Foundations for Machines-Analysis & Design_By Shamsher Prakash.pdf

(]', =

G

(3.54b)

2 a' _ k'iJ; _ 2 a'v;

(3.60a)

-k'- 2 a'- 2 a'if; ax' axaz

(3.60b)

axaz

ax

and

and

(V' 2 + k 2 )1j; = 0

87

By substituting the expressions for and if! from Eqs. (3.56) into the right-hand sides of Eqs. (3.60) at z = 0, we get

in which

-2ixaA + (2x'- k')B

=

o

(3.61a)

(3.55) and where w is equal to 2Tr!T, in which Tis the period of the propagated waves. Then his equal to 2Tri(VJ) and k is equal to 2Tri(V;T). Now, VJ and V,T r~present the wavelengths of lo~gitudinal and transverse waves, respectively. Therefore, h and k are reciprocal values of the wavelengths, and k is always larger than h. Particular solutions of Eqs. (3.54a and b) may ~:)!' assumed as (3.56a)

2

Uo = G

(3.61b)

Therefore, we obtain (3.62a) and

and

B=(~t~)(~)

(3.56b) where

2

(2x - k )A + 2ix(3B

where

(3.62b)

88

WAVE PROPAGATION IN AN ELASTIC MEDIUM

89

WAVE PROPAGATION IN A SEMI-INFINITE ELASTIC HALF SPACE

F(x)

=

(2x

2

k')'- 4x'af3

-

(3.63)

By using Eqs. (3.52), (3.56), and (3.62), we get u = ix

(2

2

X -

k') -a' 2 {3 -#< ;(X) - a e e;xx ~

F(x)

=

o

(3.67)

x may (3.64a)

be determined from Eq. (3.63). Instead of Eq. (3.67), which contains irrational expressions, we may consider the following equation, which does not contain radical signs (Barkan, 1962):

and

F(x)f(x) w =a

-(2x'- k')e-"' + 2x'e-~' · u 0 e'xx _ F(x) G

J+- x[(2x

2

k') e-"'- 2a{3e-~']e;x., F(x) dx

-

-00

(3.65a)

__!__

J+" a[ -(2x

2.,.0 -·

2

-

k') e-"' + 2x 2 e-~']e;xx F(x) dx (3.65b)

f(x)

=

k')B

=

0

(3.66a)

and

o

(3.66b)

Equations (3.66a and b) will give nontrivial solution for A and B only when the determmant of this system equals zero. This leads to

k')x'

4

( h2) K_6J= 0

K_4 - 16 1 - k k'

k

6

(3.69)

Since k is greater than h, one of the roots of Eq. (3.68) lies between 1 and +oo. It be shown that th<\:"other two roots, if real, lie between zero and h 21k'. · The first root corresponds to positive values of a and {3; and therefore, it does not satisfy the condition f(x) = 0. The last two roots make a and {3 positive and imaginary and, therefore, they do not satisfy the equation F(x) = 0. This equation has only one root, x' =A 2;t which is greater than one. Therefore, A2 is greater than k'. For a Poisson's ratio of 0.5, the real root of Eq. (3.68) is Alk = 1.04678. When vis equal to 0.25, all roots ofEq. (3.68) are real and are equal to

gn.

A2

1 4

(3.70a)

=

~ (3- YJ)

(3.70b)

=

4 (3+YJ)

and 1

(3. 70c)

Of these roots, onl~;the last one satisfies the conditions of this problem. Its value is A

(2x'- k')A + 2ixf3B =

-

(2x'- k') + 4x'af3

Velocity of Wave Propagation

-

2

in which

k'

-2ixaA + (2x 2

2

h )(x

(3.68)

Equations (3.65a) and (3.65b) correspond to the waves induced by an excihng force acting along the line x = 0, z = 0.

Free surface waves occur where they are induced by some initial excitement on the border surface. Assuming, u 0 = 0, for this case, the values of constants A and Bin Eqs. (3.61a) and (3.61b) may be determined from the followmg expressiOns:

-

k'

k

and w = _

(2x'- k 2 ) 4 -16(x 2

= k 8 [1- 8 K_22+ (24- 16 -h2)

(3.64b)

In_ order to tra~sform the exciting force into one that acts along the line x- o, z= 0 (Fig. 3.12), assume that u 0 is equal to - P(dx/2.,.). By subshtutmg th1s expressiOn mto the right-hand side of Eq. (3.64) and mtegratmg With respect to x from +oo to -oo, we obtain the following expressiOns for the displacements u and w:

u = __!!'__ 2.,.0

=

1

k = 2 V3+ Y3 = 1.087664 Similarly, if we designate ·1· A is a root of Eq. (3.68) satisfying the required conditions.

(3.71)

90

WAVE PROPAGATION IN AN ELASTIC MEDIUM

WAVES GENERATED BY A SURFACE FOOTING

(3.72) Ia)

s -wave

P-wave

u

A=_":_ VR

91

(+away)

~

R -wave

I

I

r

./'>-

in which Vu is the velocity of propagation of the Rayleigh wave under consideration, it may be seen that

vR =Ak v; for v

= 0.5,

for v

=

0.25,

(3.73) Minor tremor

VR

=

0.9553V:

VR

=

0.9194V,

Thus, it is seen that the velocity of surface waves propagation is somewhat smaller than that of shear or transverse waves. Therefore Vn is less than v;. A plot of VRIV: for different values of vis sketched in Fig. 3.10 (Richart, 1962). It is thus seen that there are three principal waves in an elastic half space. These ':av~s tra~el at different velocities. Knowing their velocities, it is easy to prediCt m which order the waves will arrive at a given point as a result of a disturbance at another point.

~

t

Major tremor--\

w

(b)

{+down)

I

I

v

1

~

\_

t

Particle motion

lei

Particle Motion at the Surface

Lamb (1904) studied in detail the surface motion that occurs long distances away from a pomt source at the surface of an elastic half space. When a pomt source acts at the surface, the disturbance spreads out in the form of ~ymmetncal annular waves. The initial form of these waves depends on the mput Impulse. When the input is of short duration, the characteristic waves shown m Fig. 3.13a develop (Richart et al., 1970). These waves have three salient features which correspond to the arrivals of the P-wave, S-wave, and Raylmgh (R) wave. The horizontal and vertical components of particle motwn are shown m Figs. 3.13a and b, respectively. At the ~urface, a particle first undergoes an oscillatory lateral dismen! on arnval of the P-wave. This is followed by another oscillation at the arnval of the S-wave after a relatively quiet period. This is then followed by an oscdlatwn of much larger magnitude when the R wave arrives. With mcreasmg distance from the source, the time interval between wave arrivals becomes greater and the amplitude of the oscillations becomes smaller. In additiOn, P-wave and S-wave amplitudes decay rapidly compared to that of an R-wave. Therefore: the R-wave is the most significant disturbance along the surface of an elas!Jc half space. At large distances from the source, this wave may be the only clearly distinguishable wave. If ~he h~rizo?tal and vertical components of the particle motion starting at pomt 1 m Figs. (3.13a and b) are combined, the locus of the surfaceparticle motion for the R wave can be drawn as shown in Fig. 3.13c. The path of the particle motion is a retrograde ellipse.

Direction of wave propagation --;.-

w

Figure 3.13. Wave system from surface point source in ideal medium. (After Richart, Hall and Woods, 1970, p. 90, Reprinted by Permission of Prentice Hall, Englewood Cliffs, N.J., After Lamb, 1904.

3.4

WAVES GENERATED BY A SURFACE FOOTING

In practice, a machine foundation generates waves in the soil. To illustrate this condition, an ideal case of a circular footing undergoing vertical oscillations at the surface of an elastic half space will be considered. The energy of the oscillating footing is carried away by a combination of P, S, and R waves. The esse\)tial features of this wave field at a relatively large distance from the source'are shown in Fig. 3.14 (Woods, 1968). The distance from the source of the waves to each wave in Fig. 3.14 has been drawn in proportion to the velocity of each wave for a medium with v equal to 0.25. T_l!e body waves(!:. a!!d S J~f.()p_ag~l_"..Qll~."'~r.IVC propagates radmlly outward

3.Iofii~-"~ ~EY-.lindjkf!l wav£_~ r~~!:...~U

of

tN.~~~es____~-~~ou~-~~r -~t:J:_. i-~~~~~!~-g~y

larger volum()( ofthemed\um as they tra el ouiwara. Therefore, ilie energy density ofep: wave.decr~ases with distiihce from the footing. This decrease

i~-~!!_~gy_q._fu!~]-~~~ or decn\ase m dtsplac~ment amplitude, ts ca1l_~~-.S~t-

'

!

·.

ii .

\

.

t"·! .Q,_, "

92

WAVE PROPAGATION IN AN ELASTIC MEDIUM

r-2

r-2

3.5

Circular footing

Geometrical damping law r-0.5

-c--ov =

0.25

Relative amplitude

Percent of

Wave Type (a)

PRACTICE PROBLEMS

.--

P·-

FINAL COMMENTS

Solutions for the velocity of the P wave and the S wave (body waves) and the propagation of the R wave (the surface wave) have been described. Importance of the wave propagation phenomenon to the study of machine foundations bas been discussed. In studying wave propagation caused by a vibrating footing, it has been assumed tbat the footing is circular and placed at the surface of a semi-infinite elastic half space. In practice, a footing is more often rectangular than circular. Also, a footing is always embedded a certain depth below the ground level. Nevertheless, the study of the problem as a simplified case, as in the preceding discussion, does not reduce its practical value. In fact, the departure in the actual results as compared with the analytical predictions based upon simplified assumptions helps in advancing the state of the art in two ways:

total energy

~ ~ r-----1----'~=:::.__j S

93

Rayleigh

67

Shear Compression

26 7 (b)

~igure .3.14. ?istribution of displacement waves from a circular footing on a hom g rsotropic, elastrc half space. (After Woods, 1968.) o eneous,

rica/ damp!ng. The amplitude of the body waves decreases in proportion to the rat1o of 11 r m wh1ch r 1s the distance from the. input source However along the surface of the half space, the amplitude decreases a~ 1fr2 Th~ amphtude of the Rayleigh wave decreases as 1 /vr (Woods, 1968). The partlcl~ motion a~sociated with the compression (P) wave is a push-pull motion m the d1rect10n of the wave front. The particle motion associated w1th the shear (S) wave is a transverse displacement that is norm~! to the d1rectwn of the wave front. However, the J>article motion f ] ....... l..f·----------.--assoc1ated w1th the R wave at the surfac re-trograde ellis(; p{....jlj-;,-Te ····-······ e o_an_eastl<;_ha .. ~p.a:<_Oe__ts~ .li____b ............... J'......J;-_K __ . .J. h shaded zones along the wave fronts for t e ?dy waves md1cate the relative amplitude of particle displacement as a function of the dip angle (the angle measured downward from the surface at the ~enter of the source). The R-wave can be described by two components, vertical and honzontal, each of which decays with depth but according t 0 separate laws. For a vertically oscillating circular energy source, e.g., a vibrating fo?tmg, on the surface of a homogeneous, isotropic, and elastic half space M11ler and Pursey (1954, 1955) determined that the distribution of total mput energy among the three elastic waves was 67 percent for the R-wave 26 per~ent for the S-wa_ve, and 7 percent for the P-wave. The fact tha; two-thuds of the total mput energy is transmitted away from a surface energy source by the R-wave, which decays much more slowly with distance than the body waves, indicates that the R wave is of primary concern for foundatiOns on or near the surface of the ground.

l. The analytical tools are refined by making more realistic assumptions

and more sophisticated analysis, and 2. Test data that may be uS<;d:in developing correction factors for the analytical solutions are generated. It is important to note, however, that most soils are neither homogeneous and isotropic nor elastic. Therefore, gaps between prediction and actual behavior can be explained partially on this count. Further, many soils occur in layers. Analytical solutions for elastic waves in layered systems have been obtained (Zoeppritz, 1919; Ewing et al., 1970). The reflection and refraction of waves are used to determine the depth of overburden in exploring soils for civil engineering projects, but the corresponding solutions have limited application in machine foundations.

PRACTICE PROBLEMS

3.1

What do you understand by (a) compression wave (b) shear wave a,nd (c) Rayleigh wa'lie? 3.2 Describe the particle motion associated with compression, shear, and Rayleigh waves propagating in a semi-infinite, homogeneous, isotropic 3.3

and elastic half space. 3 In a deposit of dry sand with G = 2. 70 and dry density of 112lb/ft , estimate the shear wave velocity at 10, 20, and 30ft below ground

level. 3.4 If the Poisson's ratio of sand is 0.3, estimate the compression wave velocity in Problem 3.3.

94

WAVE PROPAGATION IN AN ELASTIC MEDIUM

3.5 If the sand gets fully submerged, will tbe shear wave velocity be altered? Justify your answer. 3.6 Compute the new shear wave velocities in Problem 3. 3. 3.7 Discuss the effect of saturation on tbe compression wave velociiy in soils. 3.8 (a) A circular footing is vibrating in the vertical direction on the surf~ce of elastiC half space. Describe schematically the dispersion of v1bratwns and several wave fronts generated (b) Explain what you understand by geometrical tktmping.

. .

I·.

'

4 Dynamic Soil Properties

REFERENCES Ba~kan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw-Hill, New York. Ewmg, W. M., _Jardetzky, W. S., and Press, F., (1957). "Elastic Waves in Layered Media" McGraw-Hill, New York. ·

Hardin • B. 0. • and Richart.' F · E ., J r. (1963) . Elastic · wave velocJhes ·· m · granular soils. J. Soil Mech. Found. Eng. Dtv., Am. Soc. Civ. Eng. 89 (No. SM-1), 33-65. Kolsky, H. (1963). "Stress Waves in Solids." Dover, New York. Lamb, H. (1904). On the propagation of tremors over the surface of an elastic solid Ph ·t 1 Trans. Soc. London, Ser. A 203, 1-42. · as. Miller, _G. F., and Pursey, H. (1954). The field and radiation impedance of mechanical radtators on the free surface of a semi-infinite isotropic· solid Proc R Soc Londo S A 223, 521-554. · · · · n, er.

Mille~,

~··.and ~ursey, H. (1955). On the partition of energy between elastic waves in a semt-m ntte sohd. Proc. R. Soc. London, Ser. A. 223, 55-69.

?:

Prak~h, s... and R

Pori, V. K.. (1981). Dynamic properties of soils from in situ test. J. Geotech ng. Drv., Am. Soc. Ctv. Eng. 107 (No. GT-7), 943-963. '

ay!Leighd, L. M(l885). On waves propagated along the plane surface of an elastic solid Proc · · on on ath. Soc. 17, 4-11. Richart, F. E .• Jr. (1962). Foundat· 'b ,. T 863-898. ton Vl ra tons. rans. Am. Soc. Civ. Eng. 127, Part 1, Richart, F. _E., Jr., Hall, J. R., and Woods, R. D. (1970). "Vibrations of Soils and . Foundattons." Prentice-Hall, Englewood Cliffs, New Jersey. Ttmoshenko, S., and

Goodie~,

G. N. (1951). "Theory of Elasticity." McGraw-Hill, New York. J. Soil Mech Found v· A S Gv. Eng. 94 (No. SM-4), 951-979. . . lV., m. oc.

Wood~, R. D. (1968). Screemng of surface waves in soils.

Zoeppritz, K. (1919). Nachr. Konigl. Ges. Wiss. GOttingen Math._ Phys., Berlin, pp. 66--

94

.

Several problems in engineering practice require a knowledge of dynamic soil properties:, "In general, proble,p~involving the dynamic loading of soils are divided into small and large strain amplitude responses. In a machine foundation, the amplitudes of dynamic motion and, consequently, the strains in the soil are usually low, whereas a structure that is subjected to an earthquake or blast loading may undergo large deformations and thus induce large strains in the soil. A large number of field and laboratory methods have been developed for determination of tbe dynamic soil properties. The principal properties that are determined by many of these methods are: 1. Shear strength, which is evaluated in terms of strain rates and stressstrain characteristics; 2. Liquefaction parameters, such as cyclic shearing stress ratio, cyclic deformation, and pore-pressure response; 3. Dynamic moduli, sucb as Young's modulus, shear modulus, bulk modulus, and constrained modulus with corresponding spring constants; 4. Damping; and 5. Poisson's ratio. In machine foundations, an understanding of dynamic soil moduli (with corresponding elastic spring constants) and damping is frequently required. Poisson's ratio is also needed, even though it is frequently not determined. In this chapter, triaxial tests under static conditions are summarized first, followed by a detailed discussion of the laboratory and field methods used to determine dynamic soil properties. Typical values of dynamic soil moduli and damping are also presented. Shear strength determination and liquefac95

96

DYNAMIC SOIL PROPERTIES

lion parameters, are beyond the scope of this text and are discussed in detail elsewhere (Prakash, 1981).

4.1

TRIAXIAL COMPRESSION TEST UNDER STATIC LOADS

A typical triaxial test set up is shown schematically in Fig. 4.1. A cylindrical sample of soil Is placed between porous stones or metal discs. The length of the sample IS g~nerally kept as twice its diameter. The soil sample is enclosed m a thm, Impermeable membrane, which is secured to the base and top caps with rubber rings (0-rings). The 0-rings and the rubber m~mbrane Isolate the sample from the fluid in the chamber. If appropriate flmds are used to pressurize the chamber, it may not be necessary to use the membrane. The porous stones provide access to the sample for either pore-water dramage or pore-pressure measurements. Fluid pressure is applied within the chamber containing the sample. The chamber pressure IS controlled by a "cell pressure control" and is measured with a pressure gauge or a pressure transducer. The chamber pressure acts umf~rmly on the surface of the sample, including the top and bottom loadmg caps. Alternatively, the chamber pressure may be activated laterally, and the vertical pressure applied independently. . A vertical load can be applied through a loading ram, which is equipped With a Jackmg arrangement and measured with a proving ring or load cell.

Loading

mm Pressure

gauge

lL.~s,nwlc enclosed i'n

a

rubber membrane

Water

Pore-pressure gauge

TRIAXIAL COMPRESSION TEST UNDER STATIC LOADS

97

The axial deformation is usually measured by a dial gauge or an LVDT attached to the bottom of the proving ring and abutted against the top of the chamber. Triaxial shear tests permit a better control of stresses and volume changes during shear and drainage conditions as compared to other shear tests. There are basically three different types of tests that may be performed in a triaxial apparatus depending upon the drainage conditions. Drainage Conditions during Shear

Most soils are saturated at some time during the design life of a structure. Drainage conditions before and during shear influence the shear characteristics of saturated soils. In shear tests, soils are first subjected to normal or confining stress, which is usually maintained at a constant level. An increasing deviator (shear) stress is then applied. Shear tests have been devised to measure the shear characteristics of soils under three different drainage conditions as follows: 1. Unconsolidated-undrained test or "quick test": In these tests, no drainage is ~ermitted under conlli)fng pressure or during shear. Thus, the normal load is not transferred to the soil grains as intergranular pressure but exists as hydrostatic excess pore pressure. It cannot, therefore, mobilize any frictional resistance. Preventing drainage during shear prevents volume changes that might otherwise take place. 2. Consolidated-undrained test or "consolidated quick test": In these tests, soils are allowed to drain (consolidate) under applied confining pressure, and no drainage is permitted during shear. Although volume changes can occur during normal loading, these are not possible during shear. This, however, leads to development of pore pressures during shear. 3. Drained test or "slow test": In these tests, full drainage (consolidation) is allowed under confining pressure. Free drainage is also permitted during shear, so that excess hydrostatic pressure does not exist in the pores of the soil, and all stresses are integranular throughout. With the triaxial apparatus, volume changes in the drained tests are measured by the amount of water that flows into or out of a calibrated burette. In undrained tests, Jhe volume change tendencies of the soil result in generation of pore wiler pressures that can be measured by an electronic pore pressure transducer, manometer, or other suitable type of pressure gauge.

To cell pressure control Mercury

Figure 4.1. Triaxial apparatus.

Different Types of Tests in Triaxial Apparatus

The aforementioned tests can be performed in a triaxial test machine. Test interpretations are as follows:

-

200

DYNAMIC SOil PROPERTIES

98

160 Effective stress circle

'\;/

I

80

I

4

0~ 0

2

0

40

Total

~'Dense

/

120

stress circle

0

..L

~

1r

Consolidated-Undrained Test on Saturated Cohesive Soil. In a consolidated-undrained test, all-around pressure,
20

=

_

0.85

25

0 35

30

1

0

5

/

--

Loose

oens: Unconsolidated-Undrained Test on Saturated Soils. When an all around incremental stress Ll.
ILoose

10

-40 -80

0.75

0

15

'-...

Figure 4.2. Mohr plot for undrained test.

=

r
I

10

5

e0

/

~

i

15

lQ.

20

f 30

25

35

Axial strain (%)

StreS~._strain curves for undrained triaxial compression of a saturated sand. (After . _ ,, . " F1gure 4 . 3 . Leonards, 1962; Bishop and Henkel, 1957)."

~ I

I

"'

(a)

- - - Effective stress

- T o t a l stress

'

a

u31

1132

(b)

Figure 4.4. (a) Void ratio vs. confining pressure; (b) Mohr's plots for consolidated undrained tests.

99

100

DYNAMIC SOil PROPERTIES

25 20

15

,,.

..1'1'

/

iil 0'3

l7.Pore-water pressure, u..,

Chamber pressure 10 f-- varied between 30.5 1-_ and ~2.0 I~ per sq in. 5 35

v

0

sl'

I....- ......,

0

ol\

'

5

( 4.1)

4

/

;::.;::;:~dtr3

and

•

J

(4.2)

II F_igure 4.~. Plot of undrained cylindrical compresSIOn test m which pore-pressure values are observed. (After Taylor, 1948.)

2 0

son's ratio are not constants for a soil but, rather, are quantities that approximately describe the behavior of a soil for a particular set of stresses, loading conditions, and geometry. The different values of the modulus and Poisson's ratio apply for any other set of loading conditions. The terms tangent modulus and secant modulus are used frequently. Tangent modulus is the slope of the tangent to a stress-strain curve at a particular point on the curve (Fig. 4.6). The value of a tangent modulus will vary with the point selected. The tangent modulus at the initial point of the curve is known as the initial tangent modulus. Secant modulus is the slope of a straight line connecting two separate points of a stress-strain curve. Based on a linear stress-strain relationship, the following elastic constants can be defined. If a uniaxial stress a, is applied to an elastic cylinder (Fig. 4.7a), there will be a vertical compression and a lateral expansion such that

.,

5

101

ElASTIC CONSTANTS OF SOILS

5

10

15

Axial strain, AL/L 0 in percent

nitude of the pore pressure at failure, f!.ud, is approximately equal to the The inclination of the envelope of th e Mo hr c1rc · 1es of total deviator stress. . stresses give the apparent angle of internal friction, cf>,, and that of the Mohr circles the "effective" angle of interna1' · t.wn, 'Po "' F or llof effectiVe I'd stresses, d Jnc no rma Y canso~ ate clays, both envelopes pass through the origi~. In figure 4.5, the tnaxial test data for a consolidated undrained test for Boston Blue clay (Taylor, 1948) has been shown.

Drained Tests on Saturated Cohesive Soils. In a fully drained triaxial test ~he mduced pore pressures are allowed to dissipate before the stresses ar~ mcreased. Therefore, the pore pressures are zero, and the stresses in this test at all stages are effective stresses.

in which ex, eY, and ez ar~ the strains in the x, y, and z directions, respectively (considered positive when compressive), E is Young's modulus of elasticity, and vis the Poisson's ratio. If shear stresses r,x are applied to an elastic cube (Fig. 4.7b), there will be a shear distortion Yu such that

(4.3) in which G is the shear modulus.

Tangent modulus

Secant modulus

4.2

ELASTIC CONSTANTS OF SOILS

Concepts from the Theory of Elasticity

The ?ehavior of a soil is nonlinear from the time a stress is applied. For f.~~c!Ical p~rposes, the actual nonlinear stress-strain curves of a soil are

zneanzed , t.e., replaced by straight lines. Therefore, modulus and Pais-

Strain ~

Figure 4.6.

Definitions of secant and tangent moduli.

102 DYNAMIC SOIL PROPERTIES

103

ELASTIC CONSTANTS OF SOILS

( 4.5d) Uniaxial

Young's

loading

modulus

E•!!l

(4.5e)

••

(a)

and Simple

Shear modulus G Tu -"f..

shear (b)

(4.5f) The volumetric strain is given by

Isotropic compression

3<,

For a special -·case in which volumetric strain equals

av

V

Constrained

compression

modulus

..

ux:'£'5.:£TY

= 3e, =

= uz = cr0

3CTo

E

= Tyz = Tzx

= 0, the

(1- 2v)

"

The bulk modulus, Eb (Fig. 4.7c) is defined as

E

2(1 + v)

(4.4)

O"o Eb = 3e, =

ao

aviV

=

E 3(1- 2v)

1

E [O'x- v(O'y + O'J]

(4.5a)

(4.7) which gives

1

= E [O'y- v(O', + O'xl]

(4.5b)

1

E [0',- v(O'x + O'y)]

(4.5c)

( 4.6)

Still another special modulus is the constrained modulus E,, which is the ratio of axial stress to axial strain for confined compression or zero lateral strain (Fig. 4.7d). This modulus can be computed from Eq. ( 4.5c) by letting ex = ey = 0 in Eqs. 4.5a and b. Thus,

ca:or an elastic and isotropic material with all stress components acting, we employ the pnnc1ple of superposition to obtain the various strain components:

e, =

Txy

Various types of moduli.

G=

eY

and

~-,,..,-,-.,-fil Ec .!!...•

theEquations (4.l) through ( 4.3) defined the three basic constants of the odrydofbelaslJcJty: E, G, and v. Actually, only two of these constants are nee e , ecause they are related as follows:

ex=

(4.5g)

Z

s.!!

Confined

Figure 4.7.

X

modulus

(c)

(d)

av

-=e +eJ +e v

Bulk

E,

=

£(1- v) (1 + v)(1-2v)

(4.8)

Uniaxial loading and confined compression involve both shear strain and

volume change as shown below:

105

FACTORS AFFECTING DYNAMIC SHEAR MODULUS 104

DYNAMIC SOIL PROPERTIES

Condition

plasticity in terms of effective stresses. On this basis, the maximum value of 6 the shear modulus Gm" (at low shear strain of 10- ) is expressed by Eq. (4.9) (Hardin and Black, 1969):

Shear

Volumetric

Uniaxial loading

G

~(1-2v) u< E u<

Confined compression

(1

=

2

'Ymax

=

2G

+ v)(1- 2v) E(1- v)

The volumetric strain becomes zero if v

(1- 2v)

Tmax

u< u<

~

1-

1J

(1- 2v) 1- ll

max

~ 1230 OCRk ( 2 .973 - e)' 1+ e

.rg' psi

( 4.9)

in which OCR is the overconsolidation ratio, 0'0 the effective all-ar~u~d stress in psi, e the void ratio, and k a factor that depends up~n the plasUcity index of clays. Hardin (1978) recommended tha~ thiS _equatiOn be used for an anisotropic state of stress by taking u 0 = ( ui + u, + u,) /3, the mean effective confining stress. The parameter k, given in Table 4.1 is related to the plasticity index PI.

1/2. Table 4.1. Values of k k

Plasticity Index PI

4.3

FACTORS AFFECTING DYNAMIC SHEAR MODULUS

Based on the study of dynamic elastic constants, the factors on which these depend are (Hardin and Black, 1968): 1. Type of soil and its properties (for example, water content and yd) and state of disturbance; 2. Initial (sustained) static stress level or confining stress; 3. Strain level; 4. Time effects; 5. Degree of saturation; 6. Frequency and number of cycles of dynamic load; 7. Magnitude of dynamic stress; and 8. Dynamic prestrain.

In machine foundations, the magnitude of dynamic stress is usually small. The tests may, therefore, be performed at small strain levels. Also, the number of dynamic stress cycles is very large. To avoid buildup of any residual strains in the soil due to the operation of the machine, it is necessary to ensure that the vibrations of the soil-foundation system are in the elastic range. This is achieved by restricting the vibration amplitudes to small values. The effect of different factors on the''value of dynamic shear modulus is now discussed. Type of Soil and Confining Pressure. The large amount of data on the values of soil constants that had been collected was analyzed by Hardin

(1978). He developed a mathematical formulation of soil elasticity and soil

0 0.181 0.30 0.41 0.48 0.50

0 ' ,. 20 ";""·

40 60 80 >100 Source: Hardin 1978.

Equation ( 4.9) may be expressed in a more convenient form as follows: = A(OCR)k ( Gmax

F(e)

Pa

)I-n(U: )"

( 4.10)

0

By introducing p, (the atmospheric pressure), the par~meter A is dimensionless whereas U: and G in Eq. (4.9) are m lb/m., and the constant 1230 ha~ the dimen;ions (lbli';;'.') 0 ' 5 It is also desirable to change the form of the void ratio function in Eq. ( 4.9) by letting

F(e)

=

0.3 + 0.7e

2

(4.11)

in Eq. (4.10). The function F(e) is more convenient to use than the void ratio function in Eq. (4.9), but it gives about the same effect as em the range 0.4 < e < 1.2. For very large values of e, Eqs. (4.9) and ( 4.10) g~e monotonically decreasing values of Gm" whereas Eq. ( 4.9) g1ves Gm" -0 for e=2.973, and Gm"' increasing for e>2.973. Equation (4.10) will approximate Eq. (4.9) for 0.4 < e < 1.2 if one makes n = 0.5 and A = 625 ·

106

DYNAMIC SOIL PROPERTIES 0 5 ) " ]

Figure 4.8t shows a plot of [ Gm,I(OCR)k(p,ii0 versus the void ratio from laboratory and field measurements (Hardin, 1978). The elastic parameters reqmred for computation of the soil constant are k, n, and v. For most purposes, it is sufficient to use v = 0 12 =05 d 1 fk" Tbl · ,n .,an vaues o 1~ a e 4.1. For a preliminary analysis, Fig. 4.8 can be used as a gmde (Hardm, 1978). For. clean sands, Richart (1977) found that G depends on 0'0 and e. His analyl!cal express10ns for the shear modulus of clean sands are G max

= 700 (2.17- e)2 - o.s 1+e (uo)

( 4.12)

FACTORS AFFECTING DYNAMIC SHEAR MODULUS

for round-grained sands (e < 0.80), and G

= max

max

-4 -

(2.97- e) 1+ e

2 ( _

)o.s

lTo

( 4.13)

= 900 (2.17- e) l+e

2 (-

)o.3s

Uo

(4.14)

For shearing strains of 10- 4, their results agree with those of Eq. (4.12). The precoiding expressions inyatiably point out the fact that Gm, is proportional to ( 0'0 )". The value of'n has been recommended as 0.5 by most investigators. Therefore, if the shear modulus is determined at a mean effective confining pressure of ( 0'0 ) 1 , its value at any other mean effective confining pressure (0'0 ) 2 can be determined from Eq. (4.15):

l'I 3 ~0

326

for angular sands. In Eqs. (4.12) and (4.13), G and 0'0 are expressed in terms of kg/ cm 2 Both equations were originally established to correspond to shearing strains of 10- 4 or less. Equation (4.12) yields values slightly lower than those obtained by pulse tests (Whitman and Lawrence, 1963). Iwasaki and Tatsuoka (1977) determined experimentally from tests on clean sands (0.61 < e < 0.86 and 0.2 < 0'0 < 5 kg/ cm 2 ) at shearing strain amplitudes of 10- 6 that G

r;;gl li2l f712 I2J

107

2000

0

!i

•a:

(4.15)

(.)

0

....1

Effective overburden pressure only.

CJ

0.5

1.0 Void ratio

Figure 4.8.

Elastic stiffness from laboratory and field measurements· 1

lab

"It

d

"I

andclays·2 1 b 1 d b · ,sr ysan s,srts, . ' :-a ' c ean san s; 3- 1a , dense, well-graded graveiMsand with some fines; 4-lab ~~labvely u~rform clean gravels; F~-field, silty. sands, silts, and days at Ferndale, Cholame,

and

Centro srtes by Shannon and Wdson-Agbabran Associates (1976)· F field d "It d claysatsit AB dCAd '2,sans,srs,an es , , an · n erson et al. (1978}. (After Hardin, 1978.) i" In this figure, S is a dimensionless elastic stiffness parameter in Hardin's (1978) d stress-sdtrai n relation ~or inherent isotropy. For clean sands, s varies from 1200 to lr~:;~~r 1 st 1ts an cays, S vanes from 700 to 2000. '

150

a, may be used in place of 0'0 in Eq.

