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Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of earth and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. Slope stability refers to the condition of inclined soil or rock slopes to withstand or undergo movement. The stability condition of slopes is a subject of study and research in
soil mechanics Soil mechanics is a branch of soil physics and applied mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids (usually air and wat ...
, geotechnical engineering and
engineering geology Engineering geology is the application of geology to engineering study for the purpose of assuring that the geological factors regarding the location, design, construction, operation and maintenance of engineering works are recognized and accou ...
. Analyses are generally aimed at understanding the causes of an occurred slope failure, or the factors that can potentially trigger a slope movement, resulting in a
landslide Landslides, also known as landslips, are several forms of mass wasting that may include a wide range of ground movements, such as rockfalls, deep-seated grade (slope), slope failures, mudflows, and debris flows. Landslides occur in a variety of ...
, as well as at preventing the initiation of such movement, slowing it down or arresting it through
mitigation Mitigation is the reduction of something harmful or the reduction of its harmful effects. It may refer to measures taken to reduce the harmful effects of hazards that remain ''in potentia'', or to manage harmful incidents that have already occur ...
countermeasures. The stability of a slope is essentially controlled by the ratio between the available
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 materi ...
and the acting
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
, which can be expressed in terms of a
safety factor In engineering, a factor of safety (FoS), also known as (and used interchangeably with) safety factor (SF), expresses how much stronger a system is than it needs to be for an intended load. Safety factors are often calculated using detailed analy ...
if these quantities are integrated over a potential (or actual) sliding surface. A slope can be globally stable if the safety factor, computed along any potential sliding surface running from the top of the slope to its toe, is always larger than 1. The smallest value of the safety factor will be taken as representing the global stability condition of the slope. Similarly, a slope can be locally stable if a safety factor larger than 1 is computed along any potential sliding surface running through a limited portion of the slope (for instance only within its toe). Values of the global or local safety factors close to 1 (typically comprised between 1 and 1.3, depending on regulations) indicate marginally stable slopes that require attention, monitoring and/or an engineering intervention (
slope stabilization Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of earth and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. Slope stability refers to the condition of i ...
) to increase the safety factor and reduce the probability of a slope movement. A previously stable slope can be affected by a number of predisposing factors or processes that make the safety factor decrease - either by increasing the shear stress or by decreasing the shear strength - and can ultimately result in slope failure. Factors that can trigger slope failure include
hydrologic Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
events (such as intense or prolonged rainfall, rapid snowmelt, progressive soil saturation, increase of water pressure within the slope),
earthquake An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, fr ...
s (including
aftershock In seismology, an aftershock is a smaller earthquake that follows a larger earthquake, in the same area of the main shock, caused as the displaced crust adjusts to the effects of the main shock. Large earthquakes can have hundreds to thousand ...
s), internal erosion (piping), surface or toe erosion, artificial slope loading (for instance due to the construction of a building), slope cutting (for instance to make space for roadways, railways or buildings), or slope flooding (for instance by filling an
artificial lake A reservoir (; from French ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to store fresh water or it may be a natural formation. Reservoirs can be created in a number of ways, including contro ...
after
dam A dam is a barrier that stops or restricts the flow of surface water or underground streams. Reservoirs created by dams not only suppress floods but also provide water for activities such as irrigation, human consumption, industrial use ...
ming a river).


Examples

Earthen slopes can develop a cut-spherical weakness area. The probability of this happening can be calculated in advance using a simple 2-D circular analysis package. A primary difficulty with analysis is locating the most-probable slip plane for any given situation. Many landslides have only been analyzed after the fact. More recently
slope stability radar Interferometric synthetic aperture radar, abbreviated InSAR (or deprecated IfSAR), is a radar technique used in geodesy and remote sensing. This geodetic method uses two or more synthetic aperture radar (SAR) radar imaging, images to generate maps o ...
technology has been employed, particularly in the mining industry, to gather real time data and assist in determining the likelihood of slope failure. Real life failures in naturally deposited mixed soils are not necessarily circular, but prior to computers, it was far easier to analyze such a simplified geometry. Nevertheless, failures in 'pure' clay can be quite close to circular. Such slips often occur after a period of heavy rain, when the pore water pressure at the slip surface increases, reducing the effective
normal stress In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
and thus diminishing the restraining
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
along the slip line. This is combined with increased soil weight due to the added groundwater. A 'shrinkage' crack (formed during prior dry weather) at the top of the slip may also fill with rain water, pushing the slip forward. At the other extreme, slab-shaped slips on hillsides can remove a layer of soil from the top of the underlying bedrock. Again, this is usually initiated by heavy rain, sometimes combined with increased loading from new buildings or removal of support at the toe (resulting from road widening or other construction work). Stability can thus be significantly improved by installing drainage paths to reduce the destabilizing forces. Once the slip has occurred, however, a weakness along the slip circle remains, which may then recur at the next monsoon.


