Engineering Rock Mechanics

Design against stress-controlled instability 38 1

• W =3,p --,pg=m H
  H A- 2

Ovaloid

• 
  
  
  
  • cA)-A -0.5~ c
  
  
  
  

BEM

3.60~

ttt

uA = pllt /=- $1 = 3.96~

For the inscribed ellipse,

OB = p[ 0.5 (1 + (2 x 7))^1 - 11 = -0.17~ -0.15~

Square with rounded corners, k = 1

+

k= 1

W =1.25D

pA = 0.2D

Figure 20.23 Application of elliptical approximation to other excavation shapes.

Our second example concerns a square opening with rounded corners
in a hydrostatic stress field, as shown in the lower diagram of
Fig. 20.23. In this case of a hydrostatic stress field, we anticipate that the
maximum stress will be associated with the smallest radius of curvature,
i.e. at the rounded corners. Thus, using the geometry of the opening with
pA = 0.20, we take W = 1.250, and this gives oA = 3.53~. The more accurate
value determined by the boundary element method was 3.14p-again, the
approximation gives a good preliminary estimate.

Approximation to complex boundary profiles. To show how the approach can
be extended to complex boundary profiles, we show a typical underground
hydroelectric scheme machine hall geometry in Fig. 20.24. From the equa-
tions shown in Fig. 19.16, one would expect:

(a) the radii of curvature at points A, B and C are very small, and hence

(b) the radius of curvature is negative at point D, and the induced stress

the stress concentration will be very high at these points;

might also be negative, i.e. tensile.

For the appropriately inscribed ellipse, the ellipse equations give the
following values: sidewall stress = 1.83; and crown stress = 0.72.