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Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the '' stability of slopes'' of soil- and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. It is performed to assess the safe design of a human-made or natural
slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
s (e.g. embankments,
road cut In civil engineering, a cut or cutting is where soil or Rock (geology), rock from a relative rise is removed. Cuts are typically used in road, Rail transport, rail, and canal construction to reduce a route's length and Grade (slope), grade. Cut ...
s,
open-pit mining Open-pit mining, also known as open-cast or open-cut mining and in larger contexts mega-mining, is a surface mining technique that extracts rock (geology), rock or minerals from the earth. Open-pit mines are used when deposits of commercially ...
, excavations,
landfill A landfill is a site for the disposal of waste materials. It is the oldest and most common form of waste disposal, although the systematic burial of waste with daily, intermediate and final covers only began in the 1940s. In the past, waste was ...
s etc.) and the equilibrium conditions.
Slope stability Slope stability refers to the condition of inclined soil or rock slopes to withstand or undergo movement; the opposite condition is called slope instability or slope failure. The stability condition of slopes is a subject of study and research i ...
is the resistance of inclined surface to
failure Failure is the social concept of not meeting a desirable or intended objective, and is usually viewed as the opposite of success. The criteria for failure depends on context, and may be relative to a particular observer or belief system. On ...
by sliding or collapsing. 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 Safety is the state of being protected from harm or other danger. Safety can also refer to the control of recognized hazards in order to achieve an acceptable level of risk. Meanings The word 'safety' entered the English language in the 1 ...
,
reliability Reliability, reliable, or unreliable may refer to: Science, technology, and mathematics Computing * Data reliability (disambiguation), a property of some disk arrays in computer storage * Reliability (computer networking), a category used to des ...
and
economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
, and designing possible remedial measures, e.g. barriers and stabilization. Successful
design A design is the concept or proposal for an object, process, or system. The word ''design'' refers to something that is or has been intentionally created by a thinking agent, and is sometimes used to refer to the inherent nature of something ...
of the slope requires
geological Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth s ...
information and site characteristics, e.g. properties of
soil Soil, also commonly referred to as earth, is a mixture of organic matter, minerals, gases, water, and organisms that together support the life of plants and soil organisms. Some scientific definitions distinguish dirt from ''soil'' by re ...
/
rock Rock most often refers to: * Rock (geology), a naturally occurring solid aggregate of minerals or mineraloids * Rock music, a genre of popular music Rock or Rocks may also refer to: Places United Kingdom * Rock, Caerphilly, a location in Wale ...
mass, slope
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
,
groundwater Groundwater is the water present beneath Earth's surface in rock and Pore space in soil, soil pore spaces and in the fractures of stratum, rock formations. About 30 percent of all readily available fresh water in the world is groundwater. A unit ...
conditions, alternation of materials by
faulting In geology, a fault is a planar fracture or discontinuity in a volume of rock across which there has been significant displacement as a result of rock-mass movements. Large faults within Earth's crust result from the action of plate tectonic ...
,
joint A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
or discontinuity systems, movements and tension in joints, earthquake activity etc. 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. 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 In its most common sense, methodology is the study of research methods. However, the term can also refer to the methods themselves or to the philosophical discussion of associated background assumptions. A method is a structured procedure for bri ...
. Before the computer age stability analysis was performed graphically or by using a hand-held calculator. Today
engineer Engineers, as practitioners of engineering, are professionals who Invention, invent, design, build, maintain and test machines, complex systems, structures, gadgets and materials. They aim to fulfill functional objectives and requirements while ...
s have a lot of possibilities to use analysis
software Software consists of computer programs that instruct the Execution (computing), execution of a computer. Software also includes design documents and specifications. The history of software is closely tied to the development of digital comput ...
, ranges from simple ''limit equilibrium'' techniques through to computational limit analysis approaches (e.g. Finite element limit analysis, Discontinuity layout optimization) to complex and sophisticated '' numerical solutions'' ( finite-/ distinct-element codes). 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 Fracture is the appearance of a crack or complete separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity sur ...
, progressive creep,
liquefaction In materials science, liquefaction is a process that generates a liquid from a solid or a gas or that generates a non-liquid phase which behaves in accordance with fluid dynamics. It occurs both naturally and artificially. As an example of t ...
of weaker soil layers, etc.). In these cases more sophisticated
numerical modelling Computer simulation is the running of a mathematical model on a computer, the model being designed to represent the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determin ...
techniques should be utilised. Also, even for very simple slopes, the results obtained with typical limit equilibrium methods currently in use (Bishop, Spencer, etc.) may differ considerably. In addition, the use of the
risk assessment Risk assessment is a process for identifying hazards, potential (future) events which may negatively impact on individuals, assets, and/or the environment because of those hazards, their likelihood and consequences, and actions which can mitigate ...
concept is increasing today. Risk assessment is concerned with both the consequence of slope failure and the
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
of failure (both require an understanding of the failure mechanism).


