Slope stability analysis

Schematic of the method of slices showing rotation center.

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.[14][15] Some of these approaches are discussed below.

The Swedish Slip Circle method assumes that the friction angle of the soil or rock is equal to zero, i.e., ${\displaystyle \tau =c'}$. In other words, when friction angle is considered to be zero, the effective stress term goes to zero, thus equating the shear strength to the cohesion parameter of the given soil. The Swedish slip circle method assumes a circular failure interface, and analyzes stress and strength parameters using circular geometry and statics. The moment caused by the internal driving forces of a slope is compared to the moment caused by forces resisting slope failure. If resisting forces are greater than driving forces, the slope is assumed stable.

Division of the slope mass in the method of slices.

In the method of slices, also called OMS or the Fellenius method, the sliding mass above the failure surface is divided into a number of slices. The forces acting on each slice are obtained by considering the mechanical (force and moment) equilibrium for the slices. Each slice is considered on its own and interactions between slices are neglected because the resultant forces are parallel to the base of each slice. However, Newton's third law is not satisfied by this method because, in general, the resultants on the left and right of a slice do not have the same magnitude and are not collinear.[17]

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 (${\displaystyle E_{r},E_{l}}$) and shear (${\displaystyle S_{r},S_{l}}$) forces between adjacent slices constrain each slice and make the problem statically indeterminate when they are included in the computation.

Force equilibrium for a slice in the method of slices. The block is assumed to have thickness ${\displaystyle b}$. The slices on the left and right exert normal forces ${\displaystyle E_{l},E_{r}}$ and shear forces ${\displaystyle S_{l},s_{r}}$, the weight of the slice causes the force ${\displaystyle W}$. These forces are balanced by the pore pressure and reactions of the base ${\displaystyle N,T}$.

For the ordinary method of slices, the resultant vertical and horizontal forces are

${\displaystyle {\begin{aligned}\sum F_{v}=0&=W-N\cos \alpha -T\sin \alpha \\\sum F_{h}=0&=kW+N\sin \alpha -T\cos \alpha \end{aligned}}}$

where ${\displaystyle k}$ represents a linear factor that determines the increase in horizontal force with the depth of the slice. Solving for ${\displaystyle N}$ gives

${\displaystyle N=W\cos \alpha -kW\sin \alpha \,.}$

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

${\displaystyle \sum M=0=\sum _{j}(W_{j}x_{j}-T_{j}R_{j}-N_{j}f_{j}-kW_{j}e_{j})}$

where ${\displaystyle j}$ is the slice index, ${\displaystyle x_{j},R_{j},f_{j},e_{j}}$ 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:

${\displaystyle \sum _{j}T_{j}R_{j}=\sum _{j}[W_{j}x_{j}-(W_{j}\cos \alpha _{j}-kW_{j}\sin \alpha _{j})f_{j}-kW_{j}e_{j}]}$

Using Terzaghi's strength theory and converting the stresses into moments, we have

${\displaystyle \sum _{j}\tau l_{j}R_{j}=l_{j}R_{j}\sigma _{j}'\tan \phi '+l_{j}R_{j}c'=R_{j}(N_{j}-u_{j}l_{j})\tan \phi '+l_{j}R_{j}c'}$

where ${\displaystyle u_{j}}$ is the pore pressure. The factor of safety is the ratio of the maximum moment from Terzaghi's theory to the estimated moment,

${\displaystyle {\text{Factor of safety}}={\frac {\sum _{j}\tau l_{j}R_{j}}{\sum _{j}T_{j}R_{j}}}\,.}$

The Modified Bishop's method[18] 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

${\displaystyle F={\cfrac {\sum _{j}{\cfrac {\left[c'l_{j}+(W_{j}-u_{j}l_{j})\tan \phi '\right]}{\psi _{j}}}}{\sum _{j}W_{j}\sin \alpha _{j}}}}$

where

${\displaystyle \psi _{j}=\cos \alpha _{j}+{\frac {\sin \alpha _{j}\tan \phi '}{F}}}$

where, as before, ${\displaystyle j}$ is the slice index, ${\displaystyle c'}$ is the effective cohesion, ${\displaystyle \phi '}$ is the effective internal angle of internal friction, ${\displaystyle l}$ is the width of each slice, ${\displaystyle W}$is the weight of each slice, and ${\displaystyle u}$ is the water pressure at the base of each slice. An iterative method has to be used to solve for ${\displaystyle F}$ because the factor of safety appears both on the left and right hand sides of the equation.

