Comprehensive solution wedge stability - Rock Slope Engineering: Civil and Mining - page 405

arctan tan φ 1
( FS )

arcsin S z
S a

Compute s and v using equations (III.23)
and (III.24).

ψ t1 = −

(III.100)

N 1 + N 1 = T 3 ρ(rv − s) (III.112)

arctan S x
S y

N 2 + N 2 = T 3 µ(rs − v) (III.113)

α t1 = (III.101)

If N 1 < 0or N 2 < 0, ignore the results
of this section.
If N 1 > 0 and N 2 > 0, T min =

If N 1 < 0, contact is lost on plane 1
when T

(a)

T 3
If N 1 > 0 and N 2 < 0, T min = smallest of
T 1 , T 3
If N 1 < 0 and N 2 > 0, T min = smallest of
T 2 , T 3
If N 1 < 0 and N 2 < 0, T min = smallest of
T 1 , T 2 , T 3

= 0. Assume contact on plane 2

only, after application of T . Evalu-

ate S x , S y , S z , S b and Q b using equations

(III.72) to (III.76) with T = 0.

(( FS )S b − Q b )

T 2 =

( FS ) 2

(III.102)

+ tan 2 φ 2

arctan tan φ 2
( FS )

arcsin S z
S b

Example Calculate the factor of safety for the
following wedge:

ψ t2 = −

(III.103)

Plane 12345

arctan S x
S y

α t2 = (III.104)

ψ 45 70 12 65 70

α 105 235 195 185 165

All cases. No restrictions on values of N 1
and N 2 . Assume contact on both planes
after application of T .

(a)

η =+ 1

500 lb / ft 2 , H 1 = 100 ft, L = 40 ft, c 1 =

1000 lb / ft 2
φ 1 =

c 2 =

160 lb / ft 3 .

20 ◦ , φ 2 = 30 ◦ , γ

(( FS ) 2 =

χ = + B) (III.105)

(1a) T u 5

u 5 calculated from equation (III.59) .
(a x , a y , a z )

= 0, E = 0, u 1 = u 2 =

(( FS )S − Q )
χ

T 3 = (III.106)

= ( 0.68301, − 0.18301, 0.70711 )

(( FS )υi x − ρk x tan φ 1 − µl x tan φ 2 )
χ

t x =

(b x , b y , b z ) = ( − 0.76975, − 0.53899, 0.34202 )

(III.107)

(d x , d y , d z ) = ( − 0.05381, − 0.20083, 0.97815 )

(( FS )υi y − ρk y tan φ 1 − µl y tan φ 2 )
χ

(f x , f y , f z ) = ( − 0.07899, − 0.90286, 0.42262 )

t y =

(f 5 x , f 5 y , f 5 z ) = ( 0.24321,

−

0.90767, 0.34202 )

(III.108)

(g x , g y , g z ) = ( −

0.56107, 0.34451, 0.63112 )
(g 5 x , g 5 y , g 5 z ) = ( −

(( FS )υi z − ρk z tan φ 1 − µl z tan φ 2 )
χ

0.57923, 0.061627, 0.57544 )

t z =

(i x , i y , i z ) = ( − 0.31853, 0.77790, 0.50901 )

(j x , j y , j z ) = ( −

(III.109)

0.79826, 0.05452, − 0.03272 )

ψ t3 = arcsin ( − t z ) (III.110)

(j 5 x , j 5 y , j 5 z ) = ( − 0.81915, − 0.25630, − 0.09769 )

arctan t x
t y

(k x , k y , k z ) = ( 0.54041, − 0.28287, 0.77047 )

α t3 = (III.111)

(l x , l y , l z ) = ( − 0.64321, − 0.57289, 0.47302 )