. SWAMI SARAN Department of Civil Enginemng University of Roorkee Roorkee-247 667

(INDIA)

~

~) j JW' .:r-.~ ,J}j~-4-'

\.-:.

J:'" 4i~ \;S"

.5357 :OJW

13!fi/3

11!2 :~Jt;

1999 iF

Galgotia Publications pvt.ltd.

5,AnsarIRoad, Daryaganj,New Delhl-110 002 .

'"

. ...

"

,I,

,.

"'C','

;. -,'

c

'.

'0'

, "~; ,

, ,~'

>.~.:;~;~"!'ij:~,;~i::>A~~;.

.:j : ~

,;

~

'.p', ,:'..

11'

:':"~~~:O~':i;~,~:~,,~~,

.'

~

TA :.

-'

,qqq

'

Foundatio~s~,

' ,

i

('

~

\

.. . ...

, . ,'-

, };~ i-~

"-)f>;

~",;'~~

First Edition 1999

~ Reserved - 1999

No matter in full or part may be reproduced or transmitted in any form or by any means (exceptfor review or criticism) without the written permission of the author and publishers.

Though much care has been taken by the author and the publishers to make the book error (factual or printing) free. But neither the author nor the publisher takes any legal responsibility for any mistake that might have crept in at any stage.

Published by .Suneel Galgotia for Galgotia Publications (P) Ltd. 5, Ansari Road, Darya Ganj, New Delhi-ll0 002. '"" ,';.. ". ::::..',.-

, ,

,.,.

'0',"

.. :.

',.' '"

Laser Typeset by .. ADO Computer's 402 (RPS) DDA Flats Mansarover Park, Shahdara, Delhi-l 10 032. Ph : 2292708 . "..

.

Prinled al Cambridge PrintingWorks,New Dclhi-llO028 .

>. ..- "". ;."

,

PREFACE

During the last 25 years, considerable work in the area of soil dynamics and machine foundationshas been reported.Courseson soil dynamicsandmachinefoundationsalreadyexistat graduatelevelin many institutions, and its inclusion at undergraduate level is progressing fast. The author is engaged in teaching the course on soil dynamicsand machine foundationsat gr'duate level from last fLfteenyears. The text of this book has been developed mainlyout of my notes preparedfor teaching the students.The consideration in developingthe text is its lucide presentationfor clear understandingof the subject.The material has been arrangedlogicallyso that the reader can follow the developmentalsequenceof the subject with relative ease. A number of solved examples have been included in each chapter. All the formulae,charts and examples are given in SI units. Some of the material included in this text book has been drawn from the works of other autors. Inspiteof sincereefforts,somecontributionsmay nothavebeen acknowledged.The authorapologisesfor suchomissions. The author wishes to express his appreciationto Km. Lata Juneja, Sri RaJeevGrover and Sri S. S. Gupta for typing and drawing work. Thanks arealso due to the many collegues,friends and studentswho assistedin

wittingof thisbook.

.

.

The author would be failing in his duty it he does not aclaiowledge the support he received from his family members who. encouraged him through the various stages. of study and writing. The book is dedicated to author's Sonin law, (Late) Shri Akhil Gupta as a token of his love, affectionand regards to him. (Dr. Swami Saran)

:

11~f1 1.:;;-,0,...::,

':,':' "~iJ ,

..;

'.o..;oc;.""

òò×溬þôòò¢

c ,'t...,:

'. '-~~'

,.)~

, ".,.~

1"

i""" ,"",-:,,»"'"

j J ~ .,~ ;"""""""',","_'"""~r""""',"""",,,.'o_-' ~'j.'W.,~",

.

,',. ..;' f~ '.- .,-;'~O#y . ~""'t'j oS

;"'...,

,~'~:~~F~B

.:'l~

CONTENTS

. PREFACE

1.

INTRODUCTION 1.1 General 1.2 1.3 1.4

2.

THEORY OF VIBRATIONS 2.1 General 2.2 Defmitions 2.3 Harmonic Motion 2.4 Vibrations of a Single Degree Freedom System 2.5 Vibration Isolation 2.6 2.7 2.8

3.

