# Slope stability analysis

For a broader coverage related to this topic, see Slope stability.
Figure 1: Rotational failure of slope on circular slip surface

Slope stability analysis is performed to assess the safe design of a human-made or natural slopes (e.g. embankments, road cuts, open-pit mining, excavations, landfills etc.) and the equilibrium conditions.[1][2] Slope stability is the resistance of inclined surface to failure by sliding or collapsing.[3] The main objectives of slope stability analysis are finding endangered areas, investigation of potential failure mechanisms, determination of the slope sensitivity to different triggering mechanisms, designing of optimal slopes with regard to safety, reliability and economics, designing possible remedial measures, e.g. barriers and stabilization.[1][2]

Successful design of the slope requires geological information and site characteristics, e.g. properties of soil/rock mass, slope geometry, groundwater conditions, alternation of materials by faulting, joint or discontinuity systems, movements and tension in joints, earthquake activity etc.[4][5] The presence of water has a detrimental effect on slope stability. Water pressure acting in the pore spaces, fractures or other discontinuities in the materials that make up the pit slope will reduce the strength of those materials. [6] Choice of correct analysis technique depends on both site conditions and the potential mode of failure, with careful consideration being given to the varying strengths, weaknesses and limitations inherent in each methodology.[7]

Before the computer age stability analysis was performed graphically or by using a hand-held calculator. Today engineers have a lot of possibilities to use analysis software, ranges from simple limit equilibrium techniques through computational limit analysis approaches (e.g. Finite element limit analysis, Discontinuity layout optimization) to complex and sophisticated numerical solutions (finite-/distinct-element codes).[1] The engineer must fully understand limitations of each technique. For example, limit equilibrium is most commonly used and simple solution method, but it can become inadequate if the slope fails by complex mechanisms (e.g. internal deformation and brittle fracture, progressive creep, liquefaction of weaker soil layers, etc.). In these cases more sophisticated numerical modelling techniques should be utilised. In addition, the use of the risk assessment concept is increasing today. Risk assessment is concerned with both the consequence of slope failure and the probability of failure (both require an understanding of the failure mechanism).[8]

Within the last decade (2003) Slope Stability Radar has been developed to remotely scan a rock slope to monitor the spatial deformation of the face. Small movements of a rough wall can be detected with sub-millimeter accuracy by using interferometry techniques.

## Conventional methods of analysis

Most of the slope stability analysis computer programs are based on the limit equilibrium concept for a two- or three-dimensional model.[9][10] In rock slope stability analysis conventional methods can be divided into three groups: kinematic analysis, limit equilibrium and rock fall simulators.[8]

### Basic analysis

#### Method of slices

The method of slices is a method for analyzing the stability of a slope in two dimensions. 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 equilibrium for the slices.

#### Bishop's method

The Modified (or Simplified) Bishop's Method [11] proposed by Alan W. Bishop of Imperial College is a method for calculating the stability of slopes. It is an extension of the Method of Slices. By making some simplifying assumptions, the problem becomes statically determinate and suitable for hand calculations:

• forces on the sides of each slice are horizontal

The method has been shown to produce factor of safety values within a few percent of the "correct" values.

$F=\frac{\sum [\frac{c'+((W/b)-u)\tan\phi'}{\psi}]}{\sum[(W/b)\sin\alpha]}$

where

$\psi=\cos\alpha+\frac{\sin\alpha \tan\phi}{F}$
c' is the effective cohesion
$\phi'$ is the effective internal angle of internal friction
b is the width of each slice, assuming that all slices have the same width
W is the weight of each slice
u is the water pressure at the base of each slice

#### Sarma method

Main article: Sarma method

The Sarma method,[12] 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.

