370 Design and analysis of underground excavationsTaking into account both the two components aI and a2 of the apical angle, the discontinuity
stiffnesses and rock stress, we have for the limiting vertical load:
H H
P=o (k slcos2aI + k,lsin2al)sin(@l -aI) + 0 (kS2cos2a2 + k, 2 sin2a2)sin(Q2-a2)
I Dl Dl
where
D I1 =k, cosaIcosQ1+ k,lsinalsin~l and D2=k,2cosa2cos~2+ kn2sina 22 sine.If k,, >> k,, and k,, >> k,,, then we haveHosinaIsin(@,-al) Hosina2sin(g,-a2)
P= +
1 sinQl sine,which allows for different angles of friction on the two discontinuity surfaces.----H0Figure 20.12 Analysis of an asymmetric triangular roof prism.stresses, with the result that the normal forces can be obtained by
multiplying this stress by the area of the face of the block. As shown in
Fig. 20.13, the shear force is obtained from the normal force through the
application of the Coulomb friction criterion, which defines the analysis as
being one of limiting equilibrium.
In Figure 20.8, it has been seen how kinematic analysis may be used to
determine the complete geometry of the block, and hence the orientations
of the normals and the bisectors.
It is necessary to check the limiting condition on each face to ensure that
all individual frictional components are assisting in the maintenance of
stability, i.e. are negative (when reckoned in accordance with the axes
shown in Fig. 20.13). If this is not the case, it is prudent to assume that
progressive failure may take place through initial rotation of the block.
If the wedge is unstable, then the degree to which the weight is in excess
of the constraining force can be used to indicate a factor of safety and the
degree to which the block may require support.20.7.4 Use of block theory
A major advance was made by Goodman and Shi (1985) in the application
of mathematical topology to rock blocks and their removability from the
rock surrounding an excavation. The advantages of a complete mathe-
matical description of rock blocks are the ability to develop comprehensive