Stability of slopes - The Mechanics of Soils and Foundations - page 334

For drained loading the increment of work done by the internal stresses for an
increment of plastic collapse is δ W = 0 and, noting that F 1 = F 2 , the increment of

work done by the external loads for dry soil is

δ E = δ v γ V (21.16)

where V = zl cos i u is the volume of the block. Hence, equating δ E = δ W , an upper

bound is given by

δ v γ V = 0 (21.17)

Since the volume V is non-zero, the upper bound is given by 0 and hence, from

the geometry of Fig. 21.11(b), an upper bound for the critical slope angle is given by

δ v =

i u = φ c

(21.18)

Figure 21.12 shows an infinite slope whose angle with the horizontal is a lower
bound i l , and a block of soil of length l measured down the slope and depth z measured
vertically; the forces on the faces of the block are shown and, as before, the forces F 1

Figure 21.12 Equilibrium states of stress for an infinitely long slope in dry soil.