SOIL MECHANICS - CHAPTER 30 pptx

Chapter 30

FLAMANT

In 1892 Flamant obtained the solution for a vertical line load on a homogeneous isotropic linear elastic half space, see Figure 30.1. This is the two

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..................................................................................................... ....................................................................................................

..........................

.........................

.

.

...............

.....................................

.

.

.

.

.........

...

...

.........................

................................

...........

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....

.........

..... .

.

.

....

....

...

.

θ r

x

z

F

σxx

σxz

σzz

σzx

Figure 30.1: Flamant’s Problem.

dimensional equivalent of Boussinesq’s basic problem. It can be considered as

the superposition of an infinite number of point loads, uniformly distributed

along the y-axis. A derivation is given in Appendix B.

In this case the stresses in the x, z-plane are

σzz =

2F

π

z

3

r

4

=

2F

πr

cos3

θ, (30.1)

σxx =

2F

π

x

2

z

r

4

=

2F

πr

sin2

θ cos θ, (30.2)

σxz =

2F

π

xz2

r

4

=

2F

πr

sin θ cos2

θ. (30.3)

In these equations r =

√

x

2 + z

2. The quantity F has the dimension of a

force per unit length, so that F/r has the dimension of a stress.

Expressions for the displacements are also known, but these contain singular terms, with a factor ln r. This factor is infinitely large in the

origin and at infinity. Therefore these expressions are not so useful.

On the basis of Flamant’s solution several other solutions may be obtained using the principle of superposition. An example is the case of a

uniform load of magnitude p on a strip of width 2a, see Figure 30.2. In this case the stresses are

σzz =

p

π

[(θ1 − θ2) + sin θ1 cos θ1 − sin θ2 cos θ2], (30.4)

σxx =

p

π

[(θ1 − θ2) − sin θ1 cos θ1 + sin θ2 cos θ2], (30.5)

σxz =

p

π

[cos2

θ2 − cos2

θ1]. (30.6)

168