rock slope stability analysis: limit equilibrium method

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Rock Slope Stability Analysis: Limit Equilibrium Method Plane failure analysis Wedge failure analysis Toppling failure analysis

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Plane failure analysis Wedge failure analysis Toppling failure analysis . Rock Slope Stability Analysis: Limit Equilibrium Method . The block is considered to undergoes slippage along the plane for the value of ratio < 1, else it is stable. Planar Failure Analysis . - PowerPoint PPT Presentation

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Page 1: Rock Slope Stability Analysis: Limit Equilibrium Method

Rock Slope Stability Analysis: Limit Equilibrium Method

•Plane failure analysis

•Wedge failure analysis

•Toppling failure analysis

Page 2: Rock Slope Stability Analysis: Limit Equilibrium Method

Planar Failure Analysis

A block is rest on a slope having angle θ

The block is considered to undergoes slippage along the plane for the value of ratio < 1, else it is stable

Page 3: Rock Slope Stability Analysis: Limit Equilibrium Method

Plane failure analysis along a discontinuity

θ

A

B C

H

Unstable Block blockW

α

W

Geometry of a slope for plane failure

Page 4: Rock Slope Stability Analysis: Limit Equilibrium Method

Plane failure analysis along a discontinuity

Planar Failure Analysis

• the plane on which sliding occurs must strike parallel or nearly parallel (within

approximately + 200 ) to the slope face

• the failure must daylight in the slope face. This means that its dip must be smaller than the dip of the slope face

• the dip of the failure plane must be greater than the angle of internal friction angle of this plane

Page 5: Rock Slope Stability Analysis: Limit Equilibrium Method

Plane failure analysis along a discontinuity

W cosθW

W sinθ

R

Block A

sShearStresgthShearStren

Factor of safety =

s

c

tanFactor of safety =

Aw

Awc

sin

tancos

sintancos

wwcA

Factor of safety = =

Aw )sin( Normal Stress;

Aw )cos( Shear Stress ,

Page 6: Rock Slope Stability Analysis: Limit Equilibrium Method

Water is filled in discontinuities

'

2

41 gh

The effective normal stress due to present of water in the joint, is given as

Page 7: Rock Slope Stability Analysis: Limit Equilibrium Method

Tension crack present in the upper slope surface

Tension crack in upper surface of slope and in the face

Page 8: Rock Slope Stability Analysis: Limit Equilibrium Method

plane failure with tension crack

B

D

W

z

b C The depth of critical tension crack, zc and its

location, bc behind the crest can be calculated by the

following equations:

cot)cot(cot Hbc

Page 9: Rock Slope Stability Analysis: Limit Equilibrium Method

Length of discontinuities; SinCDHAD

The weight of the block;

Factor of safety =

sin

tancoswwcA

Page 10: Rock Slope Stability Analysis: Limit Equilibrium Method

Tension crack present in the slope surface

plane failure with tension crack

B

C

D

W

)tan)(tancot( bHz

Length of discontinuities; SinCDHAD

The weight of the block =

Factor of safety =

sin

tancoswwcA

Page 11: Rock Slope Stability Analysis: Limit Equilibrium Method

Compound slope with water in upper slope angle

Compound slope with a positive upper slope angle

Geometry of slope with tension crack in upper slope angle

c

Compound slopes have appreciable angle with the horizontal. High slope formation has in generally a positive upper slope angle while the shorter slope has a negative slope angle

Page 12: Rock Slope Stability Analysis: Limit Equilibrium Method

Depth of tension crack, tan)cot(tan HbbHZ c

Weight of unstable block, )cot21 2 bZbHXXHW

)cottan1( X

or

Area of failure surface, sec)cot( bHA

Driving water force, 2

21

ww ZV

Uplift water force, AZU ww21

Factor of safety =

cossin

tan)sincos(VW

VUwcA

Page 13: Rock Slope Stability Analysis: Limit Equilibrium Method

Effect of rock bolts

Geometry of slope with tension crack in upper slope and its interaction with rock bolt

