Conventional methods of slope stability analysis can be divided into three groups:
kinematic analysis, limit equilibrium analysis, and
rock fall simulators. Most slope stability analysis
computer programs are based on the limit equilibrium concept for a
two- or
three-dimensional model. Two-dimensional sections are analyzed assuming
plane strain conditions. Stability analyses of two-dimensional slope geometries using simple analytical approaches can provide important insights into the initial design and risk assessment of slopes. Limit equilibrium methods investigate the equilibrium of a soil mass tending to slide down under the influence of
gravity. Translational or rotational movement is considered on an assumed or known potential slip surface below the soil or
rock mass. In rock slope engineering, methods may be highly significant to simple block failure along distinct discontinuities. Several versions of the method are in use. These variations can produce different results (factor of safety) because of different
assumptions and inter-slice boundary conditions. The location of the interface is typically unknown but can be found using numerical optimization methods. For example,
functional slope design considers the
critical slip surface to be the location where that has the lowest value of factor of safety from a range of possible slip surfaces. A wide variety of slope stability software use the limit equilibrium concept with automatic critical slip surface determination. Typical slope stability software can analyze the stability of generally layered soil slopes, embankments, earth cuts, and anchored sheeting
structures. Earthquake effects, external
loading, groundwater conditions,
stabilization forces (i.e., anchors, geo-reinforcements etc.) can also be included.
Analytical techniques: Method of slices Many slope stability analysis tools use various versions of the methods of slices such as
Bishop simplified,
Ordinary method of slices (
Swedish circle method/Petterson/Fellenius),
Spencer,
Sarma etc.
Sarma and
Spencer are called rigorous methods because they satisfy all three conditions of equilibrium: force equilibrium in horizontal and vertical direction and moment equilibrium condition. Rigorous methods can provide more
accurate results than non-rigorous methods.
Bishop simplified or
Fellenius are non-rigorous methods satisfying only some of the equilibrium conditions and making some simplifying assumptions. This allows for a simple static equilibrium calculation, considering only soil weight, along with shear and normal stresses along the failure plane. Both the friction angle and cohesion can be considered for each slice. In the general case of the method of slices, the forces acting on a slice are shown in the figure below. The normal (E_r, E_l) and shear (S_r, S_l) forces between adjacent slices constrain each slice and make the problem
statically indeterminate when they are included in the computation. For the ordinary method of slices, the resultant vertical and horizontal forces are : \begin{align} \sum F_v = 0 &= W - N \cos\alpha - T \sin\alpha \\ \sum F_h = 0 &= kW + N \sin\alpha - T\cos\alpha \end{align} where k represents a linear factor that determines the increase in horizontal force with the depth of the slice. Solving for N gives : N = W\cos\alpha - kW\sin\alpha \,. Next, the method assumes that each slice can rotate about a center of rotation and that moment balance about this point is also needed for equilibrium. A balance of moments for all the slices taken together gives : \sum M = 0 = \sum_j (W_j x_j - T_j R_j - N_j f_j - k W_j e_j) where j is the slice index, x_j, R_j, f_j, e_j are the moment arms, and loads on the surface have been ignored. The moment equation can be used to solve for the shear forces at the interface after substituting the expression for the normal force: : \sum_j T_j R_j = \sum_j [W_j x_j - ( W_j\cos\alpha_j - kW_j\sin\alpha_j)f_j - k W_j e_j] Using Terzaghi's strength theory and converting the stresses into moments, we have : \sum_j \tau l_j R_j = l_j R_j \sigma_j'\tan\phi' + l_j R_j c' = R_j (N_j - u_j l_j) \tan\phi' + l_j R_j c' where u_j is the pore pressure. The factor of safety is the ratio of the maximum moment from Terzaghi's theory to the estimated moment, : \text{Factor of safety} = \frac{ \sum_j \tau l_j R_j}{ \sum_j T_j R_j} \,.
Modified Bishop's Method of Analysis The Modified Bishop's method is slightly different from the ordinary method of slices in that normal interaction forces between adjacent slices are assumed to be collinear and the resultant interslice shear force is zero. The approach was proposed by
Alan W. Bishop of
Imperial College. The constraint introduced by the normal forces between slices makes the problem statically indeterminate. As a result, iterative methods have to be used to solve for the factor of safety. The method has been shown to produce factor of safety values within a few percent of the "correct" values. The factor of safety for moment equilibrium in Bishop's method can be expressed as : F = \cfrac{\sum_j \cfrac{\left[c' l_j +(W_j -u_j l_j)\tan\phi'\right]}{\psi_j}}{\sum_j W_j\sin\alpha_j} where : \psi_j = \cos\alpha_j+\frac{\sin\alpha_j \tan\phi'}{F} where, as before, j is the slice index, c' is the effective cohesion, \phi' is the effective internal angle of internal friction, l is the width of each slice, W is the weight of each slice, and u is the water pressure at the base of each slice. An iterative method has to be used to solve for F because the factor of safety appears both on the left and right hand sides of the equation.
Lorimer's method Lorimer's Method is a technique for evaluating slope stability in cohesive soils. It differs from Bishop's Method in that it uses a
clothoid slip surface in place of a circle. This mode of failure was determined experimentally to account for effects of particle cementation. The method was developed in the 1930s by Gerhardt Lorimer (Dec 20, 1894-Oct 19, 1961), a student of geotechnical pioneer
Karl von Terzaghi.
Spencer's Method Spencer's Method of analysis requires a computer program capable of cyclic algorithms, but makes slope stability analysis easier. Spencer's algorithm satisfies all equilibria (horizontal, vertical and driving moment) on each slice. The method allows for unconstrained slip plains and can therefore determine the factor of safety along any slip surface. The rigid equilibrium and unconstrained slip surface result in more precise safety factors than, for example, Bishop's Method or the Ordinary Method of Slices. proposed by
Sarada K. Sarma of
Imperial College is a
Limit equilibrium technique used to assess the stability of slopes under seismic conditions. It may also be used for static conditions if the value of the horizontal load is taken as zero. The method can analyse a wide range of slope failures as it may accommodate a multi-wedge failure mechanism and therefore it is not restricted to planar or circular failure surfaces. It may provide information about the factor of safety or about the critical acceleration required to cause collapse.
Comparisons The assumptions made by a number of limit equilibrium methods are listed in the table below. The table below shows the statical equilibrium conditions satisfied by some of the popular limit equilibrium methods. •
Polygonal failure -> sliding of a nature rock usually takes place on
polygonally-shaped surfaces; calculation is based on a certain assumptions (e.g. sliding on a polygonal surface which is composed from
N parts is kinematically possible only in case of development at least
(N - 1) internal shear surfaces; rock mass is divided into blocks by internal shear surfaces; blocks are considered to be rigid; no tensile strength is permitted etc.) •
Toppling failure -> long thin rock columns formed by the steeply dipping discontinuities may rotate about a pivot point located at the lowest corner of the block; the sum of the moments causing toppling of a block (i.e. horizontal weight component of the block and the sum of the driving forces from adjacent blocks behind the block under consideration) is compared to the sum of the moments resisting toppling (i.e. vertical weight component of the block and the sum of the resisting forces from adjacent blocks in front of the block under consideration); toppling occur if driving moments exceed resisting moments ==Limit analysis==