Engineering Rock Mechanics

304 Surface excavation instability mechanisms

Line load of P per unit

length in y direction

Line load of Q per unit

length in y direction

‘Jr

0 X

T

P

t z Contour of constant

cry is semi-cylindrical

For constant ur & P

r = k sin 9 where k = -

For constant or & Q

r = k cos 0 where kq = - 2Q

TWr

2P

P TUr

(4 (b)

Figure 17.18 Boussinesq and Cerruti solutions for line loads on the surface of a

CHILE half-space.

The interest is in the application of a line load at an arbitrary angle to

the surface. This can be obtained by resolving the force into its normal and

shear components and then superposing the Boussinesq and Cerruti

solutions, respectively. After some algebraic manipulation, the radial stress

induced in the solid can be expressed with reference to the line of action

of the inclined line load as

2Rcosp

0, =

m

For values of -42 < /3 < 7d2, cos pis positive and hence the radial stress is

compressive, whereas, for angles outside this range, cos p is negative-

giving tensile radial stress.

The resulting locus of radial stress for an inclined load is shown

in Fig. 17.19. The reader should verify that, in the extreme cases of Q = 0

or P = 0, the locus would be that of the Boussinesq and Cerruti solutions,

respectively. This interpretation assists in the understanding of the con-

tribution made by the normal and shear components to the inclined

load solution. Note that the left lobe of the locus represents a tensile

radial stress and the right lobe represents a compressive radial

stress.

In applying this solution to a real rock, it would be necessary to be able

to sustain the induced tensile stress in order for the solution as shown to