Page 14: Rock Slope Stability Analysis: Limit Equilibrium Method

FOS =

sincossin

tan)cossincos(TVW

TVUwcA

Page 15: Rock Slope Stability Analysis: Limit Equilibrium Method

Wedge Failure Analysis

Geometric conditions of wedge failure: (a) pictorial view of wedge failure; (b) stereoplot showing the orientation of the line of intersection

Page 16: Rock Slope Stability Analysis: Limit Equilibrium Method

Analysis of wedge failure considering only frictional resistance

Resolution of forces to calculate factor of safety of wedge: (a) view of wedge looking at face showing definition of angles β and α, and reactions on sliding Plane RA and RB, (b) stereonet showing measurement of angles β and α, (c) cross-section of wedge showing resolution of wedge weight W.

Page 17: Rock Slope Stability Analysis: Limit Equilibrium Method

Plane failure analysis along a discontinuity

Page 18: Rock Slope Stability Analysis: Limit Equilibrium Method

Analysis of wedge failure with cohesion and friction angle

Pictorial View of wedge showing the numbering of intersection lines and planes

Page 19: Rock Slope Stability Analysis: Limit Equilibrium Method

Analysis of wedge failure with cohesion and friction angle

br

wa

r

wba

r

YBXAYCXCH

FS

tan)

2(tan)

2()(3

245

24

cossinsin

na

X

nbnai

nbnabaA.

2.

sinsincoscoscos

135

13

cossinsin

na

Y

nbnai

nbnaabB.

2.

sinsincoscoscos

Page 20: Rock Slope Stability Analysis: Limit Equilibrium Method

Analysis of wedge failure with cohesion and friction angle

Where, Ca and Cb are the cohesive strength of plane a and b, фa and фb are the angle of friction along plane a and b, is the unit weight of the rock, and H is the total height of the wedge. X, Y, A and B are dimensionless factors, which depend upon the geometry of the wedge, Ψa and Ψb are the dips of planes a and b, whereas, Ψi is the plunge of the line of their intersection.

Under fully drained slope condition, the water pressure is zero. Therefore, factor of safety of the wedge against failure is given by:

babar

BAYCXCH

FS

tantan)(3

Page 21: Rock Slope Stability Analysis: Limit Equilibrium Method

Toppling Failure Analysis

Page 22: Rock Slope Stability Analysis: Limit Equilibrium Method

Kinematics of block toppling failure

Case 1:

Case 2:

Case 3:

Case 4:

Page 23: Rock Slope Stability Analysis: Limit Equilibrium Method

Inter-layer slip test

If is the dip of slope face and α is the dip of the planes forming the sides of the blocks, then

the condition for interlayer slip is given by:

(180 − − α) ≥ (90 − ф)

or

α≥ (90 − ) + ф

Page 24: Rock Slope Stability Analysis: Limit Equilibrium Method

Block alignment test

The dip direction of the

planes forming sides of the

blocks, αd is within about 100

of the dip direction of the

slope face αf, i.e.

|(αf− αd)| <10◦

Page 25: Rock Slope Stability Analysis: Limit Equilibrium Method

Limit equilibrium analysis for toppling failure

The factor of safety can be calculated as the ratio of resisting

moments to driving moments

Page 26: Rock Slope Stability Analysis: Limit Equilibrium Method

Limit equilibrium analysis for toppling failure

Model for limiting equilibrium analysis of toppling on a stepped base (Goodman and Bray, 1976).

Page 27: Rock Slope Stability Analysis: Limit Equilibrium Method

Forces acting on the nth column sitting on a stepped base

Page 28: Rock Slope Stability Analysis: Limit Equilibrium Method

Figure 17: Limiting equilibrium conditions for toppling and sliding of nth block: (a) forces acting on nth block; (b) toppling of nth block;

Page 29: Rock Slope Stability Analysis: Limit Equilibrium Method
Page 30: Rock Slope Stability Analysis: Limit Equilibrium Method