Earthquake Loading Equivalent Dynamic Load to an Actual Earthquake Load Seismic Force for Pseudo-staticAnalysis Illustrative Examples References Practice Problems

Theory of Vibration Measuring Instruments Vibration of Multiple Degree Freedom Systems Undamped Dynamic VibrationAbsorbers Illustrative Examples Practice Problems

WAVE PROP AGATION IN AN ELASTIC, HOMOGENEOUS. . AND ISOTROPIC MEDIUM 3.1 General 3.2 Stress, Strain and Elastic Constants 3.3 Longitudinal Elastic Waves in a Rod oflnfmite Length 3.4 Torsional Vibration ora Rod of Infmite Length 3.5 End Conditions 3.6 3.7 3.8 3.9 3.10 3.11

Longitudinal Vibrations of Rods of Finite Length Torsional Vibrations of Rods of Finite Length Wave Propagation in an lnfmite, HomogeneousIsotropic, Elastic Medium Wave Propagation in Elastic, Half Space Geophysical Prospecting Typical Values of CompressionWave and Shear Wave Velocities Illustrative Examples References..'. . ; Practice Problems

1-12 I 3 6 9 12 12 12 13-66 13 14 15 18 32 36 39 48 53 64 êéóïïé 67 67 70 72 74 76 80 81 86 93 108 108 116 117

£i,.,

~

viii

4.

ëò

Soil Dynamics & Machine Foundations

DYNAMIC SOIL PRO~ER~5. 4.1 General

'-. ."' .

4.2 4.3

LaboratoryTechinques Field Tests

4.4

FactorsAffecting Shear Modulus, ElasticModulus and Elastic Constants IllustrativeExamples References PracticeProblems

ïèéóîíé 187 187 201 221 236

DYNANnCEARTHPRESSURE ëò ï

General

ëòî

Pseudo-static Methods

5.3

Displacement Analysis Illustrative Examples References

237

PracticeProblems

6.

7.

DYNAMIC BEARING CAPACITY OF SHALLOW FOUNDATIONS 6.1

General

6.2

Pseudo-static Analysis

6.3 6.4

Bearing Capacity of Footings Dynamics Analysis Illustrative Examples References Practice Problems

LIQUEFACTION OF SOILS 7. 1 General 7.2 Definitions 7.3 7.4

--.118-186 118 118 147 163 174 182 184

Mechanism of Liquefaction Laboratory Studies

DynamicTriaxial Test 7.6 Cyclic Simple Shear Test 7.7 Comparisonof Cyclic Stress Causing Liquefactionunder Triaxial and Simple Shear Conditions 7.8 StandardCurves and Correlations for Liquefaction 7.9 Evaluationof Zone of Liquefactionin Field 7.10 VibrationTable Studies 7.11 Field Blast Studies

éòë

7.12

Evaluationof LiquefactionPotentialusing Standard Penetration Resistance

7.13

Factors Affecting Liquefaction

-

îíèóîéè 238 238 238 . 249 268 277 278 îéçóííç 279 2.79 281 283 288 296 300 301 306 309 314 319 323

Contents

ix

'

324 326 332 336 339

7.14 AntiliquefactionMeasures 7.15 Studies on Use of Gravel Drains IllustrativeExamples References Practice Problems

8.

9.

.

GENERAL PRINCIPLES 8.1 General 8.2 8.3

Types of Machines and Foundations General Requirements of Machine Foundation

8.4 8.5

Perimissible Amplitude Allowable Soil Pressure

8.6 8.7

Permissible Stresses of Concrete of Steel Permissible Stresses of Timber References

FOUNDATIONS OF RECIPROCATING MACHINES 9.1 General 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

10.

Modes of Vibrationof a Rigid FoundationBlock Methods of Analysis Linear Elastic Weightless Spring Method Elastic Half-space Method Effect of Footing Shape on Vibratory Response Dynamic Response of Embedded Block Foundation Soil Mass Participating in Vibrations Design Procedure for a Block Foundation Illustrative Examples References Practice Problems

FOUNDATIONS OF IMPACT TYPE MACIDNES 10.1 General

11.

340-351 340 340 347 348 349 349 350 351 íëîóìîî 352 ~

352 353 354 370 392 394 400 402 408 419 . 420

ìîíóììî 423

Design Procedure for a Hammer Foundation

426 432

Illustrative Examples References Practice Problems

436 442 442

10.2 DynamicAnalysis 10.3

.