Measuring the Angle of Repose

The angle of repose is defined as the steepest angle of granular unconfined material measured from the horizontal plane on which the
granular Granularity (also called graininess), the condition of existing in granules or grains, refers to the extent to which a material or system is composed of distinguishable pieces. It can either refer to the extent to which a larger entity is subd ...
material can be heaped on without collapsing, ranging between 0-90°. For granular material, the angle of repose is the main factor that influences the slope's stability under different conditions in relation to the cohesiveness/
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
of the material, the grain size, and the particle shape.


Theoretical Measurement

A simple
free body diagram A free body diagram consists of a diagrammatic representation of a single body or a subsystem of bodies isolated from its surroundings showing all the forces acting on it. In physics and engineering, a free body diagram (FBD; also called a force ...
can be used to understand the relationship between the angle of repose and the stability of the material on the
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
. For the heaped material to collapse, the frictional forces must be equivalent to the horizontal component of the
gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
m g \sin\theta, where m is the mass of the material, g is the gravitational acceleration and \theta is the slope angle: m g \sin\theta = f The frictional force f is equivalent to the multiplication product of the coefficient of static friction \mu and the Normal Force N or mgcos\theta: m g \sin\theta = N \mu m g \sin\theta = \mu m g \cos\theta \left ( \frac \right ) = \mu \theta_R = \arctan(\mu) Where \theta_R is the angle of repose, or the angle at which the slope fails under regular conditions, and  \mu is the coefficient of static friction of the material on the slope.


Experimental Measurement


''Tilting Box Method''

This method is particularly well suited for finer grained material < 10mm in diameter with relatively no cohesion. The material is placed in a base of a box, which is gradually tilted at a rate of 18°/min. The angle of repose is then measured to be the angle at which the material begins to slide.


''Fixed Funnel Method''

In this method, the material is poured down a
funnel A funnel is a tube or pipe that is wide at the top and narrow at the bottom, used for guiding liquid or powder into a small opening. Funnels are usually made of stainless steel, aluminium, glass, or plastic. The material used in its construct ...
from a certain height into a horizontal base. The material is then allowed to heap until the pile reaches a predetermined height and width. The angle of repose is then measured by observing the height and
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the cone and applying the arctangent rule.


Angle of Repose and Slope Stability

The angle of repose and the stability of a slope are impacted by
climatic Climate is the long-term weather pattern in an area, typically averaged over 30 years. More rigorously, it is the mean and variability of meteorological variables over a time spanning from months to millions of years. Some of the meteorological ...
and non-climatic factors.


''Water Content''

Water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
content is an important parameter that could change the angle of repose. Reportedly, a higher water content can stabilize a slope and increase the angle of repose. However, water saturation can result in a decrease in the slope's stability since it acts as a lubricant and creates a detachment where
mass wasting Mass wasting, also known as mass movement, is a general term for the movement of rock or soil down slopes under the force of gravity. It differs from other processes of erosion in that the debris transported by mass wasting is not entrained in ...
can occur. Water content is dependent on soil properties such as grain size, which can impact
infiltration Infiltration may refer to: Science, medicine, and engineering *Infiltration (hydrology), downward movement of water into soil *Infiltration (HVAC), a heating, ventilation, and air conditioning term for air leakage into buildings *Infiltration (me ...
rate, runoff, and water retention. Generally, finer-grained soils rich in
clay Clay is a type of fine-grained natural soil material containing clay minerals (hydrous aluminium phyllosilicates, e.g. kaolin, Al2 Si2 O5( OH)4). Clays develop plasticity when wet, due to a molecular film of water surrounding the clay par ...
and
silt Silt is granular material of a size between sand and clay and composed mostly of broken grains of quartz. Silt may occur as a soil (often mixed with sand or clay) or as sediment mixed in suspension with water. Silt usually has a floury feel ...
retain more water than coarser sandy soils. This effect is mainly due to
capillary action Capillary action (sometimes called capillarity, capillary motion, capillary rise, capillary effect, or wicking) is the process of a liquid flowing in a narrow space without the assistance of, or even in opposition to, any external forces li ...
, where the adhesive forces between the fluid, particle, and the cohesive forces of the fluid itself counteract gravitational pull. Therefore, smaller grain size results in a smaller surface area on which gravitational forces can act. Smaller surface area also leads to more capillary action, more water retention, more infiltration, and less runoff.


