Geotechnical Engineering

DHARM

370 GEOTECHNICAL ENGINEERING

= q

a

z

a

z

io

1

1

1

1

2 32 2 32

+F

HG

I

KJ

R

S

|

T|

U

V

|

W|

−

+F

HG

I

KJ

R

S

|

T|

U

V

|

W|

L

N

M M M M M M

O

Q

P P P P P P

// ...(Eq. 10.41)

σz = qK. BC

where KBC =

1

1

1

1

2 32 2 32

+F

HG

I

KJ

R

S

|

T|

U

V

|

W|

−

+F

HG

I

KJ

R

S

|

T|

U

V

|

W|

L

N

M M M M M M

O

Q

P P P P P P

a

z

a

z

io

// ...(Eq. 10.42)

Similarly, if Westergaard’s theory is to be used,

KBC =^1

1

1

1

22

+F

HG

I

KJ

−

+F

HG

I

KJ

L

N

M M M M M M

O

Q

P P P P P P

a

z

a

z

io

ηη

...(Eq. 10.43)

where η =

12

22

−

−

ν
ν, ν being Poisson’s ratio.
The application of these equations in a practical problem will be very simple as the
numerical values of the various quantities are known.

10.6 Uniform. Load on Rectangular Area

The more common shape of a loaded area in foundation engineering practice is a rectangle,

especially in the case of buildings. Applying the principle of integration, one can obtain the

vertical stress at a point at a certain depth below the centre or a corner of a uniformly loaded

rectangular area, based either on Boussinesq’s or on Westergaard’s solution for a point load.

10.6.1Uniform Load on Rectangular Area based on Boussinesq’s Theory

Newmark (1935) has derived an expression for the vertical stress at a point below the corner of

a rectangular area loaded uniformly as shown in Fig. 10.15.

The following are the two popular forms of Newmark’s equation for σz:

σz =

q mn m n

mn mn

mn

mn

mn m n

(^4) mn mn
21
1
2
1
21
1
22
22 22
22
22
1
22
π^2222
++
+++
F
H
GG
I
K
JJ
++
++
F
HG
I
KJ

• ++
  +++
  F
  H
  GG
  I
  K
  JJ
  L
  N
  M
  M
  O
  Q
  P
  P
  −
  ()
  sin
  ...(Eq. 10.44)