OF MACIDNE FOUNDATION DESIGN

.

FOUNDATIONS OF ROTARY MACHINES 11.1 General 11.2 Special Considerations 11.3 Design Criteria

ììíóìêð 443 444 445

,{"

'250

(/en?)

Duration of 'Earthquake (8)

20,000

'5

'. Affected. Area

60,000:,..., ;l

J

"

: f,2()',000' 2 ,00 ,OO ',

'

-

111

111

.....

0.8

0.6:

I...

..... ,-C 111

x 0.4

u I 0

lt5 0.2 0 10

3

Conversion

1

factor,

0-3 0'1 (Ns )0.65 Tmax.

0-03

0'01

( ~S)k Tmax Fig. 1.6 : Conversion factor versus shear stress ratio

For getting the equivalent number of cycles for 0.75 'tmax'read the yalue of conversion factor (Fig. 1.6) corresponding to an ordinate value of 0.75. It comes out as 1.5. The value of equivalent number of cycles obtained for 0.65 'tmaxas illustrated in Table 1.4 is divided by this conversion factor to obtain equivalent number of cycles corresponding to 0.75 'tmaxi.e. 9.0/1.5= 6.0 cycles. Seed and Idriss (1971) and Lee and Chan (1972) developed the above concepts specifically for liquefacti~mstudies. More details of these procedures have been.discussed in Chapter 7.

-

--""" "~.

..

j'

Introduction

..

9

1.4 SEISMIC FORCE FOR PSEUDO-STATIC ANALYSIS For the purpose of determining seismic force, the country is classified into five zones as shown in Fig. 1.7. Two methods namely (i) seismic coefficient method and (ii) response spectrum method are used for computing the seismic force. For pseudo-static design of foundations of buildings, bridges and similar structures, seismic coefficient method is used. For the analysis of earth dams and dynamic designs, response spectrum method is used (IS- 1893 : 1975).

Equivalent modi fide

Q) Iv.

mercalli intensity IX and above Vl1I VII VI Less than VI

Bombay

0

~ Vo po rt

.. ". In

0

0

Blair

:

0

.: 0

'I)

Fig. 1.7 : Seismic zonesof India in seismic coefficient

method, the design value of horizontal

'ollowing expression :

a()

CJ.his obtained by the

~

ah

vhere

seismic coefficient

= ~I ao

...(1.3)

= Basicseismiccoefficient,Table 1.5

I. = Coefficient depe.ndingupon the unportance of structure,:Table 1.6

~ = Coefficient

depending upon the soil-f~undation system, Table 1.7

.

:/

+'

Ti mtZ,t

v 0 ~

>

Timcz,t

c 0

0

.... c:,I c:,I

v 0

.et Fig. 2.4: Vector representation

of harmonic displacement. velocity and acceleration

When two harmonic motions having little different frequencies are superimposed. a non harmonic motion as shown in Fig. 2.5 occurs. It appears to be harmonic except for a gradual increase and decrease in amplitude. The displacement of such a vibration is given by: Z = AI sin (0011- 91) + A2 sin (0021- 92)

..

-

D,

N .,/

...(2.7)

2A max

2Am\n

./

.,/

+' C c:,I

E

TimtZ (t)

,~

v 0

a. III

c

,--.'J'

---

'-'" ,.,

~T , b

"""'-

~ "

:'

3!

j,;I',:

. ~~ 'i; 'P1>1Flg;'2.5':Motion containi.ng a beat

---

','"

;;"

C,"

'i'j{':-;,':::;;;~,.

18

Soil Dynamics & Machine Foundations

The dashed curve (Fig. 2.5), representing the envelop of the vibration amplitudes oscillates at a frequency, called the beat frequency, which corresponds to the difference in the two source frequencies: I 1n f,(t+Zn/o>"cI)

-Z I

= en0> f,.21t/o)ncl Zz r:2 -ZI =e Znl;!"l-f,Zz ZI loge 22.-

21t;

~

...(2.44 a)

...(2.44 b) ...(2.44 c) ...(2.44 d)

~ },

",inF1':

j'/,"

" 't~..

' .~t

.o,~;

',.