#### 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.

### Limit equilibrium analysis

The conventional limit equilibrium methods investigate the equilibrium of the soil mass tending to slide down under the influence of gravity. Transitional or rotational movement is considered on assumed or known potential slip surface below soil or rock mass.[13] In rock slope engineering, methods may be highly significant to simple block failure along distinct discontinuities.[8] All methods are based on comparison of forces (moments or stresses) resisting instability of the mass and those that causing instability (disturbing forces). Two-dimensional sections are analyzed assuming plain strain conditions. These methods assume that the shear strengths of the materials along the potential failure surface are governed by linear (Mohr-Coulomb) or non-linear relationships between shear strength and the normal stress on the failure surface.[13] analysis provides a factor of safety, defined as a ratio of available shear resistance (capacity) to that required for equilibrium. If the value of factor of safety is less than 1.0, slope is unstable. The most common limit equilibrium techniques are methods of slices where soil mass is discretized into vertical slices (Fig. 2).[10][14] Results (factor of safety) of particular methods can vary because methods differs in assumptions and satisfied equilibrium conditions.[13][15]

Figure 2: Method of slices

Functional slope design considers calculation with the critical slip surface where is the lowest value of factor of safety. Locating failure surface can be made with the help of computer programs using search optimization techniques.[16] Wide variety of slope stability software using limit equilibrium concept is available including search of critical slip surface. The program analyses the stability of generally layered soil slopes, mainly embankments, earth cuts and anchored sheeting structures. Fast optimization of circular and polygonal slip surfaces provides the lowest factor of safety. Earthquake effects, external loading, groundwater conditions, stabilization forces (i.e. anchors, georeinforcements etc.) can be also included. The software uses solution according to various methods of slices (Fig. 2), such as Bishop simplified, Ordinary method of slices (Swedish circle method/Petterson/Fellenius), Spencer, Sarma etc.

Sarma and Spencer are called as 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]

Another limit equilibrium program SLIDE[17] provides 2D stability calculations in rocks or soils using these rigorous analysis methods: Spencer, Morgenstern-Price/General limit equilibrium; and non-rigorous methods: Bishop simplified, Corps of Engineers, Janbu simplified/corrected, Lowe-Karafiath and Ordinary/Fellenius. Searching of the critical slip surface is realized with the help of a grid or as a slope search in user-defined area. Program includes also probabilistic analysis using Monte Carlo or Latin Hypercube simulation techniques where any input parameter can be defined as a random variable. Probabilistic analysis determine the probability of failure and reliability index, which gives better representation of the level of safety. Back analysis serves for calculation of a reinforcement load with a given required factor of safety. Program enables finite element groundwater seepage analysis.[17]

Program SLOPE/W[18] is formulated in terms of moment and force equilibrium factor of safety equations. Limit equilibrium methods include Morgenstern-Price, General limit equilibrium, Spencer, Bishop, Ordinary, Janbu etc. This program allows integration with other applications. For example finite element computed stresses from SIGMA/W[19] or QUAKE/W[20] can be used to calculate a stability factor by computing total shear resistance and mobilized shear stress along the entire slip surface. Then a local stability factor for each slice is obtained. Using a Monte Carlo approach, program computes the probability of failure in addition to the conventional factor of safety.[18]

STABL WV[21] is a limit equilibrium-based, Windows software based on the stabl family of algorithms. It allows analysis using Bishop's, Spencer's and Janbu's method. Regular slopes as well as slopes with various types of inclusions may be analyzed.

SVSlope[22] is formulated in terms of moment and force equilibrium factor of safety equations. Limit equilibrium methods include Morgenstern-Price, General limit equilibrium, Spencer, Bishop, Ordinary, Kulhawy and others This program allows integration with other applications in the geotechnical software suite. For example finite element computed stresses from SVSolid[23] or pore-water pressures from SVFlux[24] can be used to calculate the factor of safety by computing total shear resistance and mobilized shear stress along the entire slip surface. The software also utilizes Monte Carlo, Latin Hypercube, and the APEM probabilistic approaches. Spatial variability through random fields computations may also be included in the analysis.

Some other programs based on limit equilibrium concept:

• GALENA[25] - includes stability analysis, back analysis, and probability analysis, using the Bishop, Spencer-Wright and Sarma methods.[25]
• GSLOPE[26] - provides limit equilibrium slope stability analysis of existing natural slopes, unreinforced man-made slopes, or slopes with soil reinforcement, using Bishop’s Modified method and Janbu’s Simplified method applied to circular, composite or non-circular surfaces.[26]
• CLARA-W[27] - three-dimensional slope stability program includes calculation with the help of these methods: Bishop simplified, Janbu simplified, Spencer and Morgenstern-Price. Problem configurations can involve rotational or non-rotational sliding surfaces, ellipsoids, wedges, compound surfaces, fully specified surfaces and searches.[27]
• TSLOPE3[28] - two- or three-dimensional analyses of soil and rock slopes using Spencer method.[28]