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 of analysis[19] 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.[19]

The Sarma method,[20] 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.

The assumptions made by a number of limit equilibrium methods are listed in the table below.[21]

The table below shows the statical equilibrium conditions satisfied by some of the popular limit equilibrium methods.[21]

Rock slope stability analysis based on limit equilibrium techniques may consider following modes of failures:

• Planar failure -> case of rock mass sliding on a single surface (special case of general wedge type of failure); two-dimensional analysis may be used according to the concept of a block resisting on an inclined plane at limit equilibrium[27][28]
• 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.)[28]
• Wedge failure -> three-dimensional analysis enables modelling of the wedge sliding on two planes in a direction along the line of intersection[28][29]
• 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[30][31]

Figure 3: Finite element mesh

Modelling of the continuum is suitable for the analysis of soil slopes, massive intact rock or heavily jointed rock masses. This approach includes the finite-difference and finite element methods that discretize the whole mass to finite number of elements with the help of generated mesh (Fig. 3). In finite-difference method (FDM) differential equilibrium equations (i.e. strain-displacement and stress-strain relations) are solved. finite element method (FEM) uses the approximations to the connectivity of elements, continuity of displacements and stresses between elements.[40] Most of numerical codes allows modelling of discrete fractures, e.g. bedding planes, faults. Several constitutive models are usually available, e.g. elasticity, elasto-plasticity, strain-softening, elasto-viscoplasticity etc.[39]

Discontinuum approach is useful for rock slopes controlled by discontinuity behaviour. Rock mass is considered as an aggregation of distinct, interacting blocks subjected to external loads and assumed to undergo motion with time. This methodology is collectively called the discrete-element method (DEM). Discontinuum modelling allows for sliding between the blocks or particles. The DEM is based on solution of dynamic equation of equilibrium for each block repeatedly until the boundary conditions and laws of contact and motion are satisfied. Discontinuum modelling belongs to the most commonly applied numerical approach to rock slope analysis and following variations of the DEM exist:[39]

• distinct-element method
• Discontinuous Deformation Analysis (DDA)
• particle flow codes

The distinct-element approach describes mechanical behaviour of both, the discontinuities and the solid material. This methodology is based on a force-displacement law (specifying the interaction between the deformable rock blocks) and a law of motion (determining displacements caused in the blocks by out-of-balance forces). Joints are treated as [boundary conditions. Deformable blocks are discretized into internal constant-strain elements.[39]

Discontinuum program UDEC[41] (Universal distinct element code) is suitable for high jointed rock slopes subjected to static or dynamic loading. Two-dimensional analysis of translational failure mechanism allows for simulating large displacements, modelling deformation or material yielding.[41] Three-dimensional discontinuum code 3DEC[42] contains modelling of multiple intersecting discontinuities and therefore it is suitable for analysis of wedge instabilities or influence of rock support (e.g. rockbolts, cables).[39]

In Discontinuous Deformation Analysis (DDA) displacements are unknowns and equilibrium equations are then solved analogous to finite element method. Each unit of finite element type mesh represents an isolated block bounded by discontinuities. Advantage of this methodology is possibility to model large deformations, rigid body movements, coupling or failure states between rock blocks.[39]

Discontinuous rock mass can be modelled with the help of distinct-element methodology in the form of particle flow code, e.g. program PFC2D/3D.[43][44] Spherical particles interact through frictional sliding contacts. Simulation of joint bounded blocks may be realized through specified bond strengths. Law of motion is repeatedly applied to each particle and force-displacement law to each contact. Particle flow methodology enables modelling of granular flow, fracture of intact rock, transitional block movements, dynamic response to blasting or seismicity, deformation between particles caused by shear or tensile forces. These codes also allow to model subsequent failure processes of rock slope, e.g. simulation of rock[39]

Hybrid codes involve the coupling of various methodologies to maximize their key advantages, e.g. limit equilibrium analysis combined with finite element groundwater flow and stress analysis; coupled particle flow and finite-difference analyses; hydro-mechanically coupled finite element and material point methods for simulating the entire process of rainfall-induced landslides.[45] Hybrid techniques allows investigation of piping slope failures and the influence of high groundwater pressures on the failure of weak rock slope. Coupled finite-distinct-element codes provide for the modelling of both intact rock behavior and the development and behavior of fractures.