Limit equilibrium analysis

Conventional methods of slope stability analysis can be divided into three groups:
kinematic In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with s ...
analysis, limit equilibrium analysis, and
rock Rock most often refers to: * Rock (geology), a naturally occurring solid aggregate of minerals or mineraloids * Rock music, a genre of popular music Rock or Rocks may also refer to: Places United Kingdom * Rock, Caerphilly, a location in Wale ...
fall simulators. Most slope stability analysis
computer program A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. It is one component of software, which also includes software documentation, documentation and other intangibl ...
s are based on the limit equilibrium concept for a two- or
three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
model. Two-dimensional sections are analyzed assuming
plane strain Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Pl ...
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 In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
. Translational or rotational movement is considered on an assumed or known potential slip surface below the soil or
rock Rock most often refers to: * Rock (geology), a naturally occurring solid aggregate of minerals or mineraloids * Rock music, a genre of popular music Rock or Rocks may also refer to: Places United Kingdom * Rock, Caerphilly, a location in Wale ...
mass. In rock slope engineering, methods may be highly significant to simple block failure along distinct discontinuities. All these methods are based on the comparison of
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s, moments, or stresses resisting movement of the mass with those that can cause
unstable In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
motion (disturbing forces). The output of the analysis is a
factor of safety In engineering, a factor of safety (FoS) or safety factor (SF) expresses how much stronger a system is than it needs to be for its specified maximum load. Safety factors are often calculated using detailed analysis because comprehensive testing i ...
, defined as the
ratio In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of the shear strength (or, alternatively, an equivalent measure of shear resistance or capacity) to the shear stress (or other equivalent measure) required for equilibrium. If the value of factor of safety is less than 1.0, the slope is unstable. All limit equilibrium methods assume that the
shear strength In engineering, shear strength is the strength of a material or component against the type of yield or structural failure when the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a mater ...
s 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. The most commonly used variation is Terzaghi's theory of shear strength which states that : \tau = \sigma'\tan\phi' + c' where \tau is the shear strength of the interface, \sigma' = \sigma - u is the effective stress (\sigma is the total stress normal to the interface and u is the pore water pressure on the interface), \phi' is the effective friction angle, and c' is the effective cohesion. The methods of slices is the most popular limit equilibrium technique. In this approach, the soil mass is discretized into vertical slices. 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 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 A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
. 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 Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value''. ''Precision'' is how close the measurements are to each other. The ...
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. Some of these approaches are discussed below.


Swedish Slip Circle Method of Analysis

The Swedish Slip Circle method assumes that the friction angle of the soil or rock is equal to zero, i.e., \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.


Ordinary 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. 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 In statics and structural mechanics, a structure is statically indeterminate when the equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and reactions on that structure. Mathemati ...
when they are included in the computation. For the ordinary method of slices, the resultant vertical and horizontal forces are : \begin \sum F_v = 0 &= W - N \cos\alpha - T \sin\alpha \\ \sum F_h = 0 &= kW + N \sin\alpha - T\cos\alpha \end 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 _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 = \frac \,.


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 Imperial College London, also known as Imperial, is a public research university in London, England. Its history began with Prince Albert, husband of Queen Victoria, who envisioned a cultural district in South Kensington that included museums ...
. 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 where : \psi_j = \cos\alpha_j+\frac 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 Karl von Terzaghi (October 2, 1883 – October 25, 1963) was an Austrians, Austrian Mechanical Engineer, mechanical engineer, geotechnical engineer, and geologist known as the "father of soil mechanics and geotechnical engineering". Early life I ...
.


