EARTHQUAKE-INDUCED SETTLEMENT - Geotechnical Earthquake Engineering - page 194

where cyc uniform cyclic shear stress amplitude of the earthquake

r d depth reduction factor, also known as stress reduction coefficient (dimen-

sionless). Equation (6.7) or Fig. 6.5 can be used to obtain the value of r d .

v 0 total vertical stress at a particular depth where the settlement analysis is being

performed, lb/ft 2 or kPa. To calculate total vertical stress, total unit weight t

of soil layer (s) must be known.

a max maximum horizontal acceleration at ground surface that is induced by the

earthquake, ft/s 2 or m/s 2 , which is also commonly referred to as the peak
ground acceleration (see Sec. 5.6)

g acceleration of gravity (32.2 ft/s 2 or 9.81 m/s 2 )

As discussed in Chap. 6, Eq. (7.1) was developed by converting the typical irregular
earthquake record to an equivalent series of uniform stress cycles by assuming that cyc
0.65 max , where max is equal to the maximum earthquake-induced shear stress. Thus cyc is
the amplitude of the uniform stress cycles and is considered to be the effective shear stress
induced by the earthquake (i.e., eff cyc ). To determine the earthquake-induced effec-
tive shear strain, the relationship between shear stress and shear strain can be utilized:

cyc eff eff G eff (7.2)

where eff effective shear stress induced by the earthquake, which is considered to be
equal to the amplitude of uniform stress cycles used to model earthquake
motion ( cyc eff ), lb/ft 2 or kPa

eff effective shear strain that occurs in response to the effective shear stress
(dimensionless)

G eff effective shear modulus at induced strain level, lb/ft 2 or kPa

Substituting Eq. (7.2) into (7.1) gives

eff G eff 0.65 r d v 0 ( a max / g ) (7.3)

And finally, dividing both sides of the equation by G max , which is defined as the shear
modulus at a low strain level, we get as the final result

eff ( G eff

G max ) 0.65 r d ( v 0

G max ) ( a max

g )

____ ____ ____

(7.4)

Similar to the liquefaction analysis in Chap. 6, all the parameters on the right side of the
equation can be determined except for G max . Based on the work by Ohta and Goto (1976)
and Seed et al. (1984, 1986), Tokimatsu and Seed (1987) recommend that the following
equation be used to determine G max :

G max 20,000 [( N 1 ) 60 ] 0.333 ( m ) 0.50 (7.5)

where G max shear modulus at a low strain level, lb/ft 2

( N 1 ) 60 standard penetration test N value corrected for field testing procedures and
overburden pressure [i.e., Eq. (5.2)]

m mean principal effective stress, defined as the average of the sum of the three
principal effective stresses, or ( 1 2 3 )/3. For a geostatic condition and
a sand deposit that has not been preloaded (i.e., OCR 1.0), the coefficient