Engineering Rock Mechanics

326 Design and analysis of surface excavations

Y Z x,yz Coordinates yT>z-x Coordinates

Let R = Jm

andp= R+z

Boussinesq’s problem

(T =- 3Pz3

2n~5

Cerruti’s problem

1

ux = - Q [ ~- R2+x2 (I-2v)(Rp-x2)

4rrG R3 RP2

1

• Qx [ 3x3’+(3
  4rrG R Rp

= Qx 3y2 (1 - 2u)(3R2 - x2 - (2R/p)x2)

Y 2rr~3[F- P?

Figure 18.12 Cartesian forms of the solutions to Boussinesq’s and Cerruti’s

problems.

can be superimposed, the stresses and displacements associated with any

loading of the surface can be estimated. It is only necessary to be able to

discretize the load into suitable component areas over which any normal

and shear stresses acting can be considered to be uniform, as illustrated

in Fig. 18.13.

7 8.3.2 Analytical integration over loaded areas

Here we consider only the cases of stresses and displacements in the

z-direction, for the Boussinesq solution, to demonstrate the principle of