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

• 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[29][30]
• 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.)[30]
• Wedge failure -> three-dimensional analysis enables modelling of the wedge sliding on two planes in a direction along the line of intersection[30][31]
• 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[32][33]

### Stereographic and kinematic analysis

Kinematic analysis examines which modes of failure can possibly occur in the rock mass. Analysis requires the detailed evaluation of rock mass structure and the geometry of existing discontinuities contributing to block instability.[34][35] Stereographic representation (stereonets) of the planes and lines is used.[36] Stereonets are useful for analyzing discontinuous rock blocks.[37] Program DIPS[38] allows for visualization structural data using stereonets, determination of the kinematic feasibility of rock mass and statistical analysis of the discontinuity properties.[34][38]

### Rockfall simulators

Rock slope stability analysis may design protective measures near or around structures endangered by the falling blocks. Rockfall simulators determine travel paths and trajectories of unstable blocks separated from a rock slope face. Analytical solution method described by Hungr & Evans[39] assumes rock block as a point with mass and velocity moving on a ballistic trajectory with regard to potential contact with slope surface. Calculation requires two restitution coefficients that depend on fragment shape, slope surface roughness, momentum and deformational properties and on the chance of certain conditions in a given impact.[40]

Program ROCFALL[41] provides a statistical analysis of trajectory of falling blocks. Method rely on velocity changes as a rock blocks roll, slide or bounce on various materials. Energy, velocity, bounce height and location of rock endpoints are determined and may be analyzed statistically. The program can assist in determining remedial measures by computing kinetic energy and location of impact on a barrier. This can help determine the capacity, size and location of barriers.[41]

## Numerical methods of analysis

Numerical modelling techniques provide an approximate solution to problems which otherwise cannot be solved by conventional methods, e.g. complex geometry, material anisotropy, non-linear behaviour, in situ stresses. Numerical analysis allows for material deformation and failure, modelling of pore pressures, creep deformation, dynamic loading, assessing effects of parameter variations etc. However, numerical modelling is restricted by some limitations. For example, input parameters are not usually measured and availability of these data is generally poor. Analysis must be executed by well trained user with good modelling practise. User also should be aware of boundary effects, meshing errors, hardware memory and time restrictions. Numerical methods used for slope stability analysis can be divided into three main groups: continuum, discontinuum and hybrid modelling.[42]

### Continuum modelling

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. 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.[42]

### Discontinuum modelling

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:[42]

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.[42]

Discontinuum program UDEC[43] (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.[43] Three-dimensional discontinuum code 3DEC[44] 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).[42]

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.[42]

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.[45][46] 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[42]

### Hybrid/coupled modelling

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 adopted in the SVOFFICE[47] or GEO-STUDIO[48] suites of software; coupled particle flow and finite-difference analyses used in PF3D[46] and FLAC3D.[49] 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, e.g. ELFEN,[50] provide for the modelling of both intact rock behaviour and the development and behaviour of fractures.[42]

## Rock mass classification

Various rock mass classification systems exist for the design of slopes and to assess the stability of slopes. The systems are based on empirical relations between rock mass parameters and various slope parameters such as height and slope dip.

## References

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3. ^ Kliche 1999, p. 2
4. ^ USArmyCorps 2003, pp. 1–2
5. ^ Abramson 2002, p. 1
6. ^ Beale, Geoff; Read, John, eds. (2014). Guidelines for Evaluating Water in Pit Slope Stability. CSIRO Publishing. ISBN 9780643108356.
7. ^ Stead 2001, p. 615
8. ^ a b c Eberhardt 2003, p. 6
9. ^ Abramson 2002, p. 329
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11. ^ Bishop, A. W. (1955). "The use of the Slip Circle in the Stability Analysis of Slopes". Géotechnique 5: 7. doi:10.1680/geot.1955.5.1.7. edit
12. ^ Sarma, S. K. (1975). "Seismic stability of earth dams and embankments". Géotechnique 25 (4): 743. doi:10.1680/geot.1975.25.4.743. edit
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