Discussion: Determination of critical slip surface in slope analysis

Wang, J.-J.; Lin, X.

doi:10.1680/geot.2007.57.5.481

The authors have developed a new method for determining the critical slip surface in slope stability analysis based on the limit equilibrium technique. In the new method, six unknowns—the angles α_i and δ_i+1, the forces on the slip surface N′_i and T_i, and the forces on the interslice boundary E′_i+1 and X_i+1—can be obtained from the six equations (1), (2), (3), (4), (6) and (9). It is noted that the angle δ_i+1 made by the interslice boundary with the vertical is also an unknown. In this way, it is inconvenient to solve the problem of the slope stability. Another more convenient method for determining the critical slip surface in slope analysis is developed in this discussion.

For a simpler homogeneous soil slope, the angle δ_i+1 can approximately be obtained by Rankine's theory (Rankine, 1857) as

δ_{i + 1} = \frac{π}{4} - \frac{{\bar{ϕ}}_{i + 1}^{'}}{2}

(37)

In order to obtain the factor of safety of the analysed slope conveniently, the factor of safety on the slip surface is defined as (Bishop, 1955)

F_{s} = \frac{N_{i}^{'} tan ϕ_{i}^{'} + c_{i}^{'} b_{i} sec α_{i}}{T_{i}}

(38)

The forces acting on the inclined slice are shown in Fig. 14. The equilibrium conditions in the horizontal and vertical directions for the forces acting on the slice are respectively expressed by

\begin{array}{l} - N_{i}^{'} sin α_{i} - U_{i} sin α_{i} + T_{i} cos α_{i} \\ + E_{i}^{'} sin (90 - δ_{i} - {\bar{ϕ}}_{i}^{'}) + P W_{i} cos δ_{i} - X_{i}^{'} sin δ_{i} \\ - E_{i + 1}^{'} sin (90 - δ_{i + 1} - {\bar{ϕ}}_{i + 1}^{'}) - P W_{i + 1} cos δ_{i + 1} \\ + X_{i + 1}^{'} sin δ_{i + 1} - Q_{i} - H_{i} = 0 \end{array}

(39)

\begin{array}{l} N_{i}^{'} cos α_{i} + U_{i} cos α_{i} + T_{i} sin α_{i} \\ - E_{i}^{'} cos (90 - δ_{i} - {\bar{ϕ}}_{i}^{'}) - P W_{i} sin δ_{i} - X_{i}^{'} cos δ_{i} \\ + E_{i + 1}^{'} cos (90 - δ_{i + 1} - {\bar{ϕ}}_{i + 1}^{'}) - P W_{i + 1} sin δ_{i + 1} \\ + X_{i + 1}^{'} cos δ_{i + 1} - W_{i} - V_{i} = 0 \end{array}

(40)

where Q_i is the total horizontal inertia force induced by earthquakes under seismic conditions; H_i and V_i are the horizontal and vertical forces acting on the upper surface of the slice, respectively; and X′_i+1 = c′_i+1d_i+1.

Fig. 14.

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Forces acting on an inclined slice

The equilibrium condition for the moments of all the forces on the slice about the corner point A (Fig. 14) is given by

\begin{array}{l} N_{i}^{'} l_{i} + U_{i} l_{i}^{'} + X_{i + 1}^{'} b_{i} sec α_{i} sin (90 - α_{i} - δ_{i + 1}) \\ + E_{i + 1}^{'} sin {\bar{ϕ}}_{i + 1}^{'} b_{i} sec α_{i} sin (90 - α_{i} - δ_{i + 1}) \\ + E_{i + 1}^{'} cos {\bar{ϕ}}_{i + 1}^{'} [z_{i + 1} + b_{i} sec α_{i} cos (90 - α_{i} - δ_{i + 1})] \\ + P W_{i + 1} [z_{i + 1}^{'} + b_{i} sec α_{i} cos (90 - α_{i} - δ_{i + 1})] \\ + H_{i} d_{i} cos δ_{i} + Q_{i} y_{i} - W_{i} x_{i} - V_{i} a_{i} - P W_{i} z_{i}^{'} \\ - E_{i}^{'} cos {\bar{ϕ}}_{i}^{'} z_{i} = 0 \end{array}

(41)

where l′_i and z′_i are the distances from the acting points of the forces U_i and PW_i to the point A, respectively; z′_{i + 1} is the distance from the acting point of the force PW_i+1 to the point B; x_i and y_i are the horizontal and vertical distances from the centre of gravity of the slice to the point A, respectively; and a_i is the horizontal distance from the acting point of the force V_i to the point A. All of the six distances are known for any slice of an analysed slope.

