Stress -con trolled instability mechanisms 355
- = - P ((1 + k)[(l + 42) + (1 - q2) COS 2 (x - PI]
2q - -(1 - k)[(l + q)2 cos2x + (1 - 42) cos2p]]
kPZ where q = w - - H
2Figure 19.15 Stresses induced on the boundary of an elliptical excavation in
plane strain for a CHILE material (after Bray, 1977; from Brady and Brown,
1985).
three parameter groups describing the problem-these relate to the aspect
ratio of the opening, the ratio of the in situ principal stresses, and the five
elastic moduli for a transversely isotropic material.
The cross-section through the elliptical excavation together with the salient
geometrical parameters and the field stresses are shown in Fig. 19.18(a). The
general three-dimensional stress field and the model chosen to represent the
transversely isotropic rock are shown in Fig. 19.18@). Note that the element
shown in Fig. 19.18@) represents the state of stress at a point, and the stress
components indicated represent local stresses; this is in contrast to Fig.
19.18(a) where the field stresses are indicated.
Very often, long excavations have their longitudinal axis aligned with the
strike of the plane of isotropy and therefore the problem can be simplified
by assuming plane strain and hence only having to take into account four
material properties-this is shown in Fig. 19.18(c).
These ideas are used with associated equations in connection with
discussion on zones of influence presented in Chapter 20.
-^4 where, for an ellipse, the radii of curvature are
Figure 19.16 Stresses induced on the boundary of an elliptical excavation
(aligned with axes parallel and perpendicular to the principal stresses) in
plane strain for a CHILE material (after Bray, 1977; from Brady and Brown,
1985).