(4.15)

Strain Level. The other important factor affecting soil modulus is strain level. Ishihara (1971) presented Fig. 4.9, which indicates strain levels associated with different phenomenon in the field and in corresponding field and laboratory tests. Typical variations of G versus shear strain amplitude for different types of in situ tests are shown in Fig. 4.10. The soil modulus values may vary by a factor of 10, depending upon the strain level. It is customary to plot a graph between normalized modulus (defined as G-value at a particular,,~train, divided by Gm, at a strain of 10- 6 ) and shear strain. One such plot is shown in Fig. 4.11. The shear strains induced in soil may not be precisely known (Prakash and Puri, 1981). In the case of wave propagation tests, the shear strain amplitudes are low and are assumed to be of the order of 10- 6 The shear strains induced in soil essentially depend upon the amplitude of vibration or settlement, which in turn depends upon superimposed loads, the foundation

contact area, and soil characteristics. The measured values of amplitude or settlement take care of the factors affecting them. The shear strain am-

109

FACTORS AFFECTING DYNAMIC SHEAR MODULUS

1.2,-------,,---------,-----,------,------, Magnitude of strain

ta,·'

ta,-'

w-4

10-3

10- 2

w-'

I

Phenomena

Wave propagation, vibration

Cracks, differential settlement

Slide, compaction, !iquifacation

Mechanical characteristics

Elastic

E!astic·plastic

Failure

Constants

Shear modulus, Poisson's ratio, damping ratio

Angle of internal friction cohesion

Seismic wave method

•e a" 0

in situ vibration test

·;;; f

.s

~ •E

--,

Repeated loading test Wave propagation test

~~

Dynamic shear strain 1

o E

Resonant column test

~0 f,

.0~

j~

Figure 4.11.

Repeated loading test

Figure 4.9.

No~;lized

shear modulus

(t;)·gmaJ

vs. shear strain. (After Prakash and Puri,

1980; Prakash, 1981.)

Strain levels associated with different in situ and laboratory tests. {After Ishihara,

1971.)

plitude y, may be considered equal to the ratio of the amplitude or settlement to width (Prakash, 1975; Prakash and Puri, 1977). For the case of a vertically vibrating footing, the ratio of amplitude to width yields the normal strain e, and the shear strain y, may be computed as follows: (4.16)

"oo·r---r-~--r-rr--~--,-Tl-r--,---~~==~==~---,,---.---,-,-, 0 fo(t:edv'lbrotloot•" 6 F,..e vibration to•t 0 Cyclic plate lood tost

,ooo~'---+--4-~-4-~-+--~---f-tf---~~~·~·~M;"~·oo~"~'"~"~'"~,_f-1~--~--~~tj

l;-t---r-1

1600

j

For an axisymmetric ·case, ax

S:::.i::--- r-- ...

•

= aY,

and ez is given by

1

r-

8'

=

2G(1 + v) (
(4.17)

0

O'z -

'Ye =

O'x

2G

·•:~ y, (
( 4.18)

( 4.19a)

or 'Yo E,

Figure 4.1 0. 108

Dynamic shear modulus vs. strain. {After Prakash and Puri, 1980; Prakash, 1981.)

=

( 1- "x)(l + v) ----".2'-------

(4.19b)

DYNAMIC SOIL PROPERTIES

110

If the then the If one principal

loading condition corresponds to the at rest condition (K0 case), ratio of o")a, is given by Eq. (4.7) and y0 /e, = 1. considers an extreme case of incipient failure with a, as the major stress, then ( 4.20)

and the value of y 0 /e, will depend upon the value of the angle of internal friction of soil,, and the Poisson's ratio, v. Typical values of y0 /e, are listed in Table 4.2. Table 4.2. Typical Values of y0 /E, Angle of Internal Friction, cf> (I)

v = 0.25

v = 0.33

v = 0.45

v = 0.5

(2)

(3)

(4)

(5)

25 30 35 45

0.9317 !.00 !.054 !.097

1.014 1.142 1.183 1.2!6

1.35 1.38 1.398 1.41

!.5 1.5 1.5 1.5

Source: Prakash and Puri, 1981.

111

FACTORS AFFECTING DYNAMIC SHEAR MODULUS

It was found that for most soils the time-dependent behavior at low strain levels can be characterized by an initial phase when a modulus changes rapidly with time. This is followed by a second phase when the modulus increases almost linearly with the logarithm of the time (Fig. 4.12). For the most part, the initial phase results from the void ratio changes and iucrease in effective confinement during primary consolidation. The second phase, in which the modulus increases almost linearly with the logarithm of time, is believed to result largely from any decrease in void ratio and changes in the soil structure due to a strengthening of the physico-chemical bonds in the case of cohesive soils and an increase in particle contact for cohesionless soils. This phase proceeds at a constant confining stress and is referred to as the long-term time effect. The long-term effect represents the increase in the modulus with time that occurs after primary consolidation is completed. Two methods are used to describe the long-term time effects. The long-term effect is expressed in an absolute sense as a coefficient of shear modulus increase with time, 10 . That is,

( 4.21) in which t 1' t2 are the times after primary consolidation, and !:J.G is the . change in low-amplitude shear modulus from t 1 to t 2 (Fig. 4.12). Numerically, I a equal the value of G for one logarithmic cycle of time. The long-term effect is also expressed in relative terms by the normalzzed shear modulus increase with time, Na· That is,

For values of and v in the range of interest, it is reasonable to assume that Yo= e,.

Time Effects. The effect of duration of confinement at a constant pressure on the magnitude of shear moduli was first reported by Richart (1961). It was observed later that when specimens were confined at a constant confining pressure, shear moduli measured at shearing strain amplitudes below 0.001 ,percent (commonly referred to as low-amplitude moduli at a strain of 10- or less) increased with the time of the specimen's confinement. These studies showed that the shear moduli of artificially prepared soil specimens indeed increase with the length of time a specimen is confined. More recently, sustained-pressure studies on undisturbed specimens of sands and clays have shown that this time-dependent behavior is also characteristic of natural soils (Anderson and- Stokoe, 1977). The time dependence of moduli has significant implications. In the first place, it means that duration of confinement at a constant confining pressure must be considered when performing laboratory tests. Also, if the laboratory values of tbe moduli of a given soil are to be compared, they should be compared after equal confinement times for similar drainage conditions, and these times should be equal to or greater than the time of primary consolidation.

( 4.22)

~

~ ~

1

u

-

/';G

G- LOG 10 (t 2tt 1l ao "' CONSTANT y < J0-3%

.,; ~ ~

c~ c ~

oc

~

w

~

~

w

c

~

:::; ~

2

~

"'

PRIMARY CONSOLIDATION--+- LONG-TERM TIME t 1000 EFFECT

c

~

0

10'

102

103

t1

10 4

DURATION OF CONFINEMENT (LOG SCALE) Figure 4.12. Phases of rnodulus4irne response. (After Anderson and Stokoe, 1977, copyright ASTM. Reprinted with permission.)

"'

Table 4.3. Typical values of /G and NG

Specimen Type

(kN/m')'

Low-Amplitude Shear Modulus 2 G 1000 (kN lm Y

Vacuum extruded Compacted by raining and tamping

200 to 300 70 to 280

140000 to 190000 50 000 to 180 000

Kaolinite Bentonite

Vacuum extruded

70 to 550

4000 to 170 000

Agsco sand Ottawa sand

Compacted by raining and tamping

70 to 280

50000 to 110000

Confining Pressure

Soil Type

EPK kaolinite Ottawa sand Quartz sand Quartz silt

Typical

r

G

Typical

N'G

(kNim')'

(%)

24 000 to 35 000 1400 to 5500

17 to 18 1 to 11

1000 to 8500

5 to 25

Marcuson and Wahls (1972)

2000 to 10 000

1 to 17

Afifi and Richart (1973)

Reference Hardin and Black (1968) Afifi and Woods (1971)

Dry clay

Air-dried EPK Kaolinite

Saturated EPK Kaolinite

Vacuum extruded

~~~",-,.~~~

i''""~"'c''S,--- --'-

Undisturbed a

70 to 220

SO 000 to 2600 000

2000 to 22 900

1 to 14

Silty sand Sandy silt Clayey silt Shale Boston blue clay 9 Clays 1 Silt Clay fills Decomposed marine limestone San Francisco Bay mud Dense silty sand Stiff OC' clay

Undisturbed a Undisturbedd

70 to 700 35 to 415

32 500 to 54 000 13 000 to 235 000

=7000 26 OOO!'to 23 500

15 to 18 2 to 40

Undisturbedd Undisturbedd

35 to 70 325 to 830

50 000 to 200 000 365 000 to 1300 000

4200 to 15 000 28 000 to 102 000

7 to 14 3 to 4

Stokoe and Abdel-razzak (1975) Yang and Hatheway (1976)

17 to 550 220 to 620 1280 to 1300

7600 to 150 000 45 000 to 180 000 300 000 to, 320 000

725 to 32 000 5000 to 17 000 14 000 to 26 000

8 to 22 4 to 10 4 to 8

Lodde (1977) Fugro, Inc, (1977) Fugro, Inc, (1977)

#

ltJndisturbedd Undisturbedd Undisturbedd

Source: Anderson and Stokoe, 1977, copyright ASTM. Reprinted with permission. "I a defined by Eq, (01), "NG defined by Eq, (422),

1 kN/m 2 = 0.145 psi. Nominally undisturbed. e Overconsolidated.

c

d

-"'

-~~,""""'""'''

;;1, 'i

Stokoe and Richart (1973a, b)

Trudeau et al. (1974) Anderson and Woods (1975, 1976)

115

FACTORS AFFECTING DYNAMIC SHEAR MODULUS

114

DYNAMIC SOIL PROPERTIES

in which G 1000 is the shear modulus measured after 1000 minutes of constant confining pressure (after completion of the primary consolidation). The duration of primary consolidation and the magnitude of the longterm time effect vary with such factors as soil type, initial void ratio, undrained shearing strength, confining pressure, and stress history. Figure 4.13 shows typical time-dependent modulus responses for different soils. Typical values of Ic and Nc are given in Table 4.3. The shape of tbe low-amplitude modulus-time graph at a constant confining pressure depends primarily on whether the soil is predominantly fine-grained (silts and clays) or coarse-grained (sands). Figure 4.14a illustrates typical changes in the shear modulus of a clay with time at a constant confining pressure. The two distinct phases of modulustime response are clear from this figure. First, during primary consolidation, values of the shear modulus are initially constant. They then increase rapidly and finally begin to level off. Second, during the long-term time effect, values of the modulus increase linearly with the logarithm of time. Figure 4.14b shows the vertical height change of the clay specimen during the constant-pressure confinement. By comparing the height change results with the modulus-time graph, it can be seen that the end of the initial phase in the modulus-time graph coincides with the end of primary consolidation. The point of transition in the modulus-time graph for this loading sequence is defined as the end of primary consolidation. Therefore, modulus values determined at a time before the end of the primary consolidation is complete are at an effective stress less than that assumed, because excess pore-water pressures still exist in the specimen. During the long-term time effect phase of modulus response, the shear

6G DUE TO CHANGE IN VOID RATIO

iG

,, - - - ·151

Gtooo

--+-

lONG-TERM TIME EFFECT

0,5

~

4

'' 8

L~.::.:..:..:_.:.,-.__

I

101

'

___._,__

(b) HEIGHT CHANGE

102

-:'::,---~

10 3

10

2.0 4

tO-WEEKS

DURATION OF CONFINEMENT, t (MIN) NG

SOl LS

at constant confining F•rgure 4 .14 · Typical modulus and height changes with time for day · d ·h · · )

<•>

NC CLAYS

OC CLAYS CLEAN SANOS

pressure. (After Anderson and Stokoe, 1977, copyright ASTM. Reprmte wrt permrssron.

5-20 3-10 .c

1-3

'

:' SANDS :

OL-----~----~----~----~-----' to 1 to2 toJ to4 to5

DURATION OF CONFINEMENT (MIN)

Figure 4.13. Effect of confinement time on shear modulus. (After Anderson and Stokoe, 1977, copyright ASTM. Reprinted with permission.)

modulus increases almost linearly with the logarithm of time. In Figure 4.14a, this increase was monitored for 10,000 minutes or about o~e week (Anderson and Stokoe,_ }.977). The coefficieJ~t of shear modul~s mcrease ith time I is aboui~900 psi (6200 kN I m ) , and the normalized shear W ' G' modulus increase with time Nc is about 15 percent. The value of I generally increases as the confining pressure increases. Values of N de~rease with increasing undrained shearing strength and increase with increasing void ratio for fine-grained soils (Anderson and Woods, 1976). Stress history also affects values of I 0 and Nc· Figure 4.15 shows this effect for a series of modulus-time tests conducted on one specimen. Values of the modulus were determined over approximately a

116

DYNAMIC SOIL PROPERTIES

12~-------,--------r--------r--------,

..

FACTORS AFFECTING DYNAMIC SHEAR MODULUS

117

28

AIR-DRY OTTAWA SAND 130- 50)

X

<

"'

w

.,;

y

0-

"'~ ~ ~ < 0 w z %<

w

~

190

4

~

~

10-3,

~

ON

26

c,. ,.,

JD-

~ 61000

180

~

~

"'<,l.

~

co

i::'J. G DUE TO CHANGE IN YOlO RATIO

25

Is

0

6G

N lo G .. G1000 ~"

.. LOG 10 (t 21t 1)

~

~X ~~ ~ ~

2

<

170

24 1

101

102

DURATION OF CONFINEMENT.

o,'--------~~--------~--------~.-------_j 1 1 10 10 2 10-4 103 DURATION OF CONFINEIIENT, t (MIN)

"''

0

~

"' 1725 KN/12

DETROIT CLAY

%z w~

~

::; ~

~~ We ~<

0 0% ~

"'

w

.,;

e0 = 0.5

27

~ ~ ~

0

~

X

<

0 0 .. 207 KN/111 2

'"'

t (min)

'"'

Figure 4.16. Typical modulus change with time for sand. (After Lodde, 1977; Anderson and Stokoe, 1977, cbpyright ASTM. Reprinted;;,~~th permission.)

Fi~ure 4.15. Effect _of stress history on shear modulus~time relationship. (After Anderson and Stokoe, 1977, copynght ASTM. Reprinted with permission.)

one-week period of confinement at each pressure in the following sequence· 17, 34, 60, 34, and 17psi (117, 235, 414, 235, and 117kN/m 2 ). It can b~ observed that m the overconsolidated state values of I and N d d · · ' a 0 were re uce relative to value of Ia and N 0 in the normally consolidated state. A typiCal modulus-time graph of a coarse-grained soil is shown in Fig. 4.16. It can be se~n that the shape of the modulus-time response for the ~ohes10nless soil differs significantly from that of the fine-grained soil shown m Fig. 4.14. For the sand, the primary consolidation phase is not evident Rather, the long-term time effect had begun by the time the first mea: sur~ment was made. The long-term time effect is, however, similar to that which occurs for clays. In Fig. 4.16, the linear increase in modulus with the logarithm of time was momtored for ~bout 10,000 min. Values of I a and Na for this sand are 250 psi (1725 kN /m ) and 1.0 percent, respectively. These values are much smaller than those shown for the clays in Figur<> 4.14a. Moduli m~asured -~t sheari ng strain amplitudes between 0.001 and 0.1 3 percent (stram of 10 to also increase with time. The results of a number of tests show that long-term modulus increases occur at low to mtermediate strain levels (0.001 to 0.1 percent) for stiffer clays (Lodde, 1977). Preliminary results from long-term, high-amplitude modulus tests on sand seem to indicate that long-term modulus increases occur m clean, dry sands at strain amplitudes up to 0.1 percent as well.

w- )

Because of the general similarity between the increase in moduli with time at low and high-shearing strain amplitudes, it seems reasonable to conclude that many of the factors that affect the low-amplitude modulus time response also affect the high-amplitude modulus-time response (at the start of high-amplitude cycling) (Anderson and Stokoe, 1977). Anderson and Stokoe (1977) also proposed a method, which can be used to predict the in situ shear moduli from laboratory tests after allowing for time effects.

Degree of Saturation. Biot (1956) showed that the presence of fluid exerts an important influence on the dilatational (longitudinal) wave velocity but produces only a minor effect on the shear wave velocity. The fluid affects the shear wave velocity only by adding to the mass of the particles in motion. A study of the influence of degree of saturation on the shear wave velocity for a sample of Ottawa sand shows that much of the difference between the values for the dry and saturated conditions can be accounted for by the effect of the weight of water. Therefore, it is sufficient for an evaluation of V or G fi)J cohesionless soils to consider the in situ unit weight ·' . and the effective pressure. Frequency and Number of Cycles of Dynamic Load. The effect of number of cycles and frequency of dynamic loading was investigated by Hardin and Black (1969). For number of cycles between 1 and 100, the dynamic shear modulus of dry sands was observed to increase slightly with number of cycles whereas for cohesive soils a decrease in modulus with number of cycles was observed. Low strain shear modulus was found to be practically unaffected by the frequency of loading.

119

EQUIVALENT SOIL SPRINGS DYNAMIC SOIL PROPERTIES

118

Magnitude of Dynamic Load. The magnitude of dynamic load controls the shear strain levels induced in the soil and hence the dynamic shear modulus should be expected to decrease with an increase in the dynamic load. Dynamic Prestrain. The effect of strain history on the dynamic shear modulus of sands was investigated by Drnevich, Hall, and Richart (1967) using torsional-vibration-type resonant column equipment. The soil samples were first subjected to large amplitude vibrations (dynamic prestrain) for a predetermined number of cycles and then the low-amplitude vibration modulus was determined. It was observed that the value of the dynamic shear modulus generally increased with the number of prestrain cycles, as shown in Fig. 4.17. No data is available of the effect of dynamic prestrain on the dynamic shear modulus of clays and silts.

pressure under working conditions in the field. The. sample is then subjected to an axial stress equal to the anticipated statlc stress under workmg conditions. Positive and negative values of a small mcrement . of load corresponding to the loading levels in the .field should then be apphed (Fig. 4 18) It will be seen that initially the stram mcreases after each apphcat10n . t· le But after 6 to 10 cycles the additional axial stram generally of s ress eye . ' . d' 1 · becomes negligible, and a closed loop on the loadmg-:-unloa mg eye e !S obtained. The value of the modulus may then be determmed from the slope of the line "aa" by using Eq. ( 4.23): change in unit stress Modulus= corresponding change in unit deformation

The soil behaves as an elastic material in the sense that there is a reduction in deformation when the stress is removed, but the matenal absorbs energy and hence provides material damping. . The load-deformation behavior of soils may also be represented m ter~s of equivalent ;~oil springs which_ar_~ essentially, related to th~ Young s modulus E, tlie shear modulus c;, ·and Pmsson s rat10. v. Thts concept considerably simplifies the solution of many problems a?d ts commonly used ' in analyzing the machine foundation problem and ts d~scussed belo':. The spring constant k is defined as the load per umt deflectiOn, 1.e., p

(4.24a)

k = LI.Z

ao o-0 "" 1188 psf 'Yox = 1.6 X lQ-4

=

(4.23)

612 psf

'YOx = 1.6 X

w- 4

Cyc!es of high-amplitude torsional vibration

Figure 4.17.

Effect of number of cycles of high-amplitude vibration on the shear modulus at

I I

low amplitude (C-190 Ottawa Sand, e0 =0.46, Hollow Cylindrical Specimens). (Drnevich, Hall, and Richart, 1967. © 1968 The University of New Mexico Press.)

"'

For analysis of the dynamic behavior of foundations-soil system, the soil is usually represented by equivalent springs. This concept will now be explained.

4.4

Static working stress

EQUIVALENT SOIL SPRINGS Axial strain

By way of illustration, let a triaxial test be performed on a sample of soil such that the confining pressure is as near as possible to the confining

Figure 4.18.

Ax·,·al stress vs. axial strain in a triaxial test using repeated static loading.

120

DYNAMIC SOIL PROPERTIES EQUIVALENT SOIL SPRINGS

in which P is the load on the spring (Fig. 4.19b) and I!.Z is the change in length of the spring. Next, consider a plate of area A and thickness t that is subjected to a pressure of umform intensity, p (Fig. 4.l9c). The change in the thickness of the plate, l!.t, is given by ( 4.24b) Because the total load P is pA, the above equation may be rewritten as

or P EA -=-=k

l!.t

may undergo vertical oscillations, horizontal translation only, rocking only, and yawing (torsion about the vertical axis). The corresponding conditions induced in the soil are (1) uniform compression, (2) uniform shear, (3) nonuniform compression, and (4) nonuniform shear. Therefore, the soil constant, characterizing the soil reaction below the foundation block and the corresponding elastic deformation are different in each case. The value of the equivalent soil spring for any mode of vibration may be computed from the dynamic shear modulus or Young's modulus, Poisson's ratio, and the geometry of the foundation contact area as explained in Chapter 6 (Sections 6.4 and 6.7). Barkan (1962) defined a set of soil coefficients to represent the soil reaction for any particular mode of vibration. These soil coefficients are defined below: 1. Coefficient of Elastic Uniform Compression (C,J:

(4.25)

t

in which P is the total load and k the spring constant. It ts explamed m Chapter 6 (Section 6.2) that a rigid machine foundation

121

c" =

uniform compression ( p) elastic settlement (s")

( 4.26)

';o,- '7

From the definition, the spring constant k, is then

k'

=

load elastic deformation

Therefore,

(4.27) in which A is the area of the test plate or the foundation base. 2. Coefficient of Elastic Uniform Shear ( CJ:

(a)

uniform shear

(b)

C,

=

Tav

elastic shear displacements,

( 4.28a)

p

As for the spring constant k, the corresponding spring constant kx is A

(4.28b)

~~~~~~~~

+ r'========-----V'; T (c)

Figure 4.19.

Co~cept of spring constant in soil. (a) Unloaded spring· (b) static deflecti

{c) plate as a sprrng.

'

A .

on z,

3 & 4. Coefficient of Elastic Nonuniform Compression ( C1 ) and Coefficient of Elastic Nonuniform Shear ( C",). Because the elastic deformation is not uniform over the base of the block as in cases 1 and 2, no simple definitions for C and C", can be given. However, definitions in terms of mathematical quantities and overall displacements (rotations) of the block and its geometry are given in Chapter 6 (Section 6.7). The value of C, can be determined from Eq. (4.29) (Barkan, 1962)

122 DYNAMIC SOIL PROPERTIES

c

=

u

1.13£ _1_ I - vz

VA

( 4.29)

LABORATORY METHODS

123

ultrasonic pulse test, (3) cyclic simple shear test, ( 4) cyclic torsional simple shear test, and (5) cyclic triaxial compression test.

in which 4.5.1

E =Young's modulus and

A = Area of the footing v =Poisson's ratio

The value of C" varies inversely with the square root of the f h alrea o contact of the foundation with the soil. Thus if C and C c ffi · t f 1 · . ' "' u2 are t eva ues of oe helen s o e asttc umform compression corresponding to areas A and A 2 , t en 1 ( 4.30) However, for areas greater than 10m 2, A, is taken as !Om' only. This is duhe to the fact that for large areas, soil rigidity and area effects cancel each ot er. The values of the different soil coefficients are approximately related to each other as g1ven m Eq. (4.31) (Barkan, 1962)

c,- = ! cu

( 4.3!a)

C 4, = 2Cu

( 4.3!b)

C1,=0.75C"

( 4.3lc)

and

Th~ vaflue of the ratio ratio o 2,

c.;c" depends upon the shape of area and for (alb) Ci> =!.73Cu

( 4.3ld)

may be used in calculation. Also these coefficients will vary with area as

does.

c u

It may be pointed out that the equivalent soil springs essentially depend ohn ttheffvaluEes of E and G. Therefore, they will be influenced by all factors t a a ect and G. ···

4.5

LABORATORY METHODS

The following laborato~y methods are used to determine the dynamic elasl!c constants and dampmg values of soils: (!)resonant column test, (2)

Resonant Column Test

The resonant column test for determining the modulus and damping characteristics of soils is based on the theory of wave propagation in prismatic rods (Richart et a!., 1970). Either compression waves or shear waves can be propagated through the soil specimen from which either the Young's modulus or shear modulus can be determined. Such a device consists essentially of a coil-magnet drive system, an accelerometer or velocity transducer to monitor the motion of the drive system, a linear variable differential transducer (LVDT) or other type of displacement transducer to detect a change in the vertical height of the soil specimen, and a confining chamber. In the test setup, the coil-magnet drive system is attached to a top cap, which is seated on a membrane-encased, cylindrical soil specimen. The soil specimen can,be either hollow or sol~d, depending on the capabilities of the particular test" device. The bast'''"pedestal, upon which the specimen is placed, is connected to a drainage line. Filter paper strips may be used along the length of the specimen to accelerate specimen consolidation. The top cap and bottom pedestal are usually serrated or roughened in some manner to assure good mechanical coupling between the soil and equipment. The system is generally set up so that only a hydrostatic confining pressure can be applied, although anisotropic loading conditions can be simulated in some devices. An electrical system is employed to operate and monitor. the resonantcolumn equipment. To obtain accurate shear wave velocity measurements, it is necessary to use electronic equipment (Fig. 4.20). In such a system, a signal generator supplies a sinusoidal voltage to the coils in the coil-magnet drive system. The magnetic field induced by the current in the coils interacts with the magnetic field from the permanent magnet, thereby resulting in a torsional oscillation of the drive cap and specimen. By varying the frequency of the input signal, the amplitude of vibration can be varied. An accelerometer (or velocity transducer) located on the top cap generates a voltage proportional to the amplitude of vibration of the soil-top-cap system. This signal is c~nditioned and then displayed on an oscilloscope. The amplitude and freqlency of the signal at resonance are monitored. An LVDT is used to monitor the changes in specimen height. The purpose of the test is to vibrate the soil-top-cap system at the first-mode resonance at which the material in a cross section at every elevation vibrates in phase with the top of the specimen. The shear wave velocity and shear modulus are then determined on the basis of system constants and the sizes, shape, and weight of the soil specimen (Drnevich, 1977; Drnevich et al., 1977).

DYNAMIC SOIL PROPERTIES

124

125

LABORATORY METHODS

r::::=~;t7'0(1,t)

SINE-lAVE

r--------1

VOLTitiE.TER

SINE-WAVE

GENERATOR

GENERATOR

X

X

O(l,tl

AHD AMPLIFIER

FREQUENCY COUNTER

f----.--j

J

(a) J/Jo =

{b) J!Jo = 0.5

00

"-

.J

Weightless spring

Dnvmg force '--""'' '--~-j

CHARGE AMPLIFIER

~~O~,

OSCILLOSCOPE

Rigid mass

Dashpot

~I'

ORAl NAGE

Figure 4.20. Typical electronics for resonantMcolumn device.

The resonant column technique was first applied to testing soils by the Japanese engineers Ishimoto and !ida (1937) and Iida (1938, 1940). About 20 years later, Shannon et al. (1959) and Wilson and Dietrich (1960) described new applications of the resonant column principle (Woods, 1978). It is possible to develop several versions of the resonant column test by using different end conditions to constrain the specimen. Some common end conditions are shown schematically in Fig. 4.21. Each configuration requires a slightly different type of driving equipment and methods of data interpretation. The fixed-free apparatus is the simplest configuration in terms of equipment and interpretation. Apparatus of this type was described by Hall and Richart (1963). In the fixed-free apparatus shown in Fig. 4.21a, the distribution of angular rotation 0 along the specimen is ~ sine wave, but by adding a mass with mass polar moment, 10 at the top of the specimen as in Fig. 4.21b, the variation of 0 along the sample becomes nearly linear. Later models of the fixed-free device (Drnevich, 1967) take advantage of endmass effects to obtain uniform strain distribution throughout the length of the specimen. The apparatus in Fig. 4.21d has a fixed base and a top cap that is partially restrained by a spring, which in turn reacts against an inertial mass. The configuration of the apparatus shown in Fig. 4.21c can be described as the spring-base model. The apparatus of the Shannon-Wilson device is of this type, if the spring is considered stiff compared to the specimen's stiffness. For a condition in which the spring is weak compared to the specimen, the

Specimen, nonrigid distributed mass

Rigid mass

weightless spring

,J;

0/'Fixed

Orivingfo* (c)

(d)

tic of resonant column end conditions (After Woods 1978.) Figure 4.21. Schema

. n of Fl·g 4 21c could be called free-free. In such a case, a node confi gurat10 · · · d' 'b t' o ld . 'dh . ht of the specimen and the rotatwn lstn u IOU w u will occur at m1 e1g ' . . · · 1 be 1 • ve By adding end masses, the rotatwn d1stnbut10n can a so . . · b 1 d by a b e a - sme wa . ' I ]' For K = 1 0 tests the mertml mass IS a ance made near y mear. o · ' . · 11 d be counter weight, but if !!>he changes the counterweight, an axm oa can nt applied to the specimen. . · Th hearing strain on a circular cross section Ill a torswna1 resona colun:'n ~est varies from zero at the center to a maximum at the outer edge. To stud the influence of shearing strain amplitude on she~r modulus and . y D . h (1967 1972) developed the hollow cyhnder apparatus dhampm_g, F" rne4v2IC2 the co,nfiguration of which is similar to the schematic of · l t' · not s own m Ig. · • . 4 21b The average shearing strain on any honzonta cross s~c wn IS. . F Ig. · · · · d the sheanng stram IS greatly different from the maximum or mtmmum, an

127

LABORATORY METHODS DYNAMIC SOIL PROPERTIES

126

Permanant

Soil spec'! men

Top cap

No. 28 gauge

winding wire

(b)

W

AA -:+-~· . • f t cOfutfin adapted for attenuation measurements. ( er Figure 4.23. s·chematrcs o resonan ,, -Woods 1978.)

Bottom cap

Figure 4.22. Hollow specimen resonant column and torsional shear apparatus. (After Drnevich, 1972.)

. (W d 1978) but the technique still needs to be oo s, ' shear wave attenuation perfected. .b d calibration procedure and aids for that propagates either a rod Drnevich et a!. (1977) descn e a t reducing data derived from an appara u~ compression wave or a shear wave or bot . 4 .5.2

uniform along the height of the specimen. Drnevich also increased the torque capacity of his device to produce "large" shearing strain amplitudes. Anderson (1974) used a modified "Drnevich" apparatus to test clays at shearing strain amplitudes up to 1 percent. Woods (1978) tested dense sands on the same device at shearing strain amplitudes up to 0.5 percent at 40 psi (276 kN 1m 2 ) confining pressure. Lord et a]. (1976) suggested a technique to measure. attenuation as a function of frequency in the resonant column apparatus. In the apparatus, two straight lengths of coil winding wire are embedded a vertical distance d apart in a soil specimen (Fig. 4.23a), which is excited longitudinally by short pulses of different carrier frequencies. As the waves travel past the wires, the wires move with the velocity of their neighboring soil particles. Because the wires are in a magnetic field, they generate a voltage proportional to their Velocities. The voltage is measured at two points a known distance

apart making it possible for the operator to evaluate the coefficient of attenuation. A similar concept (Fig. 4.23b) was developed to determine

Ultrasonic Pulse Test

. . enerate and receive ultrasomc waves m Piezoelectric crystals are used ~~Juce either compression or shear waves. soils, and some of them can p th t includes a pulse generator, an Stephenson (1977) described a s:J~~s (t:ansmitter and receiver). The pulse oscilloscope, ~nd two ultraso~iCt direct current pulse to the transmittmg enerator dehvers a vanable vo tage, . ulse to the time base of the g d . 1 •th a 7-V tngger P probe simultaneous y WI h dueled tests on unconfine . oscilloscope. With t~\s . apparatus, e con specimens of cohesive 'soils. . h . e is that it is difficult to identify and One of the drawbacks of this tee mqu More importantly the strain · a1 f es of the waves. ' . h interpret exact arnv lm . d with this pulse technique, are only m t e f the technique is that tests can be amplitudes, which can be achieve . -n. ·mary advantage o . · very low region. "'e pn d. t while they are still retamed m a ft seafloor se Imen s . l f performed on very so . h . e is not currently used routme y or core liner (Woods, 1978). This tee mqu the measurement of soil properlies.

llH

4.5.3

DYNAMIC SOil PROPERTIES

Cyclic Simple Shear Test

A soil element at xx, as indicated in Fig. 4.24, may be considered to be subjected to a series of cyclic shear stresses, which may reverse many times during the life of a machine foundation. In the case of a horizontal ground surface, there are no shear stresses on the horizontal plane before the foundation of a machine is installed. However, shear stresses are introduced when soil is excavated for a foundation and when the machine is installed. Normally, static stresses are also not constant at the base of an embedded foundation. Then, when the machine is operated, cyclic shear stresses are introduced. Thus, the actual problem involved with machine foundations is that there are normal as well as initial shear stresses that act on the horizontal plane. Oscillatory shear stresses are also introduced. With a surface footing, there are no initial shear stresses on the soil at the base of the foundation, but these stresses are present on all other horizontal planes below the ground surface. These conditions are different for earthquake loading, because the normal stresses on the plane remain constant while cyclic shear stresses are induced during the period of shaking. A simple shear device simulates all these loadings and consists of a sample box, an arrangement for applying a cyclic load to the soil, ·and an electronic recording system. The Roscoe (1953) device has a box for a square-shaped sample with side lengths of 6 em and a thickness of about 2 em. This box is provided with two fixed side walls and two hinged end walls so that the sample may be subjected to deformations of the type shown in Fig. 4.25. The schematic diagram in Fig. 4.25 illustrates how the end walls rotate simultaneously at the ends of the shearing chamber to deform the soil uniformly (Peacock and Seed, 1968). Kjellman (1951), Hvorslev and Kaufman (1952), Bjerrum and Landra (1966), and Prakash et al. (1973) have described this type of apparatus. Typical shear-stress-shear-strain relationships obtained during cyclic simple shear tests are shown in Fig. 4.26a. A soil exhibits nonlinear stress-strain characteristics from the very beginning of the loading cycle. For purposes of high stress-high strain loading such as that due to an earthquake analysis, this behavior can be represented by a bilinear model, as shown in Fig. 4.26b (Thiers and Seed, 1968). This bilinear model is defined by three parameters: a

a

t

t

\ \((0\'0000\\ _, Figure 4.24.

earthquake.

