Rock Slope Stability Analysis: Limit Equilibrium Slope Stability Analysis: Limit Equilibrium ......

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Transcript of Rock Slope Stability Analysis: Limit Equilibrium Slope Stability Analysis: Limit Equilibrium ......

  • Rock Slope Stability Analysis: Limit Equilibrium Method

    Plane failure analysis

    Wedge failure analysis

    Toppling failure analysis

  • Planar Failure Analysis

    A block is rest on a slope having angle

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

  • Plane failure analysis along a discontinuity

    A

    B C

    H

    Unstable Block blockW

    W

    Geometry of a slope for plane failure

  • 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

  • Plane failure analysis along a discontinuity

    W cosW

    W sin

    R

    Block A

    sShearStresgthShearStren

    Factor of safety =

    s

    c

    tan+Factor of safety =

    Aw

    Awc

    sin

    tancos+

    sintancos

    wwcA +

    Factor of safety = =

    Aw )sin( =Normal Stress;

    Aw )cos( =Shear Stress ,

  • Water is filled in discontinuities

    ='

    2

    41 gh =

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

  • Tension crack present in the upper slope surface

    Tension crack in upper surface of slope and in the face

  • 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

  • Length of discontinuities; SinCDHAD =

    The weight of the block;

    Factor of safety =

    sin

    tancoswwcA +

  • 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 +

  • 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

  • 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, 221

    ww ZV =

    Uplift water force, AZU ww21

    =

    Factor of safety =

    cossin

    tan)sincos(VW

    VUwcA+

    +

  • Effect of rock bolts

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

  • FOS =

    sincossin

    tan)cossincos(TVW

    TVUwcA+

    ++

  • Wedge Failure Analysis

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

  • 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 and , and reactions on sliding Plane RA and RB, (b) stereonet showing measurement of angles and , (c) cross-section of wedge showing resolution of wedge weight W.

  • Plane failure analysis along a discontinuity

  • Analysis of wedge failure with cohesion and friction angle

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

  • 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

    =

  • Analysis of wedge failure with cohesion and friction angle

    Where, Ca and Cb are the cohesive strength of plane a and b, a and b 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, a and b are the dips of planes a and b, whereas, i 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 +++=

  • Toppling Failure Analysis

  • Kinematics of block toppling failure

    Case 1:

    Case 2:

    Case 3:

    Case 4:

  • Inter-layer slip test

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

    the condition for interlayer slip is given by:

    (180 ) (90 )

    or

    (90 ) +

  • Block alignment test

    The dip direction of the

    planes forming sides of the

    blocks, d is within about 100

    of the dip direction of the

    slope face f, i.e.

    |(f d)|

  • Limit equilibrium analysis for toppling failure

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

    moments to driving moments

  • Limit equilibrium analysis for toppling failure

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

  • Forces acting on the nth column sitting on a stepped base

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