Denudation, weathering and mass wasting - Fundamentals of the Physical Environment - page 302

Mohr-Coulomb failure criteria

KEY PROCESSES

The parameters of the Mohr-Coulomb equation are now summarized in Figure 13.14 . Any point on the solid line
indicates the shear stress needed to exceed the shear strength related to specific values of normal stress, , cohesion,
c, and friction angle, . This line, however, shows 'ideal' intact rock mass strength (IRS), and the presence of
discontinuities and water in slopes substantially reduces shear strength. Discontinuities remove cohesion between
intact blocks. As water infiltrates the mass, any downslope component increases shear stress, v, and provides an
uplifting force u. This reduces normal stress to effective stress and may reduce the friction angle by x to a residual
value. Their collective impact is shown by discontinuous rock mass strength in dry (DRS d ) and wet (DRS w ) states
and the Mohr-Coulomb equation is modified to

+ v = c + ( - u) . tan ( - x)

or simplified to

e = c + e n . tan r

where e = effective shear stress, increased by the weight of water. This restates the balance of shear stress and

strength but has not yet formally introduced the significance of slope angle and still relates only to rock mass. However,
the modified equation applied to discontinuous rock mass is also a reasonable approximation for debris and soil on
slopes, with a little qualification. Cohesion = 0 in discontinuous rock mass and applies equally to large, loose blocks
or incohesive soil grains on a sloping surface. We can apply remaining Mohr-Coulomb criteria to such a block or
grain and see its sliding resistance , R, on a slope of known angle, given as

R = cA + W . cos B tan

where c = cohesion (zero), A is the block/grain-to-slope contact area, W = block/grain weight, B = slope angle and
= friction angle. However, part of the mass is mobilized downslope by gravity and limiting equilibrium reflects a
balance between perpendicularand tangentialforces. In the case shown in Figure 13.15 , water behind and beneath
the block, respectively, exerts shear stress and reduces effective stress.

Piezometric
(water pressure)
surface

IRS

φ

τ 1

Weight of water

DRS d

V

φ

τ

τ 2

Sliding force

DRS w

Friction
resistance

φ r

c

Uplift

u

τ 3

W sin B

Resisting
force

σ 1

σ

B

W cos B

W

Figure 13.14 Mohr-Coulomb relationship between shear
stress (), normal stress () and cohesion (c) in rock mass; see
text for explanation. The shear stress required for failure in a
rock mass with normal stress 1 falls from 1 in intact (non-
fractured) rock to 2 in dry, discontinuous rock and to 3 in wet,
discontinuous rock respectively.

Figure 13.15 Forces acting on a block on a rock slope; see
text for explanation. In wet conditions, water in discontinuities
(dark area) reduces sliding resistance by exerting a bouyancy
or uplift force, u, and adding the weight of water behind the
block, V, to sliding stress.