X

L

I v..-:;

II I

111111 A

I W//,

ru

r

l

D

'lll

Soil sample

Shearing chamber Plan view

I

I

Soil deformation Elevation

Figure 4 . 25 . Schematic diagram illustrating rotation of hinged end plates and soil deformation in oscillatory simple shear. (After Peacock and Seed, 1968.)

Shear stress

•

_L X

Shear strain

/;?5777747/ H

Idealized stress condition for an element of soil below ground surface during an

(a)

(b)

Figure 4.2&. (a) Stress-strain curve of a soil. (b) Bilinear model (After Thiers and Seed, 19&8.) 129

DYNAMIC SOIL PROPERTIES

130

(1) modulus G, until a limiting strain 'Y, is reached, (2) modulus G 2 beyond strain 'Yy, and (3) strain 'Yy· If the direction of strain is reversed, the behavior can again he determined hy using modulus G 1 until a strain change of 2y, is developed, and the modulus G 2 again controls the behavior. This pattern is continued throughout the cycle. Typical stress-strain plots taken from simple shear tests of San Francisco Bay mud were made from the records of deformation and loads versus time data for different cycles of loading. Figure 4.27 shows such plots for cycle 1, cycle 50, and cycle 200, with about 4 percent shearing strain. The decrease in peak load as the number of cycles increases is reflected by the progressive flattening of the stress-strain curves. Similar tests were performed at different peak strains, and plots of the dynamic moduli G 1 and G 2 versus peak strains are shown in Fig. 4.28. However, in machine foundations, the magnitude of dynamic (oscillatory) load is small compared to the static load (Fig. 4.18). Hence, the stress-strain loops stabilizes after 6 to 10 cycles, and no further irrecoverable deformations occur. Consequently, the modulus corresponding to that condition is adopted in machine foundation design. However, corrections for confining pressure and other factors need to be applied, as described in Sections 4.3 and 4.7. A major drawback of most of the cyclic simple shear apparatus is that they do not permit measurement or control of lateral confining pressures during cyclic loading; therefore, it is impossible to investigate in detail the effects that K0 -consolidation has on the behavior of a given soil.

40r-----.--,------,----,

"

~ 20~~~~~----+-------r------1

'0

E u

E

i

r----~~~::;;r~~~~~~~

(a) Dynamic modulus G 1 ~,·

1u

"'

20

~

'

".2" 0 "0

10

0

E u

'[' ro

>-

0

Shear

10

Peak strain,%

c

stress

131

LABORATORY METHODS

00 Peak strain,% (b) Dynamic modulus G 2

(kgfcm 2)

Figure 4.28. Effect of cyclic loading on dynamic moduli. (a) Dynamic modulus G,. (b) Dynamic modulus G2 • (After Thiers and Seed, 1968.)

4.5.4 Cyclic Torsional Simple Shear Test

(a) Cycle 1

(b) Cycle 50

(c) Cycle 200

Figure 4.27. Stress-strain curves and bilinear models for San Francisco Bay mud. (a) Cycle no. 1. (b) Cycle no. 50. (c) Cycle no. 200. (After Thiers and Seed, 1968.)

In an attempt to provide the capability of measuring confining pressur~ and controlling K 0 conditions, several investigators have developed torswnal simple shear devices. ]$ihara and Li (1972) modified a tn.axml apparatus to provide torsional straining capabilities. Although this device permits lateral stress control it has a distinct disadvantage in that the sheanng strams of a specimen bei~g tested can range from zero at the center of the specimen t? a maximum at the outside radius (Woods, 1978). Other mvestigators mcludmg Hardin (1971), Drnevich (1972), Yoshimi and Oh-Oka (1973), Ishibashi and Sherif (1974), Ishihara and Yasuda (1975), Cho et al. (1976), and Iwasaki et a!. (1977) developed hollow cylinder torsional shear apparatus. The devices developed by Yoshimi and Oh-Oka, Ishibashi and Sherif, and Cho et al. are

132

lABORATORY METHODS

DYNAMIC SOil PROPERTIES

4.5.5

based on the same concept but use much shorter specimens, and the height at the outside and inside diameters can be varied. . The apparatus (Fig. 4.22) designed by Drnevich (1972) has an advantage m that both resonant column and cyclic torsional shear tests can be performed in the same device. Ishihara and Yasuda (1975) also used a long, hollow, cyhndncal sample configuration as did Iwasaki et al. (1977). It is difficult to prepare specimens for the long, hollow, cylindrical devices, and, obviously, "undisturbed," cohesionless soils cannot be tested in this device. However, Woods (1978) was able to trim hollow, cylindrical specimens of many different types of soils. The cyclic torsional simple shear devices of Yoshimi and Oh-Oka Ishib~shi and Sherif, and Cho et al. were designed to generate uniforO:: sheanng strams throughout the wall of a cylindrical specimen whose bottom or top surface was tapered (Fig. 4.29) so that the taper would be exactly proportional to the inside and outside radii of the hollow cylinder. "Undisturbed" sp~cimens can be formed more easily for the size and configuration of this device than for the devices which require taller hollow cylindrical specimens. Although the initial shearing stress conditions in a short hollow cylindrical specimen were described as uniform by Ishibashi and Sherif (1974), Ladd and Silver (1975) noted that, because the shearing stresses on each boundary are not the same, all initial stresses cannot be uniform throughout a sample.

133

Cyclic Triaxial Compression Test

Cyclic triaxial tests have been extensively used to study the stress-deformation behavior of saturated sands and silts (Puri, 1984; Seed et al., 1986). Also, Young's modulus E and the damping ratio ' have often been measured in cyclic triaxial tests (Fig. 4.30) when strain-controlled tests have been conducted. These tests are performed in essentially the same manner as the stress-controlled tests for liquefaction studies. A servo-system is used to apply cycles of controlled deformation. Young's modulus is determined from the ratio of the applied axial stress to axial strain, and the shear modulus is computed from Eq. ( 4.4). As in all laboratory attempts to duplicate dynamic field conditions, cyclic triaxial tests have limitations, among which are the following: 1. Shearing strain measurements below 10- 2 percent are generally difficult. 2. The extension and compression phases of each cycle produce different results.(Annaki and Lee, 1977); therefore, the hysteresis loops are not symmt!'fric in strain-controlleji tests, and samples tend to neck in stress-controlled tests. '· 3. Void ratio redistribution occurs within the specimen during cyclic testing (Castro and Poulos, 1977).

Vertical stress

---X

!t

A

Compression

l

~ Boundaries indicated by heavy lines

I

~//~>-~

0H 0H~u,

""

•·

aH

I

IJH

"" '"

~

Figure 4.29. Specimen, cross section and initial stress distribution in a hollow cylindrical sample with tapered ends. (After ladd and Silver, 1975.)

I D _ _!___ Area of Hysteresis Loop - 21r Area of Triangle OAB & OA'B'

Figure 4.30. Equivalent hysteretic stress-strain properties from cyclic triaxial test. (After Silver and Park, 1975.)

134

DYNAMIC SOIL PROPERTIES 135

FIELD METHODS

4. Stress concentrations occur at the cap and base of the specimen being tested. 5. The principal stress changes direction by 90° during the test. Void ratio distribution is common to all cyclic shear tests, whereas the other limitations are related mostly to the cyclic triaxial test. It is generally agreed that (Seed et al.. 1978; Annaki and Lee, 1977) for an earthquake-type excitation of soils, cyclic simple shear or cyclic torsional shear techniques are more appropriate than cyclic triaxial compression. Wolfe et al. (1977) reported on their adaptation of the cyclic testing of a cubical device developed by Ko and Scott (1968). Their results compared favorably with cyclic triaxial compression tests, but problems remained concerning specimen preparation and appropriate testing methods to simulate _field conditions correctly with this apparatus (Woods, 1978). Silver (1981) prepared a table (Table 4.4) indicating the relative quality of each test technique for measuring dynamic soil properties.

4.6

FIELD METHODS

The following methods for determining dynamic properties of soil are in use in different parts of the world. 1. 2. 3. 4. 5. 6. 7. 8.

Cross-borehole wave propagation test Up-hole or down-hole wave propagation test Surface wave propagation test Vertical footing resonance test Horizontal footing resonance test Free vibration test on footings Cyclic plate load test Standard penetration test

Brief descriptions of these tests along with the typical setups and methods of interpretation of data are presented in the following pages. {'''[ .

Table 4.4. Relative Quality of Laboratory Techniques for Measuring Dynamic Soil Properties

4.6.1 Relative Quality of Test Results

Technique (1)

Shear

Young's

Material

Modulus

Modulus

Damping

Effect of No. of Cycles

(2)

(3)

(4)

(5)

Good

Good

Good

Good

Resonant column with adaptation Ultrasonic

pulse Cyclic

Fair Fair

triaxial

Poor

Good

Good

Good

Good

Good

Good

Good

Good

Cyclic torsional shear

Fair

Good

Cyclic simple shear

Attenuation (6)

Source: Silver, 1981.

There are several available field methods with which the dynamic soil properties and damping of soils can be determined. Salient features of these methods will now be described.

'"!;-~.:7

Cross-Borehole Wave Propagation Test

In the cross-borehole method (Stokoe and Woods, 1972), the velocity of wave propagation is measured from one borehole to another. A minimum of two boreholes are required, one for generating an impulse and the other for the sensors. In Fig. 4.31, the impulse rod is struck on top, causing an impulse to travel down the rod to the soil at the bottom of the hole. The shearing between the rod and the soil creates shear waves that travel horizontally through the soil to the vertical motion sensor in the second hole; the time required for a shear wave to traverse this known distance is monitored. There are four important considerations in conducing a cross-borehole shear wave propagation test: (1) the boreholes, (2) the seismic source, (3) the seismic receiver, and (4) the recording and timing equipment. Although a minimum of two boreholes are required, for extensive investigations and for increased accuracy, three or more boreholes should be preferred whenever possible. If boreholes are installed in a straight line, wave velocities can be calculated from the intervals of time required for passage between any >J:,wo boreholes. Thus, the necessity for precisely recording the triggering time is eliminated (Stokoe and Hoar, 1978). In addition, the boreholes must be vertical for the travel distance to be

measured properly. In general, any borehole 10m or deeper should be surveyed with an inclinometer or another logging device for determining its verticality (Woods, 1978). Impulsive sources are more often used, although both impulsive andsteady-state seismic sources are in use. The major criteria for a seismic

137

FIELD METHODS 136

DYNAMIC SOIL PROPERTIES

/

S = Source R = Receiver

(a) Up hole

(b) Down hole

/

Figure 4.32. (a) Up-hole and (b) down-hole techniques for measurement of velocity of wave propagation.

Transducer

Figure 4.31. Sketch showing propagation.