When assuming that the normal stress distribution in any of the slip surface AB, and interslice boundaries AD and BC, is linear, the total effective normal stresses acting on the slip surface and interslice boundary can be rewritten as

N_{i}^{'} = \frac{1}{2} (σ_{i}^{'} + σ_{i + 1}^{'}) b_{i} sec α_{i}

(42)

E_{i}^{'} = \frac{1}{2} ({σ_{i}^{'}}^{0} + σ_{i}^{'}) d_{i}

(43)

The distances l_i and z_i from the acting points of the effective forces N′_i and E′_i to the point A are given by

l_{i} = \frac{b_{i} sec α_{i}}{3} \cdot \frac{σ_{i}^{'} + 2 σ_{i + 1}^{'}}{σ_{i}^{'} + σ_{i + 1}^{'}}

(44)

z_{i} = \frac{d_{i}}{3} \cdot \frac{2 {σ_{i}^{'}}^{0} + σ_{i}^{'}}{{σ_{i}^{'}}^{0} + σ_{i}^{'}}

(45)

Substituting equations (42) to (45) into equations (39) to (41), and considering equation (38), yields the following equations:

\begin{array}{l} \frac{1}{2} (σ_{i}^{'} + σ_{i + 1}^{'}) b_{i} (\frac{tan ϕ_{i}^{'}}{F_{s}} - tan α_{i}) \\ + \frac{1}{2} (σ_{i}^{' 0} + σ_{i}^{'}) d_{i} sin (90 - δ_{i} - {\bar{ϕ}}_{i}^{'}) \\ - \frac{1}{2} (σ_{i + 1}^{' 0} + σ_{i + 1}^{'}) d_{i + 1} sin (90 - δ_{i + 1} - {\bar{ϕ}}_{i + 1}^{'}) \\ + \frac{c_{i}^{'} b_{i}}{F_{s}} - {\bar{c}}_{i}^{'} d_{i} sin δ_{i} + {\bar{c}}_{i + 1}^{'} d_{i + 1} sin δ_{i + 1} \\ - U_{i} sin α_{i} + P W_{i} cos δ_{i} - P W_{i + 1} cos δ_{i + 1} \\ - Q_{i} - H_{i} = 0 \end{array}

(46)

\begin{array}{l} \frac{1}{2} (σ_{i}^{'} + σ_{i + 1}^{'}) b_{i} (1 + \frac{tan ϕ_{i}^{'} tan α_{i}}{F_{s}}) \\ - \frac{1}{2} (σ_{i}^{' 0} + σ_{i}^{'}) d_{i} cos (90 - δ_{i} - {\bar{ϕ}}_{i}^{'}) \\ + \frac{1}{2} (σ_{i + 1}^{' 0} + σ_{i + 1}^{'}) d_{i + 1} cos (90 - δ_{i + 1} - {\bar{ϕ}}_{i + 1}^{'}) \\ + \frac{c_{i}^{'} b_{i} tan α_{i}}{F_{s}} - {\bar{c}}_{i}^{'} d_{i} cos δ_{i} + {\bar{c}}_{i + 1}^{'} d_{i + 1} cos δ_{i + 1} \\ + U_{i} cos α_{i} - P W_{i} sin δ_{i} + P W_{i + 1} sin δ_{i + 1} \\ - W_{i} - V_{i} = 0 \end{array}

(47)

\begin{array}{l} \frac{1}{6} b_{i}^{2} (σ_{i}^{'} + 2 σ_{i + 1}^{'}) {sec}^{2} α_{i} - \frac{1}{6} d_{i}^{2} (2 σ_{i}^{' 0} + σ_{i}^{'}) cos {\bar{ϕ}}_{i}^{'} \\ + \frac{1}{2} d_{i + 1} b_{i} (σ_{i + 1}^{' 0} + σ_{i + 1}^{'}) sec α_{i} \\ \times cos (90 - α_{i} - δ_{i + 1} - {\bar{ϕ}}_{i + 1}^{'}) \\ + \frac{1}{6} d_{i + 1}^{2} cos {\bar{ϕ}}_{i + 1}^{'} (2 σ_{i + 1}^{' 0} + σ_{i + 1}^{'}) \\ + {\bar{c}}_{i + 1}^{'} d_{i + 1}^{'} b_{i} sec α_{i} sin (90 - α_{i} - δ_{i + 1}) \\ + U_{i} l_{i}^{'} - P W_{i} z_{i}^{'} \\ + P W_{i + 1} [z_{i + 1}^{'} + b_{i} sec α_{i} cos (90 - α_{i} - δ_{i + 1})] \\ + H_{i} d_{i} cos δ_{i} + Q_{i} y_{i} - W_{i} x_{i} - V_{i} α_{i} = 0 \end{array}