''Vegetation''

The presence of
vegetation Vegetation is an assemblage of plant species and the ground cover they provide. It is a general term, without specific reference to particular taxa, life forms, structure, spatial extent, or any other specific botanical or geographic characte ...
does not directly impact the angle of repose, but it acts as a stabilizing factor in a hillslope, where the tree roots anchor into deeper
soil Soil, also commonly referred to as earth or dirt Dirt is an unclean matter, especially when in contact with a person's clothes, skin, or possessions. In such cases, they are said to become dirty. Common types of dirt include: * Debri ...
layers and form a fiber‐reinforced soil composite with a higher shear resistance (mechanical cohesion).


''Roundness of Grains''

The shape of the grain can have an impact on the angle of repose and the stability of the slope. The more rounded the grain is, the lower the angle of repose. A decrease in roundness, or an increase in angularity, results in interlocking via particle contact. This linear relationship between the angle of repose and the roundness of grain can also be used as a predictor of the angle of repose if the roundness of the grain is measured.


Applications of Angle of Repose in Science and Engineering

The angle of repose is related to the shear strength of geologic materials, which is relevant in
construction Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and com ...
and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
contexts. For granular materials, the size and shape of grains can impact angle of repose significantly. As the roundness of materials increases, the angle of repose decreases since there is less friction between the soil grains. When the angle of repose is exceeded, mass wasting and
rockfall A rockfall or rock-fallWhittow, John (1984). ''Dictionary of Physical Geography''. London: Penguin, 1984. . is a quantity/sheets of rock that has fallen freely from a cliff face. The term is also used for collapse of rock from roof or walls of mi ...
can occur. It is important for many civil and geotechnical engineers to know the angle of repose to avoid structural and
natural disaster A natural disaster is "the negative impact following an actual occurrence of natural hazard in the event that it significantly harms a community". A natural disaster can cause loss of life or damage property, and typically leaves some econ ...
s. As a result, the application of
retaining wall Retaining walls are relatively rigid walls used for supporting soil laterally so that it can be retained at different levels on the two sides. Retaining walls are structures designed to restrain soil to a slope that it would not naturally keep to ...
s can help to retain soil so that the angle of repose is not exceeded.


Slope stabilization

Since the stability of the slope can be impacted by external events such as
precipitation In meteorology, precipitation is any product of the condensation of atmospheric water vapor that falls under gravitational pull from clouds. The main forms of precipitation include drizzle, rain, sleet, snow, ice pellets, graupel and hail. ...
, an important concern in civil/ geotechnical engineering is the stabilization of slopes.


Application of Vegetation

The application of vegetation to increase the slope stability against
erosion Erosion is the action of surface processes (such as water flow or wind) that removes soil, rock, or dissolved material from one location on the Earth's crust, and then transports it to another location where it is deposited. Erosion is dis ...
and
landslide Landslides, also known as landslips, are several forms of mass wasting that may include a wide range of ground movements, such as rockfalls, deep-seated grade (slope), slope failures, mudflows, and debris flows. Landslides occur in a variety of ...
is a form of bioengineering that is widely used in areas where the landslide depth is shallow. Vegetation increases the stability of the slope mechanically, by reinforcing the soils through plant roots, which stabilize the upper part of the soil. Vegetation also stabilizes the slope via
hydrologic Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
processes, by the reduction of soil
moisture Moisture is the presence of a liquid, especially water, often in trace amounts. Small amounts of water may be found, for example, in the air (humidity), in foods, and in some commercial products. Moisture also refers to the amount of water vapo ...
content through the interception of precipitation and
transpiration Transpiration is the process of water movement through a plant and its evaporation from aerial parts, such as leaves, stems and flowers. Water is necessary for plants but only a small amount of water taken up by the roots is used for growth ...
. This results in a drier soil that is less susceptible to mass wasting. Stability of slopes can also be improved by: * Flattening of slopes results in reduction in weight which makes the slope more stable * Soil stabilization * Providing lateral supports by piles or retaining walls * Grouting or cement injections into specific zones * Consolidation by surcharging or electro osmosis increases the stability of slope