,-:)

Tlreory of Vtb",tiolU

. . Natural logarithm mk. ,decrement.

of ratio of two successive

peak

amplitudes

{i,e,

log,

(~)} is called as logarith.

1

Z\

r:-:2

~= 2x loge ~ ' As for small valuesof~, V1- ~-

or

::

1

...(2.44 e)

tbus, damping of a system can be obtained from a free vibration record by knowing the successive amplitudes which are one cycle apart. If the damping is very small, it may be convenient to measure the differences in peak amplitudes for a number of cycles, say n. In such a case, if Z" is the peak amplitudes of the n,h cycle, then Zo Zl Z2 Zn-I 0 - = - =- =...== e where~ = 2x ~ .'

Z\

Z2

Zo, Zn

Therefore,

~ = -n

Z 0 oge Zn

I

1

or ~}:

Zn

Z2 = Zo ~ .. Z"-I = eno [ Z, ] [ Z2 ] [ Z) ] [ Z" ] 1

Hence

ZJ

= -2xn

I

..

...(2.441)

Z0..

...(2.44 g)

oge.z n

Therefore, a system is over damped if ~ > 1; critically damped if ~ = 1 and under damped if ~ < 1. 2.4.3 Forced Vibrations Of Single Degree Freedom Syst~m. In many cases of vibrations caused by rotating parts of machines, th~ systems are subjected to periodic exciting forces. Let us consider the case of a single degree freedom sys~.:mwhich is acted upon by a steady state sinusoidal exciting force having magnitude F and frequency 0>(i.e. F(t) = Fosin rot). For this case the equation of motion (Eq. 2.11) can be written as : .. . 111Z + C Z + K Z = Fo Sin ro t ...(2.45) Eq-;(2.45) is a linear, non-homogeneous, second order differential equation. The solution of this equation consists of two parts namely (i) complementary function, and (ii) particular integral. The complementary function is obtained by considering no forcing function. Therefore the equation of motion in this case will be :

..

.

m Z, + C Z, + K Z, = 0

...(2.46)

The solution of Eq. (2.46) has already been obtained in the previous st?ctioIland is given by, ZI = e-O>/"(C\sinrondt+C2cosrondt) ...(2.47) Here ZI represents the displacement of mass m at any instant t when vibrating without any forcing function. . The particular integral is obtained by rewriting Eq. (2.45) as m 2:2+ C 22 + K Z2 = Fo sin rot where Z2 = displacement of mass m a~~nYinstant t when vibrating with forcing function.

...(2.48)

~

.;Y,

8111;1' "

"--H,'

26

"

"

Soil Dynamics & Mac/line Foundations

;

The,solution

of Eq. (2A8).,\~

,,',

"

.

gi'{en by'.

~

',",

,,'

" ,

'"

t,;

"

22 = AI sin 00t + A2cos 00't where AI and A2 are two, arbitrary constants.

...(2.49)

,

"...~ ,'"

.

. .:' , "

Substituting Eq. (2.49) in ~q. (2.48)

','

-

m (- At 002sin 00t - A2002cos (0t) + C (AI 00cos 00t - A2 ID;in 00t) + K (AI sin 00t + A2 cos 00t) ,",,'

""'='Fosin'ro,t':.

:

,

Considering .sine and Cosine functions in Eq. (2.50) separately, 2

(-

"

,;

"

, ,

'

(,.

'

','

",

,"~

,

'

'

,..'(2.50: ,

,'.,

.

. ...(2,51 a)

m AI 00 + KAt - CA2 00) sin 00t = Fo sin 00t

(- m A2 002+ KA2 + CA! 00).I::osffi t,~ O. 'J" From Eg. (2.51 a),

,

",'

,..(2.? 1 b)

.,

.

Al(~ - o}) - A2(~ and from Eg. (2,51 b) A{~-W

)

F.

m =

)+A2(~-w2)

0,

Q.

:

",(2,52

0

',n

a)

...{2.52 b)

=0

Solving Egs, (2,52 a) and (2.52 b), we get (K-moo2) Fo

At- -

'.)2.53

0..

2

a)

(K - mm2) + C2m2 "

and

A2 =

- CmFo

...(2,53 b) t

"

".

';(1..;>'

«(0 t - 8)

(2.,SS)

,

J! = ~1- TJ2)2+ (2T1~l

ill

38.

