Geomechanical models

a) Plane failure model (2D)
b) Wedge failure model


 

a) Plane failure model (2D)

(1) Model geometry and parameters

H = height of the slope face [m] or [ft]
d = distance of the tension crack from the slope crest [m] or [ft]
z = depth of the tension crack
fw% = filling of the tension crack (0 = no water, 1 = filled)
gr = unit weight of rock [kN/m³] or [lb/ft³]
gw = unit weight of water [kN/m³] or [lb/ft³]
bsf = dip angle of the slope face [°]
bus = dip angle of the upper surface [°]
bfp = dip angle ot the failure plane [°]
c = cohesive strength (cohesion) on the failure plane [kPa] or [lb/ft²]
f = friction angle on the failure plane [°]
e = seismic coefficient (fraction of g) = horizontal acceleration
brb = dip angle of rock bolt/cable [°]
T = tension of the cable [kPa] or [lb/ft²]
F = safety factor

(2) Comments

Two cases are considered : · the slope has a tension crack in its upper face · the slope has a tension crack in its face

(3) Results

F is computed following the plane failure analysis of HOEK & BRAY (1981, pp. 150-154).

(4) Example

H = 30 m
d = 9 m (tension crack in the upper surface)
fw% = 0 (tension crack fully drained) or 1 (tension crack filled)
g
r = 25.5 kN/m³
g
w = 9.81 kN/m³
b
sf = 60°
b
fp = 30°
c = 48 kPa
f = 30°

F = 1.347 (drained) or 0.794 (filled) z = 14.804 m

Idem with :
b
rb = 24° and
T = 1000 kPa

F = 1.695 (drained) or 1.000 (filled)

 

b) Wedge failure model

(1) Model geometry and parameters

H = slope height (referred to plane 1) [m] or [ft]
D = distance of tension crack from crest along trace of plane 1
gr = unit weight of rock [kg/m³3] or [lb/ft³]
gw = unit weight of water [kg/m³] or [lb/ft³]
a1,2,3,4,5 = dip direction of the planes 1, 2, 3, 4 and 5 [°]
b1,2,3,4,5 = dip angle of the planes 1, 2, 3, 4 and 5 [°]
c1,2 = apparent cohesion on the planes 1 and 2 = cohesive strength [kPa] or [lb/ft²]
f1,2 = friction angle on the planes 1 and 2 [°]
F = safety factor

Three additional parameters are used to indicate

(2) Comments

When selecting a pair of random discontinuities at random from a set of data, it is not known whether :

In order to resolve these uncertainties, the discontinuities are labelled 1 and 2, the upper ground surface by 3, the slope face by 4 and the tension crack by 5. This order is illustrated in the sketch above. A check on whether the two planes do form a wedge is included in the sequence of calculations.

Depending upon the geometry of the wedge and the magnitude of the water pressure acting on each plane, contact may be lost on either plane.

(3) Results

F is computed following HOEK & BRAY (1981). The calculations determine whether there is contact on plane 1 only, on plane 2 only or on both planes, or whether contact is lost on both planes.

(4) Example

H = 100 ft
D = 40 ft
gr = 160 lb/ft³
g
w = 62.4 lb/ft³
a1,2,3,4,5 = 105, 235, 195, 185 and 165°
b1,2,3,4,5 = 45, 70, 12, 65 and 70°
c1,2 = 500 and 1000 lb/ft2
f1,2 = 20 and 30°

With the tension crack
F = 1.139 (with water) or 1.736 (dry slope)
Wedge weight : 28'272'195 lb
Sliding direction : 157.732° / 31.197°

Without the tension crack
F = 1.219 (with water) or 1.846 (dry slope)
Wedge weight : 36'788'030 lb