Principles of rock slope design - Rock Slope Engineering: Civil and Mining - page 12

(a)

(b)

FS = / s

W sin p

W cos p

p

= c + tan W

s

c

s

Effective normal stress ( )

Figure 1.8 Method of calculating factor of safety of sliding block: (a) Mohr diagram showing shear strength
defined by cohesion c and friction angle φ ; (b) resolution of force W due to weight of block into components
parallel and perpendicular to sliding plane (dip ψ p ) .

this surface is given by or

τ = c + σ tan φ

(1.1)

τ s A = W sin ψ p and

Equation (1.1) is expressed as a straight line on
a normal stress—shear stress plot (Figure 1.8(a)),
in which the cohesion is defined by the intercept
on the shear stress axis, and the friction angle is
defined by the slope of the line. The effective nor-
mal stress is the difference between the stress due
to the weight of the rock lying above the sliding
plane and the uplift due to any water pressure
acting on this surface.
Figure 1.8(b) shows a slope containing a con-
tinuous joint dipping out of the face and forming
a sliding block. Calculation of the factor of safety
for the block shown in Figure 1.8(b) involves the
resolution of the force acting on the sliding sur-
face into components acting perpendicular and
parallel to this surface. That is, if the dip of the
sliding surface is ψ p , its area is A , and the weight
of the block lying above the sliding surface is W ,
then the normal and shear stresses on the sliding
plane are

τA = cA + W cos ψ p tan φ (1.4)

In equations (1.4), the term [ W sin ψ p ] defines
the resultant force acting down the sliding plane
and is termed the “driving force” ( τ s A ), while
the term [ cA

W cos ψ p tan φ ] defines the shear
strength forces acting up the plane that resist slid-
ing and are termed the “resisting forces” ( τA) .
The stability of the block in Figure 1.8(b) can be
quantified by the ratio of the resisting and driv-
ing forces, which is termed the factor of safety,
FS. Therefore, the expression for the factor of
safety is

+

resisting forces
driving forces

FS = (1.5)

cA + W cos ψ p tan φ

W sin ψ p

FS = (1.6)

W cos ψ p
A

Normal stress, σ = and The displacing shear stress τ s and the resist-

ing shear stress τ defined by equations (1.4) are
plotted on Figure 1.8(a). On Figure 1.8(a) it is
shown that the resisting stress exceeds the displa-
cing stress, so the factor of safety is greater than
one and the slope is stable.
If the sliding surface is clean and contains no
infilling then the cohesion is likely to be zero and

W sin ψ p
A

shear stress, τ s = (1.2)

and equation (1.1) can be expressed as

W cos ψ p tan φ

τ = c + (1.3)

A