Analysis methods

If the forces available to resist movement are greater than the forces driving movement, the slope is considered stable. A
factor of safety In engineering, a factor of safety (FoS), also known as (and used interchangeably with) safety factor (SF), expresses how much stronger a system is than it needs to be for an intended load. Safety factors are often calculated using detailed analy ...
is calculated by dividing the forces resisting movement by the forces driving movement. In earthquake-prone areas, the analysis is typically run for static conditions and pseudo-static conditions, where the seismic forces from an earthquake are assumed to add static loads to the analysis. Slope stability analysis is performed to assess the safe design of a human-made or natural
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
s (e.g. embankments,
road cut In civil engineering, a cut or cutting is where soil or rock from a relative rise along a route is removed. The term is also used in river management to speed a waterway's flow by short-cutting a meander. Cuts are typically used in road, rail, ...
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 of extracting rock or minerals from the earth from an open-air pit, sometimes known as a borrow. This form of mini ...
, excavations, landfills etc.) and the equilibrium conditions.
Slope stability Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of earth and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. Slope stability refers to the condition of i ...
is the resistance of inclined surface to failure 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 "safe", the condition 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 There are two slightly dif ...
,
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 * High availability * Reliability (computer networking), a ...
and
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, designing possible remedial measures, e.g. barriers and stabilization. Successful
design A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design'' ...
of the slope requires
geological Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other E ...
information and site characteristics, e.g. properties of
soil Soil, also commonly referred to as earth or dirt Dirt is an unclean matter, especially when in contact with a person's clothes, skin, or possessions. In such cases, they are said to become dirty. Common types of dirt include: * Debri ...
/
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 Wales ...
mass, slope
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
groundwater Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidated ...
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 Tension may refer to: Science * Psychological stress * Tension (physics), a force related to the stretching of an object (the opposite of compression) * Tension (geology), a stress which stretches rocks in two opposite directions * Voltage or el ...
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 br ...
. 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 invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the l ...
s have a lot of possibilities to use analysis
software Software is a set of computer programs and associated documentation and data. This is in contrast to hardware, from which the system is built and which actually performs the work. At the lowest programming level, executable code consists ...
, ranges from simple ''limit equilibrium'' techniques through to computational limit analysis approaches (e.g.
Finite element limit analysis A finite element limit analysis (FELA) uses optimisation techniques to directly compute the upper or lower bound plastic collapse load (or limit load) for a mechanical system rather than time stepping to a collapse load, as might be undertaken wit ...
,
Discontinuity layout optimization Discontinuity layout optimization (DLO) is an engineering analysis procedure which can be used to directly establish the amount of load that can be carried by a solid or structure prior to collapse. Using DLO the layout of failure planes, or 'dis ...
) 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 Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
and
brittle fracture Fracture is the 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 surfaces within the solid. If a displa ...
, progressive creep, liquefaction of weaker soil layers, etc.). In these cases more sophisticated
numerical modelling Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
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 Broadly speaking, a risk assessment is the combined effort of: # identifying and analyzing potential (future) events that may negatively impact individuals, assets, and/or the environment (i.e. hazard analysis); and # making judgments "on the ...
concept is increasing today. Risk assessment is concerned with both the consequence of slope failure and the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of failure (both require an understanding of the failure mechanism). Within the last decade (2003)
Slope stability radar Interferometric synthetic aperture radar, abbreviated InSAR (or deprecated IfSAR), is a radar technique used in geodesy and remote sensing. This geodetic method uses two or more synthetic aperture radar (SAR) radar imaging, images to generate maps o ...
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.


Limit equilibrium analysis

Conventional methods of slope stability analysis can be divided into three groups:
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
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 Wales ...
fall Autumn, also known as fall in American English and Canadian English, is one of the four temperate seasons on Earth. Outside the tropics, autumn marks the transition from summer to winter, in September (Northern Hemisphere) or March ( Southe ...
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 execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program ...
s are based on the limit equilibrium concept for a two- or
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
model. Two-dimensional sections are analyzed assuming
plane strain In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
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 () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. 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 Wales ...
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 change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s, moments, or stresses resisting movement of the mass with those that can cause
unstable In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
motion (disturbing forces). The output of the analysis is a
factor of safety In engineering, a factor of safety (FoS), also known as (and used interchangeably with) safety factor (SF), expresses how much stronger a system is than it needs to be for an intended load. Safety factors are often calculated using detailed analy ...
, 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 materi ...
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'', while ''precision'' is how close the measurements are to each ot ...
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 static equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and reactions on that structure. Math ...
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. 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, Wis 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 An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals. E ...
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 Austrian mechanical engineer, geotechnical engineer, and geologist known as the "father of soil mechanics and geotechnical engineering". Early life In 1883, he was born the first ch ...
.


