depths studied represents a range of typical aggregate
thicknesses above geotextiles. The technique used can
easily be applied to other length/width ratios and other
normalized depths.
VERTICAL BOUSSINESQ STRESS
BELOW A RECTANGLE
The vertical stress at depth below the corner of a
uniformly loaded rectangle of dimensions L B is es-
timated as (Fig. 1; Newmark 1935)
2
w 2 AoVo ⋅ Vo + 1 +
(1a)
σz =
4π V 2 + A 2 V 2
o
o
o
2 AoVo
for Ao > Z
-1
tan
2
2
Vo - Ao
2
w 2 AoVo ⋅ Vo + 1 +
Figure 1. Notation used in the estima-
σz =
(1b)*
tion of vertical stress at depth below
4π V 2 + A 2 V 2
o
the corner and center of a uniformly
o
o
loaded rectangle.
-1 2 AoVo + πfor A < Z
tan
2
o
2
Vo - Ao
and
X 2 + Y 2 + Z2
=
V2
.
Z2
where
=
Thus Iz , or the ratio of the vertical stress below the
w
applied uniform pressure
=
center of a rectangle at depth Z to the stress applied on
Z
depth below the corner of the rectangle
the rectangle (L B) at the surface, is equal to
for which the vertical stress is desired
L2 + B2 + Z 2
2
Vo =
1 2 AV ⋅ V 2 + 1 +
Iz =
Z2
(2a)
π V 2 + A2 V 2
Ao = LB
.
Z2
-1 2 AV
for A > Z
tan
2 - A2
Note that Vo is the ratio of the radius vector from point
V
Z to the corner of the rectangle to the depth of point Z.
To obtain the stress below the center of the rectangle
at depth Z, I used superposition and divided the rect-
I z = 1 2 AV ⋅ V + 1 +
2
angle into four smaller rectangles, such that
(2b)
π V 2 + A2 V 2
L
X=
-1 2 AV
2
+ πfor A < Z
tan
and
.
V 2 - A2
B
Y=
2
VERTICAL BOUSSINESQ STRESS
and A and V are defined as follows:
BELOW A CIRCLE
A = XY
Z2
The vertical stress at depth Z below the center of a
*This form of the equation is not published by Newmark
uniformly loaded circle of radius r, divided by the ap-
(1935), but it is required to avoid predicting tension (e.g.,
plied stress, is estimated as (Fig. 2; Newmark 1942)
Bowles 1988).
2
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