Civil Engineering Reference
In-Depth Information
unit weight of the soil and
z
depth below ground surface. The horizontal earthquake force
F
acting on the soil column (which has a unit width and length) is
g
)
a
(
t
z
g
)
a
max
v
0
(
a
max
F
ma
(
W
g
)
__
___
____
(6.1)
where
F
horizontal earthquake force acting on soil column that has a unit width and
length, lb or kN.
g.
W
total weight of soil column, lb or kN. For the assumed unit width and length of
soil column, the total weight of the soil column is
t
z.
t
total unit weight of soil, lb/ft
3
or kN/m
3
.
z
depth below ground surface of soil column, as shown in Fig. 6.4.
a
acceleration, which in this case is the maximum horizontal acceleration at
ground surface caused by the earthquake (
a
a
max
), ft/s
2
or m/s
2
.
a
max
maximum horizontal acceleration at ground surface that is induced by the earth-
quake, ft/s
2
or m/s
2
. The maximum horizontal acceleration is also commonly
referred to as the peak ground acceleration (see Sec. 5.6).
v
0
total vertical stress at bottom of soil column, lb/ft
2
or kPa. The total vertical
stress
t
z.
As shown in Fig. 6.4, by summing forces in the horizontal direction, the force
F
acting
on the rigid soil element is equal to the maximum shear force at the base on the soil ele-
ment. Since the soil element is assumed to have a unit base width and length, the maximum
shear force
F
is equal to the maximum shear stress
max
, or from Eq. (6.1):
m
total mass of soil column, lb or kg, which is equal to
W
max
F
v
0
(
a
max
g
)
____
(6.2)
Dividing both sides of the equation by the vertical effective stress
v
0
gives
v
0
(
v
0
v
0
)
(
a
max
max
g
)
____
___
____
(6.3)
Since the soil column does not act as a rigid body during the earthquake, but rather the
soil is deformable, Seed and Idriss (1971) incorporated a depth reduction factor
r
d
into the
right side of Eq. (6.3), or
r
d
(
v
0
v
0
)
(
a
max
g
)
max
____
___
____
v
0
(6.4)
For the simplified method, Seed et al. (1975) converted the typical irregular earthquake
record to an equivalent series of uniform stress cycles by assuming the following:
cyc
0.65
max
(6.5)
where
cyc
uniform cyclic shear stress amplitude of the earthquake (lb/ft
2
or kPa)
In essence, the erratic earthquake motion was converted to an equivalent series of uni-
form cycles of shear stress, referred to as
cyc
. By substituting Eq. (6.5) into Eq. (6.4), the
earthquake-induced cyclic stress ratio is obtained.
v
0
0.65
r
d
(
v
0
v
0
)
(
a
max
CSR
cyc
g
)
___
___
____
(6.6)
where CSR
cyclic stress ratio (dimensionless), also commonly referred to as seismic
stress ratio.