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 The Sarma method is a method used primarily to assess the stability of soil slopes under earthquake, seismic conditions. Using appropriate assumptions the method can also be employed for static slope stability analysis. It was proposed by Sarada K. ...
, proposed by
Sarada K. Sarma Sarada Kanta Sarma is a geotechnical engineer, emeritus reader of engineering seismology and senior research investigator at Imperial College London. He has developed a method of seismic slope stability analysis which is named after him, the Sarm ...
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.


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 Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
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 Limit analysis is a structural analysis field which is dedicated to the development of efficient methods to directly determine estimates of the collapse load of a given structural model without resorting to iterative or incremental analysis. For t ...
. 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 Discontinuity layout optimization (DLO) is an engineering analysis procedure which can be used to directly establish the amount of load that can be carried by a solid or structure prior to collapse. Using DLO the layout of failure planes, or 'dis ...
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 numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
. Stereographic representation ( stereonets) 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/sheets of rock that has fallen freely from a cliff face. The term is also used for collapse of rock from roof or walls of mi ...
simulators determine travel paths and trajectories of unstable blocks separated from a rock slope face. Analytical solution method described by Hungr & Evans assumes rock block as a point with mass and
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
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 property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
, 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 Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
and failure, modelling of pore pressures, Creep (deformation), 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 methods used for slope stability analysis can be divided into three main groups: Continuum mechanics, continuum, discontinuum and hybrid modelling.


Continuum modelling

Modelling of the Continuum mechanics, continuum is suitable for the analysis of soil slopes, massive intact rock or heavily jointed rock masses. This approach includes the ''Finite difference method, finite-difference'' and ''finite element'' methods that Discretization, discretize the whole mass to finite number of elements with the help of generated mesh (Fig. 3). In ''Finite difference method, finite-difference'' method (FDM) Differential equation, 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 Displacement field (mechanics), displacements and stresses between elements. Most of numerical codes allows modelling of discrete fractures, e.g. bedding planes, Fault (geology), faults. Several constitutive models are usually available, e.g. Elasticity (physics), elasticity, elasto-plasticity, strain-softening, Viscoplasticity#Elastic perfectly viscoplastic solid, 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, 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 Newton's laws of motion, motion 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 ''discrete element method, 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 Newton's laws of motion, law of motion (determining displacements caused in the blocks by out-of-balance forces). Joint (geology), Joints 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 ''discrete element method, 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 method, finite-difference'' analyses. Hybrid techniques allows investigation of piping slope failures and the influence of high groundwater pressures on the failure of weak rock slope. Coupled ''finite-''/''discrete element method, distinct-element'' codes provide for the modelling of both intact rock behavior and the development and behavior of fractures.


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.


Probability classification

The slope stability probability classification (SSPC) system is a rock mass classification system for slope engineering and slope stability assessment. The system is a three-step classification: ''‘exposure’'', ''‘reference’'', and ''‘slope’'' rock mass classification with conversion factors between the three steps depending on existing and future weathering and damage due to method of excavation. The stability of a slope is expressed as probability for different failure mechanisms. A rock mass is classified following a standardized set of criteria in one or more exposures (''‘exposure’'' classification). These values are converted per exposure to a ''‘reference’'' rock mass by compensating for the degree of weathering in the exposure and the method of excavation that was used to make the exposure, i.e. the ''‘reference’'' rock mass values are not influenced by local influences such as weathering and method of excavation. A new slope can then be designed in the ''‘reference’'' rock mass with compensation for the damage due to the method of excavation to be used for making the new slope and compensation for deterioration of the rock mass due to future weathering (the ''‘slope’'' rock mass). If the stability of an already existing slope is assessed the ''‘exposure’'' and ''‘slope’'' rock mass values are the same. The failure mechanisms are divided in ''orientation dependent'' and ''orientation independent''. Orientation dependent failure mechanisms depend on the orientation of the slope with respect to the orientation of the Discontinuity (geomechanics), discontinuities in the rock mass, i.e. sliding (plane and wedge sliding) and toppling failure. Orientation independent relates to the possibility that a slope fails independently from its orientation, e.g. circular failure completely through newly formed discontinuities in intact rock blocks, or failing partially following existing discontinuities and partially new discontinuities. In addition the shear strength along a discontinuity ('sliding criterion') and 'rock mass cohesion' and 'rock mass friction' can be determined. The system has been used directly or modified in various geology and climate environments throughout the world. The system has been modified for slope stability assessment in open pit coal mining.


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

* {{Geotechnical engineering Landslide analysis, prevention and mitigation