302 Surkrce excavation instability mechanisms4 d B4
i i A
i
i 60"VIV
to v3 c' =^25 kNh2
vi = 1.0 (arbitrary assumption) 4' = 20"
v2= vi - sin^20 = 0.3473
sin 100
VI sin 60
sin 100
v2 sin 20
sin 100
v2 sin 60
sin 100viv = VI sin 80 = 0.9848v12= ~ = 0.8794 vzV = v2 sin 20 = 0.1 188v3= - = 0.1206 vjV = v3 sin 40 = 0.0775
7 Datum
~23 = - = 0.3054 v3v(a) (b)
Figure 17.16 Virtual work applied to rock foundation instability: (a) foundation
geometry; and (b) associated virtual displacements.then vectorially adding the other virtual displacements triangle by trian-
gle. Thus, following from vl, the triangle vl - v12 - vz is completed, followed
by the triangle vz - ~23 - v3. The displacements may be determined either
by trigonometrical calculation or by drawing the polygon accurately to scale
and measuring directly.
From these various virtual displacements, the virtual work can be calcu-
lated. As an aid in considering the various components of virtual work, they
can usefully be assigned to one of two categories: external virtual work,
EVW (due to the applied forces and the weight of the rock blocks); and
internal virtual work, IVW (due to the work done by forces arising from
the strength of the discontinuities). Hence we have
EVW = (dpm + W,)vsv + Wzvzv + Wlvlv and
Ivw =(VI + v2 + v3 + v1z + vz3)c'Lcosf$'
with the result that, because EVW + IVW = 0 and solving for pus with
d = 6 and the values indicated in Fig. 17.16, pm = 1629 kN/m2.
When conducting this type of analysis one must ensure that the correct
signs are maintained for the virtual displacements associated with the
external forces: this applies for both the virtual displacement polygon and
the calculation of virtual work. An inspection of Fig. 17.16 reveals that v3v
is the only negative displacement in this example, resulting from an initial
inherent assumption that displacements upwards are positive.
Continuum analysis. Studying the sustainability of a stress distribution in
terms of the lower bound theorem of plasticity is mostly applicable to rock