(((((~itcross~bore

Impulse rod Recorder

~~~en

hole technique for measurement of velocity of wave

~1:)"

Trigger geophone

~

\

Wooden

source are: (1) It must be capable of generating predominantly one kind of wave. (2) It must be capable of repeating desired characteristics at a predetermined energy level. Velocity transducers (geophones) that have natural frequencies of 4 to 15 Hz are adequate for detecting (receiving) the shear waves as they arrive from the source. The receivers must he oriented in the shearing mode and should be securely coupled to the sides of the boring. The recording equipment should be able to resolve arrival times of up to 0.2 msec or 5 percent of the travel time. Storage oscilloscopes are also often used ..

plate

Rubber expander

plate

3-component geophone

Figure 4.33.

4.6.2

Back

Equipment and instrumentation for down~hole survey. (After Woods, 1978.)

Up-Hole or Down-Hole Wave Propagation Tests

Up-hole and down-hole tests are performed by using only one borehole. In the up-hole method, the receiver is placed at the surface, and shear waves are generated at different depths within the borehole Fig. 4.32a. In the down-hole method, the excitation is applied at the· surface·, and one or more receivers are placed at different depths within the hole (Fig. 4.32b). Both the up-hole and the down-hole methods give average values of wave velocities for the soil between the excitation and the receiver if one receiver is used, or between the receivers, if more than one is used in the borehole

(Richart, 1977). Figure 4.33 is a schematic diagram of a down-hole survey with all principal elements included (Imai and Yoshimura. 1975).

4.6.3

Surface-Wave Propagation Test

Rayleigh waves and L'llve waves can be used to determine the shear moduli of soils near the surface. The Rayleigh wave (R-wave) (Section 3.3) travels in a zone close to the surface. An electromagnetic or other harmonic vibrator can be used to generate a steady-state R-wave, and the ground surface can be deformed as shown in Fig. 4.34. A mechanical oscillator is usually set to work at approximately 10Hz. One ray is drawn away from the centerline of the oscillator. One of the geophones connected to the horizon-

138

DYNAMIC SOIL PROPERTIES

FIELD METHODS

139

Qo sin wt

(4.34) and G=V:p

f--_,._x

Figure 4.34.

( 4.35)

in which p is the mass density of the soil and v, the Poisson's ratio of the soil. Table 4.5 gives repesentative values for Poisson's ratios that can be used in lieu of the measured values.

Deformed -shape of a half~space surface. (After Woods, 1978.)

tal plates of the oscilloscope is fixed 30 em away from the oscillator along the ray so that the sensmg ax1s of the geophone is vertical. A similar geophone, connected to the vertiCal plates of the oscilloscope, is moved along this ray away from the oscillator. The second geophone is moved until the Lissajous figure on the oscilloscope screen becomes a circle. The two signals are at the same frequency and 90° out of phase. However, if the phase angle is d1ffere~t than 90~, the Lissajous figure is an ellipse, and for a zero phase angle, 1t 1s a strmght hne (Doebelin, 1966). The distance between the two geophones is measured. This distance is then a measure of the wavelength of the generated Rayle1gh wave. The test is repeated at other frequencies. The test can also be conducted by using a phase meter in place of an oscilloscope. In case of uniform soil up. to infinite depths and the Lissajous figure of a cncle, the wave length, A", of the propagating waves is given by (4.32) in which S is the measured distance between geophones. The veloCity of the Rayleigh waves v, is then given by ( 4.33a) or (4.33b) in which f is the frequency of vibration at which the wavelength has been measured. In case the phase angles corresponding to different distances between

g~ophones are recorded, a curve is plotted of th€ phase angles versus the dt~tances. From the ~urve, the distance S between the geophones is deter-

mmed for a phase difference of 90°. The remaining computations remain unaltered as above. Shear wave velocity may then be obtained from Eq (4.~. . The elastic modulus E and the modulus G of the soil medium are calculated as follows:

Table 4.5. Representative Values of Poisson's Ratio

Type of Soil Clay Sand

v

0.5 0.3 -0.35 0.15-0.25

Rock

-

~ .•

--------:c:-;;:"_ _ _ _ _ _ _ _ _ __

c:-.

':£ ',,-

The effective depth of penetration of the R-wave has been empirically related to one-half of Rayleigh's wavelength (Fry, 1963; Ballard, 1964). A procedure for monitoring the shear wave velocities with depth based on an impact of a falling weight and Spectral Analysis of the Surface Wave (SASW) has been the subject of investigation by Nazarian et al., 1983. In the field, two vertical velocity transducers are used as receivers. The receivers are placed securely on the ground surface symmetrically about an imaginary centerline. A transient impulse is transmitted to the soil by means of an appropriate hammer. The range of frequencies over which the receivers should function depends on the site being tested. To sample 50 to 100ft deep materials, the receiver should have a low natural frequency, in the range of 1 to 2Hz. In contrast, for sampling shallow layers, the receivers should be able to respond to high frequencies of 1000Hz or more. Several tests with different receiver spacings are performed. The distance between the receivers after every test is generally doubled. The geophones are always placed symmetrically about the selected, imaginary, centerline. Nazarian et al. (1983,~have shown that use of this setup reduces scatter in data collection due to the fact that the distances covered in the previous tests are always included in the next tests. In addition, at each receiver spacing, two series of experiments are performed. First, the test is carried out from one direction (forward profile) and then without relocating the receivers the same test is performed with the source on the opposite side of the receivers (reverse· profile). By running forward and reverse profiles and by averaging the data of these two tests, the effect of any internal phase shift between receivers is minimized.

140

DYNAMIC SOil PROPERTIES

The raw data obtained from the impact test is reduced with the help of a dynamic signal analyzer (DSA), and the inversion curve is obtained. From the different DSAs that are being commercially manufactured, the DSA supplied by Hewlett Packard (HP 3562A) has been used successfully, A typical shear wave profile for a site in which the velocity profiles have been determined both by !he crosshole method and SASW method show good agreement (Nazarian and Stokoe, 1984). The tally between the values measured by the two methods is strikingly close. The SASW method is economical, and less time consuming than the crossbore hole method and has the advantage of complete automation (Nazarian and Stokoe, 1984; Woods and Stokoe, 1985; Prakash, 1986). However, proper inversion techniques must be used.

4.6.4 Vertical Footing Resonance Test The block (footing) resonance test can be used for determining modulus and damping values. According to IS 5249 a test block 1.5 x 0.75 x 0.70 m high is cast either at the surface or in a pit 4.5 X 2.75 mat a suitable depth (Fig. 4.35a) and is excited in vertical vibrations. Two acceleration or displacement transducers are mounted on top of the block (Fig. 4.35b) such that they

Motor oscillator assembly

Concrete IM 1501 Depth to be

Tied

Ia) 4 50 m 1 mfmin

[§]

i'=;:J[O] 'tr

l ~

,. 2.75 m

1m min

(b)

Figure 4.35.

Setup for a block~resonance test. 141

142

DYNAMIC SOIL PROPERTIES

FIELD METHODS

sense vertical motion of the block. A mechanical oscillator, that works on the principle of eccentric masses mounted on two shafts rotating in opposite directions (Prakash, 1981) is mounted on the block so that it generates purely vertical sinusoidal vibrations. The line of action of the vibrating force passes through the center of gravity of the block. After a suitable dynamic force value is chosen, the oscillator is operated at a constant frequency, and the acceleration of the oscillatory motion of the block is monitored. The oscillator frequency is increased in steps of small values from one cycle up to the maximum frequency of the oscillator, and the signals of monitoring transducers are recorded. The same procedure is repeated for different dynamic force levels used in a test. At any force level and frequency, the dynamic force should not exceed 20 percent of the total mass of the block and motor-oscillator assembly. A form for recording results of this test is shown in Table 4.6.

because different forces cause different strain levels below the block. This is accounted for when the appropriate design parameters are being chosen. The coefficient of elastic uniform compression, C, of the soil is then determined from Eq. (4.37), (Refer to Chapter 6, Section 6.7), ( 4.37) in which fnz is the natural frequency in vertical vibrationS, m the mass of the block, oscillator, and motor, and A the contact area of the block (footing) with the soil. The value of C" from Eq. ( 4.37) corresponds to the area of the test block and for other areas; its value may be obtained by using Eq. (4.30). Knowing the value of C", E may be obtained from Eq. (4.29). Shear modulus can then be determined by using Eq. ( 4.4).

Determinaton of Coefficient of Elastic Uniform Compression of Soil C".

143

The amplitude of vibration A, at a given frequency f, is given by

A=~ 2 2 '

4tr

I

in which a, represents the vertical acceleration of the block in mm/sec 2, and f is the frequency in Hz. Amplitude vs. frequency curves are plotted for each force level to obtain the natural frequency of the soil and the foundation system tested (Fig. 4.36). The natural frequency, fu., at different force levels is different

0.20

'E

-""'

f--

0.12

"'

0.08

I

For a horizontal footing (block) resonance test, the mechanical oscillator is mounted on the block so that horizontal sinusoidal vibrations are generated in the direction of the longitudinal axis of the block. Three acceleration or displacement transducers are mounted on the side of the block with one near the top, a second near the bottom, and the third in the middle along the vertical centerline of the transverse face of the block to sense horizontal vibrations (Fig. 4.35a). The oscillator is excited in several steps, starting from rest. The signal of each acceleration pickup is amplified and monitored. The remaining procedure is the same as for vertical resonance test. Similar tests may be performed by exciting the block in the direction of transverse axis.

Determination of Coefficient of Elastic Uniform Shear of Soil, C,.

I

\

VJ

0.04

~

~~ r--o:

70'

r

"" 20

25

140'_

30

35

40

45

Frequency, (Hz)

Figure 4.36. Amplitude vs. frequency plot from vertical resonance test at raw mill site in Bhutan. (Prakash et al., 1976.)

( 4.38)

105°

__ ,y ,/ ~ 15

In a

horizontal-vibration test, the amplitude of horizontal vibrations, A"' is determined by the equation

~

I !f ·~

E

0 10

f

Horitontal Footing Resoilarice Test

1 r;

0.16

.s u .0..e"

4.6.5

( 4.36)

/

in which ax is the horizontal acceleration in the direction under consideration, and f the frequency of the horizontal vibrations in hertz. Amplitude vs. frequency curves are plotted for each force level to obtain the natural frequency, fnx, of the soil and block tested as for the case of vertical vibrations. A plot of the amplitude with the height of the block determines its mode of vibrations. The coefficient of elastic uniform shear (C,) of the soil is then determined by using Eq. ( 4.39). (For details, see Section 6.7.)

144

C ~

'

145

FIELD METHODS

DYNAMIC SOIL PROPERTIES 2

87T Yf~x (Ao + Io ± (Ao + Io) 2 - 4yA 0 I0

Y

(4.39) Load intensity

in which Y is equal to Mml Mmo• /,,x is the horizontal resonant frequency of block soil system, A 0 is equal to AIM, I0 is equal to 3.46 (II Mm 0 ), Mm is the mass moment of mertia of the block, oscillator, and motor about the horizontal axis passing through the center of gravity of the block and perpendicular to the direction of vibration, Mmo the mass moment of inertia of the block, oscillator, and motor about the horizontal axis passing through the center of the contact area of the block and soil and perpendicular to the direction of vibration, and I the moment of inertia of the foundation contact area about the horizontal axis passing through the center of gravity of the area and perpendicular to the direction of vibration. The coefficient of elastic uniform shear, C, should be corrected for area effects as for the case of C".

\

\ I I

Ses

-r

,___ _ _ ___J

Elastic rebound -

4.6.6

Free Vibration Test on Footings

(a)

Figure 4.37. (a) Load intensity vs. settlement' ill a cyclic-plate-load test and (b) Load intensity vs. elastic rebound from cyclic-plate-load test. (After Prakash, 1981.)

Free vibration tests may be performed by pulling the block and releasing it in a longitudinal direction or by hitting it with a hammer for vertical excitation. From the observed natural frequency, the C,, C,., E, and G values can then be determined. 4.6. 7 Cyclic Plate Load Test The equipment for a cyclic-plate load test is similar to that used in a static-plate load test. It is assembled according to details given in the American Society for Testing Materials (1977) or Barkan (1962) or in textbooks on foundation engineering. After the equipment has been set up and arranged, the initial readings of the dml gauges are noted, and the first increment of static load is applied to the plate. This load is kept constant for some time until no further settlement occurs or until the rate of settlement becomes negligible. The final readings of the dial gauges are then recorded. The entire load is removed and the plate is allowed to rebound. When no further rebound occurs, the readings of the dial gauges are again noted. The load is then gradually increased until its magnitude is equal in value to the next higher proposed stage of loading; the load is maintained constant and the final dial gauge readings are noted. The entire load is then r~ducect' to zero and final dial gauge readings are recorded when the rate of rebound' becomes negligible. The cycles of loading, unloading, and reloading are continued until the estimated ultimate load has been reached; the final values of dial gauge

readings are noted each time. The magnitude of the load increment is such that the ultimate load is reached in five to six increments.

Se

(b)

r

I i I

i

The elastic rebound of the plate corresponding to each intensity of loading can be obtained from the data obtained during cyclic-plate load tests, as shown in Fig. 4.37a. The load intensity versus the elastic rebound is plotted as shown in Fig. 4.37b. The value of C" can be calculated from Eq. ( 4.26). This equals slope of the plot in Fig. 4.37b. 4.6.8

Standard Penetration Test

In the standard penetration test (SPT}, a standard split spoon sampler is driven with a 140-lb hammer that falls freely through a distance of 30 in. The number of blows for 12 in of penetration of the split spoon sampler is designated as theN value. This is Nmcasured· In a design problem using N values, a correction for e~ective overburden pressure is applied (Peck et al., 1974). Although the test iS designated as a "standard" test, there are several personal errors as well as errors that are equipment based. Therefore, the use of SPT to measure any soil property has been questioned by many engineers (Woods, 1978). The "uses and abuses" of SPT have been described by Fletcher (1965}, Mohr (1966), and Ireland et al. (1970}. De Mello (1971} presented an extensive review of SPT from which it is evident that although SPT is used extensively in soil investigations, there has been no documented, carefully controlled research conducted on it. Schmertmann

146

DYNAMIC SOil PROPERTIES

{1975) described uses of SPT in estimating soil properties and pointed to some hope for more accurate uses of SPT in the future with improvements in the performance and understanding of the test. Recent careful studies by Kovacs (1975), Kovacs et al. (1977a, b), Palacios (1977), and Schmertmann (1975, 1977) have thrown new light on the potential of SPT for obtaining consistent and useful soil properties. Seed (1976, 1979) and Seed et al. {1983) presented correlations between SPT and observed liquefaction. It has been reported that Chinese engineers also use this approach in their attempts to assess the liquefaction potential of sand deposits. Skempton (1986) has presented a more recent review on the effects of overburden pressure, relative density particle size, aging and overconsolidation on the measured SPT values. Schmertmann {1977) suggested that with SPT, either only those dynamic properties for which SPT provides a direct model of the phenomenon can be measured, if at all, or the factors which control the behavior of SPT similarly control the correlative dynamic behavior. As an example of the former, pile driving is an obvious direct model; of the latter, it has been shown that the factors which affect cyclic liquefaction also affect SPT in a parallel manner (Seed, 1976). Additional advantages of using SPT to evaluate dynamic effects, according to Schmertmann (1977), are that SPT is a dynamic test for modeling a dynamic phenomenon, and it is essentially undrained for each blow in that it generates principally a shearing energy. Seed et al. (1984) have correlated the energy input of different types of SPT-hammers used in the United States, Japan, and China and have proposed correction factors for the energy input of different hammers. Imai (1977) reported a correlation between (uncorrected) N and shear wave velocity, V,(m/sec) in 943 recordings at four urban locations in Japan and was able to establish the following relationship: V

'

=

91N°. 337

( 4.40)

In arriving at the above relationship, be converted the N values over 50 or under 1 from the penetrating length at the time of 50 or 1 blows into the number of blows necessary for penetration as deep as 30 em. Prakash and Puri (1981, 1984) successfully applied the above relationship in predicting dynamic soil properties at different depths.

4.7

EVALUATION OF TEST DATA

Inthe twoprevious sections, laboratory and field methods for determining soil moduli have been described. This leaves us the task of ascertaining the strains and the confining pressures at which the moduli values have been

determined. It is customary to make a plot of G versus shear strain. It is necessary to

147

DAMPING IN SOILS

determine G at a mean effective confining pressure corresponding to the mean effective confining pressure below the prototype foundation and at a shear strain which may be induced in the soil when the foundation is subjected to dynamic load. Prakash (1981) and Prakash and 2Puri (1981) used a mean confining pressure a01 of 1 kg/em' or (1000 kN/m ) to reduce the data from different tests to a common confining pressure for comparison purpose only using Eq. ( 4.15) (4.15)

( 4.41) and _ _ vcrz ux = u = - -v y 1-

( 4.42)

The sam/;.~marks apply to arf"ivaluation of C", C
4.8

DAMPING IN SOILS

Types of Dgmping in Soils. The motion of the footing-soil system can be damped by two specific energy losses: (1) the ab_r;~_!fl~I()IIOf"nerg)'':"It~m the soil mass and (2) the dissipation of energy associ:~t.,dwitli tlie geometry of the fo"iii1dationc-soi!..J;ystem:-Thefol"mer 1S known as material or internal 4

·

~ta~~~~~~:~;n;~i~~~~~;';;~~~;:~i"t;~~-i~t;;;;~i~1~i~!~ L fo ~, Materi7d Dam in8] Damping in a freely vibrating system reduc~s the pea amplitudes after each successive cycle. If the fore~ of dampmg IS considered proportional to the velocity of the mollon, It IS called vtscous damping (Section 2.4). . . . When a cylindrical s_a,mple of soil is set into a state of free vibratiOn, as m a resonant column test:1lhe vibration decreases in amplitude and eventually disappears. This reduction in amplitude of vibration is caused by internal damping within the soil mass. The decay of vibration is similar to that described for the free vibration of a viscously damped system. It should, however be understood that the internal damping in soils is not considered to be

the

result of a viscous behavior; nevertheless, the theory for a

single-degree-of-freedom system with viscous damping is useful for describing the effect of the damping that occurs in soils.

149

DYNAMIC SOIL PROPERTIES

148

DAMPING IN SOILS

=

The decay of free vibration of a single-degree-of-freedom system with viscous damping is described by the logarithmic decrement, which is defined as the natural logarithm of two successive arriplffiides..of motion, or

z a= log, _l z,

;:

f::

I

(2.77)

II

or

a=

2-rrt;

v ' I

(2.80a)

~

I II

I

I•

;:

f::

The logarithmic decrement may be obtained experimentally from a resonant-column test by setting a soil sample into steady-state forced vibration, then shutting off the driving power and recording the decay of amplitude with time. Figure 4.38a shows a typical vibration-decay curve obtained from a resonant-column test for Ottawa sand (Hardin, 1965; Hall, 1962). The logarithmic decrement was obtained from the decay curve by plotting each amplitude against a cycle number on semilog graph paper (Fig. 4.38b). If the damping in the material produces an effect similar to that occasioned by the viscously damped free vibrations, the plot is a straight line on the semilog graph. Hall (1962) found that the damping determined from the decay of steady-state vibrations in resonant-column samples of rounded granular material is similar to viscous damping. The values of logarithmic decrement varied from 0.02 to approximately 0.20 in his tests. Hardin (1965) described further studies on resonant-column tests for evaluating the ;!
1::

f\h

I'

v

Ia I

to J-,--.-~-,-,,.,--r-ITIII 9 8 7

No. of cycles (b)

·

(4.43)

T ical free vibrationwdecay curves obtained from resonantwcolumn tests on 4 38 6~::=a ~and (a/!mplitude-time decay curves and (b) amplitude vs. cycle number plot. (After Hardin, 1965.)

in which t!:...i.L£S?.~!!lsi~!!L'!Ls)1~~~--~~s
a=9-rr(y,) \
Cr .... ,t- .,n

in which Yo is the

(4.44)

>!'··•

shearing~~{~;it~ ~~piit'~d~ ---and iT0

the mean confining

pressure (expressed in lblft'). He also suggested that Eq. ( 4.44) be used only within the limits of the shearing-strain amplitudes of 10- 6 to 10- 4 for

2

•

f

confining pressures of 500 lbift' < (T0 < 3000 lblft and for frequencieS o less than 600 cycles/sec. . . Drnevich (1967) studied the effect of high-amplitude sh~anng strams on the damping in sands. He found that: (1) no change occ\l~s m dampmg With cycles of prestrain for prestrain amplitudes less than 10 ; (2) the loga~ith. d t vari·es with (0' )- 113 within the range of 400 to 2000 lb/ft for mtc ecremen o _5 -4. d (3) an m Y all shearing-strain amplitudes between 10 and 6.0 X 10 , an

151

DYNAMIC SOil PROPERTIES

150

cycles of high amplitude prestrain increase damping in some cases to twice the original value. Significant increases in damping may be related in part to the test procedure used to control the shearing-strain amplitude. As the shear modulus increases because of the prestraining, the procedure of maintaining a constant amplitude develops a larger strain energy at each cycle. It would be expected that the hysteresis loop would then include a larger area to represent the increased damping. Richart et al. (1970) stated that the values of logarithmic decrement for sands may be as large as 0.20 and that these can be estimated from Eq. ( 4.44 ). Some additional data on internal damping for several types of soils are shown in Table 4.7. ~--------~-----······--·-"· · · · ....

Figure 4.39. Stress-strain curves for a system with hysteresis damping.

Equivalent

g

Dry sand and gravel Dry and saturated sand Dry sand Dry and saturated sands and gravels

Clay Silty sand Dry sand

Reference

0.03-0.07

Weissmann and White (1961)

0.01-0.03 0.03

Hall and Richart (1963) Whitman (1963)

0.05-0.06 0.02-0.05 0.03-0.10 0.01-0.03

Barkan (1962) Barkan (1962) Stevens (1966) Hardin (1965)

Sour~e:

Richart, Hall, and Woods, "Vibration of Soils and Foundations," Repnnted by permission of Prentice-Hall, Englewood Cliffs, New Jersey.

_

ilEu

1-l.cs-£ "

The condftion for a decaying '\ibFation is illustrated in Fig. 4.39b. Point 1 corresponds to the maximum stress in a cycle, which starts at pomt 1 and ends at point 2. It is seen from this figure that the value of 11, depends o~ whether the steady-state A" or the decaying-vibration cond1t1?n A,d IS considered when the damping values are large. In case of decaymg vibrations, the relationship between the lqgarith!Il_ic_.~_e_c_!:~~_!lt and the !l'_ecJ~C damping capacity is (Richar.!_.."!__'l~~,_]_~!O~

----- ----------"

©

( 4.46)

1970, p. 398.

Material damping in soils is also sometimes studied in terms of specific damping capacity, which is defined as the ratio of the energy absorbed in one cycle of vibration to the potential energy at maximum displacement in that cycle. The damping capacity may be expressed as a percentage or as a decimal. In a stress-strain diagram, the specific damping capacity may be represented as the ratio of (1) the area enclosed by the hysteresis loop and (2) the total area under the hysteresis loop. For the steady-state condition in Fig. 4.39a, t the specific damping capacity ll" is given by A

'

'

(a) steady-state vibrations.

Table 4.7. Some Typical Values of Internal Damping in Soils

Type Soil

DAMPING IN SOilS

( 4.45)

The term E. in Eq. (4.45) represents the strain energy described by the area under the hysteresis loop. t The horizontal scale in Fig. 4.39a is greatly exaggerated for simplicity of illustration.

in which k" represents the proportionality factor between strain energy and the square of the displacement amplitude for the nth cycle of decaymg vibration. It needs be emphasized that there is no general relat1o~sh1p il,d, and the ralio of between Acs and 11 cd• but for small values of a, A"= • 1 the proportionality constants, K~+ 1 1 K", is approx1~ately .. Sometimes the decrease in amplitude of vibratiOn w1th distance fro~ _a source that is caused by energy losses in the soil is also evaluated. This IS designated as attenuation (the energy loss as a function of distance) and IS measured in terms of -W,e coefficient of attenuation a (1 /ft). The coeffic1ent of attenuation is related to the logarithmic decrement by

z,. Va

{j= - - =Aa

(4.47)

w

in which V is the wave velocity, w the circular frequency, and A the

wavelength of the propagating wave (Richart et al., 1970). . . Attenuation needs to be distinguished from JI:~":'".t!..'.~~~~~ll_'!'f!~? (SectiOn

153

DYNAMIC SOIL PROPERTIES

152

DAMPING IN SOILS

3.4), which occurs in elastic systems because of the dispersion of wave energy from a source. Internal damping in materials may also be evaluated by measuring the angle at which the strain lags the stress in a sample undergoing sinusoidal excitation. If the soil is assumed to be a linear viscoelastic solid, the complex shear modulus G may be considered to be composed of a real and an imaginary component, each of which is a function of frequency, such that G*(w) = G 1 (w) + iG2 (w)

(4.48a)

In Eq. (4.48a), G 1 (w) is the elastic component and G 2 (w) the viscous component. The loss angle 15L is defined by

G,

tan 15 = L Gl

( 4.48b)

and it is related to the logarithmic decrement 15 and to the ratio p.w I G [see Eq. (4.43)]. Thus,

2

3 ':;~if;, Bx, B

or By,

~

( 4.49)

ratio for .,~scillation of rigid circular footing on the elastic Figure 4.40. EquJValen ampl g d © 1970 226 Reprinted by permission of Prentice~ half-space. (Richart, Hall, and Woo s, C ' P· · Hall, Englewood Cliffs, New Jersey.)

From the above discussion, it can be seen that there are several methods for measuring and describing the damping liisOITS. ..F~rthermore:··because darniJTiigin~oil~~~ii"~~i!ses·;;;;mt~I11~Iiiii5lli~~if~·o(vf6rationJEq. 4. 44], i!_~ay be convementto use different .method.s for differ~nt r"f!ges ()f amplitude ('Rlcilarlet al., 1970). Beneath machine foundations, the order llrinagnii.llde of vibrations encountered in soils is such that the logarithmic decrement should be less than 0.2.

damping constants obtained in the analog solutions (See Section 6.4) and

·

15

= 1T

tan 8L

~~9metl:i(;(J]i5a~P.i;~. It was shown in Chapter 3 (Section 3.4) that in an elastic medium, energy is dispersed as it recedes from the source of disturbance. This dispersion of energy produces response curves that have a finite amplitude of motion at resonance. This indicates that damping is present in the system. However, the assumption of an ideal elastic halfspace precludes loss of energy because of inelastic behavior of the material, which constitutes the half-space. ;This geometrical distribution of elasticw~~~_ll..~_::!\l:Jt.."s·_l.>=~-de~igf!"t"d as gf()'!'etr!f.(ll_!f:q!!'J!iiJg. · - ... ·... ·- From each solution for vibration of a footing on a half-space, -",Yi!]tle of the equivalent damping ratio !; is derived. This iS' then used in a lumpedanalys!S:-Aconvenient method for evaluating !; is to equate the peak amplitudy of motion from the half-space solution to the peak amplitude obtain(ld from the mass-spring-dashpot system and then to solve for g. Figure 4.40 wasprepared by using this method (Richart et al., 1970). Approximately the same results can be obtained by calculating !; from the

parameter..

t d

·0

the expression for critical dampmg, c = 2v'kiii

(2.26)

'

With this approach, the damping ratio is c !; = -

(2.27)

c,

Expressions for the d~mping ratio (Section 6.4) are For vertical osctllauon, 0.425 !;, =

'[lJ;

( 4.50a)

For horizontal oscillation, (4.50b)

155

DYNAMIC SOIL PROPERTIES

154

DAMPING IN SOILS

For rocking oscillations, X

( 4.50c} For torsional oscillations, 0.50 1 + 2B" in which

(4.50d}

B =(1-v} m

=

B x

(7- 8v} .!!!._ 32(1- v) pr~

= 3(1- v}

B "' and

pr~

4

z

8

( 4.50f} w

Mmo pr~

B = Mmz

( 4.50g}

2. Determine the asymptote

It must be noted that if the response of an actual machine foundation or a vibrating footing is monitored, the measured ~mp.li1J,I_d.e.s.r_e!!.ectJhe effu£!.!?!. both the material ~nd geol!l.e_t!i£lJLc.!am£i!l.g. Barkan (1962) showed that an irrcre1!se "inflie·-value of Poisson's ratio of soil leads to an increase in dissipation of energy from a vibrating foundation into the soil. Further, the damping (combined material and geometrical} properties of soil are deter· mined not only by its characteristics (inertia and elastic properties) but by the size and mass of the foundation. A large amount of the data in the literature shows that the damping constant increases with the contact area of the footing for vertical vibrations. Lorenz (1953) recommended a graphical method for determining damp· ing (combined material plus geometrical} from a forced amplitude frequency graph by exciting a footing with vertical vibrations. For frequency-dependent excitation, the maximum amplitude of motion zmax is 2

z m"'

= -c===m~0:=;ec=w~===, 2 2 2 V(k- Mw

}

+ (cw)

(2.49b}

and the procedure for determining damping is as follows: I. Draw a tangent from the origin of the coordinate system to the

amplitude frequency curve (Fig. 4.41} so that it touches the curve at w =

,

• ce c rve for the damped forced vibrations. (After Lorenz, 1953, Figure 4.41. Ty~•ca1 resonan u copyright ASTM:' Reprinted with permissionp:

(4.50h}

pr~

l/1

( 4.50e)

If

"V "M

(4.51}

(4.52) · M 1·n Eq . (4 .52} , because Z "' is measured in Fig. 4.41 and 3. D etermme (m 0 e) is a known quantity. 4. Determine k from Eq. ( 4.51). 5. Then damping constant c from ( 4.53) in which Z, is the amplitude at w

=

w,.

Factors Affecting Damping. There are several factors that affect damp~ The more important of these are: strain level, confimng press~re, vot ra~io number of cycles of oscillatory stress, and frequency of motton. In' Table 4.8, the ini'~ortance of each variable is indicated by the symbols ~ R d L (Hardin and Drnevich, 1972a). . . . , Theandam~ing factor !; increases generally with an mcreasmg stra~~ amplitude, decreasing confining pressure, '?creadsm~ vmd :a(t~~72~ldp~o. b of cycles of loadmg. Hardm an rnevtc creasmg num e~d ex ressions for determining damping in a partic~lar P~~~fe~h~~~s~derabl~ judgment is still needed to select ~value of da?'pmg P . d . blem Fortunately the chmce ts not that dtfficult to be taken m a estgn pro · •

in

and varied in machine foundation design.

156

DYNAMIC SOIL PROPERTIES

157

EXAMPLES

Table 4.8. Parameters Affecting Damping for Complete Stress Reversal Damping a Parameter

Clean Sands

(I)

(2)

Strain amplitude IT2. .··.Effective mean principal stress !

. 3. 4. 5. 6. 7. 8. 9.

:Void ratio

Number of cycles of loading

Cohesive Soils (3)

/v

v v v v

(v

\ v '

\_vj

L R L L R R

u

gradation, mineralogy 12. Soil structure 13. Volume change due to shear strain

R R

R R

(for strains less than 0.5%)

u

R

Degree of saturation Overconsolidation ratio Effective strength envelope Octahedral shear stress

Frequency of loading (above 0.1 Hz) 10. Other time effects (thixotrophy) II. Grain characteristics, size, shape,

L L L L L Figure 4.42.

Soil profile (Example 4.9.1 ).

Dynamic shear modulus at point' A is given by (3.30)

G=V ,,2 p

Source: Hardin and Drnevich, 1972a. a V means Very Important, L means Less Important, and R means Relatively Unimportant except as it may affect another parameter; U means relative importance is not clearly known at this time.

= G

120' x 1750 = 256.8 kg/em' 9.81 X 100 X 100

The effective overburden pressure at point A is given by 4.9

EXAMPLES

cr,A:

5 X 1750 = 0.8750 kg/ em' 100 X 100

EXAMPLE 4.9.1

The soil profile at a site is shown in Fig. 4.42. A cross borehole test was conducted at this site to determine the value of shear wave velocity in a small area surrounding point A (Fig. 4.42) and its average value was observed to be 120 m/ sec. Calculate the value of dynamic shear modulus G for point A. Also determine the values of G for points B, C, and Din the profile. If subsequently the water table rises to the ground surface, what will be the values of G at A, B, C, and D. Assume 'YJ = I. 75 g/ em 3 and y,., = 2.05 g/cm 3 Solution Dynamic Shear Modulus-water table 10m below ground level (as shown in Fig. 4.42}. Observed value of shear wave velocity V, =120m/sec

Dry unit weight of soil 'Yd = 1.75 g/cm 3 = 1750 kg/m

3

The values of effective overburden pressure at B, C, and D may be calculated in a similar manner. These values of cr, for different pomts are shown in column 3 of Table 4.9. The value of dynamic shear modulus at B may be ca~culated from the known value at A by using Eq. ( 4.15} and cr, in place of 0'0aud replacmg 1 and 2 by B and A, respectively: ( 4.15} or 1.75 ) (G) B = 256 ' 8 ( 0.875

0 5 '

= 363.2 kg/em

2

The value of G at points C and D may be calculated in a similar manner. These values are shown in column 5 of Table 4.9.

159

EXAMPLES

DYNAMIC SHEAR MODULUS-WATER TABLE AT GROUND SURFACE

"

-'0

c "

.0

" 0" [-< ~

~

N

2

~

(j

u

~

" t: ~ ~

s "-"' " u :;:: "

0\('f")V)OO

ocirl~~

0\00'70\ ,......, N M('f")

~

.,~

all these calculations. The values of G corrected for the effect of rise in water table at different points may be obtained by modifying the previous values for the change in effective overburden pressure by using Eq. ( 4.15). These corrected values are shown in Table 4.9, column 6.

.c ""

"'~

-~ ~ ....c " o-6

0

" 2 ::: g ..oo't:

-" '"""' 2 s "g

::E " ~a:) ~

0

~

OONM-.;:t 0('1')~~

lf)\O,.....,lrl N('f")~"i"

~

- {.)2

~

.,;

...

,., ,.,

"

tnOI.rlc),_;~r-i

E

"'>< ~

'0

:1 c

"'~ .0 ""

..a

~

s

~.,.

~~ lb"'

0

~ ~ ~

ao =

~

"'

.c"

"' '5"'

....,;

" :;:; ,_"'

158

3

=

KoO\

The values of K for different soil layers are calculated as follows: 0 For sand layers 1 and 2 K 0 = (1- sin)

~

"" ~

0\ + ii2 + ii3

iiz = ii3

"""

0;

(4.9)

The values of e, OCR, k, PI and 0'0 are needed for using this equation. The values of e, PI, and OCR are given in the problem. These values are also listed in Table 4.10, columns 2, 3, and 4, respectively. The value of 0'0 is calculated as

u

~

"'>-c Q

2

05 29 3 - e) (0'0 ) psi +e

Gm"' = 1230(0CR)k ( · ;

"~

" " i5 t:::" " ~0:: ~

~

Solution .•;_.,_,, The values of low strain dynamiC shear modulus for the soil profile can be calculated by using Eq. (4.9).

O)N

- ·B " c

EXAMPLE 4.9.2

The'soil profile for a site is shown in Fig. 4.43a. Calculate the values of dynamic shear modulus at different depths in the profile. Also plot the variation of shear modulus with depth. Assume shear strain amplitude 'Yo= 10- 6 (i.e., low-amplitude vibration case).

NV)f:-..0

Q.

~

Due to rise in water table, the effective overburden pressure decreases. The new values of effective overburden pressure for points A, B, C and D 3 are shown in Table 4.9, column 4. ')',, = 2.05 g/ cm has been used in making

oooo tri 0 tr)' 0 -

-N

For normally consoli~ted clay (layer #3) K 0 is given by the equation below (Brooker and Ireland,' 1965) K = 0.4 + 0.007(PI) for 0 s; PIs; 40 0 For the normally consolidated clay and PI= 30, K 0 is determined to be 0.61. (See also Fig. 4.49.) For over consolidated clay (layer #4), the value of K 0 is obtained from charts given by Brooker and Ireland (1965). For PI= 55 and OCR= 2, the value of K 0 is determined to be 0.95 (Fig. 4.49).

0

,..

_,.

~

. ...,...,...,...,:::;,_, 0" " ::r .

Ci(ll~(tl

(D<=:In

;.. .

" ~

2

~"

" 0.

" e:.

><-,.,..,

(!>

"

0.

<

...,..,_ ~

~. ~

0

(!> 0 -O> 0o-· "' ~ "

or """0 s· _q,

" ~

0

""" 0>1

"0. Nq[ ....S' i:t' 0 s "0. "" " "' -l " o'lr 82l ";....._,. ".... ".... "" ~· 0" 0.. " :E"" "".... :r-l "s· ~

(!>

0"

(!>

(!>

~

(!>

~

"" _,." g." " 0

0"

:E

" s·

..... 0

"

0

" s0<= 8g. '-0

"

~

"'

e

E.

Or:n'"1-

. og:=; """ " '

. ...

(!>

" 0. " +:-.-oo-

s

"

v. 0

""" -l

"='

~

0"'

tl:l

-

~

0

~

i

-c

Depth (ftl

..

N

"'

k-"'~"'

. I.

~

a ~

. if .

~§

/

e...

...

-< ~

3• ;;·

0 0 0

,.. ~

"'

0 0 0

~

!;

...3 0

"'

0 0 0

!;;.

c < ~

0 0

~

~

0 0

c

... ~

~

~

0

~ 3

•

15' ~ ~

ro ~

3

0

~

0

c ~

·"'

u

~

N 0 0 0

"¥-

ro~r~~~ , 5"00~: «9,_.<::;:; ~0 "'0 ~0~-o
..

;;; ~

•3 -c

~

"00

': . •• ' '

0

'

f' '· o o

'

•

<

,

0

"3.,.

'

·:. ~ . • ~ :'. . .~ :-:::: ~·:. ·:. < ,·.: :. .:_: . :.

0 0 0

..."

'".,_

ocoU.'. ·, ~

;.

~

;=;

V>

0 ;::

. "'.

"':::!m

~

Lateral

:::

;::

m

"

OCR

K,

Effective Vertical Pressure r:T, (psf)

Void Ratio e

z >

0

"

0"

Table 4.10. Values of Dynamic Shear Modulus at Different Depths (Example 4.9.2)

Depth (ft)

0

-<

fl- PI

':7'

Stress (7"3

=

=

(T2

Ko iii (psf)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

0 2.5 5.0 5.0 10.0 15.0 15.0 17.5 20.0 20.0 22.5 25.0

1.10 1.10 1.10 0.85 0.85 0.85 1.125 1.125 1.125 0.80 0.80 0.80

0 0 0 0 0 0 30 30 30 55 55 55

1 1 1 1 1 1 1 1 1 2 2 2

0.50 0.50 0.50 0.455 0.455 0.455 0.61 0.61 0.61 0.95 0.95 0.95

0 250 500 500 763 1026 1026 1157.5 1289 1289 1458 1627

0 125 250 227.5 347.1 466.8 625.9 706.1 786.3 1224.5 1385.1 1545.7

;f~ '~}-

Mean Effective Confining Pressure r:T0 (psf)

k

Dynamic Shear Modulus G (psi)

Remark

(8)

(9)

(10)

(11)

0 166.7 333.3 318.3 485.8 653.2 759.2 856.5 953.8 1246.0 1409.4 1572.8

0 0 0 0 0 0 0.24 0.24 0.24 0.38 0.38 0.38

0 2210.4 3126.0 4455.6 5504.4 6381.6 4538.4 4821.6 5088.0 12348.0 13136.4 13876.8

Top of layer #1 In layer #1 In layer #2 (top) In layer #2 In layer #3 (top) In layer #3 In layer #4

162

DYNAMIC SOil PROPERTIES

The value of G may now be calculated. The calculated values of G at different depths are shown in Table 4.10, column 10, and are plotted in Fig. 4.43b. EXAMPLE 4.9.3

The following tests were performed at a proposed factory site to determine the in situ dynamic properties of the soil. 1. Four steady-state vibration tests were conducted on a 1.5 x 0.75 x 0. 7 m high concrete block. The tests were conducted by subjecting the block to a vertical unbalanced force and a horizontal unbalanced force. In both cases, the tests were repeated at different settings of eccentric masses. The data on observed resonant frequencies and corresponding amplitudes of vibration are shown in Table 4.11, columns 1 through 4. The tests were conducted in an open pit, and the base of the block was at 4.0 m below ground level. The density of concrete was 2.4 t/m 3• 2. Two cyclic plate load tests were performed on 60 X 60 em plate in a pit 3 X 3 X 4.0 m deep. The typical cyclic plate load test data for Test-5 are plotted in Fig. 4.44 and the values of the load intensity and corresponding settlements are listed in Table 4.12, columns 1 through 3.

•I •I •I .i~r

-

•• I I

•I •I

.. I

I

~

~

s-s;"~

00

X

X X

X X

X X

"'v-:oo

"' "' """' ("!')vi

"'"' "'"' -::r..-.:

X-'_X

~~Vl

r-.i("<')ff)

~

~

"' 0

00

00

~~

load intensity (kg/cm 2) 0

0

2

3

4

5

--"':::::- :'--.... ~f'::...

6

~

4

8 E E c

12

w

E w

Bw

"'

16

20

24

Figure 4.44.

., ~ ~ "'',I'.,,, ~-..., ~ ' ' ' "', ~[\ ' ' -...,

s

.,. 0

"-""~>

' ',

'-....,

''

''

II

j

~~

0 ~

~

X

E 0 ;-:;

1\ ' ',

"

-....\

Settlement vs. load intensity {cyclic plate load test) (Example 4.9.3).

0 0

0

0

" <'>t--O ' 0 "' ~

0

0

"' 0 <'>t--

0

0

"' 0

"' r--

0

0

"' 0

"' r--

X

E

"'

~

II

~

'E

Jj" 163

165

EXAMPLES N values

-::t-Mt-ov;

.,.)000<")("") ViMOOO\t;'"--

('1'")7('1,....,,......,

:6:6!;:~M

rl"
~rl~NN

"'I ..,I "'I "'I "'I

--,.--1-00000

x X X X X

l.rJ\O,....;om

Figure 4.45.

'7"1.00\000

' Typical bore log of the site.(Prakash and Puri, 1984) (Example 4.9.3).

rlrlNv)v-;

l- t-- -

\0

'('f)

~ci~ciN

\ONO,.....,O\ 7V1"1"("1'")N

~ 00 t-l.rlN\00

v-ioONr-..:0

NNN--

v;

t--OV'>

0001."--0tn

o,....;,....;triM

0000 v; Mtrl'7NO

N~t--=>.0~

~

"'

o..:,....; "" OO'T ..,...,.

3. A standard penetration test was performed, and a typical borehole log for the site and the SPT values vs. depth are shown in Fig. 4.45. Water table was not encountered in the borehole, which was 12m deep. The average in 3 situ density of the soil was 2.05 g/ cm Determine from the above data 1. a plot of dynamic shear modulus vs. shear strain at a depth of 4.0 m.

2. a normalized plot of G/Gm, vs. shear strain. 3. the variation of G vs. shear strain at the depth of 2.4 m. Solution (1) Block vibration test Size of tbe block= 1.5 X 0.75 X 0.70 m 2 Area of the block A= 1.5 x 0.75 = 1.125 m Weight of the bloc~.= (1.5 x 0.75 x 0.7) x 2.4 = 1.89 t Weight of vibrator ='0.11 t Weight of block and vibrator= 2.00 t

Vertical Vibration Test: C = 4'1T

v;

v; N000\,......;\0

"

NNMviv-i

c 164

2

2 2

f nz m

A

t/

m

3

4'1T (2.00) 2 3 2 3 "= l.lZ5 x 9 _81 f"' tim = 7.154!"' tim

( 4.37)

166

DYNAMIC SOIL PROPERTIES

or

167

EXAMPLES

0', 1 = Effective overburden pressure at the depth under consideration 0', 2 = Increase in vertical pressure occasioned by the static weight of the

C" = 0. 007154[~, kg/ em 3

block

for

0'" 1 = 37.5 x 2.05

fu, = 29.75 Hz

= 0.076875 kg/ cm 2

C" = (0.007154)(29.75) 2 = 6.33 kg/cm 3

2

4q [ 2mnv'm 2 +n 2 +1 m 2 +n +2

_

These calculated values of C" are shown in Table 4.11, column 5. The values of G are calculated from Cu as follows·•

avz = 41T

mz .

c

=

"

1.13£ 2 (1- v )

G=

+ sm

1

\Ill

(4.29)

2mnv' m + n + 1 J 2 z 2 z m +n +1+mn 2

2

(Taylor, 1948).

(4.4)

+ v)

- L/2- 75 -2 m37.5-

-z-

Substituting for E in terms of G from Eq. ( 4.4) into Eq. ( 4.29), we get

B/2

C = 1.13(2G(1 + v)] _1_ " (1- v 2 ) \Ill

n=

Z

37.5

= 37.5 = 1

•S<': 2

2

G = (1- v)(VA)C" 2 X 1.13 (1- 0.33) G= 2. (Y150 x 75)C" 26

2

2

2

= 134 g/cm = 0.134 kg/cm

2

2

0'" = 0.076 + 0.134 = 0.210 kg/cm 2 Mean effective confining pressure, 0'01 , letting K 0 = 0.5

C" = 6.33 kg/ cm 3 G = 31.44 X 6.33 = 199.01 kg/cm

_


2

The calculated values for G are shown in column 7, Table 4.11.

Correction for confining pressure

=0.21

(2X0.5+1) 2 =0.14kg/cm 3 2

The value of G for the mean effective confining pressure of 1 kg/cm may be obtained as follows:

. The mean effective confining pressure 0'01 at a depth of one-half the w1dth = 75/2 = 37.5 em below the center of the blo'ck is given by for

_ _ _ (2K0 + 1) in which

(T'v

2

, _1 2 X 2 X 1V2 + 1 + 1] +sm 2 2 + 1 2 + 1 +2 2 1 2

For

Uoi -

2

0' = 4(2.4 X 70) [ 2 X 2 X 1V2 + 1 + 1 2 + 1 + 2 2 " 4'1T 2 2 + 1 2 + 1 + 2 2.1 2 2 2 + 1 2 + 1

or

= 31.44C" kg/cm

_1

+ nz + 1 + m2n2 mz + nz + 1

Figure 4.46 may be used for computing 0'" 2

E

2(1

= 76.875 g/ cm 2

3

G 1 = 199.01 kg/cm

2

1.0 )0.5 G 2 = 199.01 ( 0. 14 = 531.9 kg/cm

2

169

EXAMPLES

The values of G corrected to a mean effective confining pressure of 1 kg/cm 2 are shown in Table 4.11, column 9. Shear strain: Shear strain 'Yo is given by

U JO Ui


;

":

~

~ "

"

~

~

f"!

f"!

~

N

"!

~

~

'Yo=

~' ~~~~ ~

for amplitude= 0.185 mm 'Yo =

'-( ~

II"',.

~'

I= 0.75

~V~ VI

~; ~'<~r-~'~if~~h rh '); II I/ I I II I I 'll_.f/~/., rT I. 11111111 I 1 1 I "'~'l,.,ff$) ~

:·<(J//,

.

r; '/. I. I

II

L

I 1 I I I

X

10

.. ~

'-'<..'//

~

J//'1///.

/

"'-'<'/ / / ·'"

~

;..;;.

..0

"' N

N

// r////-

T""¥ /7 ,..'0 ::« /

~

":

'//.1/ ///_/j : /~ %~W& ~

.

_v

r::::

~

'"'"

..

- %'k" 0: ~~;

~

0 0

0


0

1,{}

0

1,{}

~

M

M

N

U JO Ui

~·

'.. "":-:."'~.'<

Mm t

=

X 0.85

20

= O 37 5 . 7 m

1.89 ( 1.5' + 0.70') 1.89 (0 3775-0 35)' 0.11 (0 85- 3775)' 9.81 12 + 9.81 . . + 9.81 · O.

= 0.04669 t m/sec-

~ ~

.

...,

2

Mass moment of inertia M mo:

N

v,.~·

0

.

Mass moment of inertia M m about an axis passing through the combined center of gravity and perpendicular to tbe plane of vibration is given by

~ ~~1~~ v 'k~@~~

~

Mmo

t

=

2

1.89 (1.5 + 0.7') 1.89 (0 35)' 0.11 (0 85)' 9.81 12 + 9.81 · + 9.81 . ,-~~.

= 0.07569 t m/secMm

2

0.04669

'Y = Mmo = 0.07569 = 0.6168

0

oi

The values of C" may be obtained as follows: t For calculations of Mm, M, 0 , and /, see Appendix 4.

168

-4

(\r:~= 0.2109 m'

= 1.89 X 0.35 + 0.11

WI/I 'I I 7 I II I I

~W,tC1rt,lfk1Vj r/ I I ///; f7"""<4·//;~~'/ '/ I ~Ill

"'

0

2.46

Height of combined center of gravity of block and vibrator L.

f

~0

=

(2) Horizontal vibration test Moment of inertia of the base contact area about axis of rotation

N

"

0.185 750

The calculated values of 'Yo are shown in column 6 of Table 4.11.

oi

5 ~

Amplitude Width

170

DYNAMIC SOIL PROPERTIES

2 C = 87T Yf:x " (A 0 + I 0 ) ± Y(A 0 + I 0 ) 2

s, = 2.38- 1.505 = 0.875 mm

-===-- = 5.5181

p 2.25 2 c.=;.:= (0. 875110 ) = 25.71 kg/em

2.0 ) ( 9.81

=3 46 (-I-)= 3.46 (0.2109) _ 0 6408 Mmo 0.07569 - · •

Substituting the values of A 0 , I0 , and yin Eq. (4.39), we get C =

8( 7T

) X

(0.6168)/~x

(5.5181 + 9.6407) ± Vc5.5181 + 9.6408)'

"

4 X 0.6168

c

2

X

5.5181

48.7005[~

t-

X

( 4.26)

The values of C,. for other data in the cyclic plate load test may be calculated in the same way and these are shown in Table 4.12, column 4. The values of G for the test and the corresponding values at a mean effective confining pressure of 1 kg/ em' may be computed in the same way as for the case of block vibration test. Similarly, the strain level, Yo, may be obtained as a ratio of the elastic deflection to the width of the plate. All these values are shown in Table 4.12, columns 5 through 9.

X

0

For, p = 2.25 kg/cm 2

1.125

A= A o m

I

( 4.39)

4yA 0l 0

-

171

EXAMPLES

( 4) Standard penetration test The Standard Penetration Test (SPT) value or N-value at 4.0 m is 13 • (Fig. 4.45). The,shear wave velocity, V, is given by

9.6408

3

" - 15.1589 ± 9.9267 tim

( 4.40)

= 9.30 f~ tim 3 = 0.00930[~

= 91 (13) 0 "337 = 215.9 m/sec

kg/ cm 3

or

2

C,. = 2(0.0093 f~) = 0.0186 f~ kg/ cm

G

3

For

2

(V,) y/g =

(215.9) X 2.05(1000) k I 2 9.81 X 100 X 100 g em

( 4.35)

= 974.0 kg/ em' fu

= 16'

c.= 4.76 kg/cm

3

_ 4 X 2.050 2 u,. = 100 X 100 = 0.82 kg/ em

The calculated values of C,. are shown in Table 4.11, column 5. The value of c. may be calculated and corrected for confining pressure in the same manner as for the case of vertical vibrations. Shear strains may also be calculated in the same manner (column 6) Table 4.11.

(3)

=

Cyclic plate load test

From the load intensity versus settlement plot for the cyclic plate load test (Frg. 4.44), we can compute the elastic settlement as follows:

_

u 01

= 0.82

(2K0 +1) 3

= 0.546 kg/ em

2

2

Gat 1 kg/cm 2 = 1317 kg/cm The associated strain level in the wave propagation test is low. It is in the • -6 range 10 . a. G vs. 'Yo·

in which

The values of G vs. shear strain, y0 , are plotted in Fig. 4.47 (Curve A). b. G/Gm, vs. y6 , i.e., normalized G/Gm, vs. Yo plot:

s, =Total settlement for a given applied pressure increment s P = Residual moved.

settlement

after the

load

increment

t Since fn is the first natural frequency, negative sign has been used.

has

been

re-

The values of G for different strain levels in plot A (Fig. 4.47) were divided by Gm,, i.e., 1317 kg/cm 2• These values for different strain levels are shown in Table 4.13, column 3, and are plotted in Fig. 4.47 (Plot B).

172

DYNAMIC SOIL PROPERTIES

EXAMPLES

173

151()0

The mean effective confining pressure 1.0

N~

E 1250

Legend 6 Block vibration test

u

" 0

1000

"'

o Cyclic plate load test

~ 3.28 tim'

o From N values

~

0

3

0.6

750

~

0

E

"

<'•i

0.4

u

i"ro

0

~

~ 2.05 X 2.4( 0.5 X32 + 1)

0.8

250

0

Shear strain 'Yo

Figure 4.47.

G vs. 'Yo and G/Gma~ vs. Yo (Prakash and Puri, 1984) (Example 4.9.3);.

a

1 '

(3)

( 4)

1 X 10 6 5 X 10' 6 1 X 10' 5 5 x 10··' 1 x 10·• 5 x 10·' 1 x 10·' 5 X 10·' 1 X 10·'

13!7 1225 1137 837 740 425 355 237 200

1.0 0.930 0.863 0.635 0.562 0.323 0.269 0.180 0.152

1121.3 1042.8 967.7 712.0 630.2 362.1 301.6 201.8 170.4

The value of G corresponding to the mean effective confining pressure of 1 kg/cm 2 ~ 642.2 (110.328) 0 ' 5 ~ 1121.3 kglcm 2• This value of G is at a shear strain level of 10· 6 and is Gm" for the depth of 2.4 m. The value of G at 2.4-m depth may be obtained by multiplying Gm" at 2.4 m with ordinates of G/Gm"x plot (Plot B, Fig. 4.47). The values of G so obtained are shown in Table 4.13, column 4, and are also plotted in Fig. 4.47, plot C. This plot shows the variation of G vs. 'Ye at 2.4-m depth for a mean effective confining pressure of l.Okg/cm 2• 4.9.~-

.

-;;..

~

1. In order to determine the in' situ dynamic properties of a soil at a proposed site for a machine foundation 10m X 8 m, the following tests were performed: (a) Vertical vibration tests on a block 1.5 m x 0.75 m x 0.70 m with different settings of the oscillator's eccentric masses. The data obtained are shown in Table 4.14, column 1 through 4. · (b) Cyclic plate load on a plate 30.5 em X 30.5 em. The elastic settlement corresponding to a load intensity of 1.0 kg/ cm 2 is 2.4 mm, which gives a c" value of 4.20 kg/cm 3 (c) Wave propagation test which gives an average value of the dynamic shear moduli, G ~ 335.5 kg/cm 2 The distance between the geophones is 5.0m. 2. The permissible amplitude of vibration is 1.0 mm. Static stress below 2 the foundation is 0.7kg/cm • 3. The water table is 2.5 m below the ground level. The unit weight of soil at the site is 1.7 g/cm 3, and the submerged unit weight is 1.05 g/cm 3 Find the value of C" for the design of the machine's foundation. The 3 assumed density of the•.,concrete in the test block is 2.4 tim •

Gmax

(2)

"'

~ 0.328 kg/cm 2

EXAMPLE

Table 4.13. G and Yo Values (Example 4.9.3) Shear G kg/em' Strain at 4.0-m G kg/em' at G depth" 1', 2.4-m depth'' (1)

~ C3

Gmax -1317 kg/cm 2 at 4.0-m depth. Gmax = 1121.3 kg/cm 2 at 2.4-m depth.

c. G vs. 'Ye at 2.4-m depth

'

N value at 2.4-m depth ~ 7

Shear wave velocity V, ~ 91(7) 0 ' 337 ~ 175.3 m/sec 2

G ~ (175.3) X 2.05 _ , - 6422 tim 9 . 81 ~ 642.2 kg/ cm 2

Solution

( 4.40)

(1) Block vibration test. By following the procedure illustrated in Example 4. 9 .3, one can analyze the block vibration test data. The values of C" for the area of the test block can be computed from the observed natural frequencies of the vertical vibrations. These values are shown in Table 4.14, column 5. The values of

175

EXAMPLES

the shear strain levels are shown in Table 4.14, column 7. The mean effective confining pressure ii0 below the test block can be computed in the2 same way, (as in Example 4.9.3) and its value is found to be 0.14 kg/cm The values of C" for ii0 = 1 kg/cm 2 are shown in Table 4.14, column 8.

Nt'--0\\0tn

--,.....,

oci0~MN ,.....,....,

"'

1

'
I

""

I

,....._,....,

""

I

(2) Cyclic plate load test. 3 The value of C" from the cyclic load test data is 1/0.24 = 4.2 kg/ cm The value of C" corresponding to the area of the test block can now be obtained.

.;-

I

,.-.,

0000 0 ,.....,.,....,

t:,

xxxxx

0

X

0'

~

N

c"'=~ C~.~1 Az

-----

3 Cuz = ~ 0.305 X 0.305 _ (1. X 0 _75 ) or , Cu 2 = 1.20 kg/em 5 42 The mean effective confining pressure at a depth of 30.5/2 = 15.25 em is found to be 0.732 kg/cm 2• The corresponding shear strain level is equal to 2 2.4/(30.5 X 10) = 7.86 X 10- 3• The value of C" for ii0 = 1 kg/cm is equal to 3 112 (1/0.732) (1.2) = 1.46 kg/cm

tnNOO-::tt---

00

00000 «::t«::t-::1"-::t«::t

00000

(4.30)

7

(3) Wave prbpagation test ·s· 2 From the wave propagation test, G = 335.5 kg/cm • The value of C" corresponding to the area of the test block is obtained from

ooc:!ifn~b

00o.rio.ri-.,i

C"

=

1.13£ (1- vz)

1

vA

( 4.29)

= 1.13 X 2G(1 + v) _1_ k /em' [1 - (0.33) 2 ] vA g

1!2 "'
Q,....;('f)('f)tr)

c:ic:iOOO

1.13

X

2 X 335.5 X (1 + 0.33) 1 2 (1- 0.33 ) Y150 X 75

= 10.67 kg/ cm 3

E

X

E u U

. o.

0<'>

"'

For the wave propagation test, the value of the dynamic shear modulus is representative of a depth of L/2, in which L is the distance between the geophones. The average effective confining pressure at a2 depth of 1/2(5/2) = 1.25 m is 1.7 x 1.25 X 100(2 X 0.5 + 1)/3 = 141.7 g/cm =i0.1417 kg/cm 2 2 The value of C" at a,~confining pressure of 1 kg/ cm is given by C" = 3 (1/0.1417) 0 · 5 x 10.67 = 28.41 kg/ cm • The shear strain level associated with the wave propagation is taken as 10-'.

(4) C" vs. 'Yo 2 The values of C" for a mean effective confining pressure of 1.0 kg/cm and for the area of the test block from the different tests and the corresponding shear strain levels are plotted in Fig. 4.48. 174

176

"'

DYNAMIC SOIL PROPERTIES 30

E

"' M

6

"" '"

~I

I

I

I

~

0

20

0

......

0

15

.E

~

0

. ro

10

·o

"'

w 0 u

o Resonance test

A Cyclic plate lo~d test

0

cw

I

I

I

I

A = 1.5 X 0.75 m 2

u

u

I

11o = 1 kg/cm 2

E

~

I

5

~

c---"- W~~~,P'opagat,on

0 10 6

I

177

and the value of C" corresponding to 0'0 = 0.726 kg/cm 2 is 17.5(0.72611)0 · 5 = 14.9 kg/cm 3•

c

E

I

25

·;;;

.E

I

OVERVIEW

I

I

-

10 5

I

I

(7) Correction for area The area of the machine foundation= 10 x 8 =80m 2• This is more than 10m2 The value of C., corresponding to 10m2 will therefore be used in the design (Barkan, 1962) and is given by

.~

I

w-3

lQ-4

C" = 14.9 (

I~w-2

1.5X0.75) 10

112

= 4.99 kg/em

3

The reader may note that no direct correction has been applied to the value of C" for effects of water table as is usually done when calculating the bearing capacity. However, this has been accounted for in calculating the mean effective confining pressure.

Strain level 'YO

Figure 4.48.

C., vs. 'Yo (Prakash and Puri, 1977) (Ex. 4.9.4).

4.1 o

(5) Value of C" needed for the design of the machine foundation The shear strain level associated with the machine operation, 'Yo, is (Prakash and Puri, 1977, 1981). 'Yo=

Amplitude Width

1.0 -4 'Yo- 8 X 1000 = 1. 25 X 10 From Fig. 4.48 the value of C" for 'Yo= 1.25 x 10- 4 is 17.5 kg/em' and corresponds to 0'0 = 1.0 kg/ em 2• This value of C" has to be corrected for the mean effective confining pressure below the foundation and the area of the block. ( 6) Correction for the confining pressure The ~ean effective confining pressure at a depth of 8/2 = 4.0 m below the foundatiOn block can be calculated as follows:

eTa= [(250 X 1.7 + 150 X 1.05) + 4 X 700 x ..

!J[ 2Ko3+ 1 ]

in which I is the influence factor obtained from Fig. 4.46, m = LJ z = 10/4=2.5, and n=814=2. The value of /=0.181. Therefore, rTo = ((250 X 1.7 + 150 X 1.05) + (4 X 700 X 0.181)][ 2 X 0; 5 + 1 2

= 726 g/cm = 0.726 kg/cm 2

J

OVERvfEw

Soil moduli under dynamic loads depend on soil characteristics, such as void ratio, relative density, stress history, preconsolidation pressure, confining pressure, and strain level. In machine foundations, the initial static stress level and pulsating stress level are generally low, and the number of stress pulses are very large. Hence, the combination of all three factors needs to be such that the soil will not experience plastic deformations or else the machine foundation will undergo progressive settlements and tilt and fail. Simple equations have been developed for use with available data to make preliminary estimates of soil mgsJ)l]i ..at.loJ1U.tmin.qmplitudes for sands and c~_,J§9..:..HJJJ In this equation, the value of K 0 , th~~o~fflclent of eaml pressure at rest, which is a function of the plasticity index and overconsolidation of clays, may be determined from Fig. 4.49. Therefore, depending upon the strain value associated in a particular machine faun· dation, a reasonable estimate of the soil modulus can be made. If the values · are determined at one confining pressure, the corresponding values at any other confining pressure can be determined with the help of Eq. (4.15). There is no universaLand unique relationship between soil moduli and strain. However, a nonillllized plot of moduli versus strain is an adequate guide for deciphering values from one site to another in similar type of soil conditions. Several laboratory and field methods for determining soil moduli and damping have been described. Time effects oo the moduli in clays have been described briefly, but more information on this particular aspect is needed.

Several comparisons have been made among the Gm,. values obtained from different tests. Cunny and Fry (1973) determined the values of Gm,.

DYNAMIC SOIL PROPERTIES

178

179

OVERVIEW

(4.54}

3. 0

v 2. 5

•

•

.............

/

';- ~32

1/ F

0

5~

v.

c

~

B 0.5

A

~

113

1---1----

--

) K,= 2o(N1''' N = Corrected value of N measured in SPT test delivering 60% of the I theoretical free-fall energy, ii0 = mean effective confining pressures psf.

r---

8

•

D

c

-~ 1.0

where

!-....

....

4

2 I

0

0

10

20

40 50 30 Plasticity index, P.l.

60

70

80

Figure 4.49. K0 as function of overconsolidation ratio and plasticity index. (After Brooker and Ireland, 1965; Lambe and Whitman, 1968.) Reprinted by permission of John Wiley, New York.

A correlation of standard penetration values with low-amplitude shear modulus in sands is particularly welcome, since the standard penetration test is invariably performed in all soil investigations. SPT values have been successfully correlated with liquefaction potential of soils. . The use of such correlations necessitated a considerable effort m Standardizing the otherwise non standard standard penetration test. (Seed et al. 1985; Skempton, 1986}. . . The detailed discussion on material and geometncal dampmg brought out the fact thah as in the resonant..•oolumn test, only material damping is obtained from laboratory tests, whereas in footing resonance tests, combined material and geometrical damping are obtained. . . Hardin and Drnevich (1972b} demonstrated that the matenal_<'."...l11_P!~g factor I; may be expressed as --~-----------~---.....-------·-<·--""

for 14 sites from laboratory and field tests. They used the steady-state, surface vibration method to evaluate G m"' in the field, but applied the resonant column test in the laboratory. The laboratory-determined shear and compression moduli were found to range within ±50% of the in-situ moduli. They observed that the cross-hole method would have given better values of shear wave velocity V, at the depths from which undisturbed samples were taken and that inclusion of the secondary time effect would have brought the laboratory cohesive-soil values nearer to the field values. The secondary time effect is negligible for sands. Stokoe and Richart (1973a, b) and Iwasaki and Tatsuoka (1977) found agreement between the resonant column and the cross-hole field test values. Prakash and Puri (1981), who obtained in-situ data on dynamic soil constants by making resonance tests on blocks, the shear modulus test, the wave propagation test, and the cyclic-plate-load test at several sites, reduced the modulus G values to a mean effective confining pressure ii0 of 100 kN /m 2 for purposes of comparison and suggested a method for rational evaluation of test data. They suggested that the value of dynamic shear modulus for analysis and design should be selected after taking into consideration the effect of important parameters influencing its value. Standard penetration values have been related with low strain shear

modulus (Gm,,) in psf for sands by Seed et al. (1986) by Eq. (4.54)

I;= l;m"(1-

cf-)

(4.55)

m"'

in which l;m,, is equal to 2k 1 17T and k 1 is ratio of the hatched area show? in Fig. 4.50 and the area of triangle abc, A (obc): There. are ma?y questiOns concerning the validity of this equatiOn, and 1ts use m a design problem. This indicates that more work needs to be done in this direction. Machine foundations generally are partially embed~ed when they are installed. This reduces the._ampiltuifeol motio-natThe resonant peaks,

increases the value of the resonant frequency, and i~creases t~e _e~~:~!~~e

damping. However, the effects on amplitude and frequency~~n··me tests depeiiaupon the mode of vibration and m~gmtude of the. motiOn as well. For motions within the range of des1gn cntena for machmery, 1t appears that thi~ reduction Jn '.l.~!?!~tude_~esulting from partial embed~e~t.Is~~-t-he order of 10 to 25 percent. The ti~~·-effe(iis- for' damping in clays were studed by Marcusson and Wahls ( 1977) with a Hardin oscillator in the laboratory part1~ularly to determine the time-dependent characteristics of the dampmg rat1o of !SOtropically consolidated specimens of kaolinite and calcmm bentomte. They used a steady-state method and the decay of free .vib~ations to obta~n the1r results. After completion of the primary consolidatiOn, they stud1ed the

180 DYNAMIC SOIL PROPERTIES

'

Gmax

a

a

'

-';+----__j d

Figure 4.50. Geometric relationship betw h and Drnevich, 1972b.) een s ear modulus and damping ratio. (After Hardin

d . . yna~mc response as a function of time for both drained d d . conditiOns. an un ramed

fo/~:~,J~i~~d a~hdat ;~ep~:~~:n1orat~o ~ecr~ased approximately 12 percent

dimensionless time ratio during se~on~n omte per logarithmic cycle of a mended that to evaluate th ~ry compressiOn. Thus, they recomshould be continued t fie efftects of lime m c!aye~ soils, at least one test . . o ve o ten limes the lime after th . e. ~nmary consohdatwn. Particular attention should be .d Instrument error when performi'ng I t pm to the possihJhty of . ong- erm tests. . The questiOn of dynamic prope t' k . . r Ies, I.e., strength parameters for earthqua e type loadmg has been discussed by Prakash (1981) No senous efforts have been d' t d · b Irec e tow~rd the determination of the Poisson's ratio of soils AI bet . . so etter correlatiOns need to be established nee~~:nbda~pmg dand the different factors that affect it, and further studies . Id e Irecte towards evaluatiOns of the . to vibrations of footings supported by 1 d_g~_<:>,:U"!!:!gt__al!lJllllg_I_e~ supported by soils that vary in stiffnes:y:~t~ ;.:pe t~a o:s welfil ~s of footings 1 · d (F' con mng pressure As alread field result ydeffxp ame . Igure 4.9), different tests in the laboratory and m t erent strams In tri · 1 t -can be developed Eff t h . b axia ests, generally intermediate strains or s ave een made to extend the strain ra . .

~=:~n~~::~:t~r:a:cE:::t: !:~! s~~ll ~~er~ediate

~~:~i~~

to values and in profile with depth from the spec! u,es. I e. etermmatwn of shear wave ra ana ysis of surface waves (SASW) appears to be a promising field method for the future. , Ladd and Dutko (1985) presented detailed testing procedure which was

OVERVIEW

181

used to determine the cyclic properties of soils at very small strains in a triaxial apparatus. Results of tests conducted using Monetery 0 sand were compared to test results obtained using resonant column apparatus. In 14 cases involving various soil types, the authors found that the ratio of the maximum shear modulus determined by either shear wave velocity measurements or resonant column apparatus to that determined using this testing methodology ranged between 0.55 to 1.26, with an average value of 0. 92. The authors have concluded, however, that the moduli and damping data obtained from resonant column apparatus are more reliable than those from cyclic triaxial apparatus, except for very dense soils. The unique benefit of using cyclic triaxial apparatus is that a complete stress-strain relationship as a function of strain can be obtained along with post-cyclic monotonic behavior. Dyvik and Madshus ( 1985) described in detail the installation and use of piezoceramic bender elements in a variety of standard geotechnical laboratory testing equipment, triaxial, direct simple shear and oedometer devices to measure G of the specimens during consolidation and/ or shear. For purposes of cg>p1parison, piezoceramiP bender elements were also installed in a resonant column device (Itardin oscillator). Simultaneous measurements of G m" were taken on five different clays at different confining stress levels and durations both for loading and unloading. The results of both techniques over a range of Gm., from 10 to 150 MPa (145.0 to 2175.0 psi) were in excellent agreement. By incorporating bender elements into these laboratory testing devices, more information is obtained on the variation of shear modulus at large strains.

In Equation (4.9), the shear modulus has been expressed as a function of iJ-0 , the mean effective confining pressures. However, Stokoe et a!. (1985) studied the effect of state of stress on the shear wave velocity and shear modulus of a dry sand in a large-scale triaxial device in which cubic soil samples measuring 7 ft on a side were loaded in isotropic, biaxial, and triaxial states of stress. Stiffness of the sand skeleton was evaluated by propagating shear waves along each principal stress axis of the sample at pressure of 10 to 40 psi. All testing was performed at !ow-amplitude strains (less than 0.001 percent) and at wave frequencies less than 3000Hz. Shear wave velocity was found to depend about equally on only two principal stresses; (1) in the direction of wave propagation and (2) in the direction of particle motion, and it,ras determined to be relatively independent of the third principal stress. ·· · Yu and Richart (1984) tested three sands under six different stress paths to study the effect of stress ratio on shear modulus of sands. They recommend modifying Hardin's equation [Eq. ( 4.9)] by replacing u 0 = (j-1 + a-, + a-, )1 /3 by

182

DYNAMIC SOIL PROPERTIES

in which a-. and ii6 are principal stresses in the direction of shear wave propagation and along the particle motion respectively. They also proposed two new equations for Gm", which depended on a function of principal stress ratios (ii1 /ii3 or ii,lii,). Thus, it would appear that our understanding of the factors affecting shear modulus is gradually improving, and some of the concepts presented in this chapter may need a revision as more information becomes available. Machine foundations may be supported on piles in some cases. The corresponding spring constants will depend upon the: (1) soil properties; (2) pile properties; (3) pile group geometry; and (4) mode of vibrations of piles. These properties are examined in detail in Chapter 12. PRACTICE PROBLEMS

1.

In a uniform deposit of sand with G = 2.7 and e = 0.7, estimate the variation of Gm" with depth from 1m to 6 m. Assume Poisson's ratio of 0.33 and water table at great depth.

2.

If the water table in problem 1 rises to the ground level, will the values

3.

of Gm., estimated above be altered? Estimate the new values. For a depth of 5 m and from the computed values of Gm" in problem 1, estimate and plot the variation of G with shear strain for strain range of 10- 6 to w - l

4.

What do you understand by (a) material damping (b) Coulomb damping (c) viscous damping, and (d) critical damping?

A clay has liquid limit of 60% and plastic limit of 30%. Estimate the variation of G m" with overconsolidation ratio of this clay at a standard confining pressure of 1 kg/ cm 2 if OCR varies from 1 to 32. 6. The shear wave velocity determined by torsional vibration of uniformly graded dry sand specimen in a resonant column device was 750 ftl sec. The longitudinal wave velocity of this soil is 1200 ft/sec. Determine: (a) Young's modulus and shear modulus if the void ratio and the specific gravity of soil solids of the specimen were 0.7 and 2.7 respectively. (b) Poison's ratio. 5.

Assume that the confining pressures (ii0 ) in both determinations was

1 t/ft 2•

7. List and discuss briefly laboratory and field methods for determination of shear modulii of soils at different strains. 8. Discuss the factors affecting soil modulii and damping. 9. In Figure 4.13, determine Ic and N 0 for 3-soils.

REFERENCES

10.

183

( ) If you had a choice to order either laboratory tests or field tests for

a determination of soil modulii for design of a machme foundatmn, . what would be your preference? Justify completely. (b) Write a small note ~n the future role of standard penetratiOn test in estimating the soil moduh1.

REFERENCES . h ar, t F · E ., Jr · (1973) · Stress history effects on shear modulus of soils. Afifi, S. E. A., an d R lC Soils Found. (Jpn.) 13(1), 77-95. "Is shear modulus of sot . Afifi S E. A., an d W 0 ds, R · D · (1971) . Long-term pressure effects 5on1460 j Mech. Found. Div. Am. Soc. Civ. Eng. 97(SM-10), 144 . . . . . . (1977) "St ndard test method for beanng capactty of , As~-D1l 4-(?2) (reapproved 1977). ASTM, American Societ~ for Testmg Matenals_ soil for static load on spread footmgs,

Soil

°

9

Philadephia, PA. . · An derson, D . G . (1974) . Dynamic modulus of cohesive soils. Ph.D. Dtssertatton,

u ·

't of

mverst y

Michigan, Ann Arbor. . --~ . d d ·1 on D G<~:· .and Stokoe, K. H ., 'Jr·-·11977) ,,, · Shear modulus: A ttme- epen en1 sot .. ASTM Spec. Tech. Pub/. STP 654, Dyn. Geotech. Test., 66-8. d r 9 d W d R D (1975). Comparison of field and laboratory shear mo u I. Anderson, D. G., an oo_ s, . .R l. h NC Am Soc Civ. Eng., Vol. 1, pp. 69-92. · · f Proc -In Situ Meas. Sod Prop., a etg ' . Woods R. D. (1976). Time dependent increase in shear modulus o Anderson, D. G., and .' c· E 102 (GT-5) 525-537. '. . . cla . I. Geotech. Eng. Dtv., Am. Soc. tv. ng. A d y D G Espana C and Me Iore, V· R · (1978). Estimatmg m sttu shear modulus p d at n erson, . ., , ., . S . lty Conf Earthquake Eng. Soil Dyn., asa ena, .. y, I 1 181-197. competent sites. Ge?tech. Eng. Dtv. pecta CA Am. Soc. Civ. Eng., June, Proc. Am. Soc. Czv. Eng. o. ' . . . , d Le K L (1977) Equivalent uniform cycle concept of soil dynamtcs. I. Annak1, M., an e, · · · ) 549 564 Geotech. Eng. Div., Am. Soc. Civ. Eng. 10 ~ (GT-6 ' ~ . t b in-situ vibration II d R .F. , Jr. (1964). Determination of sml Baar, . N shear 4691moduh at dep h y techniques. Waterways Exp. St. Mtsc. Pap. o. . . N y k . " McGraw-Htll, ew or .. D D. (1962). "Dynamics of Bases an d Foun dat tons. Barkan, . a fim'd-saturated porous sohd. I. . of elastic . waves m Biot, M. A.· {1956). Theory of propagatiOn A d n

