Engineering Rock Mechanics

308 Foundation and slope instability mechanisms

A? 7. IO As with the analytical solu-
tion, we are able to make use of
the symmetry of the
problem, and so will only ana-
lyse one quarter of the area, with
the final stresses and displace-
ments being found by
multiplying the computed results
by 4. A typical sector with-
in the quadrant under analysis is illustrated above.

at the surface and vertical stress at a depth z are

For a sector such as that shown above, the equations for displacement

and

1-v

uZ = p- rid8

2xG 1=l ,

(17.14)

(17.15)

where r is the sector radius and de is the sector included angle.

To use Eqs. (17.14) and (17.15), we decide on the number of sectors

n, compute the resulting value of de, and then compute the radius of

each sector before substituting into the equations. Computation of each

radius can be awkward, and may involve considerable geometry. In the

case shown here, we will also need to decide which of the two edges of

the loaded area is appropriate for a given sector. However, as the method

is most likely to be implemented on a computer, it is more convenient to

calculate the radius for each sector relative to the lines representing both

edges, and then take the minimum of the two:

r = min (radius relative to side, radius relative to top). (17.16)

If we examine the geometry of the area, we find that for all sectors i,

Eq. (17.16) can be written as ..

ab

where a and b are the half width and half height of the area (i.e. 5 m and

3 m for this particular example), respectively.

For the case of 5 sectors in the quarter area, the computation is as

follows:

z3

i Oi ri ride 1 -

to) (rn) (rn)

(z' + r,2)3'2

1 9 5.062 1.590 0.205

2 27 5.612 1.763 0.222

3 45 4.243 1.333 0.175

4 63 3.367 1.058 0.135

5 81 3.037 0.954 0.118

Total 6.698 0.855