Page 192 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

Figure 6.3 Problem geometry, co- to Kirsch (1898), are
ordinate system and nomenclature for
specifying the stress and displacement p a 2
a 2 3a 4
distribution around a circular excava- rr = (1 + K) 1 − 2 − (1 − K) 1 − 4 2 + 4 cos 2
tion in a biaxial stress field. 2 r r r
p a 2
3a 4

= (1 + K) 1 + + (1 − K) 1 + cos 2
2 r 2 r 4
p 2a 2 3a 4
r
= (1 − K) 1 + − sin 2
(6.18)
2 r 2 r 4
pa 2 a 2
u r =− (1 + K) − (1 − K) 4(1 − ) − cos 2
4Gr r 2
pa 2 a 2
u
=− (1 − K) 2(1 − 2 ) + sin 2
4Gr r 2
In these expressions u r , u
are displacements induced by excavation, while
rr ,

, r
are total stresses after generation of the opening.
By putting r = a in equation 6.18, the stresses on the excavation boundary are
given as

= p[(1 + K) + 2(1 − K) cos 2
]
rr = 0 (6.19)
r
= 0

Equations 6.19 confirm that the solutions satisfy the imposed condition that the ex-
cavation boundary is traction free. Similarly, for
= 0, and r large, the stress com-
ponents are given by

rr = Kp,

= p, r
= 0
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