Soil Dynamics

2.6.1. Displacement 1121.1=

Pickup. The instrument will read the displacement

& Machine - -." '-" Foundlltions .' . -. .

of the structure directly

if

I and 8 = O.The variation o{Tl~ with-~'aiid'~-is shown in Fig-.2.21. The variation of8 with 1'\

is already given in Fig. 2.14. It is seen"'tnatwneifff is" large, 1'\21.1is approximately equal to 1 and 8 is approximately equal to 180°. Therefore to design a displacement pickup, 1'\should be large which means that the natural frequency of the instrument itself'shou~d be low compared to the frequency to be measured. Or in other words, the instrument should have a soft spring and heavy mass. The instrument is sensitive, flimsy and can be used in a weak vibration environment. The instrument can not be used for measurement of strong vibrations. ,-

-t

\

I

3.0

\0

I

--

I I

. -

-

-"

2 0 ,

.

1 0

.

0 0

3.0

2.0

1.0

.

FrequClncy

ratio,

4.0

5.0

'1.

Fig. 2.21 : Response of a vibration measuring instrument to a vibrating base

2.6.2. Acceleration Pickup (Accelerometer). Equation (2.88) can be rewritten as . I X =2 (J,)n

2

1.1

Yoro sin (rot - 8) .

...(2.89)

'

The output of the instrument will be proportional to the acceleration of the structure if J.1is constant. Figure 2.13 shows the variation of J.1with 1'\and;. It is seen that J.1is approximately equal to unity for small values of 1'\.Therefore to design an acceleration pickt!p, 11should be small which means that th~' " ",.~'n'7""~:"'"""'"

.'1::'

'lr~-r\f

........--.

- W .. .'1

-..Theory of Vlbrations

65

Wt")

K

0

b

c a

-1

~

T

A l

-1

.~

Fig. 2.36 : Mass-spring system

2.10 A body vibrating in a viscous medium has a period of 0.30 s and an inertial amplitude of30 mm. Determine the logarithmic decrement if the amplitude after 10 cycles is 0.3 mm. Ans. (0.46) 2.11 A vibration system consists of mass of 6 kg, a spring stiffness of 0.7 N/m and a dashpotwith a .

dampingcoefficientof 2 N-s/m.Determine

., .

(a) Damping ratio (b) Logarithmic decrement

Ans. (0.488, 3.55)

2.12 Write a differential equation of motion for the .system shown in Fig. 2.37 and determine the . natural frequencyof dampedoscillationsand the criticaldampingcoefficient.

b

.,

w

a

r

K

Fig. 2.37 : Mass-spring dashpot system

2.13 A mass is attached to a spring of stiffness 6 N/mm has a viscous damping device. When the mass was displaced and released, the period (jf vibration was found to be 1.8 s and the ratio of consecutive amplitude was 4.2 to 1. Determine the amplitude and phase angle when a force F = 2 sin3t N acts on the system. Ans. (0.708mm, 56.4°) 2.14 A sp~ingmass system is excited by a force Fa sin (J)t. At resonance the amplitudewas measured . .

td be.100 mill..At 80%resonantfrequencythe amplitudewas measured80 mm.Determinethe d~mpingfactorof the system. Ans (0.1874)

2~r5:AssUnlfngsm~ll ~plitudes, set up differential equation of motion for double pendulum using the coordinates shown in Fig. 2.38. Show that the natural frequencies of the system as given by the equation. .

co' I

\11,.2

= ~, g.(2 vi

:1:/2)'

Determine. the ratio of the amplitudes xI/x2' ..

.. I.

;.:':!Z~,"'~.'T"""7~..

.I

t .."'. ..~ ~~...~..,.:",,!..,:.;..~

'1'1'

, )J .J ~ .:.

,:.:;.

..-'.

",.

'\""I\~

':.."h. r

~"V :; ~,.

,.

,- "-

e.

- ! ..J

,

.,t;\.~;...t-

:.>'1;'" ; 1./uJ j

;"

...

:i:

66

Soil Dynamics & Machine "Foundations

m Fig.2.38: Doublependulumsystem 2.16 A motor weighs 220 kg and has rotating unbalance of 3000 N-mm. The motor ~s running at constant sped of 2000 rpm. For vibration isolation, springs with damping factor of 0.25 is used. Specify the springs for mounting such that only 20 percent of the unbalanced force is transmitted

to the foundation. -Also determine the magnitude of the transmitted force.