~~~p;rty

Acoust Soc Am. 28, 168-191. . . h T . . 1 . . and . H enk e,I D · J · (1957) • "The Measurement of Soil Properties m T e naxxa Btshop, A. W., Test" Edwin Arnold, London, U.K. . ·k 1 . ,L., an d L an d ra, A . (1966). Direct simple shear tests on a Norwegtan qmc cay. BJerrum, Geotechnique 26(1), 1-20. h' t Brooker, E. W., and Irelan d-··H '*'•·~- · 0 · (1965). Earth pressures at rest related to stress IS ory. Can. Geotech. I. 2, No. L . · bTt J I S J (1977) Factors affecting liquefactiOn and cychc mo l l y. . (GT-6) 501-516. .· E Castro, G., and Pou_os, · · Geotech Eng. Dtv., Am. Soc. Ov. ng. 103 , . . . H h . W K (1976) Saturated sand and cyclic dynamtc tests. Cho, Y., Rizzo, P. C., and ump nes,E . . Ph"l d~lphia PA Meet. Prepr. 2752, 285-312. 1a Am. Soc. Civ. Eng., Ann. Conv. xpo., ' ' . . d dF Z B (1973). Vibratory in-situ and laboratory sml moduh compare .1. Conny, R. W., an rdy, D·. . Am Soc Civ Eng. 99{SM-12), 1055-1076. Soil Mech. Foun · tv., · · · p Pan De Mello, V. B. F. (1971). The standard penetration tes~: A state-of-the-art report. roc. Am. Conf. Soil Mech. Found. Eng., 4th, Puerto Rtco, VoL 1, 1-86.

1H5

REFERENCES 184

DYNAMIC SOil PROPERTIES

Doebclin, E. 0. (1966). "Measurement Systems: Application and Design." McGraw-Hill, Ne~ York. Drnevich, V. P. (1967). Effect of strain history on the dynamic properties of sand. Ph.D. Dissertation, University of Michigan, Ann Arbor. Drnevich, V. P. (1972). Undrained cyclic shear of saturated sand. J. Soil. Mech. Found. Div., Am., Soc. Civ. Eng. 98(SM-8), 807-825. Drnevich, V. P. (1977). Resonant column testing-Problems and solutions. ASTM Spec. Tech. Publ. STP 654, Dyn. Geotech. Test., 384-398. Drnevich, V. P., Hall, J. R., and Richart, F. E. (1967). Effects of amplitude of vibration on the shear modulus of sand. Proc. Int. Symp., Wave Propag. Dyn. Prop. Earth Mater., Albuquerque, NM, pp. 189-199. Drnevich, V. P., Hardin, B. 0., and Shippy, D. J. (1977). Modulus and damping of soils by the resonant~column method. ASTM Spec. Tech. Pub{. STP 654, Dyn. Geotech. Test., 91-125. Dyvik, R., and Madshus, C. (1985). Lab measurement of Gma~ using bender elements. Adv. Art Test. Soils Under Cyclic Dyn. Cond., Am. Soc. Civ. Eng. Conv. Detroit, 186-196. Fletcher, G. (1965). Standard penetration test: Its uses and abuses. J. Soil. Mech. Found. Div., Am. Soc. Civ. Eng. 91(SM-4), 67-75. Fry, Z. B. (1963). Development and evaluation of soil bearing capacity, foundations of structures. Waterways Exp. St. Tech. Rep. No. 3~622, Repr. No. 1. Fugro, Inc. (1977). "Sustained~Pressure studies," Fugro Tech. Dev. Program (unpublished). Hall, J. R., Jr. (1962). Effect of amplitude on damping and wave propagation in granular materials. Ph.D. Dissertation, University of Florida, Gainesville. Hall, J. R., Jr., and Richart, F. E., Jr. (1963). Dissipation of elastic wave energy in granular soils. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 89(SM-6), 27-56. Hardin, B. 0. (1965). The nature of damping in sands. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 91(SM-l), Part I, 63-97. Hardin, B. 0. (1970). Suggested methods of test for shear modulus and damping of soils by the resonant column. ASTM Spec. Tech. Pub/. STP 479, 516-529. Hardin, B. 0. (1971). Program of simple she
:'3·

(1969). Closure to vibration modulus of normally consolidated Div., Am. Soc. Civ. Eng. 9S(SM-6), 1531-1537. V. P. (1972a). Shear modulus and damping in soils, meaeffects. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng.

Hardin, B. 0., and Drnevich, V. P. (1972b). Shear modulus··and damping in soils, design equations and curves. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 98(SM-7), 667-692. Hvorslev, M. J., and Kaufman, R.I. (1952). Torsion shear apparatus and testing procedure. USAE Waterways Exp. Stn., Bull. 38, 1-76. Iida, K. (1938). The velocity of elastic waves in sand. Bull. Earthquake Res. Inst., Tokyo Imp. Univ. 16, 131-144. Iida, K. (1940). On the elastic properties of soil particularly in relation to its water content. Bull. Earthquake Res. Inst., Tokyo Imp. Univ. 18, 675-690.

Imai, T. (1977). Velocities of P~ and S-waves in subsurface layers of ground in Japan. Proc.lnt. Conf. Soil Mech. Found. 9th, Tokyo, Vol. 2, 257~260. . . M (1975) "The Relation of Mechanical Properttes of Sods toP- and · d y h' Ima 1 T an os tmura, · · U R 1 t Qya $:Wave Velocities for Soil Ground in Japan,'' Rep. RD-477, TN-07 . rawa es. ns ., Corp., Tokyo, Japan. . . Indian Standards Institution (1978). "Indian Standard Method of Test _for ~etermmatton of Dynamic, Properties of Soil," IS 5249-1978, 1st rev. lSI, New Delhi, Indm. Ireland, H. 0., Moretto, o., and Vargas, M. (1970). The dynamic penetraton test-A standard . that is not standardized. Geotechnique 20(2), 185. Ishibashi, 1., and Sherif, M. A. (1974). Soil liquefaction by torsional simple shear dcvtce. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. lOO(GT-8), 871-888. . Ishihara, K. (1971). Factors affecting dynamic Properties of soils. Proc. Asian Reg. Conf. Sotl Mech. Found. Eng. 4th, Bangkok, Vol. 2. . 'h K d L' s (1972) Liquefaction of saturated sand in triaxial torston shear test. t, . . Isht ara, . , an Soils Found (Jpn.) 12(2), 19-39. . . · d y d s (1975) Sand liquefaction in hollow cylinder torston under asu a, . · Ishthara, K., an irregular excitation. Soils Found. (Jpn.) 15(1), 45-59. . · M d rd K (1937) Determination of elastic constants of sotls by means of U · 15 67 Ishtmoto, ., an t a, · · vibration methods. Bull. Earthquake Res. lnst., Tokyo 1mp. mv. : · . . k' T d .T._atsuoka F (1977). Dynam .. ic soil properties with emphas1s on companson of Iwasa t, . , an ~ • · .··" '.:.."" h k E 6th New , laboratory t'eSts with field measuremetfis:, Proc. World Conf. Eart qua e ng., Delhi, Vol. I, 153-158. · F . d T k · y (1977). "Shear Moduli of Sands Under Cyclic M' · t Iwasaki T Tatsuoka, . , an a agt, · 1 · ' ·' 1 Sh L ear oa d•'ng ," Tech . Memo . No · 1264. Public Works Res. Inst., mts ry o T orstona Construction, Chiba-Shi, Japan. Kjellman, w. (1951). Testing of shear strength in Sweden. Geotechnique 2, 225-232. Ko, H.-Y., and Scott, R. F. (1968). Deformation of sand at failure. J. Soil Mech. Found. Div., Am. Soc. Civ: Eng. 94(SM-4), 883-898. Kovacs, w. D. (1975). Discussion of On dynamic shear moduli and Poisson's ratios of soil deposits Soils Found. (Jpn.) 15(1). . Evans 1 c and Griffith A. H. (1977a). Towards a more standardized SPT. , · ., ' 6 K ovacs, W . D ., Proc. Int. Conf. Soil Mech. Found. Eng., 9th, Tokyo, Vol. 2, 269-27 · 'ffith A. H. and Evans, J. C. (1977b). An alternate to the cathead and rope , , Kovacs, W. D ., Gn for the SPT. Am. Soc. Test. Mater. Geotech. Test. 1 1(2), 72-81. Ladd, R. s., and Dutko, P. (1985). Small strain measurements using triaxial app~ratus. Adv. Art Test. Soils Under Cyclic Dyn. Cond., Am. Soc. Civ. Eng. Conv., Detrott. 120-147. Ladd, R. s., and Silver, M. L. (1975). Discussion of Soil liquefaction by torsional simple shear device. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. lOl(GT-8), 827-829. Lambe, T. w., and Whitman, R. V. (1968). "Soil Mechanics," Wiley, New York. A (1962). "Foundation Engineering." McGraw-Hill, New York,. . d L eonards, G . · Lodde, P. J. (1977). Shear ·Bl,oduli and material damping ratios of San Francisco Bay mu · Master's Thesis, University of Texas, Austin. Lord, A. F., Jr., Curran, J. w., and Koerner, R. M. (19~6) ..New_ transducer system for determining dynamic mechanical properties and attenuatiOn m sotl. J. Acoust. Soc. Am.

60(2), 517-520. Lorenz, H. (1953). Elasticity and damping effects on oscillating bodies. Symp. Dyn. Test. Soils, Atlantic City, NJ, ASTM Spec. Tech. Pub!. STP 156, Part 2, 113-122. and Wahls H. E. (1.972). Time effects on dynamic shear modulus of ' . 9 1373 M arcuson, W. F ., Ill , clays. J. Soil Mech. Found. Div. Am. Soc. C.v. Eng. 98(SM-12), 135 ·

186

DYNAMIC SOil PROPERTIES

Marcuson, W. F., III, and Wahls, H. E. (1977). Effects of time on damping ratio of clays. ASTM Spec. Tech. Pub!. STP 654, Dyn. Geotecb. Test., 126-147. Mohr, H. A. (~966). Discussion of Standard penetration test: Its uses and abuses. J. Soil Mech. Found. D1v., Am. Soc. Civ. Eng. 92(SM-1), 196-199. Nazarian, S., Stokoe, K. H., r_r, a~d Hudson, W. R. (1983). Use of spectral analysis of surface wa_ves method for detennmatJon of moduli and thickness of pavement systems. TransportatiOn Research Board, Res. Rec. No. 930, 38-45. Nazarian, S., and Stokoe, K. H., II (1984). In situ shear wave velocities from spectral analysis of surface waves. Proc. Eighth World Con[. Earthquake Eng. San Francisco CA "ol III 31-38. ' ' • v• · •

Palacio~, A. (1977).

!he

theory and .meas~rem~nt of energy transfer during standard penetratiOn test sampling. Ph.D. Thests, Umvers1ty of Florida, Gainesville.

