Fundamntals of rock behaviour modelling and numerical modelling methods and applications in the field of rock mechanics

1. NUMERICAL MODELLING
AN EFFECTIVE TOOL
FOR
MINE PLANNING
U.Siva Sankar, M.Tech
Under Manager,
Project Planning
SCCL
Email : uss_7@yahoo.com
Modelling
Proper understanding of complex behaviour of rock mass has always
always
been difficult for reliable design and safe operation of mining
excavations.
Understanding the behaviour of rock in general and the jointed rock
rock
mass, in particular, has always been difficult for mining engineers
engineers
involved in reliable planning and design, and safe operation of mining
projects under complex and difficult conditions.
Model: It is any representation or abstraction of a system or
process. A model is an intellectual abstraction that includes purpose,
process.
reference and cost effectiveness ( Starfield & Beloch, 1986).
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2. Modelling
Various models used in Mining:
Photo-Elastic
Models
Physical
Models
Equivalent
Material Models
Models Closed Form
Solutions
Analytical
Models
Mathematical Limit Equilibrium
Models Solutions
Numerical
Models
Physical Modelling
Physical Model: It is a miniature replica of some physical systems is of use.
These are more commonly abstractions of reality. Models are used to simulate
in the laboratory the behaviour of full scale prototype
Photo elasticity is an experimental method to determine stress distribution in a
material. The method is mostly used in cases where mathematical methods
become quite cumbersome.
The photoelastic stress analysis technique depends upon the fact that certain
optical properties of most transparent material change when these materials
are subject to stress.
The model is machined from a stress birefringent material like glass or plastic,
for, e.g., tunnel represented as a circular hole in a plate
Glass, PE rubber or epoxy resin – for hard and moderate deformable
rockmasses develop stress after being loaded at boundaries and gelatin – highly
deformable rockmasses develop stress under own weight
When a polarised light passes through a stress birefringent material patterns
of coloured or black fringes are produced.
Fringes gives trajectories of principle stresses and its direction.
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3. Photo Elastic Models
Photoelastic pattern in a glass plate
model containing a central circular hole Photoelastic pattern – Concentration of
from which vertical tensile cracks have stresses in Lower part of a Slope
propagated.
Photo Elastic Models
CSIR Polariscope for Photoelastic model analysis
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4. Equivalent Material Model
Equivalent Material Model: the purpose of this model or realistic model is to
simulate in the laboratory the behaviour of full scale prototype
Elastic, plastic behaviour, viscous flow, fracture of the modeled structure can
be simulated
Selection of Model materials and loading conditions to be carefully done
Models are built on principles of dimensional Similitude
Model Materials : generally weak fabricated materials, materials are blended
to simulate stratification, jointing and other realistic geological features.
Plaster of Paris, lead oxide saw dust oil , gypsum plaster
Disadvantages are time taking, involves labor , for every study different
models are to be built.
Equivalent Material Model
Model in loading Frame ready for testing Model deformation w.r.t roof cracking
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5. Mathematical Modelling
Mathematical Model: The representation of a physical system by mathematical
Model:
expressions from which the behaviour of the system can be deduced with known
deduced
accuracy.
Analytical solutions
1. Closed Form Solutions;
These are mathematical relations between stresses and displacements for every
displacements
point in the surrounding material.
Analytical solution for stresses and displacements around a circular hole in a
circular
biaxillay loaded elastic plate (Kirsch in 1898)
(Kirsch
Analytical solution for stresses and displacements around a parallel sided slot in
parallel
an infinite elastic medium (Salomon, 1968 & 1974).
1974) .
Analytical solution for stresses and displacements around a elliptical opening
elliptical
(Brady & Brown, 1985).
1985).
Rock-support interaction analysis (Hoek & Brown, 1980)
Rock-
2. Limit Equilibrium solutions;
In this technique gravitational stresses acting on a rigid wedge or block separated
from surrounding rockmass by discontinuities are calculated and are checked against
shear resistance offered by the contact surfaces to determine whether the block can
fall or slide.
Surrounding stress field is ignored in this technique
e.g. Slope analysis, Concept of dead weight design for designing bolting in
galleries
Slope Rock Load in a gallery
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6. Numerical Modelling
In general, the numerical, or analytical, design methods are derived from the
fundamental laws of force, stress, and elasticity.
Numerical modelling techniques require far more computational power than
analytical techniques, but they are well suited to address complex geometries
and material behaviour.
Most of the Numerical modelling undertaken in the process of mine planning
and design involves using linear elastic, static, and boundary element
programs.
The speed, memory efficiency and ease of use of these codes renders them
well suited to quick design analysis.
Numerical models can represent complex geometries with a high degree of
accuracy.
Numerical Modelling
• Approach adopted in all numerical methods is to “divide the problem
into small physical and mathematical components and
Then sum the influence of the components to approximate the behaviour
of the whole system”.
• The series of complete mathematical equations formed in this process
are then solved approximately.
