Geotechnical Engineering

DHARM

384 GEOTECHNICAL ENGINEERING

Depth z = 1 m. q = 450 kN/m^2

2m

4m

PW Q

1m

SV R

TU

IV III

II

I

A¢

0.5 m

Fig. 10.26 Stress at a point inside a loaded area (Ex. 10.8)
The loaded area and the plan position of the point A′ at which the vertical stress is
required are shown in Fig. 10.26. The area is divided into four parts as shown, such that A′
forms a corner of each.

σz = qIσσ σ σIIIIIIIV++ +I I I

Area I: m = 1/1 = 1; n = 1.5/1 = 1.5

IσI =

1

4

21

1

2

1

21

1

22

22 22

22

22

1

22

π^2222

mn m n

mn mn

mn

mn

mn m n

mn mn

++

+++

++

++

+

++

++−

L

N

M

M

O

Q

P

P

.tan−

=

1

4

21151 15 1

1151115

115 2

1151

21151 15 1

1151115

22

22 22

22

22

1

22

π 22 22

×× + +

+++×

++

++

+

×× + +

++−×

L

N

M

M

O

Q

P

P

.. −

..

..
.

tan

..

..

= 0.1936

Area II: m = 1.5/1 = 1.5; n = 3/1 = 3

IσII =

1

4

215315 3 1

15 3 1 15 3

15 3 2

15 3 1

215315 3 1

15 3 1 15 3

22

22 22

22

22

1

22

π^2222

×× ++

+++ ×

++

++

+

×× ++

++− ×

L

N

M

M

O

Q

P

P

.. −

..

..

.

tan

..

..

= 0.2290

Area III: m = 0.5/1 = 0.5; N = 3/1 = 3

IσIII =

1

4

205305 3 1

05 3 1 05 3

0532

05 3 1

205305 3 1

05 3 1 05 3

222

22 22

22

22

1

22

π^2222

×× ++

+++ ×

++

++

+

×× ++

++− ×

L

N

M

M

O

Q

P

P

.. −

..

.

.

.

tan

..

..

= 0.1368

Area IV: m = 0.5/1 = 0.5; n = 1/1 = 1

IσIV =

1

4

205105 1 1

0511051

05 1 2

0511

205105 1 1

05 1 1 05 1

22

22 22

22

22

1

22

π^2222

×× ++

+++ ×

++

++

+

×× ++

++− ×

L

N

M

M

O

Q

P

P

.. −

..

..
.

tan

..

..

= 0.1202