Peacock,."'!· H., and .seed, H. B. (1968). Sand liquefaction under cyclic loading simple shear conditions. J. Sod Mech. Found. Div., Am. Soc. Civ. Eng. 94(SM-3), 689~708. Peck, R. ~··Hansen, W. E., and Thornburn, T. H. (1974). "Foundation Engineering" 2nd ed. Wtley, New York. ' Prakash, S. (1975). Analysis and design of vibrating footings. Symp. Recent Dev. Anal. Soil . Behav., Sydney, 295-326. Prakash, S. (1981). "Soil Dynamics." McGraw-Hill, New York. Prakash, S. (1986). Future trend~ in geotechnical earthquake engineering research. Paper presented to the VIII Symposmm on Earthquake Engineering, Roorkee, December. Prakash, S., and Puri, V. K. (1977). Critical evaluation ofiS-5249-1969. J. Indian Geotech Soc 6(1), 43-56. . . . Prakash, S.; and Puri, V. K. (1980). "Dynamic Properties of Soils from In Situ Tests" (unpubhshed report). University of Missouri-Rolla, Rolla. Prakash, S.,.and Puri, V. K. (1981). Dynamic properties of soils from in situ tests. J. Geotech. . Eng. Dw., Am. Soc. Civ. Eng. 107 (GT-7), 943-963. Prakash, S.,. and Puri, V. K. (1984). Design of compressor foundations: Predictions and observattons. Int. Conf. Case Hist. Geotech. Eng., St. Louis, MO, Vol. 4, 1705~1710. Prakash.' S., Nandkumaran, P., and Joshi, V. H. (1973). Design and performance of an osclllatory shear box. J. Indian Geotech. Soc. 3(2), 101 ~ 112. Prakash, S., .Ran~an, <:'"·· .saran, S., Srivastava, L. S., and Singh, B. (1976). Report on geotechmcal. mv~stlgatiOn for Penden Cement Authority cement factory at Gornto, . Bhutan, Umverslty of Roorkee, Geotech. Eng. Stud. No. 110, Roorkee, May. Pun, V. ~· (1~8.4). Li~uefa.ction behavior and dynamic properties of Ioessial (silty) soils. Ph.D. . Thests, Ctvtl Engmeenng Department, University of Missouri-Rolla, Rolla. Rtchart, F. E., Jr. (1961). Closure to Foundation vibrations. J. Soil Mech Found n· A Soc. Civ. Eng. 87(SM-4), Part I, 169-178. · · IV., m. Richart, F. E., Jr. (1977). Dynamic stress-strain relations for soils, state-of-the-art report Proc. Int. Conf. Soil Mech. Found. Eng., 9th, Tokyo, Vol. 2, 605-612. . Richart, F ..E., .~r., H~ll, 1. R., and Woods, R. D. (1970). "Vibrations of Soils and Foundattons. Prentice-Hall, Englewood-Cliffs, New Jersey. Roscoe, K. H. (1~53). An apparatus for the application of simple shear to soil samples. Proc. Int. Conf. Sozl Mech. Found. Eng. 3rd, Zurich, VoL 1, 186~191.

Schmertmann,. J. H. (1975). Measurement of in situ shear strength. Proc. Spec. Conf. In Situ Meas. Soz/ Prop., Am. Soc. Civ. Eng., Rayleigh, NC, Vol. 2, 56-138. Schmertmann, J. H. (1977). Use the SPT to measure dynamic soil propertiesLYes but ASTM Spec. Tech. Pub/. STP 654, Dyn Geotech. Test., 341-355. . ' . Seed, H. B. (1976). Evaluation of soil liquefaction effects on level ground during earthquakes.

REFERENCES

187

State-of-the-Art Paper. Proc. Geotech. Eng. Div. Specialty Sess. Liquefaction Problems in Geotech. Eng. Am. Soc. Civ. Eng., Philadelphia PA Sept., 1~104. Seed, H. B. (1979). Soil liquefaction and cyclic mobility evaluation for level ground during earthquakes. J. Geotech. Eng. Div. Am. Soc. Civ. Eng. 105(GT~2), 201~255. Seed, H. B., and Peacock, W. H. (1971). Test procedure for measuring soil liquefaction characteristics. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 97(SM-8), 1099~1119. Seed, H. B., Idriss, I. M., and Arango, I. (1983). Evaluation of liquefaction potential using field performance data. 1. Geotech. Eng. Div. Am. Soc. Civ. Eng. 109(GT-3), 458-482. Seed, H. B., Pyke, R. M., and Martin, G. R. (1978). Effect of multidirectional shaking on pore pressure development in sands. J. Geotech. Eng. Div. Am. Soc. Civ. Eng., 104, (GT-1), 27-44. Seed, H. B., Tokimatsu, K., Harder, L. F., and Chung, R. M. (1984). "The Influence of SPT Procedures in Soil Liquefaction Resistance Evaluations," Rep. No. UCB/EERC 84-15. Earthquake Eng. Res. Cent., University of California, Berkeley. Seed, H. B., Tokimatsu, K., Harder, L. F. , and Chung, R. M. (1985). Influence of SPT procedures on soil liquefaction resistance evaluations. J. Geotech. Eng. Div. Am. Soc. Civ. Eng. IU(GT-12), 1425-1445. Seed, H. B., Wong, R. T., Idriss, I. M., and Tokimatsu, K. (1986). Moduli and damping factors for &Y.namic analysis of coheslp_11~ess soils. J. Geotech. Eng. Div. Am. Soc. Civ. Eng. 112(GT-11), 1016 to 1032. .. Shannon and Wilson~Agbabian Associates (1976). Geotechnical and strong motion earthquake data from U.S. accelerograph stations, Ferndale, Cholame, and El Centro, California. U.S. Nuclear Regulatory Commission, NUREG-0029, Vol. 1. Shannon, W. L., Yamane, G., and Dietrich, R J. (1959). Dynamic triaxial tests on sand. Proc. Pan Am. Conf. Soil Mech. Found. Eng., 1st, Mexico City, Vol. 1, 473-486. Silver, M. L. (1981). Load deformation and strength behavior of soils under dynamic loading. State-of-the-Art Paper. Proc. Int. Conf. Recent Adv. Geotech. Earthquake Eng. Soil Dyn., St. Louis, MO, VoL 3, 873-896. Skempton, A. W. (1986). Standard penetration test procedures and the effects of overburden pressure, relative density, particle size, aging and overconsolidation. Geotechnique, XXXVI(3), 425-448. Stephenson, R. W. (1977). Ultrasonic testing for determining dynamic soil moduli. ASTM Spec. Tech. Pub!. STP 654, Dyn. Geotech. Test., 179~195. Stevens, H. W. (1966). Measurement of the complex moduli and damping of soils under dynamic loads. U.S. Army Cold Reg. Res. Eng. Lab., Hanover, N. H., Tech. Pub!. No. I73. Stokoe, K. H., II, and Abdel-razzak, K. G. (1975). Shear moduli of two compacted fills. Proc. In Situ Meas. Soil Prop., Am. Soc. Civ. Eng., Raleigh, NC, Vol. 1, 422-447. Stokoe, K. H., II, and Hoar, R. J. (1978). Variables affecting in situ seismic measurements. Proc. Am. Soc. Civ. En~, Spec. Conf. Earthquake Eng. Soil Dyn., Pasadena, CA, Vol. 2, 919-939. . "• Stokoe, K. H., II, and Richart, F. E., Jr. (1973a). Shear moduli of soils: In situ and from laboratory measurements. Proc. World Conf. Earthquake Eng, 5th, Rome, Vol. 1, 356-359. Stokoe, K. H., II, and Richart, F. E., Jr. (1973b). In situ and laboratory shear wave velocities. Proc. Int. Conf. Soil Mech. Found. Eng., 8th, 1973, Moscow, Vol. 1, 403-409. Stokoe, K. H., II, and Woods, R. D. (1972). In situ shear wave velocity by cross-hole method. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 98(SM-5), 443-460. Stokoe, K. H., II, Lee, S. H. H., and Knox, D.P. (1985). Shear moduli measurements under

188

DYNAMIC SOIL PROPERTIES

true triaxial stresses. Adv. Art Test. Soils Under Cyclic Dyn Co d A S · n · m. oc. Civ. Eng. Conv., Detroit, 166--185.

Ta~lor, D. W, (1948). "Fundamentals of Soil Mechanics." Wiley, New York.

5

· Thters, G. R. and Seed H B (1968) C r p, d •. • · : · yc tc stress-stram characteristics of clay. 1. Soil Me h c · oun · Dtv., Am. Soc. Ctv. Eng. 94(SM~6), 555-569.

Trud::~· !~:i~e~~ytm~nBR. v., sand cc~ristian, J. T. (1974). Shear wave velocity and modulus

. · . aston oc. tv. Eng. 61(1), 12-25. Wetssman, G. F., and White S R (1961) D · · Spec Tech Pub[ STP 305 . S . S .,· ampmg capactty of some granular soils. ASTM . · · · , ymp. 01 Dyn., 45-59. Whtt~an, R. V. (1963). "Stress"Strain Time Behavior of Soil in One-D· · !·Co ston," Rep. R. 63-2S. De C' . tmensiOna mpres~ . P· tv. Eng., M.I.T., Cambndge, Massachusetts. WhitCm.an,ER. V., and Lawrence, F. V. (1963). Discussion. J. Soil Mech. Found Div A S IV. ng. 89(SM-5), 112-115. · ., m. oc.

Wils~~~n~thD~~o~~~ti~i~;i~~y.Rp,!~. ~~O)S Ef~~t ~ consolidation Soils, Boulder, CO, 419~

435 .

· oc.

tv.

pressure on elastic and ng. Res. Conf. Shear Strength Cohesive

Wolfe, W. E. Annaki M and Le K L ( 1977) · · ' ., a ' e, · · · Sotl liquefaction in cyclic cubic test pparatus. Proc. World Conf. Earthquake Eng. 6th, New Delhi Vol. 3 2151-2156 Woods R D (1978) M ' · ' So~ E "i. easurement of dynamic soil properties-State-of-the-Art. Proc Am · rv. ng. ,pee. Conf. Earthquake Eng. Soil Dyn., Pasadena CA Vol 1 9i-180. Woods, R. D. and Stokoe K H II (1985) Sh · · ' ' . ' · . ' · ·' · allow setsmtc exploration in soil dynamics R· h 1 C' zc ar ommemorattve Lectures R D Wood Ed"t p · · ~~~· c~~c~:dgmgDs otfa_tseossionbsponsored by the Geotechnical Engine~rin.g Division 120-156. ' · · · ·• e rm, cto er 23,

C· .

SAm

YangGe~~~c~n~~gat~:'a~ A.SW. (1~76).

Dynamic response of tropical marine limestone. J. m. oc. Ctv. Eng. 102(GT-2) 123-138 Yoshimi, Y. and Oh-Oka H (1973) A · · ' . ' nng torston apparatus for simple shear tests. Proc. Int. Conf. S~il Mech Fo , d. E · un · ng., oscow, Vol. 1 Pt 2 501-506 Yu P d R" · ' . ' . , G~o~~h ~char~· F. AE. (1984). ~tress ratio effects on shear modulus of dry sands. J. · ng. zv., m. Soc. Ctv. Eng. 110(3), 331-345. ·

·

IV.

M.

Unbalanced Forces for Design of Machine Foundations An essential requirement for adequate design of a machine foundation is that the motiqn. amplitudes underoperating conditions do not exceed the specified values. The vibration am!'ifit~des depend upon the natural frequency of the vibrating system, the operating frequency, and the magnitude of the applied dynamic forces and moments (Section 2.5). The information on magnitude and characteristics of the dynamic loads imposed by the machine on the foundation is thus vital for a satisfactory design of the machine foundation system. This information is generally supplied by the manufacturer of the machine and should be procured from him. This presents difficulty sometimes since the interests of the client and the manufacturer of the machine are not in unison, and the manufacturer of the machine may not like to admit that large unbalanced forces may occur from operation of the machine supplied by him. The situation is of course different for certain types of equipment such as compaction machinery, where unbalanced forces are purposely developed. The process of computation of unbalanced forces due to operation of reciprocating and rotary machines is discussed in this chapter.

5.1

UNBALANCED FORCES IN RECIPROCATING MACHINES

The simplest examp1~ of a reciprocating machine is the basic crank mechanism (Fig. 5.1) consisting of a piston, piston rod, connecting rod, and the crank rod. The crank mechanism is used to convert reciprocating motion into rotary motion or vice versa, and, is used in internal combustion engines, steam engines, reciprocating pumps, and compressors. The reciprocating machines may be of single-cylinder type or multicylinder type in which case several cylinders are mounted according to a definite pattern on a common crankshaft. Forces generated due to a single-cylinder reciprocating machine are considered first. 189

191

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

190

UNBALANCED FORCES IN RECIPROCATING MACHINES

(5.2)

OB = OC + CB = r cos 8 + l cos

where =Angle between the connecting rod and the X axis, and 8 =Angle between crank rod and X- aXIS. t Substituting for OB from Eq. (5.2) into Eq. (5.1), we ge x = r(1-cos 8) + 1(1-cos ) p

Cylinder

ll II

B

or

--X

II JJ0

p

From triangles AOC and ACB (Fig. 5.1)

%

I•

Xp

AC = r sin 8 = l sin

'I Letting 8 = "!I

-'

_t:--

r---c--~~----------l+c----------~

·:r

-;to'_,_~

sin

=

r . sm wt

Single-Cylinder Machines

The motion characteristics of the different components of the crank mechanism (Fig. 5.1) are as follows:

1 "'(1.!__ sin2 wt-- r4 sin4 cos '" 8 I 2 1' 2

axis

h. Different points on the crank rod undergo a rotational motion about the axis of rotation through 0. c. The connecting rod AB undergoes a complex motion, the end A has a circular motion, end B moves linearly and points between A and B move along elliptical paths.

OB may be calculated from Eq. (5.2)

(5.1)

(5.5)

)

4

wt

+ ...

)

may be

(5.6)

Neglecting higher powers of r/1 in Eq. (5.6), we get, cos = ( 1- ;;, sin' wt)

(5.7)

"' f E (57) into Eq. (5.3), we get Substituting the value of cos '"' rom q. .

Unbalanced forces associated with motion of each of these components of the crank mechanism will now be considered. Forces Due to Piston Motion. Let the crank rotate With a constant angular velocity w as shown (Fig. 5.1) causing displacement xP of the piston with reference to its extreme outward position. Let the length of the connecting rod be l and crank radius r. The displacement of the piston is given by

2 112

The value of r/1 is usually small (around 0.25) and Eq. (5.5) written in a series form [Eq. (5.6)]:

a. The piston and the piston rod undergo an oscillatory motion along X

= (r +I)- OB

(5.4)

l

_ ( _ . 2 "')'/2 = (1- (~ sin wt) :.cos- 1 sm '" I

Figure 5.1. Motion of a crank mechanism.

xP

(5.3)

x =r(1-coswt)+l(1-cos)

~P = r(1- cos

r2 . 2 wt) + 21 sm w1

(5.8)

Equation (5.8) may be rewritten as 2

xP = r( 1 _cos wt) +

~~ (1- cos 2wt)

or xP = r( 1 +

;I) _r(cos wt + ;I cos 2wt)

(5.9)

193 UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

192

UNBALANCED FORCES IN RECIPROCATING MACHINES

(5.14) Equation (5.8) shows that the piston motion is periodic but not necessarily harmonic. Equation (5.9) expresses the piston motion in terms of a primary component that varies with the frequency of rotation and a secondary component that varies at twice the frequency of rotation. The magnitude of the secondary component depends upon the r II ratio and if the length of the connecting rod is large compared to the crank radius r, the influence of the secondary component becomes negligible and the piston motion may be considered harmonic. The velocity xP and acceleration xp of the piston may be obtained by differentiating Eq. (5.9):

.

t. n of radius 0 A and may be resolved into compo-

F acts along t h e dnee IO n~nts along the directions of X and

z.

.

Forces Due :a

R d

The unbalanced forces due to

Mo:i~:e 0:a~::c~~~:~~d ~;y be determined with reasonable

;l sin 2wt)

(5.10)

complex mo wn o. f the connecting rod by a concentrated accuracy by replacmg the mass M" 0 M and M should be M at A and a mass Mh at B. Masses "' b . . :e~~rmi~~d such that the resultant unbalanced force due ~o th;~mou~n~s the same as due to motion of the connecting rod. Magmtu es. o "'an b may be determined from Eq. (5.15) and (5.16) respectively.

.. = rw '( cos wt + 7 r cos 2wt) xP

(5.11)

(5.15)

iP = rw (sin wt +

The inertia force FP due to translatory motion of the piston and piston rod is thus given by Eq. (5.12):

and

-;~-:-.·7

(5.16)

Ma2-f.:·MIJ=Mcr (5.12) where I,, I, in which MP =mass of piston, piston rod and cross head . Force Due to Motion of Crank Rod. The unbalanced force due to rotary motion of the crank will now be determined. The unbalance in the crankshaft may be replaced by an equivalent mass at the crankpin A so as to produce the same inertia force as due to motion of the crank. Since all points on the crank rod move in a circular path, equivalent mass can be determined from Eq. (5.13):

(5.13)

=

distances of A and B' respectively' from the mass centre M of

the connecting rod. · d ay be at A will be undergoing a rotary motiOn an m M h b 1 f e F due to T e mass a2 in combined with the mass of the crank M,,. The un a ance ore ..; M rotation of masses at A may be obtained by replacmg M,, by (Ma, ,,) Eq. (5.14): 2

FA =(Ma1 +Ma 2 )rw =MArw

wh~~~i~rly: :~!a~:~!a~:gm~;~e a:o~~idered along with th;,rto~rna(~_;;~ the inertia force Fn may be obtamed by addmg Mb to Fn = (Mb

in which M , 1 = Equivalent mass of crank rod lumped at A M, =mass of crank rod acting at its centre of gravity K r = crank radius

and r 1 = distances of center of gravity of crank rod from 0

The inertia (centrifugal) force F, due to rotation of mass M , 1 is given by Eq. (5.14):

(5.17)

2

P

m

q.

+ MP)rw'( cos wt + ~cos 2wt)

= "'fo.1 8 rw '(cos

wt +

I cos

2wt)

(5.18)

where M = total reciprocating in ass at B. . . . The f~ce FA at any time can be resolved into Its honzontal and vertical components given by 2

FAX = MArw cos wt

and

(5.19a)

194

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

Fz = M,\rw 2 sin wt

UNBALANCED FORCES IN RECIPROCATING MACHINES

z

(5.19b)

F,

~h';Je ~x atd F, are, A·

195

respectively, the horizontal and vertical components esn tmg force Fx m the direction of X is then given by

Fx =FAX+ FB - M

-

= MArw 2 cos wt + Msrw 2 ( cos wt + y cos 2wt) 2

2

Arw cos wt

2

r w + MBrw 2 cos wt + MB-1cos 2wt

(5.20)

It is observed from Eqs (5 19b) d ( 5 20 ) crank mechan. s . .. . an . that the operation of a single I m gives nse to unbalanced forces in the direction of piston motion F motion ~ a~ also. m a directiOn perpendicular to the direction of piston · (d . , · orce m the directiOn of piston motion has both bee;~~::~:/fr~~ ~ and secondary (depending on 2w) component1'~;';:~ of the iston m . q. (5.20). The force m a direction perpendicular to that . Pd f otwn has only a pnmary component [see Eq. (5 19b)] The magmtu e o pnmary and secondary components of F can be . . . obtamed by separatmg the terms containing w and 2w in Eq. (5.20):

pl)

Counterweights

F,

Figure 5.2.

F'x - M A rw 2 cos wt + M Brw 2 cos wt

(5.21)

--,. };,.,..

Principle of counterbalancing (both horizontal and vertical components due to

rotating mass and counterweights balance).

and

(5.22) · where F'x -- th e pnmary component of F and F"- th

of Fx.

x

x -

d

e secon ary component

The unbalanced force due to rotation f h t e mass M A can be balanced by installing a mass M' known • . A as counterwe 1ght at f . dtrectwn of radius 0 A (Fig 5 2 ) Th . an ang1e o 180° Wtth the e magmtude of M' d · d. · · · from 0 are selected so that it A an tts tstance 1 opposite to that produced bya "J.tays ~~duces a centrifugal force equal and forces is known as counterbalanci::. d s process of reducmg unbalanced reducing vibrations in certain c g a~ ts used as a remedial measure for resulting unbalanced force in thea~es. . hen co~nterbala~cing is done, the (5.23) [by making MA,; 0 in Eq. (~~~~)]~ of piston motiOn ts given by Eq.

°.

2

2

Fx-M rw cos 2wt srw 2 cos wt + Ms-I

(5.23) Multicylinder Engines

The primary component of Fx is then given by Eq. (5 .24): F; = Msrw 2 cos wt

The secondary component F; is unchanged and is given by Eq. (5.22). The unbalanced force in the direction of Z becomes zero after counterbalancing. As the primary unbalanced force in the direction of piston motion has the same frequency as that of the crank rotation Eq. (5.23), it is possible to install a mass on the crank to completely eliminate or minimize this force. However, the rotation of this counterbalancing mass will increase the force in a direciton perpendicular to the piston motion. Therefore, it is impossible to completely eliminate the unbalanced forces due to operation of a single cylinder reciprocating engine which is thus inherently unbalanced, and its operation gives rise to unbalanced forces both along and perpendicular to the direction of piston motion. The force in the direction of piston motion has a primary as well as secondary component, whereas the force perpendicular to the direction of piston motion has only the primary component. In the case of multicylind11r engines, however, it is possible to mount the cylinders so as to completely eliminate the unbalanced forces. The unbalanced forces due to operation of multicylinder engines are now considered.

(5.24)

A schematic layout of a horizontal multicylinder engine is shown in Fig . 5.3a. The cylinders are located in a plane parallel to xy plane and have a linear arrangement. The cylinders are parallel to each other. The crankshaft

197

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

196

UNBALANCED FORCES IN RECIPROCATING MACHINES 2

Y T

I I

1

Direction of

rShaft

Cylinder # i

tJJe---,---j

:

1'1

Y,

X

Y!

~q---L-i-n

F, = w'

y

x-j

cos 2(wt +.a,) (5.26)

2:"

(5.27a)

r,M Ai sin ( wt + a,)

i=l

Cylinder #1

I

~

in ':hichbalanced force in the z direction due to operation of ith cylinder, F,, =~~balanced force in the x direction due to operatiOn of tth cybnder. i~sultant exciting forces transmitted to the foundatiOn due to operatlon of all the n cylinders are given by

or1 ----~~~-------1~----------~

y

r,w' cos (wt +a,)+ MBi r,

z

--3]

I

ill

+ Ms,)

Fxi = (MA,

crank rotation

2

Fx = w'

i

r,[ (MAi

+ MsJ cos(wt +a,)+

1

Ms, cos 2(wt +a,)]

(5.27b)

i~l (a)

(b)

Figure 5.3. Schematic layout of (a) multicylinder horizontal engine in xy plane (piston motion in x direction, Engine shaft is parallel to y axis.) (b) Orientation of axes.

. rs ·27) may be written as · 1, E qs,,_~: . are 1"denuca If all cylinders F = rw'MA z

is parallel to Y axis and the crank rotation is in XZ plane with the origin of coordinates at 0, the combined mass center of gravity of the engine and the foundation. The arrangement of multicylinder engine under consideration will generate unbalanced forces in the directions of X and Z. The magnitude of unbalanced force due to any cylinder will depend upon the crank angle a which defines the relative position of pistons in different cylinders at any time. For example, a = 180" for a two-cylinder engine implies that when the piston in one of the cylinders is in the extreme right piston, the piston in the other cylinder is in the extreme left position. That is, they operate out of phase. The unbalanced forces due to operation of a multicylinder engine can be computed by following the principle that was used for the case of a single-cylinder engine and then determining the resultant force in a given direction as the algebraic sum of the components of forces due to all the cylinders in that direction. Depending upon the mounting arrangement of different cylinders in a multicylinder engine, its operation may result in the generation of either the unbalanced forces or unbalanced moments or both. Let the total number of cylinders be n. The unbalanced forces due to operation of the ith cylinder may be obtained from Eqs. (5.19b) and (5.20) by replacing MA with MA,, Mn with Mn 1, r by r, and changing wt to (wt +a;). This substitution gives (5.25)

2:" sin (wt +a,)

(5.28a)

i=l

and F =rw'f(MA +Ms) X

l

i; cos(wt+ a,)+ l Ms ~1 cos2(wt+ a,)} i~l

'

(5.28b)

The exciting moments due to these unbalanced forces are given by

"

(5.29a)

M="'FY ~ X

Zl

l

i=l

"

My ="'Fx L..J Zl

(5.29b)

I

i=l

"

M z ="'FY L..J XI I

i=l

where

M = Exciting moment about the axis of X X

M = Exciting moment about the y axis y

and

M = Exciting moment abont the Z-axis

'

(5.29c)

198

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

!he exciting moments will have primary and secondary components dependmg upon the ~ature of unbalanced forces generated by the machine. The conditiOns for balancing the fo~ces and moments generated by operatiOn ~fa honzontally mounted multiCylmder engine can be established by exammmg Eqs. (5.28) and (5.29). The primary component of exciting forces Will be balanced if [from Eqs. (5.28a and b)]

199

UNBALANCED FORCES IN RECIPROCATING MACHINES

Unbalanced moments and forces in any particular case may be calculated from the mounting details of cylinders and crank angles a. The computations become simple when all cylinders are identical and crank and connecting rods are of uniform cross section. When counterweights are used to balance the exciting force due to rotating mass M A, Eqs. (5.28) become (5.33a)

F, =0

n

2.: sin (wt +a,)= 0

t=l

(5.30a) Fx= rw

and

t=l

~

[

M 8 cos(wt+ a,)+

(5.30b)

8

cos2(wt+

a,)]

(5.33b)

n

F~ = rw 2 M 8

The secondary component will be balanced if

2.: [cos (wt +a,)]

(5.33c)

i=l

2 2

n

.2.: cos 2(wt +a,)= 0 i=l

IM

The primary and secondary components of Fx are then given by

n

2.: cos (wt +a,)= 0

2

n

"' F: = -rw- M 8 ,~[cos 2(wt +a,)] 1

(5.30c)

(5.33d)

-~;. ,' :>"

The primary component of the exciting moments will be balanced if Eqs · (5.31) are satisfied:

Substituting the value ofF, from Eqs. (5.33a) in Eqs. (5.29a,b), we get (5.34a)

n

.2.:

i=l

X, sin ( wt + a ) = 0

(5.31a)

Y, cos (wt +a,)= 0

(5.31b)

I

M y =0

n

.2.: 1=1 n

2.:

Y, sin (wt +a,)= 0

1=1

(5.31c)

Substituting the values ofF; and F: from Eqs. (5.33c,d) in Eqs. (5.29c), we get M;=LF;iyi

Y,cos2(wt+ a,)=O

(5.32)

The unbalanced exciting loads due to operation of reciprocating machine may be balanced dependmg on geometerical location of the equipment arid the crank angle. · So far, we have c6nsidered the unbalanced exciting loads due to operatiOn of honwntally mounted smgle or multicylinder engines. For vertically mounted engmes (piston motwn along Z axis), the exciting loads can be computed by mterchangmg X with Z in Eqs. (5.19)-(5.24) for singlecylmder engmes and m Eqs. (5.28) and (5.29) for multicylinder engines. For piston motion along Y axis, replace X by y in the above equations.

(5.35a)

i=l

n

M"="' Z L.,; F".Y XI I

n

1=1

(5.34c)

Also

n

In the case under consideration, only the force Fx has a secondary component and the secondary component of moment will be balanced if

.2.:

(5.34b)

(5.35b)

i=l

in which, M ~ = the primary component of M z and M~ = the secondary component of M,. . .It is observed from l!qs. (5.33) and (5.34) that in case of multicylinder engines in which the rotating masses have been counterbalanced we get only the exciting forces and moments given below: 1. Exciting force along the direction of piston motion, this force has both primary and secondary components 2. Exciting moment about an axis perpendicular to the plane of the engines; this moment has both primary and secondary components

200

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS UNBALANCED FORCES IN ROTARY MACHINES

It is also observed from Eqs. (5.33c) and (5.33d) that the maximum secondary force is equal to r/1 X (maximum primary force in that direction). The absolute values of the exciting forces and moments depend upon the cr~k :ngle a. For example, for the two-cylinder engine with crank angle a- 90 (Fig. 5.4) and rotatmg mass counterbalanced, we have, from equatiOnS (5.33), F, = 0

F~ = rw Mn[cos wt +cos (wt + 90°)] = rw 2Mn[cos wt- sin wt] 2

= V2rw 2M 8 cos (wt + 45°) = 1.41 x (primary force due to single cylinder) 2

201

Unbalanced primary and secondary forces and couples due to single and multicylinder engines for different crank arrangements are summarized in Table 5.1 (Newcomb, 1951). In case all the cylinders of a multicylinder engine are not identical, the exciting loads should be calculated using Eqs. (5.27) and (5.29). Besides the forces due to engine operation, there will be additional forces due to machines being operated by the engine or vice versa. The resultant exciting loads will influence the dynamic response of the foundation. The horizontal forces due to engine operation will also give exciting moments about the combined center of gravity of the machine foundation system.

(5.36a)

2

F"x = -r w- Mn[cos 2wt +cos 2(wt + 90)] 1 rzwz = - - M 8 [cos2wt- cos2wt] = 0

5.2

UNBALANCED FORCES IN ROTARY MACHINES

Exciting force due to rotation of an unbalanced shaft (Fig. 5.5a) is given (5.36b)

1

~~milarly, it can be shown that component of primary moment

by

M; is given

M; = 1.41 x (primary component of force due to single cylinder) x D

(5.36c)

(5.37) in which m =mass of rotor and e =effective eccentricity, that is, the distance of the mass center from the axis of rotation. The horizontal and vertical components of the unbalanced force are given by

where D =distance between center line of the pistons, and 2

2

Fx

7T)J

"- r w-M8 [ 2cos2wt-2cos2 D D ( wt+z M,--

1

2

= mew cos wt

(5.38a)

2

(5.38b)

and

2

r w2 D = - -2 M 8 [cos2wt+cos2wt] 1 rzwz = - - M nD cos 2wt 1

•

F z =mew sin wt

=(secondary force due to a single cylinder) x D'

(5.36d)

-----+D/2 ---0

High-speed rotary machines are well balanced and the eccentricity e is generally very small. However, due to their high speed of rotation, the magnitude of exciting loads may be significant. Further, the effective eccentricity may increase due to wear and tear, and balancing may be necessary to keep the unbalance within tolerable limits. When two rotary machines of the same capacity and having the same unbalance are coupled together, the unbalanced mass in each of them may be in-phase as shown in Fig. 5.5b, out-of-phase'•by 180° as shown in Fig. 5.5c, or at any phase, as shown in Fig. 5.5d. The unbalanced force for the case shown in Fig. 5.5b is given by

X

F= 2mew 2

D/2

(5.39a)

The vertical and horizontal components of the unbalanced force are given

by Figure 5.4.

Two cranks at 90°,

Fx = 2mew 2 cos wt

(5.39b)

:::"'

Table 5.1.

Unbalanced Forces and Couples for Different Crank Arrangements Crank Arrangements

Forces

Couples

Primary

Single crank

_A_

Two cranks at 180° In-line cylinders Opposed cylinders

Two cranks at 90°

k

~ 1~

y

Two cylinders on one crank Cylinders at 90°

Two cylinders on one crank Opposed cylinders

~

Three cranks at 120°

). Four cylinders Cranks at 180° Cranks at 90° Six cylinders

~

r J,llt ~

+~

~~

F' without

"' "' 0

Primary

Secondary

None

None

F"

counter wts. (0.5) F' with counter wts. 0

2F"

0

0

F'D without counter wts. F' T D with counter wts. Nil

(1.41) F' without counter wts.

0

(1.41) F' D without counter wts.

(0.707) F' with counter wts. F' without counter wts. 0 with

None

Nil

F'D

(0.707) F' D without counter wts . .

(1.41) F"

Nil

Nil

counter wts.

2F' without counter wts. F' with counter wts.

0

0

'"' .. 0

0

0

0

0

. 0

Source: Newcomb, 1951. Reprinted with the permission of the ASME. D = cylinder-center distance (in) F' = Primary force F" = Secondary force

Secondary

,.~,

None

(3.46) F' D without counter wts. (1.73) F' D with counter wts. 0 (1.41) F' D without counter wts. (0.707) F' D with counter wts.

Nil

(3.46) F' D

0

4.0 F"D

''-~

0

0

0

204

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

EXAMPLES

205

The components of the moment M in horizontal and vertical directions are given by Mx

= mew

2

l cos wt

2

Mz =mew 1sin wt

(5.39e) (5.39f)

X

When masses have an orientation as shown in Fig. 5.5d, the machine operation will give rise to both an unbalanced force and a moment. For design, the worst combination of loads should be assumed to be acting on the foundation. The unbalanced force will be given by Eq. (5.39a) and unbalanced moment by Eq. (5.39d). For more than two rotors on a common shaft, combined unbalanced forces and moments may be obtained in a similar manner.

