(3) For cohesive soils as clays and peat,

The shear strain amplitude, AΕ, may be computed from

the shear modulus is related to Su as follows:

the axial strain amplitude, Ε,and Poisson's ratio as

G = K2su

(17-25)

follows:

For clays, K2 ranges from 1500 to 3000. For peats, K2

AΕ = Ε(1 + )

(17-22)

ranges from 150 to 160 (limited data base).

For the special case of saturated soils, Poisson's ratio is

(4) In the laboratory, the shear modulus of

0.5, which leads to the following:

soil increases with time even when all other variables are

G = E/3

held constant. The rate of increase in the shear modulus

AΕ = 1.5Ε

is approximately linear as a function of the log of time

d. Correlations.

after an initial period of about 1000 minutes. The change

(1) Empirical correlations from many sets

in shear modulus, ∆G, divided by the shear modulus at

of data have provided several approximate methods for

1000 minutes, G1000, is called the normalized secondary

estimating the S-wave velocity and shear modulus for

increase. The normalized secondary increases range

soils corresponding to low-strain excitation. For many

from nearly zero percent per log cycle for coarse sands

undisturbed cohesive soils and sands:

to more than 20 percent per log for sensitive clays. For

2

G =1230(21973 - e) (OCR)" (o)0.5 (pounds 1 + per

good correlation between laboratory and field

square inch)

(17-23)

measurements of shear modulus, the age of the in situ

where

deposit must be considered, and a secondary time

e = void ratio

correction applies to the laboratory data.

η = empirical constant, which depends on

e. Damping in low strain levels.

Critical

the PI of cohesive soils (table 17-4). For sands, PI = 0

damping is defined as

and η = 0, so OCR term reduces to 1.0. For clays, the

cc = 2 √km

(17-26)

maximum value is η = 0.5 for PI > 100.

where k is the spring constant of vibrating mass and m

σ0 = 1/3 (σ1 + σ2 + σ3) = mean normal

represents mass undergoing vibration (W/g). Viscous

effective stress, pounds per square inch

damping of all soils at low strain-level excitation is

(2) For sands and gravels, calculate the

generally less than about 0.01 percent of critical damping

low-strain shear modulus as follows:

for most soils or:

G = 1000(K2)(σ0) (pounds per square foot) (17-24)

0.5

D = c/c, < 0.05

(17-27)

where

It is important to note that this equation refers only to

K2 =empirical constant (table 17-5)

material damping, and not to energy loss by radiation

=90 to 190 for dense sand, gravel, and cobbles

away from a vibrating foundation, which may also be

with little clay

conveniently expressed in terms of equivalent viscous

σ0 = mean normal effective stress as in equation

damping.

Radiation damping in machine vibration

problems is a function of the geometry of the problem

(17-23) (but in units of pounds per square foot)

rather than of the physical properties of the soil.

Table 17-4. Values of Constant r Used with Equation (17-23) to Estimate Cyclic Shear Modulus at Low Strains

Plasticity Index

K

0

0

20

0.18

40

0.30

60

0.41

80

0.48

>100

0.50

(Courtesy of 0. Hardin and P. Drnevich. "Shear

Modulus and Damping in Soils: Design Equations and

Curves," Journal., Soil Mechanics and Foundations

Division. Vol 98. No. SM7. 1972, pp 667-692. Reprinted

by permission of American Society of Civil Engineers,

New York.)

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