• By definition, the computational solutions are always approximations of
the exact solution.
A numerical model code is simply capable of:
Solving the equations of equilibrium,
Satisfying the strain compatibility equations, and
Following certain constitutive equations - when prescribed boundary
conditions are set forth.
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7. Numerical Modelling
The main sources of the input data for the numerical model are, site
investigations, and laboratory and field tests.
Numerical methods will give approximate solution, but not the exact
solution of the problem.
Numerical Approaches:
The methods are categorized as Continuum, Discontinuum and
Hybrid Continuum/Discontinuum.
The Continuum assumption implies that at all points in a problem
region; the materials cannot be torn open or broken into pieces. All
material points originally in the neighbourhood of a certain point in
the problem region remain in the same neighbourhood throughout
the deformation or transport process.
Numerical Modelling - Approaches
1. Continuum methods
Finite Difference Method (FDM)
Finite Element Method (FEM)
Boundary Element Method (BEM).
2. Discontinuum methods
Fig: (a) Continuous and
(b) Discontinuous behaviour
Discrete Element Method (DEM), of Uniaxially Loaded Specimen
3. Hybrid Continuum / Discontinuum
methods
Hybrid FEM/BEM,
Hybrid DEM/DEM,
Hybrid FEM/DEM, and
Other hybrid models.
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8. Numerical Modelling - Approaches
Free Surface
Excavation
Zone or Element Excavation
Boundary Element
Finite Boundary or
Zone of influence
Fig: Boundary Method
Fig: Domain Method
Boundary Element Method (BEM):
This method derives its name from the fact that the user ‘discretizes’, or divides
into elements, only boundaries of the problem geometry (i.e., excavation
surfaces, the free surface for shallow problems, joint surfaces and
material interfaces), thus reducing the problem dimensions by one and
greatly simplifying the input requirements.
In this method the conditions on a surface could be related to the state at all
points throughout the remaining medium, even to infinity. The information
required in the solution domain is separately calculated from the information on
the boundary, which is obtained by solution of boundary integral equation.
Numerical Modelling - Approaches
BEMs are simpler and faster, but usually not powerful enough to
accommodate complex geometry and excessive variations in rock mass
properties.
Suitable for large scale mine modelling
E.g. BESOL, MUSLIM/NL
Finite Element Method (FEM):
The continuum is approximated as a series of discrete elements connected to
adjacent elements only at specific shared points called nodes. The behaviour
of each element is then described individually using exact differential
equations. The global behaviour of the material is modeled by combining all
individual elements.
Fig: Finite Element method
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9. Numerical Modelling - Approaches
FEM is perhaps the most versatile of all methods and capable of yielding the
most realistic results even in complex geo-mining conditions. Complexity in
problem formulation and requirements of long computer time and large memory
space seem to be its major shortcomings.
e.g. ANSYS, ABAQUS, NASTRAN, COSFLOW, NISA
Finite Difference Method (FDM):
The continuum is represented by a series of discrete grid
point at which displacements, velocities and
accelerations are calculated. The displacement field is
computed by approximating the differential
equations for the system as a set of difference
equations (central, Forward or backward) that Fig: Finite Difference Method
are solved discretely at each grid point. The
differential equations are approximated through the use
of difference equations.
Numerical Modelling - Approaches
FDM results into conditionally stable solution. That is, the convergence of the
solution at different stages of iteration to a true solution depends on the size of
elements and size of the load steps. It has also got the advantage of time-
stepping which allows a better understanding of the trend and mode of a
failure”.
e.g. FLAC (Fast Langrangian Analysis of Continua)
Discrete Element Method (DEM) :
The DEM for modeling a discontinuum is relatively different compared with
BEM, FEM and FDM, and focuses mainly on applications in the fields of
fractured or particulate geological media. The essence of DEM is to
represent the fractured medium as assemblages of blocks formed by connected
fractures in the problem region, and solve the equations of motion of these
blocks through continuous detection and treatment of contacts between the
blocks. The blocks can be rigid or be deformable with FDM or FEM
discretizations.
The distinct element method is ideally suited to modelling of both large scale
geological discontinuities such as faults, dykes and highly fractured
assemblages of rock blocks.
e.g. UDEC, 3 DEC
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10. Numerical Modelling - Approaches
Fig: Various Numerical Approaches
IMPLICIT and EXPLICIT SOLUTION TECHNIQUES
Once the model has been descritized, material properties are assigned and
loads have been prescribed, some technique must be used to redistribute the
any unbalanced loads and thus determine the solution to a new state of
equilibrium. The techniques used are implicit and explicit – with respect to time.
The response of a non-linear system generally depends on the sequence of
loading, and thus it is necessary that the load path modeled be representative
of the actual load path experienced by the body. This is achieved by breaking
the total applied load into increments, each increment being sufficiently small to
ensure solution convergence for the increment after only a few iterations.
Numerical Modelling - Approaches
Implicit techniques use principle of Potential energy and assemble
systems of linear equations, which are then solved by standard techniques of
matrix formulations and reduction.
Dynamic relaxation scheme described by Otter et al. (1966), and first
applied in modelling by Cundall (1971).
In this technique no matrices are formed, solution proceeds explicitly inn
the time domain – unbalanced forces acting at a material integration point
result in acceleration of the mass that is associated with the point;
The application of Newton’s law of motion expressed as a difference
equation yields incremental displacements; applying the appropriate
constitutive relation produces new set of forces, and so on marching in time,
for each material integration point in the model.
For Linear problems and problems of moderate non-linearity implicit
solutions tend to perform faster than explicit solution.
However, as the degree of non-linearity of the system increases imposed
loads must be applied in smaller increments, which implies a greater number of
matrix formulations and reductions and, therefore, increased computational
expense.
Hence highly non-linear problems are best handled by packages that employ
an explicit solution technique.
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11. Comparison of Numerical methods
Metho Advantages Disadvantages
d
BEM •Far-field
Far- condition inherently represented •Coefficient Matrix fully populated
•Only boundaries require discretizations, •Solution time increases with exponentially with
result in early solution than any other number of elements used
method •Limited potential for handling heterogeneous and
non-linear materials
non-
FEM & •Potential for easily handling material •Entire volume must be descretized, results in
descretized,
FDM heterogeneity longer solution time
•Material & geometric non-linearity handled
non- •Far-field
Far- boundary conditions must be
efficiently, especially when explicit solution approximated
is used •For linear problems explicit solutions are
•When explicit solution is used skill is relatively slow
required for user in assessing numerical •Solution time increases with exponentially with
convergence increase in number of elements in implicit solution
•When implicit solution is used matrix are technique
banded
DEM •Solutiontime increases with linearly with •Solution time much slower than for linear
number of elements used problems
•Very general constitutive relations may be •Results can be sensitive to assumed values of
used with little penalty in terms of modelling parameters
computational expense
Applications of numerical Modelling
Design of Openings, and Pillars.
Design of Supports for mine workings.
Design of pit slopes and spoil dumps and estimating their
stability.
Prediction of Main and periodic weightings in Bord & Pillar
and Longwall workings.
Analysis of support interaction vis a vis strata.
Analysis of long term stability of permanent mine excavations.
Prediction of surface subsidence over mine excavations., and
Simulating effects of blasting on stability of mine workings in
Underground as well as in opencast mines.
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12. Usage of Numerical Models
Interpretation: use of models to help us interpret field or
laboratory data.
Design: use models to compare the relative performance
of various design alternatives, with less emphasis on the
final predicted performance.
Prediction: use a model to provide a final, quantifiable
prediction of actual field behaviour.
Majority of model application to the categories of
Interpretation and Design say 90 to 95%, i.e.,
unfortunately 5 to 10% of modelling effort to prediction
Numerical Model Calibration
Fig: Information required for calibration of the Model
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13. Comparison of various Numerical Modeling Softwares
Code Source Type Use Complexity
BESOL Mining Stress Systems 2D/3D BEM Common Simple
EXAMINE Roc Science Inc 2D/3D BEM Rare Mediocre
MAP 3D Mine Modelling Ltd 3D BEM Moderate Mediocre
LaMODEL NIOSH -- 3D BEM Moderate Simple
MUSLIM/NL USBM 3D BEM Moderate Mediocre
FLAC Itasca Consultancy Ltd 2D/3D FDM Common Advanced/Complex
COSFLOW CSIRO 3D FEM Rare Advanced
Phase2 Roc Science Inc 2D FEM Moderate Simple
ANSYS ANSYS, Inc 2D/3D FEM Moderate Advanced
ABAQUS Dassault Systems FEM Moderate Advanced
Simulia Corp
PFC Itasca Consultancy Ltd 2D/3D DEM Rare Complex
3DEC Itasca Consultancy Ltd 3D DEM Rare Complex
UDEC Itasca Consultancy Ltd 2D DEM Moderate Advanced
BEFE -- 2D/3D FE Rare Complex
&BEM
ELFEN Rockfield Software Ltd 2D/3D FE Rare Complex
&DEM
Conclusions
Numerical modeling is a very promising and effective tool in
understanding the rock mass response subjected to complex loading loading
conditions. Efficient use of this tool for reliable design and fixing of strata
fixing
management problems requires a thorough knowledge of the modeling modeling
theory, scope and limitations.
Using numerical models, shield, rock strata, coal seam and goaf
interactions can be modeled effectively for different insitu loading
loading
conditions.
Proper analysis of model response is very important which requires the
requires
basic understanding of the mechanisms involved in the physical process
process
being modeled and the requirement for its numerical simulation.
Results from numerical simulation should be compared with field
measurements for back calculations and improved input data.
More experiences are needed in comparative study between numerical
numerical
simulations and other analytical methods for precise numerical simulation.
simulation.
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