depths studied represents a range of typical aggregate

easily be applied to other length/width ratios and other

normalized depths.

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

(1a)

σz =

4π *V *2 + *A *2 V 2

o

o

o

2 *A*oVo

for *A*o > *Z*

-1

tan

2

2

2

σz =

(1b)*

4π *V *2 + *A *2 V 2

o

o

o

-1 2 *A*oVo + πfor *A *< *Z*

tan

2

o

2

and

=

.

where

=

Thus *I*z , or the ratio of the vertical stress below the

applied uniform pressure

=

center of a rectangle at depth *Z *to the stress applied on

depth below the corner of the rectangle

the rectangle (*L * *B*) at the surface, is equal to

for which the vertical stress is desired

2

1 2 *AV * ⋅ *V *2 + 1 +

(2a)

π *V *2 + *A*2 * V *2

.

-1 2 *AV *

for *A *> *Z*

tan

2 - *A*2

Note that *V*o is the ratio of the radius vector from point

To obtain the stress below the center of the rectangle

at depth *Z*, I used superposition and divided the rect-

2

angle into four smaller rectangles, such that

(2b)

π *V *2 + *A*2 V 2

-1 2 *AV *

2

+ πfor *A *< *Z*

tan

and

.

2

and *A *and *V *are defined as follows:

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).