(a)

(b)

5.3

Machines such as forging hammers, punch presses, and stamping machines produce inwact or pulse-type loads. There is no method to date (1987) to define the fbrces imposed on tile\"foundation due to an impact and the variation of these forces with time. This information may however be obtained experimentally. Experimental data on impact tests on 30-cmdiameter instrumented model footing (Drnevich and Hall, 1966; Lysmer and Richart, 1966) indicates that the shape and duration of the loading pulse varies with the energy of impact and energy absorbing characteristics of supporting medium such as elastic pads. The necessity to acquire such data on prototype foundations of this type cannot be overstressed. We have so far considered the nature of unbalanced exciting loads due to operation of different types of machine. The process of calculation of exciting loads will now be illustrated with some typical examples.

m m

.t'

UNBALANCED FORCES DUE TO IMPACT LOADS

lei

m

..._ '

•

. t' (d)

5.4

Figure_ 5.5. Unbalanced forces due to rotary machines. (a) single rotor; (b) two rotors with •: ual m phase u~balance; (c) two rotors with equal unbalance with a phase difference of 180°· 1 1 two rotors with equal unbalance at any phase. '

and (5.39c) For the case of rotors shown in Fig. 5.5c, the

res~ltant unbalanced forces

due to the two masses at any time cancel out, but there is a resulting moment M gtven by M = mew 2 (l)

(5.39d)

in which l = distance between the mass center of gravities of rotors as shown m Ftg. 5.5c.

EXAMPLES

EXAMPLE 5.4.1

A horizontal single cylinder reciprocating engine is mounted on a foundation block as shown in Fig. 5.6. The center line of the piston is 30 em above the top of the block and the center of the crank lies vertically above the mass center of the combined machine foundation system. The data on the engine is as follows~'' Weight of the piston and piston rod= 10.5 kg Weight of crank assumed concentrated at the crank pin= 4.6 kg Weight of connecting rod= 5.3 kg Length of connecting rod l = 45 em Crank radius r = 15 em Operating speed of engine = 600 rpm The connect;ng rod is of uniform cross section.

206

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

(a)

When counterweights are not installed

1. F~

207

EXAMPLES

, F"x

Force in the vertical direction (perpendicular to direction of piston motion) F, (5.19) Fz = MArw 2 sin wt

(FJm" = MArw 2 = 0.739(0.15)(62.83) 2 = 436.6 kg= 0.4366 t 2.

m

Force in horizontal direction (along the direction of piston motion) Fx The primary component of force F~ in the horizontal direction is given by (5.21) (F~)m" = (MA + M 8 )rw = (0.739 + 1.340)(0.15)(62.83) 2

Foundation block

Combined

e.G.

= 1231.0 kg= 1.231 t F~

The secondary component givenc?Y

in the horizontal direction is

-~·;·-::M r 2 w~

F" =- ·

Figure 5.6. Schematics of Problem 5.4.1.

X

2

(F~)m, =

Compute the unbalanced exciting forces and moments for the design of the foundation assuming:

3.

(a) No counterweights are installed (b) Counterweights are installed. Also find the magnitude of the counterweights Solution Total rotating mass at M A due to crank and connecting rod

cos 2wt 2

2

[

(0.45)

8

10.5 + H5.3) = ,.------;~';-'---'9.81

w

600(21T) = 62.83 rad/sec 60

(b)

(5.22)

k = 264 .5 g 0 .2645 t

Unbalanced excited moments The force F, passes through the mass center of gravity 0 and has no moment about it. The unbalanced force FY along the direciton of Y is zero and therefore the exciting moment Mx is zero. The only exciting moment in this problem is MY due to the primary and secondary components of Fx and tries to cause rotation about Y axis through the center of gravity. The primary component of moment M~ is given by (Fig. 5.6) =

F~h

= (1.231)(0.3 + 1.0) = 1.60 t m

M; = F:h = (0.2645)(0.3

= 1.340 kg sec 2/m

=

l

The secondary component of the moment

=0.739kgsec 2 /m . d T o t a I mass at B due to ptston an crank rod M

2

B

M 8 r w = (1.340)(0.15) (62.83) =

M~

4.6+ t{5.3) 9.81

Operating speed of the engine

2

M; is given by

+ 1.0) = 0.344 t m

When counterweigH'ts are installed 4. The mass of the counterweight depends upon its distance from the center of rotation. If the counterweight is installed at the same distance from the center of rotation as the mass M A, then M = 2 1 MA =0.739kgsec /m5.

Exciting force Fz

F, =0

208

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

.·. .!

209 EXAMPLES

6. Unbalanced force Fx in the direction of X The primary component

F~

z

is given by 2

F; = M 8 rw cos wt

(5.24)

(F~)m, = M 8 rw = (1.34)(0.15)(62.83) = 793.5 kg= 0.7935 t 2

2

(')

The secondary component F~ is unchanged and its value is equal to 0.2645 t as calculated in part (a). 7.

Exciting moment The primary component of exciting moment M ~ is given by M; = F~h = (0.7935)(1.3) = 1.0315 t m. The secondary component of the exciting moment remains unchanged and is equal to 0.344 t m as calculated in part (a).

M;

1.0 m

j_

EXAMPLE 5.4.2

In Example 5.4.1 if the engine was mounted vertically with XZ as the plane of crank rotation and the counterweights are installed, determine the exciting forces and moments for design of the foundation. Assume that the line of motion of the piston lies along the Z axis through the center of gravity of the engine and the foundation.

(b)

Figure 5.7.

Solution Data is as given in Example 5 .4.1. The direction of the piston motion is vertical and along Z axis and counterbalancing has been done:

Unbalanced force along the direction of piston motion: F,. The primary component of unbalanced force F, is obtained from Eq. (5.24) by replacing X by Z, F; = M 8 rw 2 cos wt

(a)

2

(F;)m, = M 8 rw = (1.34)(0.15)(62.83) 2 = 793.5 kg= 0.7935 t F~

The secondary component of force 2 (M 8 r w 2/l) cos 2wt [Eq. (5.22)] "

2

r w

2

(FJm" = MB - 1- =

2

is similary given by

F~

=

2

(1.340)(0.15) (62.83) (0.45) = 264.5 kg= 0.2645 t

(b)

Unbalanced force in a direction perpendicular to piston motion

(c)

Because of counterbalancing, Fx = 0 Unbalanced exciting moments. As F; and F~ pass through 0 and Fx = 0, no unbalanced moments are generated and Mx = My = M, = 0.

EXAMPLE 5.4.3

A two-cylinder engine is mounted on a foundation so that the direction of piston motion is parallel to Z axis and plane of crank rotation is XZ as shown in Fig. 5.7. The engine is mounted symmetrically with respect to the

t f

(a) Arrangemen o era

nks a ,

= 180°·

(b) Schematic sketch showing line of action

of exciting loads.

foundation. The two cylinders are identical in all respects. The data for the · d ers ts · as gtven · below · The connecting rods are of umform cross cyhn section. The crank angle a = 180". Weight of piston and piston rod= 5.2 kg Weight of crank assumed at crank pm = 2.1 kg Weight of connecting rod= 4.2 kg Length of connecting rod l = 45 em Crank radius r = 15 em Operating speed of engine = 1500 rpm Compute the unbalanced exciting loads for design of the foundation for the engine. Assume n~~~counterbalancing has been done. Solution

4.2) 2.1+ ( T ' · M=0428kgsec/m Total rotatmg mass A 9.81 · 4.2) 5.2+ ( 2 2 · · =0 ·744kgsec /m Total rectprocatmg mass M n -9.81

210

UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS

Operating speed of engine=

w

=

2

'lT(~~OO)

= 157.07 sec- 1

REFERENCES

Unbalanced forces Fx 1. Unbalanced force Fx 1 due to cylinder #1

(c)

2.

Crank angle a = 1800 (a)

Fx 1 = M Arw sm

Unbalanced forces and moments

Fx 1 = (0.428)(0.15)(157.07) sin wt = 1583.8 (sin wt) kg

The direction of piston motion is along Z axis. The unbalanced forces replacmg X With Z m Eqs. (5.21) and (5.22). Similarly the unbalanced forces Fx perpendicular to direction of piston motion are obtained by mterchangmg Z and X in Eq. (5.19b).

= 1.5838(sin wt) t 2.

1.

Primary component

F;, = (MA + Ms)rw

2

F;,

cos wt

(From 5.21) (cos wt) kg

= 4.3771 (cos wt) t 2.

F;,

Secondary component F 11

zl

2

=

r w MB - 1

due to cylinder #1

(d) Unbalanced moments The only exciting moments in this case results from the fact that F;, and F' are equal in magnitude but act in opposite directions and thus form :couple that causes rotation about y axis. The moment of this primary couple M; is given:l>r' (Fig. 5.7) M; = F;,(D) = 4.377(0.5) cos wt = 2.188 cos wt t m

2

COS

Resultant forces Fx Fx = Fx 1 + Fx 2 = 1.5838 (sin wt- sin wt) = 0

due to cylinder #1

= (0.428 + 0.744)(0.15)(157.07) 2 cos wt = 4337.1

Unbalanced force Fx 2 due to cylinder #2

Fx 2 = (0.428)(0.15)(157.07)' sin (wt + 180") = 1.5838(-sin wt) t 3.

F;

Unbalanced forces

(From 5.19b)

wt 2

F, along the direction of piston motion in this case are obtained by

(b)

211

The foundation should be designed for the following exciting forces and

2wt

(From 5.22)

2

2 0 ) (0.15) = ( .744 (0. 4S) (157.D7) cos 2wt = 9.17 (cos 2wt) kg

moments: F, = F; = 1.8354 t (secondary)

MY= M; = 2.188 t m (primary)

= 0.9177 (cos 2wt) t 3.

Primary component F::Z due to cylinder #2

REFERENCES

F;, = (0.428 + 0.744)(0.15)(157.07) 2 cos(wt + 180") Drnevich, V. P., and Hall, J. R., Jr. (1966). Transient loading tests on a circular footing. J. Soil

= 4.3771( -cos wt) t 4.

Secondary component F'::,_ due to cylinder #2 2

F'::,_ = (0.744)

~~·.~~))

(157,07) 2 cos 2(wt + 180")

=0.9177 (cos2wt)t 5.

Resultant primary component

F;

F; = F;, + F;, = 4.3771 6.

(cos wt- cos wt) = 0

Resultant secondary component F~ = F~ 1 + F~

=

0.9177(cos 2wt) + 0.9177 cos 2wt

= 1.8354 cos 2wt t (F;)"'"' = 1.8354 t

Mech. Found. Div., Am. Soc. Civ. Eng. 92 (SM-6), 153-157. Lysmer, J., and Richart, F. E., Jr. (1966). Dynamic response of footings to vertical loading. I. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 92 (SM-1), 65-91. Newcomb, W.K. (1951). Principles of foundation design for engines and compressors. Trans. ASME 73, 307-312, 313-318.

MODES OF VIBRATION OF A RIGID FOUNDATION BLOCK

6

Reciprocating machines are probably the oldest machines used by mankind. The classical example is a crank mechanism (Fig. 5.1) which is used to convert translatory motion into mtary motion and vice versa (Section 5.1). Typical examples of reciprocating machines are steam engines, internal combustion engines (e.g., petrol, diesel, and gas engines), pumps and · compressors. These machines may consist of a single cylinder and a piston which may be single acting or double acting, may consist of multicylinders with pistons operating in a regular pattern and mounted on a common crank. The pumps and compressors belonging to the category of reciprocating machines may be of the single-stage or multiple-stage type depending on whether the total compression is developed in one or more than one operation. The multiple-stage arrangement may consist of either several piston-cylinders operated by a common engine or several different engines. The direction of piston movement may be horizontal or vertical. Most reciprocating machines have operating speeds that are smaller than 12001500 rpm. Reciprocating machines operating at higher speeds are sometimes encountered. The foundations for reciprocating machines usually consist of rigid concrete blocks that have openings for mounting the machines. The machines may be mounted directly on a concrete block or on suitably designed elastic pads. Block foundations resting on springs are also sometimes used.

DESIGN REQUIREMENTS

The general criteria for ensuring long-term satisfactory performance of a machine foundation with respect to static and dynamic stability are set forth

in Chapter 1 (Section 1.2}. The foundation requirements for reciprocating machines with respect to dynamic stability are as follows: 212

There should be at least 30 percent difference between the operating speed of the machine and the natural frequency of the soil foundation system. 2. The amplitudes of vibration must be less than the specified permissible values. 3. In the soil, the resultant stresses occasioned by the combined action of static and dynamic loads should not exceed the permissible values.

1.

Foundations for Reciprocating Machines

6.1

213

The natural frequency of the foundation soil system is strongly influenced by the base contact area of the foundation, its geometry, mass, depth of embedment, and stiffness and damping properties of the soil. Besides these factors, the amplitudes of vibration are influenced by the unbalanced forces and moments associated with the machine's operation. It is mentioned in Chapter 4 that soils undergo increasing strains when subjected to combined static and dynamic loads, but if the magnitude of dynamic load is very small compared to the static load, as is the case with most reciprocating machines, the increase in strain after the first few applications of the dynamic load is negligible. Therefore a soil may b.e·:'!ssumed to behave as an elastic material under such loading conditions (Fig. 4.18). The fact that permissible static soil pressures below machine foundations are generally smaller than the corresponding soil pressures beneath ordinary footings, helps to ensure the above behavior. Therefore, the residual or plastic settlement of a soli below an adequately designed machine foundation is generally negligible.

6.2

MODES OF VIBRATION OF A RIGID FOUNDATION BLOCK

It may be assumed that rigid foundation experiences only rigid body displacements. Therefore, under the influence of superimposed forces and moments a rigid concrete block can vibrate in six dtfferent modes (Ftg. 6.1): 1.

2. 3. 4.

5. 6.

Translation along the Z axis Translation along the X axis Translation along the Y axis Rotation around the X axis (pitching) Rotation around Jhe Y axis (rocking) Rotation around"the Z axis (yawing)

Any movement of the block can be resolved into these six independent displacements. Hence, the block has six degrees of freedom (or modes of vibration) and six corresponding natural frequenctes. Of the stx modes, translation along the Z axis and rotation around the Z axis can occur independently of any other motion. However, translation along the X or Y axis and corresponding rotation about the Y or X axis, respectively, always

214

FOUNDATIONS FOR RECIPROCATING MACHINES Vertical

ELASTIC HALF-SPACE METHOD

215

V'

t

Torsion (yawing)

0'

0'

•o

~+X

Longitud ina I

z Figure 6.1.

METHODS OF ANALYSIS

Several methods are used in analyzing the vibration characteristics of block foundations, they are as follows: Elastic half-space method, 2. Linear elastic weightless spring method 3. Linear elastic weighted spring method, and 4. Empirical methods 1.

Because only the first two are used in practice, these are described in detail here. ··

6.4

~

~

00

Modes of vibration of a rigid block foundation.

occur together and are called coupled modes. Therefore, in analyzing rigid block foundations, one is concerned with four types of motion of which two (translation along the Z axis and rotation around the z axis) are independent, and two (translation along the X axis and rotation around the Y axis and vice versa) are coupled and occur simultaneously. The nature of soil reactions that come into play for the different modes of vibration are different, as already discussed in Section 4.4. 6.3

w

Figure 6.2. Lamb's problems for steady-s!ate o~cillating force or ~ulse loa_ding acting at a point (three-dimensional) or along a line (two-dimensional). (a) For verhcalload•ng at the surfa.ce. (b) For horizontal loading at the surface. (c) For vertical loading within the body. (d) For horazon~al loading within the body. (After Lamb, 1904. After Richart, Hall, and Woods, "Vibrations of S?ds and foundations/'© (1970, p. 193. Reprinted by permission of PrenticeRHall, Englewood Chffs, New Jersey.)

tropic, sellf\cinfinite body, whi~~_,is referred to as an elastic half-spac~. Mathematical solutions for computJng the response of a footmg vibratmg m different modes have been obtained by several investigators. Lamb's (1904) study of the response of an elastic half-space excited by a periodic vertical force acting along a vertical axis is the first investigatiOn m this area. This problem, which is also known as "dynamic Boussinesq loading", was analyzed as a two-dimensional wave propagation case. The study was then extended to cover conditions of vibratiOn occasiOned by a honzontal oscillating force acting on the surface and by a vertical or horizontal line load acting at any point within an elastic medium. These cases are Ill~strated m Fig. 6.2. Lamb also demonstrated how a series of vertical penodic forces havmg different frequencies could be combined and replaced by a single force acting on the surface. Lamb's solution covered hoth steady-state .and transient cases for calculation of surface displacement. Lamb also obtamed solutions for the response of a vibrating foundation by considering it as a problem of three-dimensional wave propagation. · Vertical Vibrations of a Footing

The problem of a ~brating rigid, circular footing resting on the_ surface of an elastic half-space was exammed by Reissner (1936, 1937). His solution for the vertical displacement at the center of the footing was based upon the solution obtained by Lamb in 1904. In this problem, the vertical displacement can be expressed by

ELASTIC HALF-SPACE METHOD

(6.1)

The elastic half-space method idealizes the machine foundation as a vibrating mechanical oscillator with a circular base resting on the surface of the ground. The ground is assumed to be an elastic, homogeneous, iso-

in which

216

FOUNDATIONS FOR RECIPROCATING MACHINES ELASTIC HALF-SPACE METHOD

P, =the magnitude of the oscillating force, w =forcing frequency (rad/sec)

3. Parabolic

G =the dynamic shear modulus of the medium r, = radius of the footing f1 and [, = Reissner's displacement functions.

(6.6)

Reissner (1936) observed that the displacement functions t, and /; were v of the medium and the fre~uenc/ of the exc1tmg force. The dimensionless terms were defined by him as follows:

depe~dent on the Poissons' ratio a = wr

,

/p = wro- 2wfro

oVa

v -1.7""" ·'

and

m

'

(6.2)

W

b=-=P r'o Ysro'

217

(6.3)

in which

ao =dimensionless frequency ratio

V, = shear wave velocity m =mass of the footing (including the mass of the machine) b =mass ratio y,. = unit weight of soil p = unit mass density of soil

Figure 6.3 shows the amplitude frequency response of a typical footing for the three types of pressure distribution for b = 5 and v = ~. Parabolic and uniform pressure distributions produced a higher displacement than a rigid base. This shows the importance of pressure distribution below the base of the footing (Richart and Whitman, 1967). Improvement on computed response can be made if the amplitude response is based upon a weighted average vertical displacement (Housner and Castellani, 1969). The effect of change in the Poisson's ratio of the elastic half-space on tbe steady-state vibration response of the footing for the case of a rigid base pressure distribution is shown in Fig. 6.4 (Richart and Whitman, 1967). The valups of the displacementJunctions, [, andf2 , Eq. (6.1) for values of v = 0.25, 0~33 and 0.5 for tbe ihr"ee types of pressure distribution and for a0 = 0 to 1.5 were obtained by Sung (1953b). The amplitude frequency response for the case of constant force excitation and frequency-dependent excitation and for different values of mass ratio b, and for v = 0.25 are shown in Fig. 6.5 (Richart, 1962). Figure 6.5 brings out the effect of mass ratio on the peak response amplitude. A high mass ratio (greater height of footing and smaller contact radius) implies a large amplitude of vibration for

It can be seen from Eq. (6.3) that the mass ratio essentially describes the relatiOn between the mass of the vibrating footing and a certain mass of the elastic half-space. Reissners' solution was extended by Quinlan (1953) and Sung (1953a), both of whom considered the effect of three types of vertical contact pressures bel?w the base occasioned by an oscillating vertical force. The pressure distnbutiOns considered were

1. Rigid base (T

z

=

p

eiwt 0

27TroVr~-r2

az=O

(6.4)

for r>r 0

2. Uniform (6.5) Figure 6.3. Effect of pressure distribution on the theoretical response of vertically vibrating rigid footing (After Richart and Whitman, 1967.)

ELASTIC HALF-SPACE METHOD

Rigid

b

~I~

~

5

1.0

o~------t;------~------~-0.5

1.0

1.5

Figure 6.4. Effect of Poisson's rati th . I (After Richart and Whitman, 1967.)o on eorebca response of a vertically vibrating footing.

Vibration amplitude = Az

b~4o ~4

a given set of conditions. Another significant point revealed when one compares Figs. 6.5a and b with Figs. 2.10a and 2.13a is that the general shapes of the response curves are similar. The curves for the low values of mass ratio b correspond with the curves for the high damping ratios. This implies that the vertical vibrations of a rigid body on an elastic half-space are damped. This damping occurs as a result of the energy being dissipated into the elastic half-space by the elastic waves radiating away from the vibrating body. This loss of energy is referred to as geometrical damping (see Fig. 3.14). The values of the displacement functions, {1 and { 2 , were computed by Sung on the assumption that the pressure distribution remained unchanged with frequency. Bycroft (1956) evaluated the weighted average of dis, placements beneath the footing and established displacement functions for rigid base pressure distribution (Fig. 6.6). For the static case, a0 is equal to zero, and f 2 also equals zero, The static displacement, Z,, due to a vertical load P0 is given by

l

(6.7a)

~T

1//7/.////7/ Po

219

Equation (6.7a) helps to define the value of the equivalent spring constant k, for vertical vibrations as follows:

= constant

.;:I~

0

4Gr 0

(6.7b)

k, = (1- v)

".Ef u

"'•

~

..§

"E ro ~

0.3 . . - - - - . . - - - - - , - - - - - - - , 0.3

c•

.Q ~

c

•E

0.2

~ I

0

< •c

"g_

0.]

0.2

,

0

__..;>.._-----.j]/4

E 0

"

0 1.5

.___.::::..-c-~:-1112 0

114 112

1.5

(,)

. ~ ;•g~!e 6.5. Ampli~ude versus frequency relations for vertical oscillation of a rigid oo .;~g ~n an elast~c halfMspace (v = ! ). (a) For constant amplitude of exciting force exc1 mg orce amphtude dependent on exciting frequency. (After Richart, 1962 .) 218

circular (b) For ·

Frequency ratio, ao

Figure 6.6. Displacement functions for a rigid circular footing vibrating vertically on the surface of an elastic half~space. (After Rycroft, 1956.)

220

FOUNDATIONS FOR RECIPROCATING MACHINES

The basic solutions of Reissner were modified by Hsieh

Hsieh's Analog.

(19~2) for the purpose of obtammg an equation for vertical vibrations simllar to the equation for damped vibrations of a single-degree-of-freedom system. A ng1d, Circular, weightless disc of radius r resting on the surface of an elastic half-space was first considered (Fig. 6.7a). This disc was subjected to a vertical oscillating force

ELASTIC HALF-SPACE METHOD

in which

c.

0

Q = Qoeh<>~

221

Gro ( -[2 = --;;;--

!'1 + !'2

r; , 17':.

)

= -;;0

(

- [2

v Gp !'1 + !'2

)

and

(6.8)

(6.13)

The vertical displacement is given by

(6.9) By differentiating Eq. (6.9) with respect to time, one gets

Thus, both c, and k, · are dependent upon a 0 and v. Next Hsieh also considered a rigid cylindrical footing of total weight W placed on the surface of an elastic half-space and excited by a vertical periodic force P (Fig. 6.7b). The equation of motion of such a system is given by 2

dz wQ eiwt dt = G~ (if; - [,)

W d z g dt 2

( 6.10)

0

This leads to

(6.12)

=

P- Q

(6.14)

By substitut1hg Q from Eq. (6.1'i):into Eq. (6.14), one obtains

f 1 wz- f z dz ~

=

Qow Gr

(f'1 +f')e'w' 2

2

W d z dz P iwt --+ e 2 c -+kz=P= g dt ' ' dt 0

0

or =

(6.15)

Equation (6.15), which is known as Hsieh's Analog, illustrates that vertical vibrations of a footing on an elastic half-space can be represented in terms of an equivalent damped spring-mass model with the difference that both the spring constant and damping are frequency dependent.

~~ u; + t~l 0

(6.11)

Lysmer's Analog. Lysmer and Richart (1966) proposed a simplified mass-spring-dashpot analog known as Lysmer's Analog for calculating the response of a rigid circular footing subject to vertical oscillations. They also defined a new displacement function F as

p

Weightless rigid disk

Q

4

Block mass = m

F= - 1 -v

G, v,

p

B,

+z Q =

dz

Czdt + Ia)

kzz (b)

Figu~e 6:7. Parameters in Hsieh's equations. (After Hsieh, 1962. Published by Thomas Telfo d

Publications.)

.

F 1 + tF2

(6.16)

The components of F are practically independent of v, as shown in Fig. 6.8. Lysmer also defise,d a modified mass ratio as

Q G, v, p

f=

r

1-v

1-v m

= -4- b = -4- - , pro

(6.17)

in which B.= modified mass ratio for vertical vibrations. By using the values of F and B, Lysmer and Richart (1966) developed the response curves shown in Fig. 6.9. The effect of frequency ratio on the variation in damping and spring factors was studied. It was observed that frequency-independent constant

223

ELASTIC HALF-SPACE METHOD

values of these quantities could be used in the frequency ranges of practical interest. The spring constant was taken as equal to its static value and is given by

'

t

4Grb k =-z 1- v

112

.;: I

114

""•

(6.18)

which is same as Eq. ( 6. 7b) and damping could be represented by

0

3 .4r; _ r-:::7"

c

(6.19)

c.= (1- v) v pG

0

~

E 0 u

By using the above values of spring and damping constants and the theory of vibrations (Chapter 2), natural frequency "'"' is determined as w

1.0

1.5

Dimensionless frequency, a0

~ii~~::.~-~~ ~riation of modified displacement function with Poisson's ratio. (After Lysmer and

6

nz

=

'V!7S T:i

(6.20)

The responss;; curves between the magnification factor, M, and a 0 , shown by the dotted lines in Fig. 6.9 wef<;'.''obtained. These are close to the exact solutions obtained with the elastic half-space model. The equation of motion for the Lysmer's Analog may thus be written as

3.4r! mi + ( _ v) 1

4Gr

vPGi + (1 _

~) z

=

P

(6.21)

The damping ratio ~' is obtained as 3r-----~r------,-------,------~ - - Half-space theory

0

"'G'""'-~

(v =

- - - - Simplified analog

""::; II

1 3}

2

Bz

=

or

5

::;,"

(6.22) Resonance occurs only when B, "'0.3, and the following approximate formulas for resonanct< condition for frequency-independent constant force excitation were establr~hed: w 2 Dimensionless frequency, a 0

;igure 6·?· ,Response of a rigid circular footing to a vertical force developed by a constant orce exc1tahon. (After Lysmer and Richart, 1966.)

222

nz

V, yB,-0.36 =-

ro

(6.23a)

Bz

and (6.23b)

225 224

FOUNDATIONS FOR RECIPROCATING MACHINES

The amplitude at operating frequency can be obtained by using theory of vibration [Chapter 2, Eq. (2.44b)]: A =

P,

' k,V (1- r

2 2 )

+ (2/j) 2

P, k,{[1- (wlw,,)']' + (2~,w/w,j}ll

2

(6.23c)

ELASTIC HALF-SPACE METHOD

7-8v

-m -

7-8v

w --,

B x = ~--=---,32(1- v) pr; - 32(1- v) 'YJ 0

(6.25)

The dimensionless frequency factor a ox is equal to wr"yprG. The expressions for the equivalent spring and damping factors are as follows: The equivalent spring (6.26)

For a frequency-dependent exciting force, which is normally the case with forces associated with machine operation, the resonant frequency is given by

(6.24a) The maximum vibration amplitude for frequency dependent exciting force is given by A

=

'

_m_,e c::-::~,;;B~,== m 0.85y B,- 0.18

and the equivalent damping _ 18.4(1- v) 2, f::(J ex7-Sv ravPv The damping ratio ~x is given by ~

(6.24b)

(6.27)

ex

0.2875

CC

~

=-=-X

(6.28)

The equation of the analog for sliding is in which me= unbalanced rotating mass and e =eccentricity _of mass "m " from the axis of rotation. e

The mass in the above analog is the total mass vibrating on the surface of the elastic half-space. The shape of the magnification factor M versus a for

(6.29) By comparison with Equations (2.37) and (2.11),

constant force excitation in Fig. 6.9 shows that the peaks at ;esonancea are

relatively flat, and significant damping is associated with the vertical mode of

(6.30)

vibration.

The curves for the magnification factors M x versus nondimensional frequen-

Sliding Vibrations of a Footing

As pointed out in Section 6.2, the sliding and rocking vibrations of a rigid block foundation are coupled and occur simultaneously, but for simplicity it IS necessary to study the cases of pure sliding and rocking vibration first. The information on natural frequencies of pure rocking and sliding is used to compute response of foundations undergoing simultaneous rocking and shdmg. It Will be shown later that the natural frequency in sliding alone is very close to the lower natural frequency of combined rocking and sliding. Arnold et al. (1955) and Bycroft (1956) presented analytical solutions for horizontal translation of a rigid circular disc resting on the surface of an elastic half-space and excited by a horizontal force,,~px = Ptiiwt. Their results were expressed in terms of the dimensionless frequency ratio a and mass ratio b; the solution was valid for all values of v. In a manne; similar to Lysmer's solution, Hall (1967) developed an analog between the elastic

cy factor aox derived through the elastic half-space solution are compared with the curves for the analog solution in Fig. 6.10 (Hall, 1967). The flat peaks on the curves in Fig. 6.10 indicate that the mode of vibration for horizontal sliding is also associated with significant damping as in the case of vertical vibrations. It may be mentioned here that the expressions for sliding along the y axis are also similar. Rocking Vibrations of a Rigid Circular Footing

The problem of pui'b rocking vibrations was analyzed by Arnold et al. (1955) and Bycroft (1956). They assumed that the vertical pressure below the footing varied according to (6.31)

half-space solution and an equivalent, damped-spring-mass system.

Hall's Analog.

Hall (1967) defined the modified mass ratio for sliding as

in which MY is the exciting moment in the ZX plane that causes rotation

227 FOUNDATIONS FOR RECIPROCATING MACHINES

226

ElASTIC HAlF-SPACE METHOD Mysinwt

Bx

=

r-"\

5 - - Exact solution

---Analog solution

- - Exact sqluf1on ---Analog solution

3

2

:£' ;'l"

"' c 0

"'•u

2

~

'2

00

•

" o.sot__L-~L-~-~-~o~.s~-L-~~-L-_L-~-~~k--~~

"''

1 frequency factor for rocking only of a Figure 6.11. Ma~nification f~ctorlvfersus l(~~ns•o~a~~s 1967. © 1968 The University of New rigid circular footmg on elastic ha -space. er ' Mexico Press.) d'

a~

Figure 6.1 0. Response of a rigid circular footing on an elastic half space for pure sliding. (After Hall, 1967. © 1968 The University of New Mexico Press.)

about tbe Y axis, and cf> is the angle of rotation. Figure 6.11 illustrates the geometry of the problem and sbows a plot of the magnification factor, M 1 vs. dimensionless frequency factor ao.P for different values of the inertia ratio B 1 (analogous to mass ratio in case of translation.) The inertia ratio Bq, is defined as

B = 3( 1 - V) _M_mo = ~ _M-"'m"'-o('--1_-_v-'-) 5 " 8 pr o 8 'Y, 5 -r

g

(6.32)

0

in which Mmo is the mass moment of inertia of the foundation and machine about the axis of rotation (in this case they axis, not shown). It may be seen in Fig. 6.11 that the response curves are characterized by relatively sharp peaks compared to the case of vertical vibrations. Hence smaller damping is associated with the rocking mode of vibration (see Fig. 4.40).

·

· . ring-dashpot model that . . . he Hall (1967) proposed an eqmva1ent mass-sp . could be used to evaluate the response of a rockmg ngtd footmg ond t b y surface of an elastic half-space. His analog for rockmg ts represente

(6.33) in which

kq, =spring constant for rocking cq, =damping constant

.,,

The terms k• and

c• c;n be computed as follows: 8Gr~

kq,

= 3(1- v)

(6.34a)

and Cq,

= (1-

v)(1 + B 4J

(6.34b)

229 228

FOUNDATIONS FOR RECIPROCATING MACHINES

ELASTIC HALF-SPACE METHOD

For critical damping, (6.35) and the damping ratio for rocking l;q, is given by

(; = ,S;_ = 1

0.15

(l+B,p)VB:;

c1"

The undamped natural frequency of rocking Wn
wn
(6.36)

is given by

={If.

