76 Ground improvement by deep vibratory methods
The total earthquake-induced settlement in a sand deposit is the sum of the
settlements of layers below the water table and the settlements of the layers
above:
S
s
s
total
w et
dry
=
∆
+
∆
∑
∑
(3.27)
Shear strain γ
eff
10
−3
10
−3
10
−2
10
−4
10
−4
10
−5
10
−5
σ′ m = 0.1 tsf
γ eff ⋅(G eff /G max )
2
1
0.5
0.2
Figure 3.31 Plot to determine the induced strain in sand deposits resulting from
M  =  7.5  earthquake. (After Tokimatsu, K. and Seed, H.B., J. Geotech. Eng.,
ASCE, 113, 861, 1987.)
Table 3.7 Magnitude scaling factors for stress ratios causing liquefaction in saturated
sand (third column) and for strain ratios in dry sand (fourth column)
Magnitude M
Representative stress
cycles at 0.65 τ max
Scaling factor MSF1
for  stress ratios
Scaling factor
MSF2 for strain ratios
8.5
26
0.89
1.25
7.5
15
1.0
1.0
6.75
10
1.13
0.85
6
5
1.32
0.6
5.25
2–3
1.5
0.4
Source: Tokimatsu, K. and Seed, H.B., J. Geotech. Eng., ASCE, 113, 861, 1987.