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.


Sarma method

The Sarma method, proposed by Sarada K. Sarma of
Imperial College Imperial College London, also known as Imperial, is a public research university in London, England. Its history began with Prince Albert, husband of Queen Victoria, who envisioned a cultural district in South Kensington that included museums ...
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.


Rock slope stability analysis

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 * 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.) * Wedge failure ->
three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
analysis enables modelling of the wedge sliding on two planes in a direction along the line of intersection * 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

A more rigorous approach to slope stability analysis is limit analysis. Unlike limit equilibrium analysis which makes ad hoc though often reasonable assumptions, limit analysis is based on rigorous plasticity theory. This enables, among other things, the computation of upper and lower bounds on the true factor of safety. Programs based on limit analysis include:
OptumG2
(2014-) General purpose software for geotechnical applications (also includes elastoplasticity, seepage, consolidation, staged construction, tunneling, and other relevant geotechnical analysis types). * LimitState:GEO (2008-) General purpose geotechnical software application based on Discontinuity layout optimization for plane strain problems including slope stability.


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 In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior. ...
.
Stereographic Stereoscopy, also called stereoscopics or stereo imaging, is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision. The word ''stereoscopy'' derives . Any stereoscopic image is ...
representation (
stereonet In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thr ...
s) of the planes and lines is used. Stereonets are useful for analyzing discontinuous rock blocks. Program DIPS allows for visualization structural data using stereonets, determination of the kinematic feasibility of rock mass and statistical analysis of the discontinuity properties.


Rockfall simulators

Rock slope stability analysis may design protective measures near or around structures endangered by the falling blocks.
Rockfall A rockfall or rock-fallWhittow, John (1984). ''Dictionary of Physical Geography''. London: Penguin, 1984. . is a quantity of Rock (geology), rock that has fallen freely from a cliff face. The term is also used for collapse of rock from roof or wa ...
simulators determine travel paths and trajectories of unstable blocks separated from a rock slope face.
Analytical solution In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. C ...
method described by Hungr & Evans assumes rock block as a point with mass and
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
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.


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 Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ve ...
, non-linear behavior, in situ stresses.
Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
allows for material deformation and failure, modelling of
pore pressure Pore water pressure (sometimes abbreviated to pwp) refers to the pressure of groundwater held within a soil or rock, in gaps between particles ( pores). Pore water pressures below the phreatic level of the groundwater are measured with piezometers ...
s, 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. User also should be aware of boundary effects, meshing errors, hardware memory and time restrictions.
Numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...
s used for slope stability analysis can be divided into three main groups: continuum, discontinuum and hybrid modelling.


Continuum modelling

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
fracture Fracture is the appearance of a crack or complete separation of an object or material into two or more pieces under the action of stress (mechanics), stress. The fracture of a solid usually occurs due to the development of certain displacemen ...
s, e.g.
bedding plane In geology, a bed is a layer of sediment, sedimentary rock, or volcanic rock "bounded above and below by more or less well-defined bedding surfaces".Neuendorf, K.K.E., J.P. Mehl, Jr., and J.A. Jackson, eds., 2005. ''Glossary of Geology'' (5th ed ...
s, faults. Several constitutive models are usually available, e.g. elasticity, elasto-plasticity, strain-softening, elasto-viscoplasticity etc.


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 In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
are satisfied. Discontinuum modelling belongs to the most commonly applied numerical approach to rock slope analysis and following variations of the DEM exist: * 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 A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
are treated as [boundary conditions. Deformable blocks are discretized into internal constant-strain elements. Discontinuum program UDEC (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. Three-dimensional discontinuum code 3DEC 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). 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. 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. 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


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; 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. 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.


See also


References


Further reading

* * * * * * * * * Coduto, Donald P. (1998). ''Geotechnical Engineering: Principles and Practices.'' Prentice-Hall. * Fredlund, D. G., H. Rahardjo, M. D. Fredlund (2014). ''Unsaturated Soil Mechanics in Engineering Practice.'' Wiley-Interscience. * * * * * * * * * *


External links

{{Authority control Landslide analysis, prevention and mitigation