(6.37)

mo

The analog solution is shown in Fig. 6.11 by dotted lines along with the elastic half-space solution and is in rather close agreement with it. Torsional Vibrations of a Rigid Circular Footing

0

The problem of torsional vibrations (rotation about the Z axis) of a circular footing resting on the surface of an elastic half-space was analyzed by Reissner (1937) and Reissner and Sagoci (1944). The horizontal displacement in the case of a rigid footing varies linearly from the axis of rotation as shown in Fig. 6.12. The inertia ratio, s., for this case may be defined as (6.38) in which Mm, =polar mass moment of inertia of the footing around the vertical axis of rotation. The analog solution for the case of torsional vibrations may be expressed as follows (Richart et al. 1970):

br:fifiro Gof=~

• . 1 f nc factor at resonance for torsional Figure. 6.12. lnert~ahratioHv~rsusd ~:~:·o~~i~ra;~~~eof ~oils and Foundations,"© 1970, P· vibrations (After RIC art, a an I d crff N Jersey) 215. Repr.inted by permission of Prentice-Hall, Englewoo ' s, ew .

The undamped natural frequency

wn.;

of the torsional vibrations is given by (6.41a)

The amplitude of vibrations A'' is given by

(6.39) in which tf; = angular rotation of the footing around the vertical axis of k~, =equivalent soil spring constant for torsional vibrations, and M,e'w' =horizontal exciting moment acting about the Z axis. The spring constant k 1, and the damping constant c 1, are given by (Richart and Whitman, 1967)

(6.41b)

rotation, c"' =damping constant for torsional vibratit?ns,

k,,

=

16

3

3

Gro

1.6r~y'Gp c1, = 1 + B,,,

( 6.40a) {6.40b)

in which the damping ratio (; 1• is given by I; = " (1

0.5

(6.42)

+ 2B,1.)

effective damping in case of As in the case for rocking vibrations, t he torsional vibrations is small (see Fig. 4.40).

230

FOUNDATIONS FOR RECIPROCATING MACHINES

Coupled Rocking and Sliding Vibrations of a Rigid Circular Footing

Equation (6.43) max then be written as

It has already been. stated (Section 6.2) that either rocking or sliding

Rx

alo~e IS

an tdeal condttton. In actuahty, the motion of a footing excited by a honzontal force or a vertical moment involves both rocking and sliding. Figure 6.13 tllustrates the conditions of a rigid circular footing that rests on the surface of an elastic half-space and is excited by a vertical moment, M,(t) = M,e'w' and a horizontal force Px(t) = Pxe'w' acting at its center of gravity. Let the footing's center of gravity lie on the vertical axis which passes through the center of the circular base, at a height L above the surface of the half-space. The motion of the footing may be expressed in terms of the translation of the center of gravity x and the rotation angle q,. The equation of motion may be obtained by considering the limiting eqmhbnum of the exciting and resisting forces and moments in terms of Newton's Second Law. The horizontal resisting forceR at the base is given X by (6.43) in which

X0

=displacement at the base and is given by

xo

=X-

(6.44)

Lcf>

231

ELASTIC HALF-SPACE METHOD

=

cxi + kxx- Lex¢- Lkxcf>

=

0

(6.45)

Similarly, the moment M R, which represents the moment due to the resistance of the elastic half-space, may be written as ( 6.46) The equation of motion for sliding is (6.47) arid rocking ( 6.48) in which M m us the mass momentp.frinertia of the foundation about an axis that passes through the system's center of gravity and perpendicular to the plane of vibrations. By substituting MR and Rx from Eqs. (6.46) and (6.45), respectively, in Eq. (6.48), one obtains

+ •

}i"

+x

~+Px (b)

I

/~

l_ 0 X

I

I

I

llr-

•

(a)

• 1-1

I

7

I I

I I r

11

'I

I!

!

~Fx 1:

(6.50a)

'N

I

I

~

I

r-

+

I I

J(--

and

0

.~

(6.50b)

l '/

-A'r£0 I

(c)

Figure 6.13. Coupled rocking and sliding vibrations of a rigid circular block on an elastic halfMspace. (After Richart and Whitman, 1967.)

Equations (6.47) and (6.49) demonstrate that coupling of the two motions, i.e. sliding and rocking, takes place because the center of gravity of the footing and the point at which the horizontal reactive force R x of the elastic half-space is applied are not the same. If L equals zero, there is no coupling effect, and sliding and rocking are independent. Particular solutions of Eqs. (6.47) aud (6.49) may be obtained by substituting

in wbich A and B are arbitrary constants. When a footing rests on an elastic half-space, the values of both the spring constant and damping coefficients are frequency dependent and must be calculated at any given frequency before the above equations can be

solved. However, if the spring constant and damping are assumed to have a frequency independent constant value as in the case of analog solutions for

232

FOUNDATIONS FOR RECIPROCATING MACHINES

sliding a~d rocking, Eqs. ( 6.47) and ( 6.49) can be easily solved. The natural freq~encws of coupled rocking and sliding are obtained by making the forcmg functiOns Px · e'w' and M/w' in Eqs. (6.47) and Eq. (6.49) e ual to zero. Th1s leads to: q ( 6.5la) and

ELASTIC HALF-SPACE METHOD

233

w, 1 and w, 2 are the two natural frequencies of the soil-foundation system undergoing vibrations due to combined rocking and sliding. The rocking and sliding may be in-phase or out-of-phase depending on the value of operating frequency w and the two natural frequencies w, 1 and w, 2 • This point is discussed in detail in Section 6.7. Damped amplitudes of rocking and sliding occasioned by an exciting moment MY can be cbtained as follows:

A

_ _M Y [(w2nx )2 + (2
x

(6.55a)

(6.51b) By substituting Eqs. (6.50a,b) into Eqs. (6.51a,b) and rearranging the terms, the frequency equat10n IS obtamed as given below (Prakash and Puri 1980, 1981b) ' 4 [ wnd

_

w2

2

w,d (

n

+4[~xwnxwnd 'Y

+ (I) nx 2

4< <

_

Sx S Wnx (t)n)

y

y

(w2 -w2 )+ nd

rt

2

2

W nx W n] + ~~""""""

(6.55b) The value of A(w 2 ) is obtained from Eq. (6.56)

2

y

~q,W,q,W,d

( 2-

y

(t)I!X

2 ]2-0

(l)nd)

(6.52) in which

Damped amplitudes of rocking and sliding occasioned by a horizontal force Px are given by Eq. 6.57(a,b) ( 6.53)

w,d = damped natural frequency in coupled rocking and sliding. ~x = damping ratio for sliding vibrations

1;1

=

px Ax=-M m m

damping ratio for rocking vibrations

If ~x

= {;1 = 0, that is, when there is no damping in the system then Eq. (6.52) reduces to ' 2

4-

w,

Wnx

2

w,

(

2

+ Wnq,) + y

2

2

W,xwn1>

y

=

0

(6.54a)

in which w, is the undamped natural frequency o"r the system. Because the effect of damping on natural frequency is small one may calculate the undamped natural frequency for coupled rocking and sliding by usmg Eq. (6.54a). Solving Eq. (6.54a) as a quadratic in w~, we get w2

nJ,2

1

= _

2y

[w 2 + (J)n(l)Nx] /IX

and

{J)/JX

(6.54b)

(6.57b) In case the footing is subjected to the action of a moment and a horizontal force, the resulting amplitudes of sliding and rocking may be obtained by adding the corresponding solutions from Eqs. (6.55) and (6.57). Rigorous solutions for Eqs. (6.47) and (6.49) can be obtained for both elastic half-space and Halls' analog by using numerical techniques on a high-speed digital computer. .For numerical solutions, frequency-dependent stiffness and damping can also be considered.

234

FOUNDATIONS FOR RECIPROCATING MACHINES

235

EFFECT OF FOOTING SHAPE ON VIBRATORY RESPONSE

So far in our discussion, we assumed that the footing has a circular contact area. Mostly the footings are either rectangular or square. We will now consider the effect of footing shape of its dynamic response. 0.16

6.5

EFFECT OF FOOTING SHAPE ON VIBRATORY RESPONSE

0.12

~ I

The natural frequency of a footing is influenced by its shape. Elastic half-space theory was developed for an oscillator with a circular contact area. Its applicability to footings of rectangular and square base areas has also been investigated. The problem involving vertical oscillations of a rigid, rectangular footing on the surface of an elastic half-space was analyzed by Sung (1953b), who used the earlier solution of Lamb (1904). Kobori (1962) and Thomson and Kobori (1963) obtained the displacement functions, [ 1 and / 2 for the displacement at the center of a uniformly loaded rectangular surface area. Elorduy et al. (1967) obtained solutions (in terms of the displacement functions f 1 and / 2 ) for vertically vibrating rectangular (alb= 2) and square (alb= 1) footings resting on the surface of elastic half-space for a typical case of v = ! and compared them with the solutions of Sung (1953a) and Bycroft (1956), who based their calculations on an equivalent circular area. Their solutions are shown in Fig. 6.14 for the cases of square and rectangular footings, respectively. Because these displacement functions are practically the same, it is acceptable to use solutions of a rigid circular base having an area equivalent to a given square or rectangular contact area for alb up to 2.0 for approximate response calculations. Similarly, for rocking or torsional vibrations of rectangular or square footings, one may compute an equivalent radius of a circular footing so that the moment of inertia of the given footing about the axis of rotation is the same as that of an equivalent circular footing about the same axis. Thus, the equivalent radius ro, may be calculated as follows: For translation along the Z, X, or Y axis,

5

..........

0.08

""

-(2

- - - Elorduy et ~1:-~igid rectangular}-._ 1 - - - - Bycroft-R1g1d c1rcular v - 4 --Sung-Rigid circular

0.24 ~------,-------,-------,

'0

=

ba') ( 37T

'

(6.58b)

'\;

:-"\

~ I

0

'Y-;:~-/

0.12

', "

<

'-

-fz

'

a

- ~ 2 b

' 1.0

0.5

+ Rectangle

'- '-' '

0

~12

ea/2 o/2

b/2

~

'\

-~---~

1.5

(b)

in which b = width of the foundation (parallel to the axis of rotation for rocking)

=r =(ab(a'61T+ b') )II'= (21")11' ol)t

" ~"(I"

11

For torsional vibrations about Z axis,

o

·''>

Fi ure 6 .14. Displacement functions for vertical. vibration. of rigid squar~ an~ rectangular fo!tings. (After Richart, Halk and Woods, 11 Vibrat1ons of Sm~s and Foundations, © 1970, P· 212. Reprinted by permission'-of Prentice-Hall, Englewood Chffs, New Jersey.)

For rocking about Y or X axis,

r

1.5

1.0

0.5

( 6.58a)

roo/= ro

..............................

1T

and

(6.58c)

a = length of the foundation (perpendicular to the axis of rotation for rocking).

236

FOUNDATIONS FOR RECIPROCATING MACHINES

For footings with length to width ratios greater than 6, an ideal twodimensional condition can be assumed and the footings treated as strip footings which can be analyzed by applying Quinlan's (1953) method. The above discussion implies that a footing of any shape whether it is a circle or a rectangle, will respond similarly as long as the areas of the footings are the same. A footing having an area of 8.0 m 2 will have an equivalent radius of 1.6 m for translational oscillations regardless of its shape, which may be a circle with a diameter of 3.2 m or a rectangle with length of width ratio of 4.0. However, the actual response of these two footings may not be the same. It has been found that two footings of different shapes will not behave identically, even though the equivalent radius based upon equal areas for translational modes are the same (Chae, 1969). In an accurate analysis, footing shape must be taken into account. Based upon his experimental observations (Chae, 1969) suggested that the concept of equivalent circular areas may be used to predict natural frequencies but that the perimeter characteristics should be taken into account for reasonable predictions of the amplitudes. The problem of vertical vibrations of rectangular footings has also been studied by Dasgupta and Rao (1978), who used a three-dimensional finite element model and three different vertical pressure distributions at the footing soil interface. However, their model is too complex for the design of ordinary machine foundations. Besides, no comparison with actual observations has been reported. It is a common practice to transform area of any shape to an equivalent circle of same area (for translational modes) or equivalent moment of inertia (for rocking or torsional modes) (Richart and Whitman, 1967; Whitman and Richart, 1967). Dobry and Gazetas (1986) and Dobry et a!. (1986) have suggested that this procedure of using the concept of equivalent radius has limitations and the foundation shape defined by the aspect ratio a/ b has a significant influence on dynamic stiffness and damping values, especially in cases of long foundations.

6.6

Vertical Vibrations

Bycroft (1956) considered the problem of a vertically vibrating rigid circular footing resting on an elastic layer. He. assum~d a ngtd~base-type distribution of vertical pressure below the footmg (this assumption IS not strictly correct) and computed values of average static_ displacement for different values of layer thickness ratio Hlr 0 in which HIS the th1ckness of the elastic layer as shown in Fig. 6.15. The values of the ratiO of average static displacement (Z,li for the layered case, to the value of a stahc displacement (Z,)~ when H-'>oo (elastic half-space) and the Hlro ratiO are plotted in Fig. 6.15. The values of the ratio of a spnng coeffiCient (k,), for the layered case, to the value of (k,)" for the elastic half-space, versus the H f r value are also plotted in the same figure. Th1s figure shows clearly the stiff~ning effect that an underlying rigid layer has on the vertlcal m~twn of a footing, which is obvious from the reduced statlc displacement and mc~eased spring coefficient. For large values of H/r 0 (>4), the hehavwr approx1mates that of a footing on an elastic half-space. For lower values of H fro ( <1), the motion is signijicantly affected by the stiffness of the_ underlymg ng1d layer. Bycroft (1956fconsidered the case of.a weightless ng1d Clfcular d1sc (B, :" 0) resting on the surface of a half-space, and Warburton (1957) _obtamed solutions for the case of B, > 0. Both of them noted that amphtudes of motion become infinite at resonance for the case of B 2 equal to zero. Th1s occurs because of the resonance of the layer itself , which acts as a column

VIBRATIONS OF A RIGID CIRCULAR FOOTING SUPPORTED BY AN ELASTIC LAYER \

In elastic half-space theory it is assumed that the medium 'is homogeneous. Actual soil deposits are layered and in certairi" cases, the response of a footing may be affected by the stiffness or rigidity of underlying layers of soil. The problem of a vibrating footing on a layered medium has received the attention of several investigators (Reissner, 1937; Warburton, 1957; Arnold eta!., 1955; Bycroft, 1956). It is assumed that the footing rests on the surface of an isotropic, homogeneous, elastic material of thickness H that extends to infinity horizontally. This layer rests on a semi-infinite body of infinite rigidity.

237

VIBRATIONS OF A RIGID CIRCULAR FOOTING

0,2

(kz)I,;,,I~

Y...... __(k,)oo

(kz}o;, =

'

4Gr0 (I - ,)

-----------OL__L__j2___L__4L_~--~6~-L--~8~_L~~~--~~~ H

'o' Figure 6.15.

Static displacement and spring constant for vertical loading of a rigid circular

footing on an elastic layer (After Rycroft, 1956.)

238

FOUNDATIONS FOR RECIPROCATING MACHINES

of elastic material fixed at the base, free at the top, and restrained against lateral deformations on the side. In this condition, it behaves like a rod of elastic material. The conditions involving the vibrations of such a column of soil have already been discussed in Chapter 3 (Section 3.1). Vibrations in higher modes are possible for such a case. For a footing which has a certain weight in which B, > 0, the amplitudes are finite at resonance, but are amplified by the underlying hard layer. The increase in amplitude at resonance occurs because the underlying hard layer obstructs the transmission of the wave energy away from the vibrating footing and reflects back a part of this energy into the layer. This results in reduced geometrical damping. The energy is finally dissipated by transmission in a horizontal direction. Kuhlemeyer (1969) discussed the transmission of wave energy into a layered medium. An estimate of the amplitude magnification at resonance for a vertically vibrating footing on anelastic layer is given in Table 6.1. The magnification factor, ML in Table 6.1 is defined as ( 6.59) in which (Z,) 1 =dynamic amplitude of the elastic layer and (Z,)oo =static deflection (elastic half-space). The term (Z,)oo is given by (Z) = (1- v)P, s oo 4Gro

239

VIBRATIONS Of A RIGID CIRCULAR FOOTING

It may be noted here tbat a hard rock underlying a relatively thin layer of elastic soil may cause large amplification of the vertical amplitudes because of the reflection of energy back into the soil. Special care must be taken m the design of a machine foundation for such a case.

Torsional Vibrations

The problem of torsional vibrations of a rigid circular footing on an elastic layer overlying a rigid layer was investigated by Re1ssner (1937), Arnold et al. (1955) and Bycroft (1956). The results provided by Bycroft (1956) on B, vs. a0 for different values of H/r are shown in Fig. 6.16. Based upon model tests as well as theoretiCal inve~tigations, Arnold et al. (1955) observed that as the. elastic layer becomes thinner in comparison with the radius of the footmg, effective damping is decreased and is less than that for a torsionally vibrating footing on an elastic half-space. Bycroft (1956) demonstrated that as the H/ro becomes sm}tller, the natural fr$quency of the torsional vibrations approaches the natural frequency of ''rod of radius r o and length H v1bratmg in torsion as a column fixed at the base and free at the top. Cases of torsional vibration of such columns have been discussed in Chapter 3 (Section 3.1).

(6.60)

By using the magnification factors given in Table 6.1 for an appropriate value of b, the amplitude at resonance can be computed by using Eqs. (6.59) and (6.60).

Table 6.1. Magnification Factors for Vertical Vibration of Rigid Circular Footing Supported by an Elastic Layer (v

= l) ML for

H ro

Rigid

b=O

1 2 3

00

4

00

00

1

00 00

b=5

b = 10

b =20

b =30

5.8 8.0 4.7 3.4

11.4 16.1 9.5 5.9

20.5 30.6 23:7 15.6

28.9 40.8 36.0 27.9

1.21

1.60

2.22

2.72

Source: Richart, Hall, and Woods, "Vibrations of Soils and

Foundations," © 1970, p. 234. Reprinted by permission of Prentice-Hall, Englewood Cliffs, New Jersey.

""•

figure 6.16. lntertia ratio vs. dimensionless frequency at resonance for torsional vibrations of a rigid circular footing on an elastic layer. (After Rycroft, 1956.)

240

6.7

FOUNDATIONS FOR RECIPROCATING MACHINES

241

LINEAR ELASTIC WEIGHTLESS SPRING METHOD

t

LINEAR ELASTIC WEIGHTLESS SPRING METHOD

Pz sin wt

P, sin wt

The linear elastic weightless spring method is sometimes used for analysis of machine foundations and utilizes the concept that the displacement of a loaded foundation resting on the surface of a soil can be determined by simulating the soil with a set of independent linear elastic springs that can produce equivalent reactive forces to the displacements developed. This concept commonly known as the elastic subgrade reaction theory has been described by Hayashi (1921), Heteyni (1946), and Terzaghi (1943, 1955). The idea of using elastic springs was extensively developed by Barkan (1962) for the purpose of predicting the dynamic response of machine foundations. His concept is based upon the following simplifying assumptions:

m

m /

Ia I

(b)

Pz sin wt

m

1. The foundation block is infinitely rigid compared with the soil.

2. 3. 4. 5.

The soil underlying the foundation is weightless. The soil can be simulated by linear elastic springs. Damping in the soil beneath a foundation may be neglected. The foundation is resting on the surface of the soil.

These assumptions make it possible to represent the foundation-soil system with an equivalent mass-spring system in which the mass represents the foundation and machine, and the spring represents the elasticity of the soil. The methods for determining the elastic soil springs for different modes of vibration have already been discussed in Chapter 4 (Section 4.4). Vertical Vibrations

For the purpose of computation, consider a situation in which a foundation block rests on the surface of the ground and is excited by the vertical unbalanced force P,(t) generated by the operation of a machine (Fig. 6.17a). Let the unbalanced force be represented by P,(t) = P, sin wt and the idealized equivalent spring-mass system by Fig. 6.17b. If the center of gravity of the foundation and machine and the centroid of the base area of the foundation in contact with the soil lie on f vertical line that coincides with the line of action of the exciting force P,, then the foundation will vibrate vertically only. Because the foundation block is assumed to be rigid, its displacement may be defined by the displacement of its center of gravity, and the vibrating mass may therefore be considered to be a concentrated

point mass. This assumption justifies the model shown in Fig. 6.17c. The problem of a vertically vibrating foundation is thus reduced to the analysis

lei

figure 6.17. Vertical vibrations of a rigid block: (a) actual case, (b) soil replaced by equivalent spring kz, and (c) equivalent model.

of a vibrating centered mass that rests on a spring, and the theory. of vibrations for an undamped single-degree-of-freedom system can be applied (Section 2.3). The equation of motion for the system in forced vtbrat10n ts therefore written as (neglecting damping m Sect10n 2.5) ( 6.61) in which m = mass of the foundation and the machine z = vertical displacement of the foundation with respect to equilibrium

.

position;

~::

k =equivalent spring constant of the soil . for vertical vibrations; w =frequency of operation of the machme. It is shown in Chapter 4 (Section 4.4) that the value of k, can be determined from

k,

=

C,A

( 4.27)

242

FOUNDATIONS FOR RECIPROCATING MACHINES

243

LINEAR ElASTIC WEIGHTlESS SPRING METHOD

in which

Px sin wt

Px sin wt

A = area of contact of the foundation with the soil. Equation (6.61) may thus be written as (6.62) Therefore, the natural frequency of a vertically vibrating system is given by wnz =

[

c A]'/2 ~

rad/sec

(6.63a)

or /,

nz

1 [cmA]'

=-

27T

-"-

lei 1 '

Hz

Figure 6.18.

Block foundation that slides only, and its equivalent model.

(6.63b)

in which Therefore, Eq. (6.65a) may be written as w," =The circular natural frequency (undamped) of the soil foundation

mi +

system in vertical vibration (rad/sec); f"' =Natural frequency of vertical vibrations (Hz)

c·';tf =

PX sin wt

;

(6.65b)

The frequency of sliding vibrations of the system is The amplitude of the vertical vibrations A, is given by A = _P_,'-,;Csi::::n_w~t~ , m( Wnz 2 _ z) W

(CA) m

wnx

=

/,

=-

112

rad/sec

-'-

(6.66a)

(6.64a)

or the maximum amplitude of motion is given hy

or (6.64b)

1 (C,A) 27T m

nx

--

112

Hz

(6.66h)

in which =The circular natural frequency of the sliding vibrations in radians/ sec and =The natural frequency of the sliding vibrations in cycles/sec or Hz. /,nx wnx

Sliding Vibrations

Consider a horizontal unbalanced force, Px(t) = Px sin wt, to act on a block foundation as illustrated in Fig. 6.18. The vibrations of the foundation in this case are analogous to vertical vibrations and may be expressed in terms of Eq. (6.65a):

The amplitude of the sliding vibrations is given by A

(6.65a) in which

x =sliding displacement of the foundation, from its equilibrium position; and A =the base area of the foundation; and

kx =the equivalent spring constant of the soil in sliding; and is given by (4.28b)

x

=

,

px

z

m(w nx - w )

(6.67)

The expressions for frequency and amplitude in sliding are thus similar to the expressions for vertical vibrations. Rocking Vibrations

Consider now only the rocking vibrations induced in a fo~ndati?n bl~ck · · moment M Y (t) • Here , M y (t) = M y sm wt m whtch by an externally excttmg

244

FOUNDATIONS FOR RECIPROCATING MACHINES

245

LINEAR ELASTIC WEIGHTLESS SPRING METHOD

M, denotes the moment acting in the XZ plane (Fig. 6.19a). The footing is symmetrical about the Y axis, and the center of mass of the foundation and the machine and the centroid of the base area lie on a vertical line and in the plane of the moment. The displaced position of the foundation is shown in Fig. 6.19a, and its rotation is cf>. The equation of motion may be obtained by applying Newton's second law of motion as explained below.

Because the displacement angle cf> is small, tan cf> = cf> in radians and the moment

1. Moment M; occasioned by the inertia of the foundation is given by

L = the distance between the center of gravity and the axis of rotation, and W =the weight of the foundation.

M=-M ;;.. 1 mo
(6.68)

in which Mmo is the moment of inertia of the mass of the foundation and machine with respect to the axis of rotation. 2. Moment Mw occasioned by the displaced position of the center of gravity of the foundation is given by

Mw = WL tan cf>

(6.69)

(6.70) in which

3. Moment MR occasioned by the soil reaction. Consider an element dA of the foundation area in contact with the soil and located at a distance l from the axis of rotation (Fig. 6.19b). The soil reaction depends upon the displacement at the point under consideration and varies from zero at the center of rotation to a maximum value at the edge of the fo9ting. The soil reaction over the elementary area dA is given by r,,,/" dR =

(6.71)

c.zq, dA

in which C =the coefficient of elastic nonuniform compression. The reactive " moment dM R occasioned by the soil reaction dR is given by Initial position 2

dMR = ldR = -C,I c/> dA Displaced position

If the foundation does not lose contact with the soil, then the soil reaction will be as shown in Fig. 6.19b, and the total reactive moment MR against the foundation area in contact with the soil is given by MR =-

(a)

I"

r c,/cf>

dA =- c.q,

I I'

dA = - c,)cf>

(6.72)

in which I is the moment of inertia of the foundation area in contact with the soil with respect to the axis of rotation. "~(

4. The exciting moment My(t) = M, sin wt. The equation of motion may therefore be written as

- Mm,;j, (b)

Figure 6.19. Rocking vibrations of a rigid block: (a) Block under excitation due to an applied moment (b) Soil reaction below the base.

+ WLcf>- C"lcf> + M, sin wt =

0

or

(6.73)

246

FOUNDATIONS FOR RECIPROCATING MACHINES

The natural frequency of the system is given by wn = (

C I- W£)112 "'M rad/sec mo

(6.74a)

LINEAR ELASTIC WEIGHTLESS SPRING METHOD

247

in which h is the height of the point above the base where amplitude is to be determined. It may be further noted that for a footing that rocks about Y axis and has dimensions, a and b along the X and Y axis, respectively, I is given by

or

ba 3 12

1=_ 1 (C1 I-WL)';' fn- Z1r M Hz mo

(6.74b)

in which w"'" =circular natural frequency of rocking vibrations (rad/sec) fn =natural frequency (Hz)

In Eq. (6.74a,b), the value of WL is negligible compared to the value of c I and may be neglected. This leads to "' w

n

=(C•I)II' M mo

(6.74c)

By comparing Eq. (6.73) with Eq. (6.61), and neglecting WL one obtains k1 =

c1 I

(6.79)

The response of the footing to rocking is thus affected by the dimension of the footing perpendicular to the axis of vibration, and this principle may be used to an advantage in proportioning the foundations undergoing rocking vibrations. Rocking vibrations occur mostly in machines that are mounted on high pedestals and have unbalanced horizontal forces and exciting moments. Torsional (Yawing) Vibrations A foundation is excited in torsional vibration when it is acted upon by a horizontal moment, M,(t) = M,.,.~in wt, around the vertical axis that passes through the center of gravity of"ihe foundation, and the position of the foundation at any time may be defined in terms of the angle of rotation
(6.75)

in which

z

k = Equivalent spring for rocking vibrations.

The amplitude of rocking vibrations A A

=

1

Mz sin wt

is given by

MY Mmo(w~- w 2 )

(6.76)

Th~ effect of ro~king is to increase the amplitudes of the vertical and t" honzontal vtbratmns. The maximum amplitude of vertical castoned by rocking is given by mo !On oc-

X

y

(a) lsometeric view

(6.77) in which a. is the dimension of the footing perpendicular to the axis of rotatmn. Stmtbriy, the contribution of rocking, towards the horizontal amphtudes ts given by A x)

(b) Plan

Figure 6.20. Torsional vibrations of rigid block:

(6.78)

(a)

Development of nonuniform shear below the base.

Block subjected to horizontal moment. (b)

248

FOUNDATIONS FOR RECIPROCATING MACHINES

the soil will thus be a nonuniform shear, which may be defined in terms of a coefficient of elastic nonuniform shear C1,. As for rocking vibrations, the equation of motion for torsional vibrations may be expressed as (6.80) in which M m< = mass moment of inertia of the machine and foundation about the

vertical axis of rotation (polar mass moment of inertia) and J" =the polar moment of inertia of the foundation's base area. The natural frequency of the torsional vibrations is given by (6.81a)

249

liNEAR ELASTIC WEIGHTLESS SPRING METHOD

assumed that the center of gravity of the machine and foundation and centroid of the foundation base area are located on a vertical axis. Figure 6.21 shows a foundation that is excited by the following forces and moments , referred to the combined center of gravity of the foundation and the macQ.ine: 1. Vertical force, P"(t) = P" sin wt, 2. Horizontal force, PJt) = P, sin wt, and 3. Moment, My(t) =MY sin wt If the origin of coordinates is located at the center of gravity, 0, the following displacements of the foundation need to be considered (Fig. 6.21): 1. Displacement in the vertical direction z, 2. Displacement in the horizontal direction x 0 at the base and 3. Rotation of the base

or

!,,<

=

1 (

-2 71"

cM"' J' ) 112 Hz m<

(6.81b)

The equatio~~ of motion referrecf't6 the center of gravity may be written by applying d'Alemberts principle as follows:

By comparing Eq. (6.80) with the equation of motion of a vertically vibrating footing [Eq. (6.61)], one finds that the spring constant for torsional vibrations (non-uniform shear conditions at the base) is given by

-mi + Z,=O

(6.85)

-mi+X,=O

(6.86)

-Mm'¥ -"+M=O 1

(6.87)

(6.82) in which

The amplitude of torsional vibrations, A • is given by

A,,= M

M" ( 2

mz WmJ!- W

2)

( 6.83)

and the horizontal displacement A;, occasioned by torsion by Initial position

Ah

=

rA,1,

(6.84)

Displaced position

in which r = horizontal distance of the point on the foundation from the axis of

yawing

_l__ •

Vibrations Accompanied by Simultaneous Rocking, Sliding, and Vertical Displacement In ?'ost situa~ions, a machine foundation will simultaneously slide, rock, and VIbrate vertically under the action of corresponding exciting forces. It is

Figure 6.21.

Block subjected to the action of simultaneous vertical P"(t), horizontal P,..(t)

forces and moment Mr(t).

250

FOUNDATIONS FOR RECIPROCATING MACHINES

251

LINEAR ELASTIC WEIGHTLESS SPRING METHOD

z

(6.93)

X,= pr?jection of all external forces acting on the foundation, on the X

6. Moment M 2 occasioned by the horizontal soil reaction x 1 is given by

Z, =projection of all external forces acting on the foundation on the axis ' aXIS

M, = sum of all external moments acting on the foundation and lying on the XZ plane, and M m = mass moment of inertia of the machine and foundation about an axis passing through combined center of gravity and perpendicular to the plane of vibrations.

M 2 = C,AL(x

~

L)

( 6. 94)

7. Moment Mn occasioned by the soil resistance [as in Eq. (6.72)] ( 6. 95)

By substituting the value of the forces and moments in Eqs. (6.85) to (6.87), one obtains

At any time t, the following forces will act on the foundation: l. Weight W of the foundation and machine. Projection on the X axis . equals zero and on the Z axis equals Z 1 = ~ W.

mi + C.Az = P,(t)

z,

mx + C,Ax ~ C,AL =

2. Soil reaction occasioned by settlement of the foundation under the action of tbe static weight W:

( 6.88)

in which

z" =elastic settlement caused by the weight of the foundation and machine

A = the area of the foundation

3. Soil reaction Z 3 at any time t occasioned by the displacement z of the foundatwn measured from the equilibrium position is given by ( 6.89) 4. Horizontal soil reaction X 1 at the base is

(6.96)

Px(t)

(6.97)

Mm¢ ~ c,ALx + (C•I ~ WL + C,AL ) = My(t)

( 6.98)

2

·;!c·'·:~

Equation (6.96) contains only the terms of z, and in no way depends upon Eqs. (6.97) and (6.98). Hence it follows that the vertical vibrations of the foundation occur independently of any other motion. Equations (6.97) and (6.98) contain both x and q, and are interdependent. Therefore, sliding and rocking are coupled modes. Because the vertical vibrations of a foundation occur independently of any other vibrations, the treatment given earlier in this article for vertical vibrations will hold in the present case also. A solution for simultaneous rocking and sliding vibrations will now be obtained. Natural Frequencies of Coupled Rocking and Sliding. The system that is considered here is a two-degree-of-freedom system. The solutions for natural frequencies are obtained by considering the free vibrations of the system, and, therefore, the forcing functions in Equations (6.97) and (6.98), may be replaced by zero. From this, one obtains

(6.90) in which

X0

( 6.99)

is given by

and X 0 =X~

(6.91)

L

in which x =horizontal displacement of the center of gravity. By substituting this value of X 0 in Eq. 6.90 one obtains X,=~ C,A(x ~

L¢)

the foundatwn-machme-system is given by

Mm¢ ~ C,ALx + (C4,I ~ WL + C,AL 2 )

=

0

(6.100)

Particular solutions of these equations may be assumed to be

(6.92)

5. Moment M 1 occasioned by the displacement of the center of gravity of

'ii'J:.

X sin (w t +a)

(6.101)