(48)

From equations (46)–(48) it is clear that, for the ith slice, four unknowns need to be solved: F_s, σ′_{i + 1}, $σ_{i + 1}^{' 0}$ and α_i. For the whole slope the number of all unknowns is (3n − 1)—one for F_s, (n − 1) for σ′_{i + 1}, (n − 1) for $σ_{i + 1}^{' 0}$ and n for α_i—and the number of all equations is 3n, so that the problem can be solved.

For the first slice (Fig. 1) the effective normal stress at the starting point of the slip surface is obtained from the Mohr circle. If there is no surcharge in the starting point, the effective normal stress is given by

σ_{1}^{'} = \frac{c_{1}^{'} cos ϕ_{1}^{'}}{F_{s}}

(49)

Considering a value for the factor of safety F_s, all other unknowns σ′₂, $σ_{2}^{' 0}$ and α₁ can be obtained. Then, for the ith slice from 2 and (n − 1), the unknowns can also be obtained using the given value of F_s. For the last slice the effective normal stress $σ_{n + 1}^{' 0}$ at the ending point of the slip surface is also obtained from the Mohr circle. If there is no surcharge in the ending point, the effective normal stress is given by

σ_{n + 1}^{' 0} = \frac{c_{n}^{'} cos ϕ_{n}^{'}}{F_{s}}

(50)

Hence the unknowns of the last slice are only F_s, σ′_{n + 1} and α_n. A new value for the factor of safety F_s can be obtained from equations (46)–(48) for the last slice by comparing the calculated value for the factor of safety with the value given above, considering another new value for the factor of safety, and repeating the computation until the problem is solved. The critical slip surface and critical factor of safety are also obtained.

Authors' reply

S. K. Sarma

D. Tan

The authors would like to thank the discussers for their interest in our work and their discussion of our paper. It is clear that the limit equilibrium technique of slope stability analysis can be modified in different ways to obtain solutions. Having gone through the discussion material, we would like to make the following comments on the discussers' note.

We know that, given α_i and δ_i+1, the factor of safety/critical acceleration can be determined as in the Sarma (1979) method. In the discussers' method α_i is an unknown and δ_i+1 is a function of α_i for each slice, and therefore they need one extra equation for each slice compared with the traditional methods. It seems that they adjust the angle α_i so that the resulting normal stresses match with the linear stress distribution both along the slip surface and along the interslice boundaries. Therefore they will get a solution, but in the absence of numerical examples it is not clear whether the solution will be acceptable. Also, it is not clear what friction angle they are going to use for defining the interslice angle when either the factor of safety is unknown or/and when the slope is non-homogeneous.

Their definition of E′_{i + 1} is not appropriate. If E′_{i + 1} represents the inclined force, then X_{i + 1} is a component of E′_{i + 1} and should not appear in their equation or in their Fig. 14. If E′_{i + 1} represents the normal force acting on the interslice boundary, then it should not be inclined.

The discussers' assumption that X_{i + 1} = c_{i + 1}d_{i + 1} + E′_{i + 1} tan ϕ_{i + 1} is not proper. Because the discussers are doing a factor of safety analysis, the interslice boundary should not be in limiting equilibrium (i.e. a factor of safety of unity). We believe that the factor of safety in interslice boundaries should be greater than or equal to the factor of safety on the slip surface, F_s, as discussed by Janbu (1973).

The more severe deficit of the discussers' work is that, for a given slip surface and interslice boundaries, their method cannot give a factor of safety or critical acceleration. Because the calculation of factor of safety is dependent on σ′_i and $σ_{i + 1}^{' 0}$ ⁠, for a given slip surface and interslice boundaries it can give an infinite number of factors of safety.

Unfortunately, the discussers do not provide any numerical examples to prove their method; furthermore, they do not discuss non-homogeneous cases. It would have been interesting to see some results compared with ours.

REFERENCE

Rankine

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