.

Ans. (Ka = 931.22 kN/m, 26.3 kN) 2.17 A small reciprocating machine weighs 60 kg and runs at a constant speed of 5000 rpm. After it was installed, it was found that the forcing frequency is very close to the natural frequency of the system, What dynamic vibration absorber should be added if the nearest natural frequency of the system should be at least 25 percent from the forcing frequency? 6 Ans. (15.3 kg, 4.2 x 10 N/m) 2.18 A mass of 1 kg is to be supported on a spring having a stiffness of 980 N/m. The damping coefficient is 6.26 N-s/m. Detelmine the natural frequency of the system. Find also the logarithmic decrement and the amplitude after three cycles 'if the initial displacement is 0.3mm. Ans. (31.14 rad/s, 0.628, 0.0456 mm) 2.19 A machine having a mass of 100 kg and supported on springs of total stiffness 7.84 x has an unbalanced rotating element which results in a disturbing force of 392 N at a 3000 rpm. Assuming a damping factor of 0.20 determine (a) the amplitude of motion due to the unbalance, (b) the transmissibility, and (c) the transmitted force. Ans. (0.043 mm, 0.148, 2.20 The static deflection of the vibrometer mass is 20 mm. The instrument .when attached chine vibrating with a frequency of 125 cpm records a relative amplitude of 0.03 mm. for the machine, (a) the amplitude of vibration,

105 N/m speed of

58.2 N) to a maFind out

(b) the maximum velocity of vibration, and , (c) the maximum acceleration of vibration. Ans. (0.0576 mm, 0.754 mmlsec, 9.86 mmlsec2), DD . "

,::~,"'.,':"'.":"'V

,.,'", .."

.

",..

,,;,,)'

~" 2 ; "vt

,1'01 l\'1'.I

o.

".

" ,

,.,,(3.25)

.

,

"

'.',

74

,,":

'."':

"'.

"

""".

'

"","

"'"

.,

Soil Dynamics & Machine' Foillldations

,

From Eq. (3.23),

or

Therefore,

oT

t. '

- =GI- a2a OX p ax2 '..

...(3.26)

a2a ,= pI cia P ax2

P 8t2

ia - 0 cia

or

a? - P a2e -

,

ox2

'~':'

at", -

Vs

,J3.27)

"

2 a2e

'~

..-.

. --zax

,J3.28)

--

"" is the shear wave velocity of the material of the rod. 3.5 END CONDITIONS Free End Conditions, Consider an elastic rod in which a compression wave is travelling in the positive ,r-direction and an identical tension wave is travelling in the negative x-direction (Fig. 3.6 a). Wh;n the t\\/O\vaves pass by each other in the crossover zone, the portion of the rod in which the two waves are superposed has zero stress wit:.1:l twice the particle veioc.~tyof either wave (Fig. 3.6 b). After the two waves have passed the crossover zone the stress and velocity return to zero at the crossover point and both the -:ompressive and tensile waves return to their initial shape and magnitUde (Fig. 3.6 c). It will thus be seen that on the centre line cross-section, the stress is zero at all time. This stre~s condition is the same as that \\hi-:h exists at the free end of the rod. By removing one-half of the rod, the centre line cross-section can be considered a free end (fig. 3.6 d). Hence 'it can be seen that a compre,ssion wave is reflected from a free l'l1d as a tension wave of the same m,agnitude and shape. Similarly, it can be observed that a tension wave IS re Ikcled from a free end as a compression wave of the same magnitude and shape.

t

----

Id-:

Vc.

0

oo ien r=o sion

-v

Xt0

"0

c

(a) Compression and tension wave~ travelling in opposite directions

~~ 00 ~

0-'=0

~,.,,' .

U = 2uO .

y

~Xt1

--q-nrn '---00 ,

Vc

(b) Waves at the crosso~'er zone Fig. : 3.6 : Elastic waves in a rod with free end conditions

..,'...,

'"

,..' --,

(...Contd.)

l4~'

\~~)'"

f'ave.Propagation

,'" "'.~

:,.

, y.~:,~.

"~ ;i0: