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BOUNDARY ELEMENT METHODS FOR MINE DESIGN
by
BARRY HUGH GARNET BRADY
M.Sc.(Qid), M.Sc.(Eng.)(London), D.I.C.
A thesis submitted to the University of London
(Imperial College of Science and Technology)
for the Degree of
Doctor of Philosophy in the Faculty of Engineering
July 1979
ABSTRACT
The subject addressed in the thesis is the design of
mine structures in hard rock generated by underground
mining methods.
Issues to be resolved in the design of supported mine
structures are identified, and currently available techniques
for analysis and prediction of the performance of these
structures are reviewed briefly. Fundamental and operational
limitations of the various techniques are assessed. The
inherent advantages of Boundary Element Methods for mine
design applications are discussed.
Several different formulations of the Boundary Element
Method are presented. Indirect formulations for analysis of
stress and displacement distributions around long openings
inclined in a triaxial stress field are described. It is
shown that concentrated singularities, which form the basis
of an indirect formulation for analysis of problems involving
long, narrow, parallel-sided openings, can be constructed
readily by coupling line load singularities. An indirect
formulation for three-dimensional analysis of tabular
orebody extraction is developed by taking account of the
procedures established in the two-dimensional, complete
plane strain analysis of long slits.
A direct formulation of the Boundary Element Method
is developed for the complete plane strain analysis of
structures in non-homogeneous media. The main advantage
of the direct formulation over the indirect formulations
is shown to be the capacity to handle a wide range of
excavation cross-sectional geometries.
A simple technique is established for estimation
of pillar and mine stiffness properties, using the Boundary
Element direct formulation. The technique is applied, in
2
conjunction with data obtained from the literature, to
assessment of the stability of pillars in a series of
hypothetical stoping layouts. It is demonstrated that
pillar stability is sensitive to the pattern of natural
fractures in the rock mass. It is concluded that the
absence of field data on the post-peak performance of hard
rock masses prevents proper evaluation of the proposed
technique for pillar stability analysis.
3
ACKNOWLEDGEMENTS
The author records his gratitude to the people who
advised him during the execution of the work reported in
the thesis, and assisted him during thesis preparation.
He would like to thank his supervisor, Dr E.T.Brown,
for general advice and guidance throughout the work
programme, for information and discussion on the strength
and deformation characteristics of rock masses, and for
critical assessment of the draft of the thesis.
He is grateful to Dr J.W. Bray for the interest taken
in the work, for unpublished information on a number of
topics, and for discussion on a wide range of issues in
mechanics.
The author was fortunate to have a number of prolonged
debates with Dr G.Hocking and Dr J.O.Watson on problems
associated with the Boundary Element Method.
Colleen Brady provided consolation during several
desperate stages of the enterprise.
The author recognizes the achievements of Miss
Jennifer Wills and her assistants, who typed the thesis.
The work was conducted during the author's tenure of
a Lectureship in Rock Mechanics in the Department of Mineral
Resources Engineering. He is grateful to Professor R.N.
Pryor and the Imperial College of Science and Technology
for providing a rewarding teaching and research environment.
Some of the work reported in the thesis was conducted
in the course of a project at Imperial College supported by
member companies of the Australian Mineral Industries
4
Research Association Limited. The author thanks the
Management of Mount Isa Mines Limited for permission to
use information on operations at the Mount Isa Mine, Australia.
The author is pleased to record the useful advice
and generous assistance given by Mr Barrie Holt and
members of his section in production of the thesis.
5
CONTENTS
Page
ABSTRACT 2
ACKNOWLEDGEMENTS 4
LIST OF FIGURES 11
LIST OF TABLES 16
NOTATION 17
PREFACE 20
CHAPTER 1. INTRODUCTION
1.1 Underground mining methods
1.2 Techniques for design of supported mine structures
1.3 Energy changes accompanying underground mining
1.4 Stability of mine pillars and mine structures
1.5 Information for design of stable pillars
CHAPTER 2. THE BOUNDARY ELEMENT METHOD FOR
ELASTOSTATICS
2.1 Principles and limitations of the method
2.2 Indirect Boundary Element formulations
2.3 Direct Boundary Element formulations 64
2.4 Displacement Discontinuity Method 70
2.5 Required developments in Boundary Element solution procedures 77
6
22
24
32
38
49
53
58
7 Page
CHAPTER 3. COMPLETE PLANE STRAIN AND COMPLETE
PLANE STRESS
3.1 Problem specification and definitions 79
3.2 Plane strain 81
3.3 Complete plane strain 81
3.4 Complete plane stress 87
CHAPTER 4. INDIRECT FORMULATION OF THE
BOUNDARY ELEMENT METHOD FOR
COMPLETE PLANE STRAIN
4.1 Description of method of analysis 91
4.2 Antiplane line and strip loads 95
4.3 Boundary Element solution procedure 98
4.4 Validation of Boundary Element program 101
CHAPTER 5. INDIRECT FORMULATION OF THE
BOUNDARY ELEMENT METHOD FOR NARROW
EXCAVATIONS AND COMPLETE PLANE STRAIN
5.1 Objectives and scope of work 105
5.2 Development of singularities for modelling contiguous parallel surfaces 108
5.3 Optimum distribution of singularities for modelling single slits 113
5.4 Boundary Element solution procedure 121
8 Page
5.5 Validation of Boundary Element program 126
5.6 Assessment of slit modelling procedure 133
CHAPTER 6. THREE-DIMENSIONAL ELASTIC ANALYSIS
OF TABULAR OREBODY EXTRACTION
6.1 Problem description for three- dimensional analysis 135
6.2 Development of compressive and shear singularities 138
6.3 Imposed distributions of singularity intensity on excavation segments 144
6.4 Three-dimensional Boundary Element solution procedure 151
6.5 Validation of Boundary Element program 154
6.6 Assessment of slot modelling procedure 164
CHAPTER 7. DIRECT FORMULATION OF THE BOUNDARY
ELEMENT METHOD FOR COMPLETE PLANE
STRAIN
7.1 Objectives in development of direct formulation 168
7.2 Establishment of boundary constraint equations 169
7.3 Solution of boundary constraint equations 176
7.4 Boundary stresses 177
7.5 Displacements and stresses at internal points 180
7.6 Symmetry code 182
9
Page
7.7 Validation of Boundary Element program 185
7.8 Use of higher order singularities in the Boundary Element algorithm 188
7.9 Non-homogeneous media 195
7.10 Appraisal of Boundary Element direct formulation
CHAPTER 8. MINE DESIGN APPLICATIONS OF THE
BOUNDARY ELEMENT METHOD
8.1 Preliminary considerations
8.2 Design problems requiring complete plane strain analysis
8.3 Study of pillar stability
8.4 The Mount Isa lead orebodies
CHAPTER 9. SUMMARY AND CONCLUSIONS
REFERENCES. 256
APPENDIX I. Stresses and displacements induced by a point load in an infinite, isotropic, elastic continuum (Kelvin Equations)
APPENDIX II. Stresses and displacements induced by infinite line loads in an infinite, isotropic, elastic continuum
APPENDIX III.Stresses and displacements due to infinite strip loads
205
208
208
213
234
252
266
267
269
Page
10
APPENDIX IV. Stresses and displacements due to infinite line quadrupoles and dipoles
APPENDIX V. Stresses and displacements due to a point hexapole and a point shear quadrupole
APPENDIX VI. User information and input specifications for Boundary Element programs
272
274
276
LIST OF FIGURES
Ficture No. Description
1.1 (a) Pre-mining conditions in a body of rock; (b) tractions and displacements induced within the surface Sr
1.2 Correlation between frequency of rock bursts, ground conditions and rate of energy release during mining (from Cook, 1978)
1.3 (a) Complete stress-strain curve for brittle rock; (b) schematic representation of loading of a rock specimen in a conventional testing machine; (c), (d) performance characteristics for the testing machine and the specimen (from Salamon, 1970)
1.4 Schematic representation of pillar loading by the country rock, and cases of stable and unstable pillar loading (from Starfield and Fairhurst, 1968)
1.5 Replacement of underground pillars (a) by equivalent forces (b) (from Salamon, 1970) .
1.6 Stress-strain curves for specimens of Tennessee Marble with various length/diameter ratios (from Starfield and Wawersik, 1968)
2.1 (a)'Surface S* subject to imposed tractions or displacements; (b) Surface S inscribed in a continuum; (c) Discretized surface S
2.2 (a) Surface S subject to imposed tractions or displacements; (b),(c) Distributions of normal and shear singularities on S; (d),(e) Normal and shear singularity intensities on element of surface S
2.3 (a),(b) Load cases for establishment of Boundary Integral Equation; (c) Method of handling singularity in range of integration
2.3 Boundary conditions on coupled half spaces for generation of normal displacement discontinuity Dz (after Crouch, 1976b)
11
12
Figure No. Description
2.5 Boundary conditions on coupled half spaces for generation of shear displacement discontinuity Dx (after Crouch, 1976b)
3.1 Plane (px, pz' pzx) and out-of-plane (p
xY Y ' p z) stress components for a long opening excavated in a medium subject to a triaxial state of stress
4.1 (a) Long excavation in a medium subject to initial stress; (b),(c) Resolution into component problems; (d) Geometric parameters for discretized problem
4.2 Uniformly distributed transverse, longitudinal and normal strip loads ,and geometric parameters determining the effect of strip loads on element . j at the point i (xi , zi )
4.3 Problem geometry for determining stresses and displacements due to an infinite, Y-directed line load.
4.4 Stress distribution around a circular hole in a triaxial stress field, from Boundary Element analysis and analytical solution
4.5 Excavation-induced displacements around a circular hole in a triaxial stress field, from Boundary Element analysis and analytical solution
5.1 Discretization of long narrow opening into segments
5.2 Resolution of real problem into uniformly stressed medium and subsidiary problem
5.3
Construction of compressive quadrupole singularity
5.4
Construction of shear quadrupole singularity from counteracting couples
5.5 Construction of antiplane dipole from opposing line loads
5.6 Stress distribution in the plane of a slit, and parabolic and elliptical distributions of singularity intensity
5.7 Stress and displacement distribution in the plane of a slit in a uniaxial compressive field (a),(b) and quasi-elliptical distribution of singularity intensity (c)
13
Figure No. Description
5.8 Geometric parameters determining influence coefficients for uniformly loaded element (a) and edge element (b)
5.9 Stress and displacement distributions around a slit in a uniaxial compressive field, modelled with three segments
5.10 Stress distribution along ray AB for a slit, in a unit shear field
5.11 Displacement distribution over excavated area, and stress distribution in pillar area, for row of slits in a uniaxial field
6.1 Isolated pillar generated during room-and-pillar mining
6.2 Single narrow opening in a medium subject to triaxial loading (a), discretization into segments (b), and a typical excavation segment(c)
6.3 Construction of a compressive dipole (a), and a compressive hexapole from three dipole singularities (b)
6.4 Construction of a shear dipole (a), and a shear quadrupole from counteracting shear dipoles(b)
6.5 Axes of symmetry for a square excavation, along which elliptical variation of singularity intensity is inferred
6.6 Distributions of singularity intensity over internal, edge and corner excavation segments
• 6.7 Stresses in the plane of, and perpendicular to
the plane of a penny shaped crack in a uniaxial field (a),(b), displacement distribution over the crack(c), and singularity distribution which models crack formation (d)
6.8 Distribution of shear stress around square openings with various span/height ratios in a unit shear field
6.9 Stress and displacement distributions around square openings with various span/height ratios in a uniaxial compressive field
6.10 Stress and displacement distributions around a square room with a central square pillar in a uniaxial compressive field
14
Figure No. Description
7.1 slice of the surface of an opening in a medium subject to triaxial stress, and problem specification for complete plane strain analysis
7.2 Load cases for establishing boundary integral equation
7.3 Geometric parameters for determination of directional derivatives of displacement at excavation boundary
7.4 Load cases for determining displacements at internal points in the medium
7.5 Problem specification for an opening which is symmetric about the Z-axis
7.6 Stress distribution around a circular hole in a triaxial stress field
7.7 Displacement distribution around a circular hole in a triaxial stress field
7.8 Displacement and stress distributions around a narrow excavation in a uniaxial stress field
7.9 Displacement and stress distributions around a narrow excavation in a longitudinal shear stress field
7.10 Problem specification for a non-homogeneous medium
7.11 Stress distribution in and around a solid cylindrical inclusion in a triaxial stress field
7.12 Stress distribution around a circular hole in a circular inclusion in a medium subject to plane strain
8.1 Problem geometry for assessing the significance of the antiplane component of complete plane strain
8.2 Representation of interaction between country rock and pillar and country rock and abutment in a supported mine structure
8.3 Application of uniformly distributed load at a pillar position to determine mine local stiffness
8.4 Pillar and mine performance characteristics based on convergences at the centre line of the pillar
15
Figure No. Description
8.5 Pillar and mine performance characteristics, based on convergence at the centre line of the pillar and average convergence over the loaded strips at the pillar position
8.6 Method of estimation of the effective abutment width
8.7 Abutment performance characteristic, and mine performance characteristics in the abutment area
8.8 Stope and pillar layouts in a tabular orebody to achieve an extraction ratio of 0.75
8.9 Central pillar and corresponding mine performance characteristics for mining layouts shown in Figure 8.8
8.10 Elastic/post-peak stiffness ratios determined in field and laboratory tests on rock specimens
8.11 Variation of the pillar stability index (negative value) with pillar width/height ratio
8.12 General cross-section (looking North) through the northern part of the Mount Isa Mine
8.13 Cross-section through narrow lead orebodies showing crown pillars generated by cut-and-fill stoping
8.14 Mining layout for extraction of adjacent thick sections of lead orebodies
8.15 Bbundary stresses at the centre of the stope back, and incremental rate of energy release, during the up-dip advance of an isolated cut-and-fill stope
8.16 Zones of overstressed rock generated in the final crown pillar of the cut-and-fill stope shown in Figure 8.15
8.17 Extent of zones of failure in a crown pillar generated by open stoping (from Fabjanczyk (1978))
8.18 Zone of tensile stress indicated by elastic analysis of mining layout, and assumed zone. of de-stressing for subsequent analysis
8.19 Strength/stress ratios in M671 and L690 pillars
LIST OF TABLES
Table No. Description
1.1 Elastic post/peak stiffness ratios ( A / A' ) for specimens with various diameter/length ratios
5.1 Comparison of stresses calculated using Boundary Element Method and closed form solution, around slit in a triaxial stress field
6.1 Analytical and numerical solutions for (a + 6 ) in the plane of a penny-shaped crack in ar
uniaxial compressive stress field
7.1 Comparison of boundary stresses around a circular hole in a uniaxial field, determined from closed form solution, and Boundary Element program with simple and higher order singularities
8.1 Sidewall boundary stresses for circular and elliptical holes in a triaxial stress field, determined by conventional plane strain and complete plane strain analysis. Hole axis sub-parallel to intermediate or minor principal stress direction
8.2 Boundary stresses for circular and elliptical holes in a triaxial stress field, determined by conventional plane strain and complete plane strain analysis. Hole axis sub-parallel to the major principal stress direction
8.3
Mine local stiffness and pillar stiffness for 12m wide pillar in 8m thick orebody
8.4
Abutment width and mine local stiffness in abutment area, for various stope. spans
8.5
Pillar and mine stiffness properties in stoping blocks with various pillar widths and width/height ratios, at constant extraction ratio of 75%
16
NOTATION
ENGLISH
Symbol Quantity Represented
[Ai] row vector of influence coefficients
[A] matrix of influence coefficients
Bx Papkovitch-Neuber function
C,c crack half-width, crack radius
Dx shear displacement discontinuity magnitude
DZ normal displacement discontinuity magnitude
E Young's Modulus
EP
pillar modulus
F traction due to unit solution integrated over range of element
G Modulus of Rigidity
H pillar height
k1 mine local stiffness
K1 mine modulus
[K] matrix of stiffnesses at pillar positions
px,pxy etc. components of pre-mining stress field
PZ pillar axial load
q element fictitious load intensity
QZ strength of point or infinite line normal quadrupole singularity
✓ length of radius vector (two dimensions)
length of radius vector (three dimensions)
surface area
convergence at pillar position
17
S
S
UI
W r
W' r
W s
W
S x strength of point or infinite line shear
quadropole singularity
t component of surface traction
T component of surface traction induced by unit solution
u component of displacement
U component of displacement induced by unit solution
component of displacement induced by unit solution integrated over range of element
energy released by excavation
volume rate of energy release, dWr dV
strain energy stored by excavation
pillar width
GREEK
Symbol • Quantity Represented
Papkovitch-Neuber function
unit weight
shear strain
convergence (unrestrained) at pillar position
volumetric strain
normal strain
Lamē's Constant
pillar stiffness in elastic range
18
Y
Y
Y
A
A
A
19
X' pillar stiffness in post-peak range
[A] matrix of pillar stiffnesses
v Poisson's Ratio
a normal stress
T shear stress
d,X harmonic functions
n pi
E summation
PREFACE
The need for sound procedures for the design of the
rock structures created by underground mining increases with
the scale of mining operations, and with the requirement to
realize the maximum potential of mineral deposits. The
trend to increased depth of mining will require, in the
future, the general implementation of design techniques
firmly based on the principles of mechanics.
In Chapter 1, different types of mine structures
are described, and the primary Rock Mechanics issues in
mine design are defined. The question of stability of a
mine structure is examined, and the techniques for assessment
of mine stability are discussed. It is suggested that the
Boundary Element Method represents the most promising
technique for analysis of stability of supported mine
structures generated during the mining of orebodies in hard
rock environments.
The principles of the Boundary Element Method are
discussed in Chapter 2. The concept of complete plane strain
is introduced in Chapter 3. This allows Boundary Element
Methods of stress analysis to be applied to the general
mining situation, where the long axis of mine openings is not
coincident with a pre-mining principal stress direction.
The development of several versions of the Boundary
Element Method, designed to handle various mine structural
configurations, is described in Chapters 4-7. The techniques
may be applied to design in massive and tabular orebodies
for which isotropic elastic behaviour of the rock mass may
be assumed, including the case where orebody elastic
properties are different from those of the country rock.
In Chapter 8, one version of the Boundary Element
Method is used to develop a technique for evaluation of
20
the parameters required to assess mine stability. In
addition, a case study of a mining operation is used to
demonstrate the practical application of the selected
Boundary Element technique to the determination of the
stress distribution in a mining layout, and to assess the
Energy Release Rate during mining. The mining implications
of the results are discussed.
21
CHAPTER 1
CHAPTER 1 : INTRODUCTION
1.1 Underground Mining Methods
The basic objective in the design of an underground
mine structure is to achieve safe and efficient extraction
of a high proportion of the in-situ ore reserve. The
particular mining method chosen for the exploitation of an
orebody is determined by such factors as its size, shape
and disposition, the distribution of values within the
orebody,and the geotechnical environment. The last factor
describes such issues as the in-situ mechanical properties
of the orebody and country rocks, the structure of the rock
mass, the pre-mining state of stress and the groundwater
distribution in the area of influence of mining. The
range of mining methods available to handle these diverse
conditions has not changed significantly in principle in
this century. Changes in mining practice that have
occurred reflect increases in the scale of operations and
improvements in working techniques through mechanization.
The emergence of Rock Mechanics as a mining technology
represents recognition of the need for sound design and
planning of highly capitalized, large scale extraction
operations.
The conventional classification of underground
mining methods, such as that discussed by Thomas (1973),
is on the basis of the type and degree of support
provided in the mine structure created by_ore extraction.
The categories of mine structure recognized by Thomas,
and examples of the mining methods which generate them,
are:
A. Naturally supported structures (open stoping, room
and pillar mining);
B. Artificially supported structures (cut and fill
stoping, shrinkage stoping);
22
23
C. Caving structures (block caving, sub-level caving).
From a Rock Mechanics point of view, the distinction
between mining methods, and the structures they generate,
may be made on the basis of the displacements induced in
the country rock, and the energy re-distribution which
accompanies mining. For supported methods of mining,
the objective is to restrict displacements of the country
rock to elastic orders of magnitude, and to maintain as
far as possible the integrity of both the country rock
and the unmined remnants within the orebody. This typically
results in the accumulation of strain energy in the
structure, and the mining problem is to ensure that unstable
release of energy cannot occur. In caving methods, the
objective is to induce large scale displacements which prop-
agate through the country rock overlying an orebody. Energy is
dissipated in the caving rock mass, by slip, crushing and
grinding. The mining requirement is to ensure that steady
displacement of the caving mass occurs, so that the mined
void is self-filling, and unstable voids are not generated
in the body of the caving material. The aim is therefore
to achieve a steady rate of energy dissipation.
Irrespective of whether a supported or caving
method of mining is employed, there are four basic Rock
Mechanics objectives in the design of a mine structure:
(a) to ensure the stability of the structure as orebody
extraction proceeds;
(b) to preserve unmined ore in a mineable condition;
(c) to protect major service openings until they are
no longer required;
(d) to provide secure access to safe working places.
These objectives are not mutually independent: the
typical design problem is to find the stope or block
excavation sequence which satisfies these objectives
simultaneously, and fulfils various other operational
requirements. The realization of the design objectives
requires, in addition to a knowledge of the geotechnical
conditions in the mine area, the capacity for determination
of stress and displacement distributions in a mine
structure, for various operationally acceptable extraction
sequences.
From the discussion of the strategies pursued in
supported and caving methods of mining, it is clear that
fundamentally different analytical techniques are required
for the design of the different types of structures.
Numerical methods suitable for the design of caving
structures have been described by Cundall (1971) and
Hocking (1977). The concern in this thesis is with the
development and assessment of practically acceptable methods
of analysis for the design of supported structures in hard
rock mines, with particular emphasis on naturally supported
structures.
1.2 Techniques for Design of Supported Mine Structures
The issues to be decided in the design of a mine
structure include stope dimensions, pillar dimensions,
pillar layout, stope mining sequence, pillar extraction
sequence, type and timing of placement of backfill, and
the overall direction of mining advance. The range through
which some of the parameters may vary, such as stope
widths, may be limited by the dimensions or properties of
the orebody. On the other hand, questions regarding stope
and pillar extraction sequence typically may only be
resolved after consideration of a wide range of options,
in which yeotechnical concerns are assessed along with
operational and economic factors. The area in which
Rock Mechanics has the most readily identifiable impact
is in stope and pillar design, and pillar layout. It is
in this area that attention is concentrated.
24
25
A common procedure followed in the past in the
design of a mining layout has been to follow precedents
established by experience and observation of the
performance of other mines,working under similar geotech-
nical conditions to those in and around the orebody for
which a layout is to be established. Although this
procedure has led typically to acceptable extraction
performance and operating conditions, it probably represents
over-design of a mine structure. It also causes lack of
recognition of the specific problems associated with the
extraction of a particular orebody,and inhibits the develop-
ment of efficient methods for handling these problems.
In attempts to establish a more appropriate basis
for design, physical models have been used to evaluate the
performance of different mining layouts. Mathews and
Edwards (1969), for example, describe the construction
and testing of large models of the 1100 copper orebody at
the Mount Isa Mine, Australia, using the methods and
loading rigs discussed by Jagger (1967). The results of
these tests were generally consistent with the observed
performance of mine pillars generated in the early stages
of extraction of the orebody, and thus produced useful
data for modifications to the initial design. However,
as a general rule, the expense and time required to design,
construct and test models which represent the prototype
in sufficient detail precludes their routine application.
In addition, the laws of similitude are rarely, if ever,
properly satisfied. The development of numerical modelling
techniques, with the capacity to analyse different rock
structures with a range of material properties quickly
and economically, has made physical modelling largely
redundant.
In the design of a stope and pillar layout, different
criteria determine the performance of stope spans and
pillars, and therefore different methods of analysis may
be required to assess the performance of these elements
26
of the mine structure. Irrespective of the methods of
analysis used, the requirement is to ensure that the
conditions for stability of pillars and stope spans are
satisfied simultaneously. Aniterative procedure is
generally involved in achievement of this requirement.
The techniques available for the estimation of stable
roof or hangingwall spans in stopes are limited, consid-
ering the mining significance of the problem. Obert et
al.(1960) suggest the use of elastic beam and plate theory
for design of roof spans in stratiform orebodies. The
approach is open to criticism on the basis of the necessity
to assume a finite tensile strength for the rock mass,
and the unknown end or side loads applied to the beam or
plate. Rock mass classification schemes such as those proposed
by Barton et al. (1974) and Bieniawski (1976) are
codifications of established practice in the design of
unsupported spans in jointed rock, and represent a
regression to design by precedent. Voegele (1978)
describes the use of the quasi-rigid block model of
jointed rock,developed by Cundall (1971), for the assessment
of stable excavation spans in a jointed rock mass. The
results of Voegele's work suggest that this type of model
presents the most promising approach for direct determina-
tion of stope span stability, from a knowledge of the rock
material properties, rock structure and the pre-mining
stress field. The development of a modelling procedure
for jointed rock,based on the conventional relaxation
techniques described by Southwell (1946), as opposed to
the dynamic relaxation employed by Cundall, has been
reported by Stewart (1979). This procedure is designed
to achieve more efficient solutions to the equilibrium
distribution of forces and displacements in a blocky
assemblage,by elimination of the time steps used in the
dynamic procedure.
The attention that has been devoted to the design
of pillar support rather than to stope span design
reflects the more serious mining implications of pillar
failure. Bunting (1911) proposed a procedure for pillar
design in flat lying, tabular orebodies that is now
identified as the Tributary Area method. Pillar load Pz
is estimated from the area Ao within a rectangle lying
in the plane of the orebodywhose edges are the centre lines
of adjacent stopes, the depth Z below ground surface and
the unit weight y of the overburden. The average axial
pillar stress āz is given by
= Pz
= A
A YZ
z A Ap
where. A. is the pillar plan area.
Pillars are designed to ensure that pillar strength
S, which is determined experimentally, exceeds the average
axial pillar stress by an appropriate factor of safety.
Alternative statements of the Tributary Area method by
Duvall(1948), Denkhaus (1962), S alamon (1967), and Agapito
and Hardy (1975) have been mainly concerned with procedures
for estimation of the pillar strength to be used in the
calculation of the factor of safety.
According to Pariseau(1975), the effect of pillar
size and shape on strength was recognized in 1907. Since
then a substantial amount of testing has been performed
to establish parameters which describe, for various
lithologies, the relatioriship'between pillar strength (i.e.
crushing load/area), volume and shape, expressed in
terms of pillar width/height ratio. Testing of large
coal specimens has been reported by Greenwald et al.
(1939, 1941) and by Bieniawski (1968), Wagner (1974)
and Van Heerden (1975). In more recent test programmes,
increased attention has been given to measurement of the
elastic and post-peak deformation characteristics of
large specimens. Considerable discussion has centred
27
on the appropriate boundary conditions to be applied
during loading of specimens. The loading procedure
described by Cook et al. (1971), and applied by Wagner,
appears superior to the others. With this procedure, the
natural boundary conditions between the specimen and the
country rock are maintained, and loads are applied by
forcing apart the walls of a slot cut at the mid-height
of the specimen using a constant displacement jacking
system.
Testing of large specimens of hard rocks is, made
difficult by the high load capacities required of jacking
systems. Successful tests have been reported by Jahns
(1966) on cubic specimens of iron ore with volumes up to
1m3, Gimm et al.(r966) on iron ore and shale specimens,
up to 2m2 in area and 1.5m high, and De Reeper (1966) on
a single 1m3 specimen of iron ore. Richter (1968)
conducted tests on iron ore, sandstone and shale specimens
with side-lengths up to 2.15m, and Pratt et al. (1972)
conducted tests on large tetrahedral samples of diorite.
In all cases the measured strengths of the large specimens
were significantly lower than uniaxial compressive strengths
of the various rock materials determined by standard
laboratory tests. In the case of Richter's tests, for
example, the strength of large iron ore specimens was
18 times lower than the values determined on laboratory
specimens.
Bieniawski (1975) has summarised the results of
large scale strength tests. He has shown that for cubic
specimens, the strengths of iron ore (Jahns), diorite
(Pratt et al.)and coal (Bieniawski) approach limiting
values apparently characteristic of each rock mass, at
cube side lengths of lm. The suggestion is that for
the rock masses tested, a volume effect on rock mass
strength disappears at this specimen size, but it is
unlikely that this applies to all rock masses. It appears
from the experimental results that pillar strength may be
28
then expressed by equations of the form
S = A + B f (W,H)
where the constants A, B and the functional relationship
f may be determined from the experimental data for the
particular rock mass.
The attraction of the Tributary Area method of pillar
design lies in its simplicity. However, it is applicable
only where the number of pillars is large and pillars are
of uniform size. It disregards the effect of location
of a .pillar within a panel or stope block, and it takes
no account of the stresses acting in the plane of the orebody.
A numerical technique for estimating pillar stresses
in tabular orebody extraction, which overcomes the
deficiences of the Tributary Area method, is based on
analysis of the displacement distribution induced by mining
and resisted by the pillars and abutments. The approach
is derived from the original suggestion by Hackett (1959)
that a mined opening in a tabular orebody could be treated
as a narrow slit or slot. Berry (1960) and Berry and Sales
(1961) used this assumption in the analysis of surface
displacements induced by longwāll mining of coal seams.
Its application to the determination of stresses and
displacements in mine structures has been pursued by
Salamon (1964) who called it the Face Element Method,
Starfield and Crouch (1972), and Crouch (1973) who
called it the Displacement Discontinuity Method. The
mined area is divided into rectangular elements, over each
of which a uniform convergence (closure) and ride are
assumed to occur between hangingwall and footwall.
Convergence and ride at the pillar positions and in
unmined areas is resisted by pillar normal and shear
stiffnesses. The procedure is to find the distribution
of convergence and ride over the mine area which produces
29
30
the known values of traction or displacement on excavation
surfaces. The numerical implementation of the method of
analysis therefore resembles the Boundary Element Method,
which is described in Chapter 2. Pillar stresses
estimated from the analysis may be compared with pillar
strengths obtained from the testing programmes described
above, to determine factors of safety against failure.
An electrical analogue for solution of the stress and
displacement distribution in mining layouts, based on Face
Element theory formulated by Salamon (1964), has been
described by Cook and Schumann (1965) and an analogue-digital
hybrid system by Fairhurst (1976).
The limitations on the Displacement Discontinuity
Method of analysis for pillar design in tabular orebodies
arises from the implicit assumption of homogeneous stress
within the pillar, and therefore failure to take account
of the effect of confinement developed in wide pillars.
The development of the Finite Element Method by Turner
et al.(1956) and its subsequent improvement as described
by Zienkiewicz (1977) and others provided a potentially
powerful technique for pillar design analyses. There
have been numerous applications of the method to the
assessment of pillar performance. Representative examples
are provided by Heuze and Goodman (1970), Blake (1972),
Mathews (1972) and Agapito (1974). Pariseau (1975)
has described a method of pillar design, based on the
Finite Element Method, which aims to take account of the
development of failure zones in pillars.Brittle and
elastic-perfectly plastic modes of failure of the rock
mass were modelled, and it was possible to model the
propagation of failure to either the attainment of a
stable state of stress in a pillar, or to final collapse
of the pillar. In spite of the sound intentions of the
procedure described, the work illustrates the inherent
deficiencies of the Finite Element Method for the design
of rock structures, other than simple geometries consisting
of a few excavations. In Pariseau's case,a single pillar
and its adjacent stopes were modelled, and it was necessary
to select quite unrealistic boundaries to the problem
area. In general, a complex mine structure in an
irregularly shaped orebodyis not modelled adequately
using the Finite Element Method, due to the arbitrary
boundaries and boundary conditions which must be defined
for the problem domain, and typically the necessity to
use a coarse mesh to represent a structure.
The application of the Boundary Element Method of
stress analysis developed by Bray (1976a) to the assess-
ment of the observed performance of a pillar in a hard
rock mine has been described by Brady (1977). It was shown
that, provided it was possible to establish a failure
criterion for the rock mass by retrospective analysis of
local rock failures, the performance of a pillar could be
predicted satisfactorily using a plane strain method of
analysis,based on assumed elastic behaviour of the rock
mass. The Boundary Element Method does not suffer from
the limitations of the Finite Element Method associated
with the necessity to define arbitrary boundaries to a
problem area; infinite boundaries to the problem area
are modelled implicitly. The method also makes less
demand on computer resources than the Finite Element
Method. The suggestion from the initial study was that
further development of Boundary Element methods could
provide efficient and practically acceptable procedures
for pillar design.
The techniques for pillar design which have been
described are based on the proposition that pillars must
operate in their elastic range, below the rock mass
strength, to achieve satisfactory performance of a mine
structure. However, the prime design requirement is to
maintain stability in a structure. Exceeding the rock
mass strength in pillars need not necessarily result in
uncontrolled collapse or instability of the structure, but
merely cause local crushing and load re-distribution
31
in the structure. Design to achieve stability must there-
fore be based on different principles from those applied
to avoid local failure in a structure. The assessment of
stability involves consideration of energy changes assoc-
iated with mining, the distribution of energy in the
structure, and the energy required to crush pillars.
1.3 Energy Changes Accompanying Underground Mining.
The significance of the re-distribution of energy
which occurs when openings are excavated underground was
first discussed by Cook (1965). The phenomenon of rock-
bursts in deep mines was described in terms of unstable
release of energy resulting from the unfavourable shapes
and methods of excavation used in longwall mining of gold
reefs. The more general significance in mine design of
energy changes due to mining has been considered by Fair-
hurst (1976), while Crouch and Fairhurst (1973) discussed
bursts and bumps in coal mining in terms of energy released
at various stages of extraction of a seam. Bray (1979)
has noted shortcomings in the procedure used by Cook (1976)
to estimate strain energy changes induced by mining. These
deficiencies are associated with failure to take account
of work terms associated with induced displacements remote
from excavations. The following discussion is intended
to provide a simple appreciation of energy re-distribution
induced by mining activity in an elastic rock mass.
Figure 1.1(a) shows a cross- section through a prism
of rock in a medium subject to field stresses px,pz in which
it is proposed to excavate an opening whose surface is S.
Prior to excavation, the surface S is subject at any point
to tractions tx, tz . The process of excavating the rock
within S reduces the tractions on S to zero, which is equivalent
to inducing traction txi' tzi on S, induces displacements
uxi, uzi on S, and induces tractions and displacements txr'
tzr' uxr' uzr' on the surface Sr of the prism. Induced
tractions and displacements are shown in Figure 1.1(b).
32
Suppose the rock within S is excavated in such a way as to
reduce gradually the tractions applied to S. According to
Love (1944) the work done by the country rock (i.e. the rock
exterior to S) on the rock within S is given numerically by
Wi = JS ( txi uxi + tzi uzi) dS
The work done on the rock within Sr by the rock
exterior to Sr may be estimated from the displacements
on Sr and average tractions txra, tzra with which they
are associated. The average tractions are given by
txra = k (2txf + txr)
tzra = ~ (2tzf + tzr )
where txf, tzf are tractions associated with the
field stresses. Thus the work done on the rock within Sr
is given by
We fSr (txra uxr + tzra uzr ) dS
To estimate the work done We on the exterior surface
Sr when an opening is excavated in an infinite body, it is
necessary to determine the limiting value of We as the sur-
face Sr becomes infinitely remote from the opening. The
evaluation of We is straightforward for simple excavation
shapes such as a circular hole and a narrow slit. As noted
by Jaeger and Cook (1976), difficulties arise with irregular
excavation geometries, for which numerical solutions must
be obtained for induced tractions and displacements, due to
the poorly behaved functions which are involved in the solu-
tion for the displacements.
33
34
(a) (b)
FIGURE 1.1: (a) PRE-MINING CONDITIONS IN A BODY OF -ROCK; (b) TRACTIONS AND DISPLACEMENTS INDUCED BY
EXCAVATION WITHIN THE SURFACE S.
The increase in strain energy, or the Stored Energy
Ws, induced in the rock contained between the surfaces S
and Sr is given by
Ws = We -141.
The Stored Energy Ws represents increased potential
energy which is stored in regions of stress concentration
around the opening. The source of this induced strain energy
is the gravitational and tectonic fields operating in the
rock mass. A net reduction in the gravitational potential
energy, for example, can be expected to accompany the exca-
vation of an underground opening. The significance of
the Stored Energy Ws is that one would expect the
stability of an opening to depend on the volume of
rock subjected to increased stress, and the magnitude
of stresses in the affected volume. This suggests that
Ws might be useful as a criterion for the local stability
of a mine excavation or structure.
The situation considered above involved gradual
reduction in the tractions tx, tz,originally applied to
the surface S of the excavation. When the excavation
is created suddenly, for example by blasting, the support-
ing forces acting on the boundary of the excavated region
are suddenly removed. Energy equivalent to the work which
would have been done against the gradually reducing support
forces,Wi,is released into the country rock and is identified
as the Released Energy Wr- It is expressed as kinetic energy
and dynamic strain energy at the excavation surface, and re-
sults in the generation of strain waves in the medium. Dynamic
stresses are therefore associated with the Released Energy.
The mining significance of the Released Energy is
that although the rock mass may be able to sustain the
static stresses around an opening, superposition of the
dynamic stresses associated with the Released Energy may
be sufficient to cause failure. Processes which could lead
to failure of the rock mass during dynamic loading are
direct failure in compression, reduction in normal stresses
on planes of weakness, leading to a reduction in shear
strength, increase in shear stresses on planes of weakness, and generation., of tensile stresses. The suggestion is
therefore that an objective in the design and excavation
of an opening should be to control the Released Energy.
In mining an orebody it is unusual for complete
stopes to be excavated instantaneously, and the Stored
Energy Ws and the Released Energy Wr are themselves of
little direct significance. The interest is instead in
35
the total strain energy, rather than the induced strain
energy, and its distribution in the mine structure, and
the energy release rate for increments of extraction at
particular stages of mining. Further, although it is
possible to determine numerically the total strain energy
stored in and around a mine structure, at this stage there
appears to be no direct way in which this can be used to
assess the stability of the structure. Indirect methods
of assessment of stability, derived from strain energy
considerations, must be used. However, it appears that
the volume rate of energy release, dV , as the volume of
the mined void increases, can be related directly to both
local instability and to ground conditions in working areas
in stopes. Hodgson and Joughin (1967) analysed data on
the incidence of damaging rockbursts in South African gold
mines, and demonstrated a good correlation between the fre-
quency of rockbursts and the energy release rate. More
recent work by Cook (1978) indicates a deterioration of
ground conditions in longwall stopes as the rate of energy
release increases. The information is summarised in Figure
1.2. The inference is that the energy release rate may be
used as a basis for evaluation of different mining layouts
and extraction sequences, and as a guide to the type of
local support required in working areas. The Face Element
Method of stress analysis described by Salamon (1964) has
been used to calculate the energy release rate, and to evaluate
potential problems associated with various mining layouts,
such as those generating remnant pillars.
In an investigation of the origin of coal mine
bumps, Crouch and Fairhurst (1973) concluded that bumps
were caused by unstable releases of energy during yield
of coal pillars. A method of analysis similar in principle
to the Face Element Method was established which allowed
the energy release rate associated with pillar yielding
to be calculated directly. This could be used to assess
the relative merits of different extraction sequences, and
to identify potential problems during extraction,following
any selected sequence.
36
1sc9L!9 0 1 lia 40 60
1 Slight_ 1
100 80
Rate if
1
Moderate
Severe
Energy Releafe
1 1
120 14(
(MJ/m3)
_I
Extreme
37
2.0 •
7. 1 -5
L m a, 1 .0 c
° 0.5 v- 0
m a L
rn
0
c
C 0
ftl L 0 •L a N a)
U
CL
FIGURE 1.2: CORRELATION BETWEEN FREQUENCY OF ROCK BURSTS, GROUND CONDITIONS AND RATE OF ENERGY RELEASE DURING MINING. (FROM COOK, 1978)
The difference between the South African approach,
and that adopted by Crouch and Fairhurst,is that whereas
the former is based on energy released by unrestrained
displacement of a newly excavated surface, the latter
considers the release of energy initially stored in the
country rock, and released by virtue of a pillar's
inability todissipate,during yield,all the locally
available energy. In this case the major concern is
therefore with identification of the factors which
determine whether a pillar will deform in a stable
manner when its strength is exceeded.
1.4. Stability of Mine Pillars and Mine Structures
Mine pillar layouts must be designed so that the
possibility of uncontrolled collapse of pillars does
not arise. The most frequently applied methods of pillar
design seek to maintain stability by ensuring that they
operate within their elastic ranges of performance. Un-
certainties concerning the in-situ strength of rock
suggest that, in general, this criterion for pillar
stability cannot be satisfied unless pillars are over-
designed. The effect of using even moderate factors
of safety in pillar design on the volume extraction ratio
obtained from cal seams at increasing depths below sur-
face has been described by Salamon (1967). This has led
to the application of criteria other than the usual strength
criterion in attempting to design an intrinsically stable
mine structure.
The possibility of instability in a mine structure
arises when the strain energy stored locally in the struc-
ture exceeds the total energy required to crush the pillar
support. Recognition by Cook (1965) that rockbursts
represented a problem of stability arose from considera-
tion of the complete stress-strain behaviour for brittle
38
rock. He subsequently discussed the significance of
the failing portion of the complete stress-strain
characteristic for rock on pillar stability (Cook,
1967). Techniques for the assessment of pillar and mine
stability, based on the ideas proposed by Cook, have been
developed by Starfield and Fairhurst (1968) and Salamon
(1970), and these are now described. In the discussion that
follows, it is assumed that the country rock is continuous
and linearly elastic, and that any non- linear behaviour is ccnfined to the pillars.
39
e (a) (b)
Load P
Displacement S
(c)
(a)
FIGURE 1.3: (a) COMPLETE STRESS-STRAIN CURVE FOR BRITTLE ROCK; (b) SCHEMATIC REPRESENTATION OF LOADING OF A ROCK
SPECIMEN IN A CONVENTIONAL TESTING MACHINE; (c), (d) PERFORMANCE CHARACTERISISTICS FOR THE
TESTING MACHINE AND THE SPECIMEN. (FROM SALAMON, 1970).
The curve ABCD in Figure 1.3(a) represents a typical
complete stress-strain curve for a brittle rock specimen
tested in a stiff machine. AB represents the elastic range
of performance of the specimen, CD the failing regime.
In a conventional testing machine, a compression test
may be terminated by violent failure of the specimen at
point C. The conditions which determine whether unstable
failure will occur,.or stable post-peak deformation along
the curve CD will be observed, may be established by
consideration of the idealized loading system shown in
Figure 1.3(b). The loading machine is represented by a
spring whose stiffness is k, through which an applied
load is transmitted to the rock specimen. If the vertical
deflections of points 01 and 02 under an applied load ps
are y_ and S respectively, the relationship between load
and spring compression is given by
Ps = k(y-S) (1.1)
This defines the load line or performance characteristic
for the spring, shown in Figure 1.3(c).
The complete performance characteristic for the rock
specimen is given by
P = f(S) (1.2)
For equilibrium between the spring and the rock
specimen,
Ps Pr
or k(y-S) = f(S) (1.3)
The state of equilibrium defined by equation (1.3)
and illustrated by point E in Figure 1.3(c) will be
stable if, when no extra energy is supplied to the system,
no further compression can be induced in the specimen.
40
No energy enters the system if 01 is fixed; i.e. y is
constant. Considering a virtual displacement AS imposed
at 02, the work done by the spring and the work done on
the rock during the virtual displacement are given by
DW s = (P + '- Ps) AS (1.4)
AWr = (P + r) AS
From equations (1.1) and (1.2),
APs
= - kLS
APr = f' (S) AS (1.5)
= XAS
where A is the slope of the performance characteristic
for the rock specimen at the equilibrium position for the
spring-specimen system.
The condition for stable equilibrium stated above
requires that during the virtual displacement, LWr>OWs;
i.e. from equations (1.4) and (1.5),
1(k +X) AS2 >0
Thus the criterion for stability of the system at
any stage of loading is that
k + X >0 (1.6)
The variation of specimen stiffness A throughout
the operating convergence range is shown in Figure 1.3(d).
Since the spring stiffness k is positive, and specimen
stiffness is positive in the elastic range of specimen
performance, equation (1.6) confirms that the spring-
41
specimen system is stable in this phase of loading. In
the post-peak range, specimen stiffness is negative, and
the possibility of unstable failure of the specimen
depends on the relative values of k and A. Unstable
failure cannot occur at any stage if the spring stiffness
and the minimum stiffness Am exhibited by the specimen in
the post-peak range satisfy the relationship
>0
This condition defines a state of intrinsic stability
in the loading of the specimen through the spring. The
limiting condition for stability during loading occurs
when the _performance characteristic for the spring
becomes tangent to that for the specimen. This occurs
when
k + A = 0
Starfield and Fairhurst (1968) proposed that equation
(1.6) be used directly to establish the stability of indi-
vidual pillars in a mine structure, and therefore to assess
the overall stability of the structure. Mine pillars are
loaded by mining-induced displacement of the country rock.
The country rock therefore replaces the spring in the
idealized loading system described earlier,and the stiff-
ness of the loading system is defined by the mine local
stiffness, kl, at the pillar position. Referring to
Figure 1.4(a), a pillar in a simple mining layout has been
replaced by a jack applying load to the country rock at
the pillar position. If the load (P) exerted by the jack
is decreased, convergence (S) at the pillar position will
increase.
Assuming that the convergence distribution at the
pillar position can be represented by a single value,
the load-convergence line or performance characteristic
for the country rock at the pillar position is shown in
42
Locally Stored Energy
JACK LOAD P
(a)
Local Energy Deficiency
Load Excess P Local Energy Load
P
43
S' Convergence S at Pillar Position
(b)
Convergence S
Convergence S
(c)
(d)
FIGURE 1.4: SCHEMATIC REPRESENTATION OF PILLAR LOADING BY THE COUNTRY ROCK, AND CASES OF STABLE AND UNSTABLE PILLAR LOADING (FROM STARFIELD AND FAIRHURST, 1968) .
Figure 1.4(b). Mine local stiffness at the pillar
position is defined by
AP kl = - DS
and is therefore positive by definition.
At any particular convergence, say S', of the
country rock at the pillar position, the area under the
load-convergence line, defined by ABC, is a direct measure
of the energy stored locally in the country rock and
available to do work in crushing the pillar. Figure 1.4(c)
illustrates a case where the energy available in the
country rock exceeds the energy required to crush the
pillar, due to the low mine local stiffness. In this
case the stability index, kl + A, is zero just beyond
the peak in the pillar load,-convergence curve, and the
pillar fails in an unstable manner. Figure 1.4(d)
represents the condition where kl + A is greater than
zero, for which the post-peak deformation of the pillar
is stable. For a complete mine structure, the criterion
for stability is that the stability index is greater than
zero for all pillars.
Mine local stiffness at any pillar position is
dependent on the stiffness of all other pillars in the
structure. To determine if a structure is intrinsically
stable, the minimum mine local stiffness kimi at each
pillar position i must be assessed, and this can be done
by assuming that all other pillars in the structure have
been removed. Suppose that the minimum post-peak stiff-
ness of a pillar is Ami. If, for all pillars, the
relationship
Xmi> - k lmi
44
is satisfied,the structure is intrinsically stable.
The procedure proposed by Salamon (1970) for analysis
of the stability of a mine structure is somewhat more
complicated than that suggested by Starfield and Fairhurst.
Figure 1.5(a) represents a set of stopes and
pillars in a mine panel. In Figure 1.5(b) the pillars
have been replaced by a set of loads which are statically
equivalent to the action of the pillars on the country
rock. It is assumed that a convergence S. at any pillar ~
position i can be used to represent the convergence
distribution at that position. The convergence S. at any ~
pillar position can be regarded as the superposition of
two separate convergences: that which would occur in the
absence of pillars (Yi)' and that due to the action of the
the pillar loads on the country rock (8 .). The latter e~
45
contribution to the net convergence is actually a divergence.
Salamon has shown that for n pillars, the relationship
between pillar loads and convergences may be expressed by
where
[p] = [K] ( [r] - [s]) (1. 7)
[p], (r] , [s] are column vectors, of order
n, of pillar loads P., and convergences ~
Yi
and 8 i
[K] is a square matrix, of order n, of stiffness coefficients.
(a)
(b)
FIGURE 1.5: REPLACEMENT OF UNDERGROUND PILLARS (a) BY EQUIVALENT FORCES (b) (FROM SALAMON, 1970)
46
The Reciprocal Theorem requires that [K] be symmetric,
and the individual stiffness coefficients are all real.
Consideration of the strain energy induced by the pillar
loads, and the theory of gsadratic forms, requires that [K]
be positive definite. The stiffness matrix [K] is therefore
real symmetric positive definite.
The conditions for stability and instability in the
mine structure are established following a procedure similar
to that considered for the loading of a laboratory specimen.
By imposing virtual convergences at the pillar positions,
and considering the work done by the country rock, and the
work necessary to compress the pillars, the condition for
stability is found to be
1 [AS] T ( [K] + [A]) [As] >0 (1.8)
where [AS] is the column vector of virtual conver-
gences, and [tS]T is its transpose,
[A] is the matrix of pillar stiffnesses, of order n.
The leading diagonal of the pillar stiffness matrix
is composed of the individual pillar stiffnesses, Ai, and
all other terms are zero.
Equation (1.8) implies that the structure is stable
if the matrix ([K] + [A] ) is positive definite. The
condition for stability then is that all principal minors
of the determinant 1K + Al be positive.
The condition for instability is derived by consid-
ering the increases in convergences Lyi, ASi and load
increments which are induced at pillar positions for a
small increase in the area mined. Equation (1.7)
indicates that pillar load increments are related to
convergence increments by the equation
47
[DP] _ [K] ( [Dr] - [Ds] ) (1.9)
The load increments must satisfy the load-convergence
relationships for the pillars; i.e.
[DP] _ [A] [Ds] (1.10)
Equations (1.9) and (1.10) yield the relationship
( [K] + [A] ) 61s] = [K] [Dr] (1.11)
Equation (1.11) indicates that unique values for the
convergences cannot be determined if the matrix
([K] + [A]) is singular. Therefore the condition for
instability is
(K + AI 0 (1.12)
The method proposed by Salamon for assessment of
mine stability is basically identical to that proposed
by Starfield and Fairhurst, in that in each case a criterion
for stability is established which involves implicitly
the energy distribution in the mine structure. This basic
equivalence may be used to obtain the relationship between
the mine local stiffness kli at pillar position i, the
mine stiffness matrix [K] and the pillar stiffness matrix [A] , through the respective conditions for instability.
given by equation (1.12) and kli + Xi = 0. It can thus be shown that
kli IK + Al. (1.13)
IA 1 ii
where IK + Ali denotes the determinant IK + AI with zero substituted for ai
IA I11is the co-factor of the term kii + Ai in the determinant IK + AI
48
The condition for intrinsic stability developed by
Salamon assumes a set of identical pillars, for which the
minimum post-peak stiffness is am. Salamon shows that
intrinsic stability is assured if
Am
AC c (1.14)
where ac is the largest root of the characteristic
equation 1K + XII = 0, where [I] is the identity matrix,
and A is a scalar quantity. The roots of the
characteristic equation are real and negative.
The quantity -ac represents the lowest value of the
mine local stiffness that can be achieved at any position.
in the mine structure for any type of pillar performance.
It is noted that the pillar position where th'e minimum
value of the mine local stiffness occurs is not specifically
identified.
Although Salamon's approach to analysis of stability
of a mine structure is more complicated to implement than
that proposed by Starfield and Fairhurst, it is valuable
in that it provides a method of assessing, through equa-
tion (1.13), whether techniques for determining pillar
and mine local stiffnesses are compatible. It also led
Salamon to propose a method for the design of stable
mining layouts in stratiform orebodies. The procedure
is to divide the orebody into panels separated by barrier
pillars. Panels and barrier pillars-are to be dimensioned
such that the role of pillars within panels is merely to
maintain the integrity of roof spans between pillars, and
may therefore be allowed to yield. In this case it is
necessary that each panel should be intrinsically stable.
The suggested method of design involves:
(i) design of the panel pillars using a factor
of safety against failure of unity;
(ii) determining whether the panel is intrinsically
stable on the basis of fAm>Ac, where f is a
suitable factor of safety for pillar stiffness.
If it is found that fAm<Ac , the options are to
increase Am, for example by increasing the width/height
ratios of pillars, or to decrease Ac by reducing the width
of the panel.
Moves to implement Salamon's design philosophy in
the mining of South African coal seams are implied in
papers by Cook et al.(1971), Wagner (1974), Van Heerden
(1975), and Oravecz (1977). With increasing depths of
metalliferous mining, it is to be expected that a design
philosophy similar to that discussed will be adopted,
with the added requirement to achieve extraction of whole
or part of the major pillars. It is therefore worthwhile
to review briefly the analytical techniques available and
the rock mass properties required to allow effectuation
of the principles of design of an intrinsically stable
structure.
1.5 Information for Design of Stable Pillars
The basic information required for the design of
a set of stable pillars consists of the post-peak
stiffness of pillars, and the mine local stiffness at
pillar positions in the mine structure.
The notion that the post-peak deformation of a pillar
can be described by a stiffness is a simplification which
is introduced for the sake of convenience. The non-
homogeneity of stress distribution in a pillar, and the
changes in stress distribution which accompany the develop-
ment of fractures, suggest that post-peak stiffness may
be a function of the system as a whole rather than an
inherent property of the pillar. The notion is retained
49
since it presents the most useful method of making a
first estimate of pillar stability.
Direct determination of the complete load - conver-
gence behaviour of mine pillars presents significant
practical difficulties. All measurements to date have
been made on model pillars, either on small intact speci-
ments tested in the laboratory, or on large specimens
tested in the field. Starfield and Wawersik (1968) reported
the results of tests conducted in a stiff compression
machine on cores of Tennessee marble with various diameter/
length ratios, and on a pillar-like specimen cut to
simulate the boundary conditions imposed on a mine pillar
in-situ. Stress-strain curves for the tests are shown in
Figure 1.6. Assuming that the post-peak stiffness of a
specimen can be represented by a single value, A' , the
post-peak performance of a pillar may be defined conveniently
by the ratio A/a', where A is the pillar stiffness in the
elastic range. Values of A/a' for specimens of various
diameter/length ratios are given in Table 1.1. It is
noted, for clarification, that in the convention used here,
post-peak stiffness V is negative. Thus increasing values
of V correspond to a change from steep to flat post-peak
load-deformation curves.
50
a
IO
)0
St•.M.a' arVot
FIGURE 1.6 STRESS-STRAIN CURVES FOR SPECIMENS OF TENNESSEE MARBLE WITH VARIOUS LENGTH/DIAMETER RATIOS. (FROM STARFIELD AND WAWERSIK, 1968).
Table 1.1. Elastic/ Post-peak Stiffness Ratios (A /a') for Specimens with Various Diameter/Length
Ratios (from Starfield and Wawersik (1968)).
D/L A/a'
0.5 -0.23
0.67 -0.93
1.0 -2.08
2.0 -5.20
2.0 -5.46 (Model Pillar)
The most comprehensive testing of large specimens
has been conducted on South African coal. The procedures
used and the results obtained are discussed by Cook et al.
(1971), Wagner (1974), and Van Heerden (1975). The results
follow the general trend observed in Table 1.1, that the
post-peak stiffness increases as the width/height ratio
increases. A summary of the experimental results is pro-
vided in Chapter 8. There appears to have been no attempt to measure the post-peak stiffness of large field specimens
of rock types other than coal, although, as noted previously,
the strength of large specimens has been measured for a
number of lithologies.
The post-peak load-deformation behaviour of intact
rock is controlled by the pattern of fracturing which devel-
ops in the specimen. It is to be expected therefore that
jointing and other natural fractures in a rock mass will
exercise a significant and possibly dominant role on the
post-peak performance of a mine pillar. This postulate is
supported by the result of tests reported by Brown and
Hudson (1972) on unjointed and jointed specimens constructed
from a rock-like material. For specimens which were square
in plan, with width/height ratios of 0.5, values of the ratio
51
A/A' were - 0.75 for the unjointed material, -3.81 for
a block-jointed specimen with joints parallel and per-
pendicular to the specimen axis, and -4.33 for an hexa-
gonally jointed specimen. These and other results reported
by Brown (197 0) suggest that any jointing will increase
the post-peak stiffness of a pillar, and that jointing in
a pillar oriented to favour slip will lead to ductile
rather than brittle behaviour of the pillar.
Considering the problem of estimating mine local
stiffness, a method of determining this parameter for
pillars in mining layouts in a stratiform orebody has
been described by Starfield and Wawersik (1968) using a
numerical procedure based on the Face Element technique.
An increment of convergence is imposed at a pillar position,
and the load increment required to maintain this convergence
calculated. The mine local stiffness kl may then be calcu-
lated directly as the ratio of load and convergence incre-
ments. The authors also define a mine local modulus K1 by
K1 = k H l Ti
where H and W are pillar height and width respectively.
Mine local modulus K1 may be compared with the post-
peak modulus of a pillar to assess stability. The procedure
used by Crouch and Fairhurst (1973) for determination of
mine local stiffness is similar to that described by Starfield
and Wawersik..
Mine structures generated during extraction of
metalliferous orebodies (other than stratiform orebodies)
are in general more irregular than those for which the
established methods of estimating mine local stiffness are
applicable. Of the numerical methods of analysis that can
be considered for this application, the Boundary Element
Method seems most appropriate, since the problem to be solved
involves determination of displacements induced at pillar
positions by loads applied in a mine structure by the pillars.
52
CHAPTER 2
CHAPTER 2 : THE BOUNDARY ELEMENT METHOD FOR ELASTOSTATICS
2.1. Principles and Limitations of the Method
The intention in this chapter is to idehtify the
premises on which the Boundary Element Method is based, and to
review briefly the different versions of the method which
have been developed. The method is appraised more from an
engineering than a precise mathematical viewpoint, since
this provides a useful physical appreciation of the notions
that are exploited in the method.
It was observed in Chapter 1 that effective handling
of a number of mining excavation design problems requires
the capacity to determine stress and displacement
distributions in a rock structure. Prior to the
development of the Boundary Element Method, the numerical
techniques available for this design activity were the
Finite Difference and Finite Element Methods. These
involve either a numerical approximation of the governing
differential equations for the medium, or a discretiz-
ation of the body into sets of connected elements. The
usefulness of these differential methods is restricted by
practical limitations which arise because of the necessity
to consider a problem domain defined by a volume of the
rock mass. The size of the numerical problem to be
solved is determined by the volume of the problem domain.
The result is that as the physical size of the problem
domain increases, the size of the numerical problem
frequently exceeds the capacity of even the largest
computers.
Solutions to the infinite and semi-infinite body.
problems presented by the design of rock structures have
been achieved by formulating methods of analysis in which
a problem is specified in terms of the conditions imposed
53
at the surfaces of excavations. These integral or
Boundary Element methods are based on the assumption of
elastic, and typically linearly elastic, behaviour of
the rock mass. The characteristic of these methods is
that the numerical size of a problem increases in
proportion to the surface area of excavations, and the
volume of the problem domain is not considered
explicitly in the analysis. A direct result of this
is that Boundary Element Methods allow finite and
infinite body problems to be treated with equal
facility.
There are basically two different versions of
the Boundary Element Method, identified by Brebbia.and
Butterfield (1978) as indirect and direct formulations.
A third version, called the Displacement Discontinuity
Method by Crouch (1976a),is a direct or an indirect
formulation, depending on the geometry of the problem
being analysed. The formal equivalence of indirect and
direct formulations has been demonstrated by Brebbia
and Butterfield (1978). The formulations differ in the
procedures used to construct relationships between the
tractions and displacements on excavation surfaces.
Figure 2.1(a) shows a cross-section of the surface
S* of a long excavation in an infinite, isotropic
elastic continuum which is subject to imposed traction
components txi, tzi, or imposed displacement
components uxi, uzi, at any point i on the surface.
The requirement is to obtain solutions for stresses and displacements in the medium which satisfy the
differential equations of equilibrium and the stress-
strain relationships for the material, and which
satisfy the imposed boundary conditions on the surface
Si'. As shown by Love (1944) for a two-dimensional problem the displacements ux, uz , in the medium must
54
5
/ t Z; I I
~XI
Zi
satisfy the Navier equations:
( X + G) DA + G 02ux = 0
( A + G) DA + G V2uz = 0 3z
(2.1)
In Figure 2.1(b), the trace of a surface S,
geometrically identical to S*, is shown inscribed in
an infinite medium. If the surface S is subject to
the same conditions of traction and displacement as
S* in Figure 2.1(a), the distribution of stress and
displacement in the region exterior to S will be the
same as the distributions in the region exterior to
S*. This follows from the uniqueness theorem proved
by Love (1944), involving in this case identity of
the total strain energy in the regions exterior to
S* and S. Physically, the identity of the stress and
displacement distributions in the regions exterior to
S* and S may be established by making a cut around S.
Since the tractions on S must be in equilibrium, the
material within S may be removed to generate the
surface S* without disturbing the equilibrium in any
way.
55
(a)
(b)
(c)
FIGURE 2.1: (a) SURFACE S* SUBJECT TO IMPOSED TRACTIONS OR DISPLACEMENTS. (b) SURFACE S INSCRIBED IN A CONTINUUM. (c) DISCRETIZED SURFACE S.
It is clear that the same conditions of identity
of stress and displacement distribution would apply if
a finite body were defined by the surface S* in Figure
2.1(a), and the region interior to S in Figure 2.1(b)
were considered as the problem domain. The solutions
of the infinite and finite body problems therefore
differ only in the specification of the normal to the
surface at any point.
The identity of the problems shown in Figures
2.1(a), (b) allows the real problem (Figure 2.1(a))to
be analysed through the continuum problem (Figure 2.1(b)).
In all formulations of the Boundary Element Method,
the solution procedure involves dividing the surface S
into a set of discrete boundary elements. In the two-
dimensional formulations reported to date, a curved
surface is represented by a set of linear elements, as
indicated in Figure 2.1(c). In a properly posed problem,
either tractions or displacements are specified on any
element. The procedure then is to use the known surface
values to determine the unknown values of traction or
displacement on each element. This is achieved by
establishing either a formal relationship between
element tractions and displacements, in the case of the
direct formulation, or by relating surface tractions
and displacements through a set of fictitious quantities,
in the case of the indirect formulation.
In both Boundary Element formulations, a knowledge
is required of fundamental or singular solutions to
the field equations for elastostatics of the type given
in equation 2.1. An example of such a solution is that
for the problem of a point load in the interior of an
infinite elastic solid, due to Kelvin, and quoted by
56
Love (1944). Expressions for stresses and displacements
induced by a point load are given in Appendix I. The
importance of singular solutions such as Kelvin's solution,
or the analogous infinite line load solution for
two-dimensional space, is associated with the facility
with which other singular solutions to the field
equations may be constructed from them. Some examples of
these nuclei of strain, or.,higher order singularities,
are given by Love (1944).
The principle of superposition is the foundation
of the Boundary Element Method. Solutions to problems
are obtained by superimposing stresses and displace-
ments induced by selected singularities. The type of
singularity used in any Boundary Element algorithm is
perfectly arbitrary, and the choice is made on the basis
of trial and error, the most appropriate singularity
being that which best suits the geometry of the type of
problem that the algorithm is required to analyse.
Banerjee and Butterfield (1977) note that the choice of
singularity must take account of the specific surface
area, i.e. thesurface area/volume ratio, of the problem.
In all cases, superposition of the stresses and
displacements induced by different types and
distributions of singularities requires that the medium
be at least piecewise linear elastic, and for
simplicity in implementation, linear elastic. The
former situation requires that a body be divided into
discrete cells and thereby increases the numerical size
of a problem.
The capacity to specify and solve a problem in
terms of surface geometry, and surface values of traction
and displacement, presents a number of major advantages,
in addition to those discussed above, over differential
methods. Some of the advantages are as follows.
57
(i) Because only the surfaces of excavations are
discretized, any errors associated with the discretiz-
ation are restricted to the immediate vicinity of the
problem surfaces. This contrasts directly with
differential methods, where discretization errors occur
throughout the volume of the problem domain.
Discretization errors in Finite Element Methods have
been discussed by Gallagher (1977).
(ii) The field variables of stress and displacement
are obtained directly, without the need for numerical
differentiation.
(iii) Values of the field variables are calculated
only at points of interest in the medium, and these
points are nominated by the program user. This limits
the amount of redundant data generated during analysis.
The following review of the various Boundary Element
formulations is intended to indicate the different
solution procedures used in each formulation, and to
describe briefly the historical development of the
various formulations.
2.2. Indirect Boundary Element Formulations
Figure 2.2(a) shows the trace of the cross-
section of the surface .S of a long excavation inscribed
in an infinite elastic medium subject to plane strain
conditions of loading. At any point on the surface it
is convenient to express the imposed tractions or
displacements relative to local axes for the point,
defined by the normal axis N, directed into the medium,
and the L axis, tangent to the surface at the point.
The objective in the indirect formulation is to find
suitable approximations to distributions of singularities
which, when applied over the surface S, produce the known
58
surface values of traction or displacement. The
singularities may be, for example, infinitesimally
spaced forces applied transverse and normal to the
boundary. Suppose that at any point j on the
surface, the initially unknown intensities of the
distributions required to satisfy the boundary
conditions on S are Q1(j), Qn(j), as shown in Figures 2.2(b),
(c). If i is any point in the medium, stresses and
displacements due to the singularity distributions are
obtained by superposition of the stress and displace-
ment components induced by load increments on small
elements of the surface, dS. Expressed relative to
the global reference axes the components may be written
axi =Jl''n Ql(j) + Fni(i,j) Qn(j)} dS S
azi = f{F12(i,j) Q1(j) + Fn2(i,j) Qn(j)}dS S
Tzxī 1{F13(i,j) Qi(j) + Fn3(i,j) Qn(j)}dS S
uxi = J 14(i,j) Q1(j) + Fn4 (i,j) Qn(j)} dS
(2.2)
uZ1 = f 15(i,j) Ql(j) + Fn5(i,j) Qn(j)}dS
where the form of the kernel functions Flietc.
is determined by the particular singularities distrib-
uted over the surface S.
Suppose the surface S is divided into a set of
k discrete elements, of which element j, subject to
singularity intensities qlj, qn., is representative,
as shown in Figure 2.2(d),(e). Equations (2.2) may
then be written in the discretized forms
59
(a)
60
(b)
(c)
(d)
(e)
FIGURE 2.2 (a): SURFACE S SUBJECT TO IMPOSED TRACTIONS OR DISPLACEMENTS. (b),(c): DISTRIBUTIONS OF NORMAL AND SHEAR SINGULARITIES ON S. (d), (e): NORMAL AND SHEAR SINGULARITY INTENSITIES ON ELEMENT OF SURFACE S.
k
axi = E ( alij qlj + a q )
j=1
k
QZ1 = E (.clij qlj + cnij qnj )
j=1
61
k
Tzxi .= E (flij q13 + fnijgn3 ) j=1
(2.3)
nj
(U 1j q + Uxi qnj ) uxi E xi lj
j=1
k '
uz i = E (ul Z1 q + Un zl qnj )
j=1
where the coefficientsalij etc. are obtained by
integrating the functions F11 etc. for the unit solutions
over the range of each element.
Equations (2.3) may be expressed in matrix
notation in the form
axi = [Ai] [q] (2.4)
uxi [Uxi] [q]
with similar expressions for azi, Tzxi, uzi.
In equations (2.4), [Ai] , [Uxi] are row vectors, of
order 2k, [q]is a column vector, of order 2k.
k
In establishing a relationship between the element
singularity intensities and the known surface values of
traction or displacement, equations similar to (2.4)
may be written by taking the point i as the centre
of an element j, and using as reference axes the local
axes for element j. Noting that
tli T nli
t Q
ni ni
and by considering as point i the centre of each
boundary element j in turn, two sets of simultaneous
equations may be constructed:
[t] = [G] [q] (2.5)
[u] = [H] [q]
where [t] and [u] are column vectors, of order 2k
[G] and [H] are square matrices, of order 2k.
In constructing the coefficient matrices [G] and
[H], two types of integrals must be considered: the
case where the point i lies in the range of element j,
when it is necessary to determine the limiting value
attained as the point approaches the element; the
general case, where point i does not coincide with
element j, when the integrals may be evaluated from
closed form solutions, or by quadrature methods.
For a properly posed problem, where either tractions
or displacements are specified on any boundary element,
equations (2.5) provide sufficient information, in
principle, to determine a set of element load
intensities which satisfy the known boundary conditions.
62
For the case of mixed boundary conditions, appropriate
terms must be interchanged between the [t] and [u] vectors,
and corresponding rows interchanged between the [G] and
[H]matrices, to establish a set of 2k equations in 2k
unknowns.
It is noted that the set of element load intensities,
[q], has no direct physical significance, but once it has
been determined, it can be used, by applying equations
(2.4), to obtain stresses and displacements at any point
in the medium. Also, it is possible to eliminate the
fictitious load vector from equations (2.5), to obtain
a relationship between element tractions and displace-
ments:
[t] = [G] [H] -1 [u] (2.6)
Equation (2.6) is a boundary constraint equation.
It defines a formal relationship between tractions and
displacements for the surface of a body, similar to that
established in direct formūlations of the Boundary Element
Method. It thus represents an ad hoc proof of the equiv-
alence of direct and indirect formulations.
The first indirectBoundary Element algorithms
developed along the lines of that described above were
proposed by Jaswon (1963) and Symm (1963), who described
procedures for the solution of steady state potential
problems, and noted the possibility of applying the
technique to problems in elastostatics. It was also
applied to the solution of torsion problems by Jaswon
and Ponter (1963).
Solution procedures for two-dimensional isotropic
elasticity have been proposed independently by a number
of researchers. Massonet (1965) used the solution of
a line load on an infinite half space as the singularity
63
for developing equations (2.4), and examined finite body
problems to assess the validity of the method. Oliviera
(1968) and Bray (1976a) used line load singularities in
an infinite medium to provide the kernel functions in
equations (2.2), Oliviera assuming linear variation of
fictitious load intensity with respect to element intrinsic
co-ordinates, and Bray uniform (strip) loading of elements.
Considering three-dimensional elasticity, Diest et
al. (1973) established the basis of an indirect formulation
using point load singularities, but did not indicate if
the proposed solution procedure had been implemented. Clarke
and Thompson (1976) used uniform distributions of point
loads over rectangular elements. Butterfield and Banerjee
(1971) used Mindlin's solution (1936) for point loads
acting in a semi-infinite elastic medium to develop a
method for analysis of the load-displacement behaviour of
pile groups.
Indirect formulations for two-dimensional elasticity
and transversely isotropic and orthotropic material proper-
ties have been developed by Tomlin and Butterfield (1974),
for near-surface structures, and Eissa (1979), for under-
ground structures, using the unit solutions given by de
Urena (1966) and Lekhnitskii (1963). Krenk (1978) has
described a method for determination of stress concentra-
tion around holes in transversely isotropic sheets.
2.3 Direct Boundary Element Formulations
The objective in direct formulations is to establish
and solve equations which relate tractions and displace-
ments on the surface of a body, allowing unknown surface
values of traction or displacement to be determined
directly from the known values. The solution procedure
is developed from Betti's Reciprocal Theorem (Love, (1944)),
which is used to construct integral equations for points
on the discretized boundary of a problem domain. A brief
outline of the procedure,for a homogeneous medium only,
64
is given below, as a detailed description is provided in
Chapter 7.
Figure 2.3(a) represents the trace S of the cross-
section of a long opening inscribed in an infinite medium.
At any point on S, either tractions or displacements
may be specified. In the absence of body forces, the
surface S may be considered to be loaded by tractions tx,
tZ, which produce displacements ux,uZ, at the surface.
Figure 2.3(b) shows a surface identical to that in
Figure 2.3(a), but in this case a line load, directed in
the X-direction and of unit intensity/unit length in the
Y-direction, is applied at a point i. Suppose the tractions
and displacements induced at any point on S are TXI, TXl,
UXI, UXl. The Reciprocal Theorem may be applied to the
load systems illustrated in Figures 2.3(a), (b). Inte-
grating around the surface S and the singularity at i
yields the boundary integral equation:
uxi + J (Txiux + T iuZ)dS = I(txUxi + tZUZI)dS ( 2.7)
x
S S
Equation (2.7) is Somigliana's identity for the
load point and the surface. A similar expression
involving uzi is established by considering a Z-directed
unit line load at the point i.
65
(a)
(b)
(c )
FIGURE 2.3 (a),(b): LOAD CASES FOR ESTABLISHMENT OF BOUNDARY INTEGRAL EQUATION. (c): METHOD OF HANDLING SINGULARITY IN RANGE OF INTEGRATION.
Suppose surface S is divided into a set of k elements,
subject to imposed tractions txj, tzj, or imposed displace-
ments 11 x3., 12 z.3, at a point on element j. Establishment
of a boundary constraint equation requires that the load
point i be moved on to the surface S, and that some
assumptions be made about the variation of traction
and displacement over the range of each element j.
This latter condition allows equation (2.7) to be
expressed in discretized form. The simplest procedure
is to assume that tractions and displacements are uniform
over each element j, and to write the discretized integral
equation for the centre of the element. Equation (2.7)
may then be approximated by
E (Tx Xluxj + Tciuz.)dS. = E
J (t x3+ tz.UZ1)dS. (2.8)
j-1 Sj j-1 S. 3
where Si denotes the surface of element j. When the
load point i lies in the range of the integration, the
Cauchy Principal Value of the integral is taken. This
is evaluated by considering the singularity to be sur-
rounded by a semi-circular surface, as shown in Figure
2.3 (c), and finding the limiting value of the integral
of the traction over the surface of the semi-circle as
the radius 6 tends to zero. If the surface S is smooth
at i, for a unit line load the limiting value is 1, and equation (2.8) may be written
1uxi + E ( (TXluxj + Tuz.)dS. = E (txjUXl + tz .UX
J
i)ds.
j=1 ~5. '=1S.
J J (2.9)
The initial term, 1 u;i, is frequently called the
free term. Since txj, tzj,uxj, uzj_ are assumed constant
over each element, equation (2.9) and the similar equation
established using a Z-directed unit line load at point i
may be written in the form
66
t xJ
tzj
(2.10) u j X k
E j=1
k E
j=1 u j Z
Fxi Fxi xJ zJ
Fzi Fzi _ xJ zj
xi . UI UI xJ zJ
UIZ1 clzi xJ z3._
MOP
67
where FXV = Jr TxidS.
UIX3 = J UxidS. etc.
S. J
For terms of the type FXi the free term is included
implicitly in the value of the integral.
By taking as the load point i the centre of each
boundary element in turn, k equations similar to equation
(2.10) are obtained. These may be expressed in matrix
notation as
[F] [u] = [UI] [t] (2.11)
representing a set of 2k equations in 2k unknowns. In
a properly posed problem 2k surface values will be prescribed.
These may be substituted into equation (2.11), which, after
any necessary rearrangement, may be solved directly for the
unknown surface values.
Displacements at internal points in the medium are
obtained by substituting the values of the complete set
of surface tractions and displacements into the discretized
form of equation (2.7). Expressions for stresses at internal
points are obtained by partial differentiation of the ex-
pressions for displacements to obtain expressions for the
strain components in terms of surface tractions and dis-
placements, and application of the appropriate stress- strain
relationships. Substitution of the values of surface trac-
tions and displacements in the discretized form of these
equations yields the values for the stress components at
the point.
S. J
The first implementation of the solution procedure
described above was reported by Rizzo (1967) for problems
in two dimensional isotropic elasticity. It was subse-
quently extended by Rizzo and Shippey (1968) to handle
non-homogeneous problems involving solid elastic inclu-
sions, and also took account of cases in which the stresses
were induced by mismatches of stress or displacement across
interfaces between inclusions and the host medium. Ricca-
della (1972) developed a method of two-dimensional isotropic
elastic analysis in which tractions and displacements were
assumed to vary linearly over each element. The boundary
constraint equation was established by taking the collo-
cation points at the ends of the elements, rather than
the element centres. As in Rizzo's method, discontinui-
ties in traction and displacement occur between adjacent
elements. Wardle and Crotty (1978) used linear variation
of element traction and displacement in a method for the
analysis of non-homogeneous media, in which inclusions
are embedded in an infinite medium. Comparison of results
of analysis of problems using the Boundary Element program
with those obtained from Finite Element analysis of the
same problems showed reasonable agreement. Differences
between the results were most marked near the interfaces
between the infinite region and the inclusions.
In all the two-dimensional methods discussed, linear
elements have been used to represent the discretized surface.
The integrations of the kernel functions required to deter-
mine the coefficients of the square matrices in equation
(2.11) have been performed analytically.
The development of a boundary constraint equation
for three-dimensional elasticity follows that for two
dimensions, with unit point loads being applied in three
orthogonal directions at nodes of elements defining the
surface of interest. The kernel functions to be inte-
grated to establish the boundary constraint equations
are therefore those for traction and displacement given
by the Kelvin solution. Cruse (1969) implemented a
68
three-dimensional solution procedure using plane
triangular elements to describe the surface, and
assuming constant traction and displacement over
each element. The boundary constraint equation was
constructed for nodes at the centroids of the elements,
so that the free term could be determined readily. An
improved formulation described by Cruse (1974) allowed
for linear variation of traction and displacement over
the triangular elements. This resulted in nodes being
located at the corners of elements, and introduced a
problem in evaluation of the free term in cases where
a node occurs at a corner or edge of the surface of a
body. The difficulty was resolved by noting that for
a point load applied to the exterior of a surface
inscribed in a continuum, the integral of the tractions
over the surface and the free term must sum to zero. In
both formulations, Cruse used analytical methods to establish
the matrix coefficients in the boundary constraint equation,
but Gaussian quadrature was used in a further development
of the method for axi-symmetric problems (Cruse et al.
(1977)) .
The most advanced implementation of a three-dimen-
sional, direct formulation has been described by Lachat
and Watson (1976). The geometry of a curved quadrilateral
boundary element is expressed in terms of quadratic shape
functions of the element intrinsic co-ordinate system,
and traction and displacement may vary linearly, quadra-
tically or cubically with respect to element intrinsic
co-ordinates. The discretized boundary integral equation
is written for 4,8 or 12 points of the quadrilateral
elements, depending on the functional variations imposed,
and the free term is evaluated implicitly following the
scheme suggested by Cruse (1974). Gaussian quadrature
is used to integrate the kernel function - shape function
products over the surface of each element, as analytical
integration is impossible. A body may be sub-divided into
elastic sub-regions, at the interfaces between which extra
69
integral equations are written to take account of the
conditions of continuity of displacement and stress at
these surfaces. The objective in dividing the body into
distinct regions was to produce a banded matrix, allowing
reduction in computer central memory requirements in
solution of the boundary constraint equation.
The application of Lachat and Watson's program.
to determination of the stress distribution around tunnel
intersections has been reported by Brown and Hocking (1976)
and Hocking (1978). These demonstrated the facility with
which the program may be used to analyse excavation shapes
which would present some difficulty in mesh generation if
analysed with Finite Elements, and are intractable if not
analysed by some numerical method.
2.4 Displacement Discontinuity Method
The Displacement Discontinuity Method may be classed
as a direct or indirect formulation of the Boundary Element
Method, depending on the geometry of the excavation analysed.
For excavations which are modelled as narrow slits it is
a direct formulation, as the problem is solved through
sets of equations relating tractions and displacements
on excavation surfaces. For other types of problems it
is an indirect formulation. The method differs from all
other formulations discussed earlier in the type of singu-
larities which are employed in the solution procedure.
The technique used in construction of the singularities is
of basic interest. The method itself is of interest be-
cause it was developed from procedures designed specifically
to handle mining problems, and the concepts established in
the development anticipated those applied in other Boundary
Element formulations.
The method originated from work on methods for pre-
diction of surface displacements induced by extraction of
70
coal seams in which excavations were treated as slits
in the rock medium. The idea that a mined area in a
coal seam could be modelled as an infinitely thin slit
was introduced by Hackett (1959). He used solutions
obtained by Westerga and (1939) and Sneddon (1946) for
cracks in an infinite, elastic, isotropic medium to
calculate displacements and stresses induced by mining
long, wide panels. The assumption was made that the
surfaces of a mined opening were traction - free after
excavation. Although Hackett was unable to take account
of the traction-free ground surface, he established that
if realistic values were assumed for the Young's Modulus
of the rock mass, displacements of comparable magnitude
to those measured in the field would be induced remote from
the excavation.
A procedure for taking proper account of the
boundary conditions at the ground surface was reported
by Berry (1960). He was able to obtain an exact solution
for the case where complete closure occurred over the
mined span, and approximate solutions for the cases of
no closure and partial closure. The value of the solution
for complete closure was that it produced an upper bound
for surface displacements induced by mining. These were
found to be independent of rock elastic properties. As measured
displacements were found to be greater than the upper bound
solution from the isotropic model, Berry concluded that the
assumption of isotropic elasticity was untenable. Two- and
three-dimensional analyses of single seam extraction in a
transversely isotropic half space, assuming both complete
and partial closure over the mined area, were reported by
Berry and Sales (1961, 1962) . In the case of complete
closure, the analysis was equivalent to introducing a
constant displacement discontinuity over the mined area
of the seam. Suitable choice of compliances for the rock
mass resulted in satisfactory matching of calculated and
measured surface subsidence and strain profiles.
71
Application of the closure distribution over a mined
area to calculation of the stress distribution in the ground
was first reported by Berry (1963) and Salamon (1963). The
Face Element Method proposed by Salamon discretized the mined
area into a set of segments, over each of which constant
closure occurred, so that each segment was equivalent to
a normal displacement discontinuity in the medium. The
analysis was subsequently extended to include relative shear
displacements between the hangingwall and footwall of
openings (Salamon, 1964). Analogue and digital techniques
for analysis of single seam extraction were established
by Salamon et al. (1964), Cook and Schumann (1965), Star-
field and Fairhurst (1968) and Starfield and Crouch (1973) .
Crouch (1976a) reported the generalization of the Displace-
ment Discontinuity Method to handle problems other than
narrow excavations, and also provided a clear description
of the method of construction of normal and shear dis-
placement singularities.
Crouch(1976b)considered an isotropic medium subject
to plane strain in the X-Z plane. Expressions for stresses
and displacements induced by strip normal and shear dis-
placementdiscontinuities, of magnitudes Dz and Dx, in an
infinite medium were developed from displacement potentials
in the following way. According to Timoshenko and Goodier
(1951) the general solution for displacements satisfying
the field equations (2.1) is given by
ux = Bx — 4 (11u) ax (x Bx + z Bz +8) (2.12)
1 uz - B - 4(1-v) az (x Bx + z Bz +(3
)
where Bx, Bz,8 are harmonic functions, called the
Papkovitch - Neuber functions. Particular Papkovitch-
Neuber functions were chosen by Crouch so that the plane
z=0 is, in one case, free of shear traction, and in another
case, free of normal traction. By choosing, in the first case,
72
B = 0 Bx
73
Bz = - 4 (1-v) az
a = — 4(1-v) (1-2v)$
(2.13)
where d is an harmonic function, displacements and stresses
are given by
ux = (1-2v) + zaxaz
uz = - 2(1-v) 2A + z (2.14)
ax = 2G (az + za )
a 2 05 az = 2G (3z2 - z āz
a' s TzX = —
2Gz axaz2 It is noted that, on the plane z=0,
finite.
a
3
TZx =0 if aXaZ is
Equations (2.13) can be used to construct solutions
for particular problems by suitable choice of the potential
function 0. Suppose the problem is to develop a displace-
ment discontinuity Dz over the range -a x 4a on z=0 in an
infinite body, where
Dz = lim uz (x, z) - lim uz (x, z) z+0- z-► 0+
From the symmetry of the problem, the uz displacement
components are equal in magnitude, but opposite in sign.
The problem may then be considered as two half spaces,
joined at the boundary z = 0, as shown in Figure 2.4,
provided the following conditions are satisfied at the
boundary of the half space z40 :
TzX = 0 — co < x< co, z = 0
uz = 1 Dz Ixl.a, z = 0 (2.15)
uz = 0 Ix1>a, z = 0
uZ 0, TZ =0 r Z TZXO , uZ D /2 -a.-~ tzx 0 . uZ=-5Z/2
74
FIGURE 2.4 BOUNDARY CONDITIONS ON COUPLED HALF-SPACES FOR GENERATION OF NORMAL DISPLACEMENT DISCONTINUITY Dz (AFTER CROUCH, 1976b)
As shown earlier, the condition Tzx =0 is satisfied
for any arbitrary harmonic function 0 satisfying equations
(2.14), while the second of equations (2.14) yields
uz = - 2(1-v) 3z z = 0
The boundary conditions for uz in equations (2.15)
will be satisfied if 0 is chosen such that
ass _ 1 az 4(1-v) Dz lx1<a, z = 0
0 ixl>a, z = 0
These conditions are satisfied if the harmonic
function 0 is defined by
D
az _ - 4~ (1-v) { tan-1 (xZa) - tan-1 (x-a)}
By integration, O is found to be given by
Dz 0 (x,z) _ .
47(1-v) {z tan-1 (xZa) - z tan-1 (x-a) +
(x+a)ln{(x+a)2 + z 2}1/2 - (x-a)ln{(x-a)2 + z20}
(2.16 )
Displacements and stresses at any point in the
medium induced by the displacement discontinuity can
then be determined using equations (2.14).
A similar procedure was used to obtain the harmonic
function which provides the solution for a shear displace-
ment discontinuity Dx in an infinite medium, illustrated
in Figure 2.5. In this case the Papkovitch - Neuber
functions are taken as
B x = 0
Bz = - 4(1-u) ax
S = - 8(1-v)2 fāX dz
where V2X= 0
FIGURE 2.5 BOUNDARY CONDITIONS ON COUPLED HALF-SPACES FOR GENERATION OF SHEAR DISPLACEMENT DISCONTINUITY Dx (AFTER CROUCH, 1976b)
75
76
Displacements and stresses are given by
2 -2(1-v) az - z
2
- (1-2v)āX - z ax z
a 2x a jx 2G (2axaz + z axaz2 ) 3 ax
-2Gz axāzr
Tzx = 2 G (aā + z a
) aZ
Considering the boundary conditions on the half-
space defining half the anti-symmetric problem, the
required expression for X is found to be
Dx X(x, z)- 4ff (1-v) {z tan-1 (xZa)-z tan-1 (xZa)+(x+a) In {(x+a) 2+z2}
- (x-a) ln { (x-a) 2 + z211/21 (2.17)
The identity of the expressions for 0 and X is
noteworthy. It is also noted that the expressions for
x induced by the normal displacement discontinuity and
for Tzx induced by the shear displacement discontinuity will therefore also be identical, apart, of course, from
the magnitudes of the displacement discontinuities which
occur in the expressions.
The direct relationships between closure (Dz) and
induced normal stress Z, and ride (Dx) and induced shear
stress TzX have been used by Crouch in direct formulations
for determining stresses and displacements around narrow
excavations in an infinite medium. In addition, solutions
have been developed from equations (2.16) and (2.17) for
the harmonic functions which describe the displacements
ux =
uz =
ax =
az
=
and stresses induced by displacement discontinuities
in a semi-infinite medium, using the method of images
introduced by Berry (1960). These have been used in
the analysis of problems associated with the mining of
narrow orebodies in faulted ground (Crouch (1979)), and
for modelling edge cracks in semi-infinite solids (Crouch
(1976b)).
The Displacement Discontinuity Method, when applied
to finite, semi-infinite and infinite body problems other
than thin slit and crack problems, involves finding the
magnitudes of the displacement discontinuities which,
when disposed around the surface of the body, generate
the known values of traction or displacement on the
surface of the body. The procedure in this case is
similar to that described earlier for indirect Boundary
Element formulations. The method has been applied
successfully to such problems as a circular disc
subject to diametral compression, a circular hole in
a biaxial field, and near-surface excavations in a
semi-infinite medium.However, for problems such as
the circular hole in a biaxial field; the numbers of
boundary elements were greater than those used in
other indirect solutions of similar problems.
2.5 Required Developments in Boundary Element Solution
Procedures.
The review of the various formulations of the
Boundary Element Method has shown that the technique
is well developed for analysis of a range of problems
in two- and three-dimensional elasticity. However,
there are a number of areas in which further work is
justified, from the following considerations. Firstly,
the work by Hocking (1978) suggests that the potential
application of comprehensive three-dimensional analysis
to the design of the complex mining layouts is limited
by computer resource requirements. For even simple
problems, job execution times become excessive. The
77
inference is that plane strain methods or similar simple
methods of analysis are required for the majority of
mining problems, in which the establishment of an excava-
tion sequence is the prime objective. Exceptions to
this rule involve, for example, the design of major
permanent openings, permanent pillars, and shaft bottom
layouts, when the expense and effort of three-dimensional
analysis are justified. Secondly, all the plane strain
Boundary Element methods discussed previously apply to
the case where the long axis of excavations is parallel
to a pre-mining principal stress direction, which limits
their practical application. The need is therefore to
bridge the gap between established two-dimensional and
three-dimensional methods of analysis.
In the discussion of direct formulations it was
noted that all solution procedures reported to date
have used point load or line load singularities to
establish the perturbations of traction and displacement
for construction of the boundary integral equation. Only
in the Displacement Discontinuity Method are singularities
specifically constructed to exploit the geometry of the
problem being analysed. It is possible that singularities
other than the usual fundamental solutions may be employed
profitably in direct formulations. Similar considerations
apply in indirect formulations. Construction and assessment
of the performance of different types of singularities is
therefore of basic interest in the development of efficient
Boundary Element formulations.
78
CHAPTER 3
79
CHAPTER 3: COMPLETE PLANE STRAIN AND COMPLETE PLANE STRESS
3.1 Problem Specification and Definitions
Underground openings are excavated in a medium
subject to initial stress, and the problem is to determine
the distribution of total stresses and excavation-induced
displacements in the medium surrounding the excavations.
Hocking (1976) used the three-dimensional Boundary Integral
Program described by Lachat and Watson (1976) to determine
the stress distribution around openings with various
length/cross-section dimension ratios. He demonstrated
that when the length/cross-section dimension ratio exceeded
2.5, the stress distribution around the opening in the
central section approached that for plane strain. It was
also shown that the plane strain solution gives an effective
upper bound to stress magnitudes, indicating the value of
plane strain analysis for underground excavation design.
An implicit assumption in the typical formulation
of plane strain methods of analysis is that a pre-mining
principal stress acts parallel to the long axis of excavation.
This condition will not be satisfied generally, as excavations
may be arbitrarily inclined in a triaxial stress field.
As shown in Figure 3.1, it is necessary to take account of shear stress components acting parallel to the long axis
of the excavation.
z
FIGURE 3.1 : PLANE (px, pz, pzx) AND OUT-OF-PLANE (pxy, pyz)
STRESS COMPONENTS FOR A LONG OPENING EXCAVATED IN A MEDIUM SUBJECT TO A TRIAXIAL STATE OF STRESS
80
In this and subsequent sections concerned with
complete plane strain, the reference axes used in the
analysis are denoted X, Y, Z, with the Y-axis parallel to
the long axis of excavation/ as shown in Figure 3.1. Openings
are excavated in an isotropic elastic continuum subject to
known field stresses px' py' pz' pxy' pyz' pzx' The
standard geotechnical convention is adopted, taking
compressive stresses and contractile strains positive, with
the sense of positive normal stresses defining the sense
of positive shear stresses in the normal way. Excavation-
induced displacement components (us, Uy, uz) at a point
(x, y, z) are taken positive if directed in the positive
directions of the co-ordinate axes.
In determining the resultant (total) stress
distribution around an opening in a medium subject to initial
stress, the options are to perform the analysis in terms
of either
(a) the stresses induced when the opening is excavated
in the stressed medium, or
(b) the total stresses induced when the unstressed
medium containing the hole is loaded, by applying
the field stresses to the medium.
In case (a), induced stress components are related
to excavation induced displacement components, and the
distribution of total stresses is obtained from the
distribution of induced stresses by superposition of the
field stresses. Case (a) is considered here, as it models
the real situation directly.
The specification of complete plane strain
described below was originally proposed by Bray (1976b).
Another specification is given by Zienkiewicz et al.(1978),
who call it quasi-plane strain. The description of complete
plane stress is the author's work.
81
3.2 Plane Strain
If (ux, uy, uz) are induced displacement components at a point (x,y,z), and the Y-axis is parallel to the long
axis of excavation, the usual specification of plane strain
is that the displacement component in the Y direction is
zero, and displacement components ux, uz in the X-Z plane
are functions of position co-ordinates x, z only; i.e.,
u = 0 Y
au E _ --~ ay -p Y
aux x _ z 0 ay ay
(3.1)
au au
- _~ x Yxy ( ax + ay )
au au z Yyz
= ( az + ay ) = 0
The specification of plane strain, as given by
equations (3.1) requires that induced stresses Xy , Tyz be zero at all points in the medium. When a long opening
is excavated in a triaxial stress field, in general an
induced component of traction in the Y direction, ty, must
be taken into account to satisfy the boundary conditions
on the excavation surface. The induced surface traction
component t is directly related to the field stress
components pxy, pyz, and induced stresses Txy, Tyz . The
conditions for plane strain, as specified by equations,(3.1)
can be satisfied only if the field stress components pxy,
pyz are zero, requiring that the Y direction be a principal
stress direction at all points in the medium.
3.3 Complete Plane Strain
The essential notion in the plane strain concept
is that conditions of induced displacement and stress are
= 0
au - - --Y = 0 ay
au 2 az
E y
Cz
82
identical in all planes perpendicular to the long axis of
an excavation in the medium. Thus a reasonable definition
of plane strain, when the Y-axis is parallel to the long
axis of the excavation, is that induced displacement
components ux, uy, uz at any point (x, y, z) are functions
of x, z only. In this case,
aux auz _ __X - ay — ay - ay - 0
(3.2)
and E x
E = 0 y
' Ez' Yxy' Yyz Yzx are, in general, non-zero.
This definition of plane strain allows the six
induced stress components to be non-zero at all points in the
medium, and the induced surface tractions tx, ty, t to be non-zero.
A complete plane strain problem may be resolved
into two subsidiary, decoupled problems. Since ux, uy, uz
are functions of (x,z) only, induced strains are defined by
ax
(3.3)
au Y = - xy _a
ax
au Yyz = - az
Yzx
( az + ax ) aux auz
From the stress-strain relationships for
isotropic elasticity,
83
ax = A A + 2 G ex
T = G Yxy etc.
where 0 is the volumetric strain (=x + c z)
A complete plane strain problem therefore
involves two components:
(a) the plane problem, involving stress components
ax, ay, az, T zx ' and displacement components
ux , uz I
(b) the out-of-plane problem, involving stress
components Txy, Tyz and displacement component
uy .
The plane problem is the usual problem considered
in plane strain analysis. The out-of-plane problem has been
called antiplane strain by Filon (1937). Antiplane problems
have been considered in detail by Milne Thomson (1962).
The following general analysis of complete plane strain,
due to Bray (1976b), is included here since it is complemented
by the discussion of complete plane stress.
In general, there are three strain compatibility
equations of the form
a 2 cx ac __ a 2YxY ay + ax axay
and three of the form
a 2 c 2 ayaz (
_ aYyz aYzx aYxy āx ax + ay + az
From equations (3.3), it is seen that for complete
plane strain, three of the compatibility equations are
satisfied identically, the others reducing to
326X a2EZ 321, zx
a + ax — = axaz
a ayyz ayxy
ax (- ax + a z ) =
aYxy aYyz = āz ( az + ax )
0
0
84
In excavating an opening in an elastic medium,
no body forces are induced, and therefore the three
differential equations of equilibrium are of the form
aax at aT Zx
ax + ayy + az = 0
In complete plane strain, noting that terms of
the type aYxy etc, are identically zero, the equilibrium Dy
equations reduce to
aa at x zx ax + az
aT aT —Y? __HZ
az + ax
aT ac zx z
ax + az
= 0
= 0
= 0
Finally, the stress-strain relationships for
isotropic elasticity and complete plane strain become
_ (1—V2) v Ex = E
{ax 1-v 6z}
Ey = 0
e = (1-v2) { a - y Q } z E z 1-v x
Yxy = 1 G
Txy
Yyz = 1 Tyz
1 = Yzx G Tzx
(3.10)
a 2 Ō etc.
Equations (3.4), (3.7), (3.9) relate to the plane
problem, and using the appropriate stress-strain relation-
ships from equations (3.10), it can be shown in the usual
way that the distribution of ax' az' Tzx in the X-Z
plane satisfies the biharmonic equation
v"$ = o
where 0 (x, z) = Airy stress function
ax x az2
Thus the plane problem may be solved by finding
a suitable stress function and satisfying the imposed
boundary conditions for the problem.
Equations (3.5), (3.6) and (3.8) provide the
information necessary for determining the distribution of Txy, Tyz in the X-Z plane, i.e. solution of the antiplane
problem.
The change in a quantity q, defined by
aYxy aYyz q =
- az ax
between any two adjacent points in the X-Z plane
is given by
dq = dx + S dz (3.11)
and from equations (3.5) and (3.6), = 0, = 0,az
Thus dq = 0, or q is constant throughout the
85
medium.
86
ay ay Remote from the excavation, --I- z_ XL = 0,
axaz and hence q=0 throughout the medium.
i.e. ayxy ayyZ
az ax or
a-LY = DTyz
az ax
From equation (3.8)
a xy aT
ax — az
Thus 'pxy , Tyz satisfy the Cauchy-Riemann conditions
and are therefore conjugate harmonic functions of (x,z); i.e.,
a 2
( ax2 a
2 Z2 ) Txy = 0
( a2 a2
ax2 az TyZ = 0
Therefore, the determination of the stress
distribution associated with the field stress component
pxy, for example, is exactly analogous to the determination
of the velocity components in the irrotational, solenoidal
flow of fluid past obstacles having the same cross-sections
as those of the excavations. Stress components at any point
due to pxy can thus be determined from a potential function
X, say
ax whereTxy āx
a
Tyz — - az
where X is chosen to satisfy the boundary conditions
for the problem and remote field conditions. Similar
considerations apply to the stress distribution associated
with the pyz component of the field stress.
0
87
In principle, then, the stress distribution
around long openings in a triaxial stress field can be
determined if a stress function and a potential function
can be found which will satisfy the boundary conditions in
the X-Z plane. It is noted also that the analogy between
potential and velocity for a hydrodynamic problem, and
displacement and stress for the antiplane problem, provides
a convenient method of establishing closed-form solutions
for displacement and stress distributions around openings
subject to antiplane strain.
3.4 Complete Plane Stress
The following analysis maintains for eomplete
plane strain and complete plane stress the identity of stress
distribution which exists between simple plane strain and
plane stress.
Considering a plate with faces normal to the Y-axis,
the normal definition of plane stress is:
(a) field stresses py, pry, pyz are zero on the faces
of the plate;
(b) induced stresses 6y, Txy, Tyz are zero
throughout the plate.
Thus Yxy ,pyz are zero throughout the plate,
and induced stresses ax az Tzx are functions of x,z only;
i.e.,£x.
cy, €z, YzX,are functions of x,z only. Inspection
of the six strain compatibility equations indicates that
three of the equations can be satisfied only if sy is a
linear function of x and z. As it is reasonable to assume
that this condition could be satisfied only in exceptional
circumstances, it is necessary to accept that, in general,
displacement components ux, uy, uz are functions of x, y, z.
Thus at any point in the plate, all six induced strain
(and stress) components are, in general, non-zero.
88
A state of complete plane stress may be defined
by a plate subject to stresses px, py, pz' pxy' pyz' pzx on its face and edges. This state of loading may exist
in a mine structure which is extensive in two dimensions.
Love (1944) suggests that if the plate is sufficiently thin,
it is useful to consider average displacement components
obtained -by integrating the components, at any point (x,z)
across the thickness of the plate and dividing by the
thickness of the plate- From this averaging process,
displacement components TT, ūy , uz are obtained which
are functions of x ,z only. It is noted that this averaging
process is valid for any thickness of plate, but the greater
the thickness, the less value may be attached to average
stresses determined from the average displacements.
Average strains are defined by
s = x
E = y
Dux a.x
- a ay 0
auz (3.12)
Ez - az
Y a 1Y
_ xy — — ax aū
yz az
_ au all x z
Yzx __ _
( az + ax ) Average stresses are defined by expressions of
the form
x = X + 2GEx
Txy = G yxy etc.
Ex =
TY
=
Ez =
Ē {ax - Ni (ay + Tr )1)}
Ē CaQy —v(rz + ax)}=0
Ē {az -v(Qx + ā)}
and thus
89
1 Y _
_ xy G xy
(3.13)
Yyz = 1 G Tyz
_ __ 1 Yzx zx
The first three equations of the set (3.13) give
(1-v2) — v Ex E ( cx 1-v oz )
__
_ (1-v2 ) (az v- cr ) z E z 1v x
All average induced stresses are independent of
y, and no body forces are induced by excavation. Thus the
following equilibrium equations can be established:
aāx aTzx ax + āZ
__ aTyz
ax + az
aT aQ
zx + z = 0
ax az
Finally, the average strains Ex, Ez, Yxy
etc. cannot vary independently throughout the plate, but
must be related through the average displacements. The
conditions for compatibility of the average strains are
obtained from equations (3.12). The equations
0
0
90
a l s X ay e
a ls a2Y + y — xy
ax2 axay
a ls
az2
a2E 27„
aye ayaz
a (_ aī zx + aīxy + aYyz Dy ay az ax
are satisfied identically. The remaining conditions for
compatibility of average strains are:
als Y a ls a 2 z x zx + _ ax2 aZ2 aXaz
a~ aY āx ~ _ axz + a
_z )-
a aXY + ate)= 0 az az ax
Comparing conditions to be satisfied for the case
of complete plane stress, i.e., equilibrium equations,
stress-strain relationships, and strain compatibility
equations, with those to be satisfied for the case of complete
plane strain, it is seen that the conditions are in all
respects identical. Therefore, with equivalent boundary
conditions on excavation surfaces, the distribution of
stress in the X-Z plane for complete plane stress will be
the same as for complete plane strain.
0
CHAPTER 4
CHAPTER 4: INDIRECT FORMULATION OF THE BOUNDARY ELEMENT METHOD FOR COMPLETE PLANE STRAIN
4.1 Description of Method of Analysis
The objective is to determine the distribution
of total stresses and induced displacements around long,
parallel openings excavated in an elastic medium subject
to a triaxial state of stress. Figure 4.1(a) shows a slice
of unit thickness, of a long opening which is to be
excavated in an infinite elastic continuum. Reference
axes X, Y, Z are oriented as shown, with the Y-axis
parallel to the long axis of the excavation. Relative to
these axes, the pre-mining stresses are p , p , p x y , xy p , p The excavation surface is denoted S*, and at yz zx' some point i on the surface it is subject to imposed
tractions txfi' tyfi' tzfi' or imposed displace-ments uxi' uyi, uzi. The problem illustrated in
Figure 4.1(a) may be regarded as the superposition of two
separate loading systems:
(i) the continuous medium subject to the field
stresses, as shown in Figure 4.1(b);
(ii) the continuous medium, free of field stresses,
in which is inscribed a surface S geometrically
identical to S* and subject to tractions txi'
tyl, t21, or displacements uxi' u yi' uzi'
as illustrated in Figure 4.1(c).
Figure 4.1(d) shows the trace of S on the X-Z plane, which
may be represented by discrete linear elements. The
orientation of a representative element i is defined by the
angle ai between the Z-axis and the outward normal through
the centre of the element,
91
(a)
(b)
92
X
Projection of S on X—Z plane
tzi ) L
k txi N
uxi 7 Normal to surface uz~ at Element i Element
J
(c)
(d)
FIGURE 4.1 : (a) LONG EXCAVATION IN A MEDIUM SUBJECT TO INITIAL STRESS;
(b),(c) RESOLUTION INTO COMPONENT PROBLEMS; (d) GEOMETRIC PARAMETERS FOR DISCRETIZED PROBLEM
93
Assume that tractions and displacements are
uniform over each element, and may be represented by the
values taken at the centre of each element. Using the
superposition scheme indicated in Figure 4.1, the tractions
on element i of the surface S are related to the imposed
surface tractions on S* by the equations
txi txfi - px sinsi - pzX cossi
tyi
• t
yfi - pxy sinsi - pyz cossi (4.1)
tzi = tzfi - pzX sinsi - pz cossi
Thus if imposed surface tractions txfi' tyfi' tzfi are
specified, the tractions txi' tyi' tzi on surface S
represent tractions which must be induced in an otherwise
unstressed medium to simulate generation of the surface
S* by excavation in the stressed medium. Induced stresses
on element i are related to the induced tractions by
the expressions
t xi
t yi
tzi
• aXl sins.
ZX 1 + T. COSsi
sins. + Tyzi cossi TXyi
• TZX. sins. + azl cossi
These equations confirm that the complete plane strain problem
may be solved in terms of the decoupled plane and anti-
plane problems, due to the decoupling of the plane
( x , a z ' T zx ) and antiplane ( Txy , Tyz ) stress components. The solution to the plane problem is
described by Bray (1976a). Parallel solutions for the
plane and antiplane problems are described here, using
uniform strip loads on elements. Expressions for the
transverse and normal strip loads illustrated in Figure
4.2(a), (c), are given in Appendix III.
✓✓r
4✓r ✓i'r
rrr ✓ A', Load intensity
Ari.9' = qy /unit area
X
i (xi,zi )
✓✓r
(d)
Element j j 2
(b)
X
Load intensity = qx /unit area
Y
X
Load intensity = qz/unit area
(c)
(a)
94
FIGURE 4.2 : UNIFORMLY DISTRIBUTED TRANSVERSE, LONGITUDINAL AND NORMAL STRIP LOADS, AND GEOMETRIC PARAMETERS DETERMINING THE EFFECT OF STRIP LOADS ON ELEMENT j AT THE POINT i(xi,zi)
95
4.2 Antiplane Line and Strip Loads
The solution of the antiplane problem requires
expressions for stress and displacement induced by a
uniformly distributed -longitudinal strip load, as illus-
trated in Figure 4.2(b). These have been obtained from
the Kelvin solution for stresses and displacements due to
a point load in an infinite medium, quoted by Love (1944)
and stated in Appendix I. Figure 4.3 shows a line load,
of intensity Q/unit length, directed parallel to the Y
axis. Integration of the expressions for a Y-directed
point load yields the following expressions for stresses
and displacement at the point i(x., z.) in the X-Z plane: 3.
6 = = Q = T = 0 x y z zx
T = Q xi 1
xy y 27 (4.2)
T = Q 1 z1 yz y 27
1
Uy = -Q 1 ln ri Y 27TG
(4.3)
u = u = 0 x z
where r 2 = x.2 + z.2 i 1 1 It is noted that an infinite constant is intro-
duced in integrating the point load solution for displace-
ment to obtain the displacement due to the longitudinal
line load. This is exactly analogous to the hydrodynamic
problem discussed by Batchelor (1970), in which the
potential due to a line source is derived from the
potential due to a point source. Thus equation (4.3)
can only be used to find the relative displacement between
any two points in the medium. The expressions given in
Appendix II for stresses and displacements due to an X-
directed line load have also been obtained by integration
of the Kelvin solutions. Infinite constants are associated
with the expressions for displacement components, so that
dy. Line load
Intensity Q.. /unit length Load Element, Y
Magnitude Qydy.
X
r. 1 - i(xi3O,zi )
Y
Z
96
FIGURE 4.3 : PROBLEM GEOMETRY FOR DETERMINING STRESSES AND DISPLACEMENTS DUE TO AN INFINITE, Y—DIRECTED LINE LOAD
1 Tyz
qy 2~ re]
2 (4.4)
in this case also it is possible to determine only relative
displacements between points in the medium. This issue is
briefly discussed by Banerjee (1977).
Expressions for stress and displacement components
due to uniformly distributed transverse, longitudinal and
normal strip loads, as illustrated in Figure 4.2(a), (b),
(c), are obtained directly from the line load solutions
by integration over the width of the loaded strip. Stresses
and displacement induced at a point i (xi,zi) by the
longitudinal strip load of intensity q on element j are
given by
Txy = qyīr [ln r )2
97
1 uy = - qy 2nG [x ln r - x + z.0 ]
2
where x = x. - x. 1 ~
and the geometric parameters are as defined in Figure
4.2(d). Considering the expression for the Tyz stress
component induced by a longitudinal strip load q applied
in the X-Y plane, and restricting attention to the plane
of the element,'it is observed that Tyz has a constant limit-
ing value of 1/2qy over the range of the element, and rapidly
falls to zero outside the range of the element. The
same behaviour is true of the cz and Tzx stress
components induced by the normal and transverse strip
loads. This means that if some required tractions
tx, ty, ; ar e established at a point under a strip
loaded element, such as the centre of the element, the
same tractions will exist over virtually the complete
width of the element.
98
4.3 Boundary Element Solution Procedure
The aim in the indirect formulation of the
Boundary Element Method is to determine the magnitudes of
a set of element loads which satisfy the known conditions
of traction or displacement on the surface S illustrated
in Figure 4.1(c). For reasons which will be apparent later,
surface tractions are assumed specified on S. The surface
S is divided into k boundary elements, and local axes
L,N are established for each element, as shown in Figure
4.1(d). At the centre of any element i, tractions tli,
tyi, tni are known. Suppose uniformly distributed transverse,
longitudinal and normal loads, of magnitudes q1j, qyj, qnj
are applied to element j. For the plane problem, the comp-
onents of stress and displacement induced at the centre
of i, expressed relative to the local axes for element
i, by the loads q1j, qnj, on element j can be expressed
by equations of the form
Qlij = alij qlj + anij qnj
• [alij anij] q1
J qnj
qnj
where the coefficients alij etc. are calculated
from the appropriate expressions in Appendix III, and
transforming from the local axes for element j to the
local axes for element i.
Stresses and displacements induced at the centre
of element i by all element strip loads are obtained by
superposition of the components induced by the individual
element loads; i.e.
- [Ul' Uni ] li li qlj
~q ali = E Lalij anij ] lj
j=1
= [Ai] [ q ] L gn7
where [A.] and [q] are row and column vectors, of order 2k.
Thus stresses and displacements induced at the centre of
element i by the strip loads on all elements may be
expressed by the equations
99
ali = [Ai] (q]
a = [Bi] [q] yi
ani = [Ci] [ q ]
Tnli [Fi] [ q ]
uli = Luh il [q]
uni Din J.] [g]
Noting that tli
(4.5)
Tnli and tni ani, we may write
tli
t ni
k
j=1
flij fnij qlj
clij cnij qnj
(4.6)
By considering the centre of each element in turn, k
equations similar to equation (4.6) may be established,
and these may be written
[T] [q]] = [t] (4.7)
For the antiplane problem, and proceeding in a similar
way to that above, stress and displacement components
induced at the centre of element i by the longitudinal
strip loads are given by
T lyi = [pi] [qy]
Tyni = [Ei] Lqy] (4.8)
100
u i Y = [Uyi] [qy ]
Since tyi = Tyni , and by considering the centre of
each element in turn, we may write
[TY ] [qy ] = [ty ] (4.9)
where the matrix [Ty] is formed from k row
vectors [Ei] .
Equations (4.8) and (4.9) are solved independently
to determine the sets of element loads [q] and [q ] which
produce the known tractions at the centre of each element.
The coefficient matrices [T] and (a. ] are both fully populated, but in each case the leading diagonal is
dominant. For most problems, the sets of equations may
be solved readily by Gauss-Siedel iteration (Fenner, 1974)
with no over-relaxation. Convergence of the solutions is
usually achieved in less than 20 cycles. Exceptions to
this will be discussed below.
Having solved for the element loads, all induced
boundary stresses and displacements may be calculated
using equations (4.5) and (4.8). Total stresses expressed
relative to element local axes are obtained from the
induced stresses by superposition of the field stresses.
The recalculation of the known boundary stresses implicit
in this procedure is used to determine if a satisfactory
solution has been found for the element loads.
Induced stresses and displacements Q , u xi xi
etc.at any internal point i in the medium are calculated
from equations similar to (4.5) and (4.8), except that
the coefficients in the vectors [A] , [Di] etc. are calculated by transforming stress and displacement
components induced by unit strip loads on element j
from the local axes for element j to the global X, Y, Z
axes. As for the boundary stresses, total stresses are
obtained from the induced stresses by superposition
of the field stresses.
4.4 Validation of Boundary Element Program
Validation of the solution procedure used in the
Boundary Element program requires demonstration that the
distributions of total stresses and induced displacements
calculated for a particular problem geometry with the
program agree with those calculated from analytical solutions
for the same problem geometry. Analytical solutions for
stresses and displacements around a circular hole subject
to plane and antiplane strain are given by Jaeger and Cook
(1976).
Figures 4.4 and 4.5 show the distribution of
stresses and displacements around a long opening of circular
cross section in a triaxial stress field. The pre-mining
stress field components, relative to the hole local axes, are
px = 0.397p, p = 0.429P. p = 0.924p, pxy = 0.116p,
pyz = 0.208p, pZx = -0.042p, where p = 0.04G. Thirty-
five boundary elements of equal length were disposed around
the complete circumference of the opening. This number was
chosen as it represents a realistic number of boundary
elements that could be used in practice for each opening
in modelling multiple openings in non-symmetric excavation
layouts. Figure 4.4 shows that the principal stress
magnitudes calculated with the boundary element program are
virtually identical with those calculated from the analytical
solution. Principal stress directions calculated by the
101
0
102
2.5
2.0 -
1 .0-
a/ p
Px
=.0.397p
PY =0.429p
Pz
=0.924p
PxY =0.116p
PYz = 0.208p
Pzx = -0.042p a
1 /p (on z=0.0)
ANALYTICAL SOLUTION
COMPUTED BY BOUNDARY ELEMENT METHOD
2/p (on z = 0.0)
2/p (on x=0.0)
0 O
a 1 /p (on x=0.0
0-0
1 .0
a p
DISTANCE FROM CENTRE OF HOLE (x/r)
(a)
1.0 2.0 3.0 4.0
-0.2 DISTANCE FROM CENTRE OF HOLE (z/r)
(b)
FIGURE 4.4 : STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A TRIAXIAL STRESS FIELD, FROM BOUNDARY ELEMENT ANALYSIS AND ANALYTICAL SOT,[TTTON
-3.0 (a)
1.0 uz/r x 10-3 (on z = 0.0)
1.0 2.0 3.0
DISTANCE FROM CENTRE OF HOLE (x/r)
ux/r x 10-3 (on z = 0.0)
u /r x 10-3 (on z=0.0) COMPUTED BY BOUNDARY 0 A ELEMENT METHOD
ANALYTICAL SOLUTION
4.0
2.0_
F- z W
0 a E 0 U
F-
w E w U Q J a
-10.0- 0
0 W U
0 W
u/r x 10-3 (on x=0.0)
1.0 2.0 DISTANCE FROM CENTRE OF HOLE ( /
u-/r x 10-3 (on x=0.0)
(b)
uz/r x 10-3 (on x=0.0)
3.0 4.0
103
FIGURE 4.5 : EXCAVATION-INDUCED DISPLACEMENTS AROUND A CIRCULAR HOLE IN A TRIAXIAL STRESS FIELD, FROM BOUNDARY ELEMENT ANALYSIS AND ANALYTICAL SOLUTION
two methods were also found to be practically identical.
From Figure 4.5 it is seen that the Boundary
Element analysis has slightly overestimated the displacement
components. Departure from the result calculated from
the analytical solution is highest immediately adjacent
to the boundary of the hole, but is never greater than about
2.5%. The agreement between the two sets of results can
be improved by increasing the number of boundary elements.
A number of other problems was analysed using
the boundary element program, to check specific sections of
the program code, such as that related to the antiplane
problem exclusively. Excellent agreement was obtained with
the results calculated from the analytical solutions, except
in cases where the problem involved excavation cross-
sections with low area/perimeter ratios. In particular,
narrow, parallel-sided slits produced difficulties. The
rate of convergence in the iterative solution for the
element loads was slow, and this was due to the occurrence
of numerically large terms in the coefficient matrices,
off the leading diagonal. Use of Gaussian elimination in
place of the iterative routine apparently did not solve the
problem (Watt, 1978).
It was concluded that the source of difficulty
was coupling between adjacent elements on opposite sides
of narrow excavations, and that a possible solution to
the problem was to take account of this interaction
explicitly. The possibility of designing singularities for
effective handling of particular problem geometries was
noted in Chapter 2. The inference was therefore that an
indirect formulation for narrow excavations should be
based on singularities which exploit the proximity of the
close-spaced parallel sides of these openings.
104
CHAPTER 5
CHAPTER 5: INDIRECT FORMULATION OF THE BOUNDARY ELEMENT METHOD FOR NARROW EXCAVATIONS AND COMPLETE PLANE STRAIN
5.1 Objectives and Scope of Work
Tabular and lenticular orebodies are common and
industriallyimportant sources of ore. Typical methods of
working these orebodies generate long openings in which the
span is many times the working height. It was noted in
Chapter 4 that the boundary element method described there
did not allow satisfactory analysis of the stress and dis-
placement distribution around narrow, parallel-sided
openings. The difficulty was considered to be associated
with the interaction between the fictitious loads on
immediately opposite elements defining the hangingwall and
footwall sides of the excavations being modelled. The
solution to this problem was of wider interest, because of
the possibility of using the results in modelling the
behaviour of geological features such as faults, which might
be considered as thin inclusions in the rock medium.
In the review of previous Boundary Element work
in Chapter 2. it was observed that the standard approach to
the analysis of extraction of tabular orebodies models the
mined area as an infinitely thin, parallel-sided slit. The
most recent description of the analysis of tabular orebody
extraction based on this approach is due to Crouch (1976).
The published results suggest that rather large numbers of
elements are required to model adequately the excavation
of a single slit, indicating that further examination of
the problem is justified.
In this work, mined openings in a tabular orebody
are modelled as narrow slits in the usual way. The interaction
between the adjacent parallel surfaces of an excavation is
taken into account by the development of singularities which
formally exploit the close coupling of these surfaces.
105
106
Distributions of these singularities have been sought which
allow the known features of the stress and displacement
distributions around narrow, parallel-sided openings to be
achieved with reasonable numbers of boundary elements. The
analysis allows the crrebodyto have any arbitrary orientation
relative to the pre-mining stress field.
Referring to Figure 5.1(a), ABCD represents the
proposed cross-section of a long, narrow opening to be
excavated in an elastic, isotropic medium. The local axes
for the excavation are X,Y,Z,with the Y axis parallel to
the long axis of the excavation, and the Z axis perpendicular
to the plane of excavation. The pre-mining stress components
relative to these axes are px' Py' pz' pxy' Pyz' pzx'
Suppose that the surfaces AB, CD are subject to equal but
opposite tractions, after excavation of the material within
ABCD. The excavation may be divided into a number of
segments, of which PQRS represents one (Figure 5.1(b)). The
upper and lower rectangular surfaces of this segment
constitute two boundary elements for the excavation. The
objective then is to achieve the required traction conditions
on PQ and RS. Due to the symmetry of the problem about the
plane through the mid-height of the orebodyparallel to the
X-Y plane, it is sufficient to achieve the required final
conditions on one surface, say CD, of the excavation.
Z
Z
(a)
(b)
FIGURE 5.1: DISCRETIZATION OF LONG, NARROW OPENING INTO SEGMENTS
107
The problem being considered is one of complete plane
strain, and thus may be resolved into plane and antiplane
problems. Consider the plane problem, shown in Figure
5.2(a). This may be regarded as the superposition of a
medium subject to homogeneous stressespx' pz' pzx' shown
in Figure 5.2(b), and the surface A'B'C'D' inscribed in a
continuum, shown in Figure 5.2(c), subject on C'D' to
tractions tx, tz and induced displacements ux, uz. The
tractions and displacements on A'B' are equal in magnitude
but opposite in sense to those on C'D'. If imposed surface
tractions on CD are txf, tzf, excavation-induced tractions on C'D' are
tx = txf pzx
tz = tzf - pz
For the antiplane problem, if the imposed surface
traction on CD is t f, the excavation - induced traction Y on C'D' is
ty = tyf pyz
If the excavation-induced displacement at a point
on C'D' is uy, then t and u represent tractions and dis-
placements which must be induced in a medium stress-free
at infinity to solve the antiplane component of the
complete plane strain problem.
Stating the problem in terms of induced stresses,
the plane problem requires stress components Tzx, a
z to
be induced on C'D', equal in magnitude to tx, tz, and
the antiplane problem requires stress component Tyz, equal in magnitude to ty, to be induced on C'D'.
t t z
4 u uz
(c)
108 Pz
X • zx Pzx
A B
fi
1
(a) • (b)
FIGURE 5.2: RESOLUTION OF REAL PROBLEM INTO UNIFORMLY STRESSED MEDIUM AND SUBSIDIARY PROBLEM
5.2 Development of Singularities for Modelling
Contiguous, Parallel Surfaces.
The suggestion that singularities for modelling
parallel, interacting surfaces could be developed by
coupling of line loads was made by Bray (1976c). The types
of singularities which result from the coupling process
resemble the "nuclei of strain" described by Love (1944).
Coupling of line load singularities to form dipole singular rities is also considered by Timoshenko and Goodier (1951).
Singularity for Control of Normal Stress Component az
Figure 5.3(a) shows a line load acting at the
point j( 0, 0) in an infinite medium, and directed in the
Z-direction,with an intensity Pz/unit length. The az
component of stress induced at the point i (xi, zi) is
given by
6z = Pz {(1-2v)-
+ 2 z3 } 4~ (1 _v)
r2 r
4
= Pz f1 (x,z)
where x = x. - x.
z = zi - z.
r2 = x 2 + z
Z Z
(a) (b)
Z Z (c) (d)
FIGURE 5.3: CONSTRUCTION OF COMPRESSIVE QUADRUPOLE SINGULARITY
z
X
P
109
X
i (x i ,z i )
X
Pz
Q n 1 Q V u r i x 1-v Qz
— r+-
1
In Figure 5.3(b), opposing Z-directed line loads
are shown acting at the points (0, - Sz7.) (0, Sz.)
The az stress component induced at i by these opposing
line loads is given by of of
az = Pz {f -zJ . i - (f + az 1 )}
1 azj i j azj
af = 2 Pz 1 z ~ az
= Q ~ af 1
az
If the quantity Q' is held constant as S z-->0, Q z constitutes a pair of coupled forces without moment, or, by
analogy with electrostatics, adipole. Thus, a dipole of
intensity QZ unit length induces a az stress component
given by
= -- az 4nQ(1 v) r2 { 1 - 2v + 4 (1+v) Zz 8 ~a } (5.1)
r r
Expressions for the five other stress and dis-
placement components induced by the dipole QZ can be
obtained in a similar way.
In coupling the opposing line loads to form a
concentrated singularity, the objective was to find a method
of controlling the value of az on surfaces which were
in virtual contact. It was found possible to do this with
distributions of dipole singularities, but there also resulted
extraneous disturbance to the stress distribution in the
medium. The requirement was to find a singularity which
could be used to control the az
stress component over
closely coupled surfaces without inducing lateral thrust near
the ends of these coupled surfaces. Referring to Figure
5.3(c), the limiting value of the thrust FX generated on
the surface AB as both (5 0 and Sx 0 was found to
be v q', where q' = 2p S . The conclusion from 1-v z z z z
this was that the concentrated singularity Qz should be
developed by the superposition of the vertically polarised
dipole QZ and a horizontally polarised dipole QX of
intensity v 0', 1-v z
110
111
as shown in Figure 5.3(d). The function of the horizontally
polarised dipole is to suppress the lateral thrust developed
by the vertically polarised dipole. The quadrupole shown
in Figure 5.3(d) is a centre of compression in the medium
without lateral thrust. The horizontally polarised dipole
Q' at pointj induces a az component of stress at point i given by
a z Qx 1 {1 + 2v 4Tr(1-v) r2
4v x2 8x2z2}
r2 r4
(5.2)
Superposition of the az stress components induced by
QZ and Q, yields the expression for the az stress component
induced by the quadrupole singularity:
a = Q (1-2v) 1 {1 + 4z2 - 8z41 Z Z 4Tr (1-v) 2 r2 r2 r4
(5.3)
Expressions for the other stress and displacement components
due to the normal singularity Qz are given in Appendix IV.
Shear Singularities
The normal singularity Qz is required to generate
a normal stress component az in the medium to realize the
induced tz component of traction on boundary elements.
Singularities are also required to induce tx and ty components
of traction on the boundary elements. Considering the tx
component initially, the requirement is to induce a Tzx
stress component in the plane representing the coupled
surfaces without inducing extraneous thrusts or similar
effects in the medium. Figure 5.4(a) illustrates a pair
of opposing line loads which constitute a shear dipole S,.,
if the distance 2Sz is sufficiently small. This clearly
has associated with it a rotational action, as well as a
shearing action. Figure 5.4(b) illustrates superimposed
shear dipoles S x and SZ which have opposing senses of
rotation. Together they constitute the shear quadrupole Sx,
which is a shear centre without moment. Using the expressions
for stress components induced by line loads in an infinite
elastic continuum, it is found that the Tzx stress component
induced by the shear quadrupole of intensity Sx is given by
112
1 26z.
Sx S = S' + S' -~~ " - —I x x z I X L A• L
i (x i ,z1 )
z z
(a)
(b)
FIGURE 5.4: CONSTRUCTION OF SHEAR QUADRUPOLE SINGULARITY FROM COUNTERACTING COUPLES
777.
Y-Polarised Line Dipole, Intensity Sy/Unit Length
z FIGURE 5.5: CONSTRUCTION OF ANTIPLANE DIPOLE FROM OPPOSING
LINE LOADS
X
r
S 27T r2 r2 1 1 { 1 2z2 } (5.5)
113
T = S 1 1 ZX x 2Tr (1-v) r 2
{1
8z2 8z4 }
r 2 r4
(5.4)
The similarity between equations (5.3) and (5.4) is
noteworthy. Expressions for the other components of stress
and displacement due to the shear quadrupole Sx are given
in Appendix IV. It can be seen from Figure 5.4(b) that
the shear quadrupole Sx is equivalent to two superimposed
normal dipoles: a compressive dipole polarised in a direction
inclined at -45° to the X axis, and a tensile dipole, equal
in magnitude to the compressive dipole, polarised in a
direction inclined at +45° to the X axis. The singularity
is therefore a centre of pure shear disturbance in the
medium.
The normal singularity Qz and the shear singularity
Sx are required for the solution of the plane problem.
The shear singularity for the antiplane problem is developed
from the coupled opposing line loads shown in Figure 5.5.
The Tyz component of stress at the point i (xi, zi) due to
the infinite line dipole of intensity Sy/unit length is
given by
T = yz
Expressions for the other stress and displacement
components due to Sy are given in Appendix IV.
5.3 Optimum Distribution of Singularities for Modelling Single Slits
In modelling the excavation of an opening with
boundary elements, the procedure is to determine the required
intensities and distributions of singularities which produce
the known values of surface traction or displacement on
the boundary elements. It is therefore of interest to
determine the ideal distribution of a singularity which
will simulate the excavation of a single narrow opening in a
unit stress field, e.g. pz = 1.0. Simulation of the
az
az
a x
=
=
=
x. 1 (x12-c2)
1
0
- 1.0
(5.6)
excavation of non-symmetric layouts of multiple openings
will then require the application of modified forms of
the ideal distribution for a single opening. These may
be achieved by various distributions of the singularity
over a suitable number of excavation segments.
Figure 5.6(a) shows a slit of width 2C excavated
in a medium in which the pre-mining stress is pz = 1.0.
The upper and lower surfaces of the slit are to be traction
free after excavation. As shown by Muskhelishvili (1953),
for a narrow slit in a uniaxial field, total stresses at
a point i (x1,0) in the plane of the slit are given by
114
Infinite discontinuities occur in the distribution
of the az
stress component near the slit ends, as indicated
in Figure 5.6(a).
A uniform distribution of normal quadrupole
intensity over the range of the slit, whose intensity is
adjusted such that the induced stress az = -1.0 at the
centre of the slit, represents the simplest possible method
of satisfying the conditions z (total) = 0.0 on the slit
surface. However, this gives rise to stresses in the medium
greater than required by the closed form solution. Increasing
the number of segments over which uniform distributions
of quadrupoles were applied to eleven,and satisfying the
condition a z (total) = 0.0 at the centre of each boundary
element, still produced a stress distribution in the abutment
significantly different from the closed form solution, as
shown in Figure 5.6(a). Other distributions of singularity
(J zIp z
3.5
3.0
2.0
1 --- --- -I I
2C ~ I ---
I
• XI
I I z L ______ -.J
tpz =1.0
tl 11 Uniformly Loaded Segments
G Closed Form Solution
o Parabol ic Distribution on Single Element
115
1.0 ~ __________________ ~ __________________ ~ __________________ ~
1 .0 1 .5 2.0
Distance from Centre of Sl it (x/C)
(a)
2.5
T
L,~_x. l( ~~. qo
L J J .. 2C .. ... 2C
(b) (c)
FIGURE 5.6: STRESS DISTRIBUTION IN THE PLANE OF A SLIT, AND PARABOLIC AND ELLIPTICAL DISTRIBUTIONS OF SINGULARITY INTENSITY
..
116
intensity were examined,involving linear variations of
intensity over segments. It was found that infinite
discontinuities in the distribution of cZ were associated
with either discontinuity in the quadrupole distribution,
or discontinuity in the first derivative of the distribution.
This suggested that if a single, loaded segment were to be
used to model a single slit, the prime requirement was to
find a distribution such that the distribution function and
its first derivative with respect to the segment intrinsic
co-ordinate were defined over the range of the segment.
The simplest distribution of singularity intensity
that satisfies the required symmetry about the Z axis, is
continuous and possesses a definite first derivative at
all points in the range of the distribution, is shown in
Figure 5.6(b). If x-3
is the intrinsic X co-ordinate for
the slit, with the centre of the slit as origin, the parabolic
distribution illustrated is described by the expression
x.2 qZ (x~) = qo ( 1 - IxiISC (5.7)
This distribution of quadrupole intensity produced
the abutment stress distribution shown in Figure 5.6(a), and
did not model the required stress discontinuity over the
excavation abutment. Bray (1976c) showed that the elliptical
distribution shown in Figure 5.6(c) satisfies the boundary
conditions over the range of the slit and simultaneously
satisfies the stress distribution in the abutment area.
This distribution is described by the expression
qZ (x.) = (10 (C2- )c.2)1/2 Ix.I<C (5.8)
For an opening in a uniaxial field, the required
quadrupole intensity is given by
qo (1-2vr
Determination of stresses and displacements induced by the
4C (1-v) 2
x. qz (x.) = qo {a + (1-a) (1
C2 3
2
(5.9)
117
elliptical distribution at a point i (xi, zi) not in the plane
of the slit involves the evaluation of integrals of the form
C
IF (C 2- x.2 ) 1/2 f (x,zi) dx.
wherex=xi - x~
and the function f(x, zi) describes the stress or displacement
component induced by the unit point singularity. A method
of evaluating this type of integral analytically was not
established. The alternatives were to use a numerical method,
such as Gaussian Quadrature, or to seek an approximation to
the elliptical distribution which gave rise to more tractable
expressions for integration. The latter coursewas adopted.
The elliptical distribution may be represented by the
superimposed parabolic and uniform distributions shown in
Figure 5.7(c), and is described by the equation
The parameter"a" may be chosen to achieve satis-
factory correspondence between the stress distributions
induced by the different quadrupole distributions. Integrals of
the form fq(x) f(x,zi) dxj may readily be evaluated in
terms of simple functions of x and z i. It has been found
that a value of a = 0.188 allows equation (5.9) to provide
a satisfactory approximation to the elliptical distribution,
which is therefore described by
x.2 qz (x~ ) = qo { 0.188 + 0.812 (1 - ) } I x . k 4C
C
The suitability of the quasi-elliptical distribution
for modelling a single slit in a uniaxial field is indicated
in Figure 5.7. Figure 5.7(a) shows cx and az stress components
in the plane of the slit calculated from the closed form
solution and using the quasi-elliptical distribution. In
this case, the boundary condition az (total) = 0 was
satisfied at the points x./ C = - 0.5 . The stresses
2.2 118
2.0
o Quasi-Elliptical Distribution
X Ell i pt i ca 1 0 i s t r i bu t ion
1 .0
O.O¥-~-¥~~m-~-4-4~~--------------____ ~ __________________ ~
2.0 3.0 Distance from Centre of Sl it (x/C)
-1 .4 (a)
p /G=O.Ol z
0.81"1 __ ~ [::J C.F.S.
0 Q/E Distn. 0.6
1 qo _1 J ~ =0. 188q T 0 0.2 0.4 0.6 0.8 1 .0 .. 2C -
x· (c) J (b)
FIGURE 5.7: STRESS AND DISPLACEMENT DISTRIBUTION IN THE PLANE OF A SINGLE SLIT IN A UNIAXIAL COMPRESSIVE FIELD (a), (b), AND QUASI-ELLIPTICAL DISTRIBUTION OF INTENSITY (c)
119
outside the range of the slit are virtually identical, for
the different methods of estimation, both close to and
remote from the slit edge. It is noted that the boundary
condition az = 0 is satisfied at only the two selected
points over the range of the slit. The required boundary
stress ax = - 1.0 is also achieved at these points only.
The maximum intensity qo of the quasi-elliptical quadrupole
load was 4.94 for the case C = 1, compared with the value
for the ideal distribution of 4.50.
Sneddon and Lowengrub (1969) show that the uz
component of displacement induced by the excavation of a
narrow slit in a uniaxial stress field (pz = 1.0)is given
by x.2 1
u = + (1-v) (1 -Z ) z - G C2
where the positive and negative signs refer to
the upper and lower surfaces of the slit respectively.
In Figure 5.7(b), displacements calculated using the quasi-
elliptical distribution are compared with those calculated
from the closed form solution. The agreement is satisfactory.
It is noted that the closed form solution requires that z
vary elliptically over the excavation, i.e., the displacement
distribution matches the quadrupole distribution. Corres-
pondence between the distributions of displacement and
quadrupole intensity is found also for the quasi-elliptical
distribution. This therefore suggests that the quasi-
elliptical distribution is a valid approximation in all
respects to the ideal distribution.
The inference from these results is that in
modelling an unsymmetric layout of narrow openings in a
uniaxial field, the distorted elliptical distributions of
quadrupole intensity which will be required to satisfy
the boundary condition az = 0 around each opening may
be achieved by
120
(i) dividing the excavation into a number of segments
or elements;
(ii) for the segments nearest the abutment of each
opening, imposing quadrupole distributions which
have the form of half the quasi-elliptical
distribution shown in Figure 5.7(c);
(iii) applying uniformly distributed quadrupole intensities
to the interior segments;
(iv) adjusting the magnitudes of the quadrupole loads
on each segment to satisfy the boundary conditions
at the centres of the elements.
As was observed above, infinite values of az are
associated with discontinuities in qz (x j) or its first
derivative. The proposed procedure for modelling the ex-
cavation of narrow openings therefore results in infinite
discontinuities in stress between adjacent elements, and
the realization of imposed boundary conditions at the centres
of elements only. Thus it is necessary to demonstrate
that adoption of the proposed discretization procedure does
in fact result in satisfactory determination of stress and
displacement distributions around narrow openings.
Sneddon and Lowengrub (1969) give the following
expressions for stresses in the plane of a single slit,width
2C, centre (0,0), in the stress fields pzX = 1.0 and pyz = 1.0:
x . __ 1
Tzx (x 2-C2)
IXiI C
1 (5.10)
T z Y
x. 1
(xi2-C 2 ) 1
121
Thus, the distribution of TzX and Tyz in the plane
of the slit is exactly the same as that of az for a
uniaxial compressive field. Comparison of equations (5.3),
(5.4), (5.5) for the case where z } 0 shows that, for
the point singularities, 6z . TzX , Tyz all vary according
to 1 . The inference therefore is that as for the normal x2
singularity Qz, elliptical distributions of the shear
singularities Sx and Sy are required over the excavation to
match the known stress distributions outside the range of
the slit, given by equations (5.10). As for the normal
quadrupole distribution, the ideal elliptical distributions
in each case may be approximated closely by superimposed
constant and parabolic distributions.
5.4 Boundary Element Solution Procedure
The problem is to determine the distribution of
total stresses and excavation induced displacements around
a set of long,narrow, parallel-sided openings in a tabular
orebody. An opening representative of the set is shown in
Figure 5.1(a). Suppose the excavations are divided into
a total of k segments. The requirement is to determine the
quadrupoleloads which, when applied to each segment,
induce known stresses on the elements defining the face of
each segment.
The expressions for stress and displacement
components due to the normal and shear singularities, given
in Appendix IV, can be integrated analytically to give
expressions for these quantities due to uniform distributions
of normal and shear singularities over excavation internal
segments, and due to quasi semi-elliptical distributions
over edge segments. For example, for the segment shown in
Figure 5.8(a), with a uniformly distributed normal quadrupole
intensity, the az stress component induced at the point i
by the quadrupole loading on the segment j is given by
122
1
a . r x+ 2z.2 x 1 L z1 _ q z 47(1-v)2 r2 1 r 2
where x = x - x. i J
r2 = (X. - X.)2 . + z 2 1
(5.11)
The az component of stress induced by the
distribution of quadrupole loading shown in Figure 5.8(b)
is given by the expression
6zi • = q (1-2y ) [ _
0.188 ( x + 2z.2 x ) + 0.812 1 k ( x + 2z2 x ) z 4 Tr (1-v) 2 r2 1 r4 k3 1
r2 1 r4
2z 2 2z.4
+ k (ln r- + 1 )+ 3z.2 _
2 r2 r4 1 r2 1
where k1 = x
j2 - 2xj1x. - xi2 + 2x~ 2xi
k 2 = 2 (x ) i - x3 2
)2 k 3
x]2 1 r4
XJ*
Z (a) Z (b)
FIGURE 5.8: GEOMETRIC PARAMETERS DETERMINING INFLUENCE COEFFICIENTS FOR UNIFORMLY LOADED ELEMENT (a) AND EDGE ELEMENT (b)
123
In general, induced stress and displacement
components at a point i due to the particular normal
quadrupole load on segment j can be written as
cxij = agzij qzj
ayij __ b qz. qZJ
6Z1j = cgzlj qzj
Tzxij = fgz1j qz3
zj uxi] =
Uxi qzj
zj uzi
j __ Uzi qzj
where the coefficients agzij etc. are evaluated
from integrals of the appropriate expressions in Appendix IV.
Similarly, stress components induced at i by a shear
quadrupole load sx on segment j are given by expressions of
the form
a = asx.. s . etc. x1J 1J XJ
The procedure followed subsequently is exactly the
same as that described in Chapter 4 for strip-loaded elements.
Stresses and displacements induced at point i by all singularity distributions on all segments are obtained by the
superposition of the
distributions on the
k
axi = E j=1
k E
j=1
stress components
various segments;
(asxijsxj + agz..gz.)
[ asxlj agzij ] [sxj]
qzj
induced by the
for example
= [q]
where [Ai] is a row vector, of order 2k , and [q] is the column vector of segment quadrupole loads, of order 2k.
For the plane problem, stress and displacement com-
ponents at point i induced by the complete system of segment
quadrupole loads are given by
= [Ai] [q]
= [Bi] [q]
= [Ci] [q]
= [Fi] [q]
= [UxiJ [q]
(5.12)
uxi
uZ1 = [Uz1~ [q~
For the antiplane problem, stress and displacement
components induced at point i by shear dipole distributions
of intensity sy on the various segments are given by
Txyl = [D1] [S y]
T yzi
u yi
= [Ei] [sy]
= [Uyi] [sy]
(5.13)
Considering point i as the centre of a boundary
element, the requirement is that the stress components
czi induced by the segment loads be equal to Tzxi' Tyzi' traction components txi, t,1, tzi induced at these points
by excavation. The system of equations to be solved is
simplified if one takes account of the limiting forms
taken by the expressions for stresses induced by the various
distributions of singularities as the point i approaches
the plane of the distributed singularities. For a uniform
124
distribution of intensity, the compressive singularity q z
induces az
and TzX stress components at point i(xi,zi)
given by
1 _ (1-2v) _ X + 2z. 2 x qZ 4Tr (1-v) 2
r
L r 2 1 r~ 2
(1-2v) r 2zi3 z1 ~ 1 _
Tzxi qz 41.1.(1-v)2 IL — r2 2
For the limiting situation, where zi 0, the
expressions become
1 (1-2v) r 1
6zi qz 4Tr (1-v) 2 L x 2
T = 0 zx
Similarly, az
and TzX stress components induced
by a uniform shear quadrupole distribution of intensity sx
are given by
1
r
Z . 2z.3
6Zi SX 2Tr (1-V) L r2 + J.
1 _ 1 r 2z2. x - x 1
T zX1 = sx 2 Tr (1-v) L 1r r J2
In the limit, as zi 0, these expressions become
aZ. = 0
1 T zxi - - sx 2Tr (1-v) Lx ] 2
Thus, in the plane of the slit, the az
and Tzx
stress components are controlled independently by the
segment loads qz and SX
respectively, i.e. there is
complete decoupling between the normal and shear tractions
induced by the compressive and shear singularities.
125
cZ1
126
Similar considerations apply to semi quasi-
elliptical distributions. Therefore, to determine the
magnitudes of the unknown segment loads which satisfy
the known boundary conditions for the plane and antiplane
problems, it is sufficient to solve the equations
[TX] [sx] = [tx ] (5.14)
[Ty] [sr] = Cty] (5.15)
[Ta] [qz] = [tz] (5.16)
where [tx ], [ty ] , [tZ ] are the column vectors, of order k, of known induced surface tractions, and [Tx],
[T], [TZ] are square matrices, of order k, of influence
coefficients for unit singularity intensities on the
various segments.
The [Tx], [Ty] and [Ti] matrices all show pronounced dominance of the leading diagonals, and therefore equations
(5.14), (5.15), (5.16) can be solved readily by Gauss-
Siedel iteration. Typically, less than 10 iterative
cycles are required to achieve satisfactory convergence.
Having solved for the set of element loads which
induce the known tractions on excavation surfaces, equations
(5.12) and (5.13) are used to calculate induced boundary
stresses and displacements, and induced stresses and dis-
placements at selected internal points in the medium.
5.5 Validation of Boundary Element Program
In validating a Boundary Element Method of analysis,
the objective is to demonstrate that satisfactory agreement
can be obtained between the numerical solution to a particular
problem, and an independent solution, such as a closed form
solution, to the same problem. As a general rule, the
requirement is that this agreement be achieved using a
127
reasonable number of elements. The necessity to use high
concentrations of elements to obtain satisfactory corres-
pondence between a numerical solution and the closed form
solution may be indicative of a deficiency in the numerical
method of analysis. In validating the Boundary Element
Method developed in this work, there is the further require-
ment to demonstrate that an adequate solution is obtained
by satisfying the boundary conditions at the centre of each
element, despite the infinite discontinuities in stress
which occur between adjacent elements.
A number of simple excavation geometries have been
examined with these requirements in mind. Figures 5.9(a),
(b) show stresses in the plane of and perpendicular to the
plane of a narrow opening excavated in a medium in which
the stress field is pz = 1.0. Three segments of equal
length were used to model the excavation. Close agreement
is demonstrated-between the boundary element solution and the
closed form solution of the same problem due to Sneddon (1969),
except for the small discrepancy in the distribution of
az shown in the immediate footwall area. A comparison
between the boundary element solution for displacements
over the excavation, and the closed form solution, is given
in Figure 5.9(c)). Again, satisfactory agreement is observed,
bearing in mind the number of excavation segments used.
Similar close agreement was noted for stresses and displace-
ments around slits excavated in media in which the pre-
mining stress fields were pzX = 1.0, and pyz = 1.0. In
both cases three segments of equal length were used to model
the excavation. The suggestion is therefore that the gross
variations in stress over an excavation associated with
discontinuities between segment singularity distributions
have a very localized effect, and may be ignored.
Figure 5.10 shows the variation in stress components
along a ray AB drawn from near the end of a slit excavated
in a medium in which pzX = 1.0. Close agreement is demon-strated between the boundary element solution and the closed
form solution due to Sneddon. Three segments were used to
(a) 1.0-
128
X 2C
Analytical solution Z o This work
C3- ZiPZ(on z=0.0)
0 0 cry/PZ(on z=•0.0)
,ky crx /Pz (on z = 0.0 )
1.0 2.0 3.0 Dist . from centre of slit ( X/C
4'V
3.0
6/pz
2.0
1.0
Analytical solution Cl This work
-u /C -2
x10
1.0
0.8
0.6
0.4
0.2
i1.0 1.O Pz G- 1000
0.2 0.4 0.6 0.8 1.0 Dist. from centre of slit (X/C) (c)
FIGURE 5.9: STRESS AND DISPLACEMENT DISTRIBUTIONS AROUND A SLIT IN A UNIAXIAL COMPRESSIVE FIELD, MODELLED WITH THREE SEGMENTS
129
4.0
3-0
o/p 20 zx
1.0
0.05C
1.0-- 2c —41A
B —Analytical solution o This work
1 Z
Tzxipzx
0
6x, pzx -1.0
Crzipzx Dist . along A B from A ( ) 1.0 2.0
FIGURE 5.10: STRESS DISTRIBUTION ALONG RAY AB FOR A • SLIT IN A UNIT SHEAR FIELD
130
represent the excavation. The results indicate that the
method adequately predicts the state of stress at points
where no particular advantage could be associated with the
symmetrical disposition of loaded segments with respect to
the point. It is seen that the high stress gradient near the
end of the opening is reproduced adequately in the boundary
element analysis.
Table 5.1 compares the results of boundary element
and closed form solutions for stresses along the ray AB
illustrated in Figure 5.10, for a narrow opening excavated
in a triaxial stress field. The field stress components were
px = 0.347, py = 0.74, pz = 0.913, pxy = 0.25,
pyz = -0.20, pzx = -0.15. The slit was modelled with three excavation segments. The agreement between the two sets
of results is satisfactory.
Finally, Sneddon and Lowengrub (1969) give the
solution for stresses and displacements around an infinite
row of pressurized collinear cracks in an unstressed medium,
from which can be obtained the solution for stresses and
displacements around an infinite row of collinear cracks
in a stressed medium. Eleven equal length cracks were
used in the boundary element analysis to represent the
infinite row, with the pillar width between each pair of
slits equal to half the slit width. Each slit was modelled
with three segments. The three central slits in the row
are shown in Figure 5.11(a). The displacements over the
central slit, shown in Figure 5.11(a), indicate good agree-
ment between the results obtained from the different solu-
tions, even quite close to the end of the slit. Similarly,
the stresses in the pillar area determined with the different
methods of analysis show acceptable agreement, as illustrated
in Figure 5.11(b).
In all the trial problems considered, the agreement
between the boundary element solution and the closed form
solution could be improved by increasing the number of
segments used to model each opening.
Table 5.1 Comparison of stresses calculated using boundary element method and closed form solution, around slit in a triaxial stress field. (Reference line AB is illustrated in Figure 5.10)
Field stresses are px = 0.347, py = 0.740, pz = 0.913, pxY = 0.250, pyz = -0.200,
p = 0.150. zx
ay
B.E.1
x y
B.E.1 C.F.S.2' B.E. C.F.S. B.E.
az
C.F.S.
T xy
B.E. C.F.S.
T yz
B.E. C.F.S.
T zx
B.E. C.F.S. ,
0.0 2.452 2.428 1.793 1.781 3.021 2.994 0.251 0.250 -0.662 -0.656 0.499 0.492
0.2 0.408 0.397 0.982 0.980 1.819 1.823 0.333 0.335 -0.320 -0.320 0.329 0.322
0.4 0.263 0.258 0.857 0.854 1.465 1.465 0.309 0.310 -0.261 -0.260 0.210 0.205
0.6 0.234 0.229 0.808 0.806 1.296 1.293 0.295 0.295 -0.237 -0.236 0.163 0.158
0.8 0.234 0.232 0.782 0.781 1.197 1.194 0.286 0.286 -0.224 -0.224 0.140 0.136
1.0 0.241 0.241 0.768 0.767 1.132 1.128 0.279 0.279 -0.217 -0.217 0.128 0.125
1.4 0.261 0.262 0.753 0.753 1.053 1.050 0.270 0.270 -0.209 -0.209 0.119 0.118
1.8 0.278 0.279 0.747 0.746 1.009 1.006 0.265 0.264 -0.205 -0.205 0.118 0.118
2.2 0.292 0.293 0.743 0.743 0.982 0.979 0.261 0.261 -0.203 -0.203 0.120 0.120
2.6 0.302 0.303 0.742 0.741 0.965 0.961 0.259 0.259 -0.202 -0.202 0.123 0.123
3.0 0.310 0.311 0.741 0.740 0.953 0.945 0.257 0.257 -0.201 -0.201 0.126 0.126
1. Boundary Element Method
2. Closed Form Solution
t•- 2C -•i X CP
Analytical solution o Boundary element solution
- -Analytical solution (single slit) 0.6
-u /C x10-2
0.2
132
0.2 0.4 0.6 0.8 1.0 Dist. from centre of slit (X/C)
4.0
3.0
az/Pz
2.0
.1.1 1.2 1'3 1.4 1.5
Dist. from centre of slit (X /C ) FIGURE 5.11: DISPLACEMENT DISTRIBUTION OVER EXCAVATED AREA,
AND STRESS DISTRIBUTION IN PILLAR AREA, FOR ROW 'OF SLITS IN UNIAXIAL FIELD
1.0
133
5.6 Assessment of Slit Modelling Procedure
The results reported above show that stresses and
displacements around long, narrow, parallel-sided openings in
a triaxial stress field may be determined accurately by an
indirect formulation of the boundary element method. The
singularities used in the solution procedure have been designed
to take account of the relative proximity of the close,
parallel boundaries of narrow excavations. The distributions of
singularity intensity imposed over segments defining the
ends of an excavation allow effective handling of the high
stress gradients which occur adjacent to slits and narrow
excavations.
The approach adopted in this work for analysis of
the mining of narrow orebodies is directly comparable with
that described by Crouch (1976). In the latter case, the
solution to narrow excavation problems is obtained by using
uniform (strip) dislocations as the singularities applied
over excavation segments. Referring to Figure 5.8(a), if
a uniform normal dislocation(closure) of magnitude Dz is
applied over a segment j with range xj1, xj2 on the X axis,
Crouch shows that the az stress component induced at
the point i (xi, zi) is given by
1 2G r X + 2Z.2 X az __ _ Dz
47r (1-v) r2 1 r" 2
where x = x. - x. 1 ~
Apart from the difference in pre-multiplier, this
expression is equivalent to that for the az stress component
induced by a strip normal quadrupole, as given in equation
(5.11). Similar considerations apply to the other components
of stress and displacement induced by the different types
of strip quadrupoles and strip dislocations used in the
solution of the plane problem.
The difference between the approach adopted by
Crouch and that used here is that in the former case a
mining layout in a tabular orebody is modelled as a series of openings separating pillars which are subject
to one-dimensional compression and shear. In the current
work, the objective has been to determine the triaxial
states of stress which develop in mine pillars, to be
used, in conjunction with the failure criterion for the
rock mass, for assessment of the possibility of rock
mass failure in the body of pillars.
134
CHAPTER 6
CHAPTER 6: THREE-DIMENSIONAL ELASTIC ANALYSIS OF TABULAR OREBODY EXTRACTION
6.1 Problem Description for Three-Dimensional Analysis
The Boundary Element Method described in Chapter
135
5 is designed to handle mining layouts in a tabular or
lenticular orebody which generate long rooms and rib
pillars. A geometric limitation on the applicability of
the analysis is the requirement that the lengths of
openings be significantly greater than the spans. The
work by Hocking (1976) suggests that a stope length/span
ratio greater than about 2 may be analysed satisfactorily
using plane strain methods, and there are many mining
situations where this condition will apply. However,
mining layouts which generate scattered pillar remnants,
and typical room-and-pillar mining in a tabular orebody,
are not amenable to analysis on the assumption of complete
plane strain. Figure 6.1 represents the basic element of
such a mine structure, with a pillar supporting spans of
adjacent country rock. This unit may occur regularly or
irregularly throughout the mining area.
c c Section AA Section BB
r-----------, I I I I I I I I
A ~------~~----+_--~~------_r~ Plan
B
FIGURE 6.1: ISOLATED PILLAR GENERATED DURING ROOM-ANDPILLAR MINING
136
In extending the complete plane strain Boundary
Element Method to allow determination of total stresses
and mining induced displacements in this type of mine
structure, it is simpler initially to consider the
single excavation shown in Figure 6.2. Referring to
Figure 6.2(a), ABCD and EFGH represent the upper and
lower boundary surfaces of a mined opening in a tabular
orebody. The height of the excavation is small compared
with either stope span, and the assumption is made that
the opening may be modelled as an infinitely thin slot.
The excavation may be divided into a number of rectangular
segments, shown in plan in Figure 6.2(b). The surfaces
PQRS and TUVW represent boundary elements of a particular
segment, as illustrated in Figure 6.2(c). The procedure
then followed is exactly analogous to that described for
complete plane strain. Orebody local axes X,Y,Z are
established with the Z axis perpendicular to the plane
of the orebody. Referred to these axes, the pre-mining
stress field is described by components p , p , pz, x y z The infinite medium is assumed to be pxy' pyz' pzx'
isotropic and elastic, and the boundary surfaces of the
excavation are assumed to be traction-free after excava-
tion. Due to the symmetry of the excavation geometry,
the problem may be solved by considering final conditions
on one surface of the excavation, e.g. the surface EFGH.
Finally, the problem may be regarded as the superposition
of two separate load systems: an infinite medium subject
to the known field stresses, and a surface in a continuum
subject to tractions tx, ty, tz, where
tx pzx
ty pyz
tz = -pz
The objective then is to find distributions of singularities
which, when applied in an infinite continuum, induce
tractions tx, ty, tz on a surface representing the boundary
of an excavation.
S P
Q R
B C
D lb)
pz
137
(a)
FIGURE 6.2: SINGLE NARROW OPENING IN A MEDIUM SUBJECT TO TRIAXIAL LOADING (a), DISCRETIZATION INTO SEGMENTS (b), AND A TYPICAL EXCAVATION SEGMENT (c)
138
6.2 Development of Compressive and Shear Singularities
The development of singularities for controlling
the surface tractions in the three-dimensional analysis of
stress and displacement distribution around a thin slot is
simplified significantly by taking account of the conclusions
from the two-dimensional analysis. Consider initially the
singularity required to induce the az stress component in
the infinite medium. The dipole singularity Q'z illustrated in Figure 6.3(a) is formed by a pair of collinear, opposing
point forces of magnitude Pz acting near the point j
(x , y, z), with separation 26z . Expressions for
stresses and displacements induced at the point i (xi,yi,zi)
by a point force Pz operating at the point j (x , y., z) are
given in Appendix I. For example, the az stress component
is given by
a P
- z z 8Tr (1-v)
{(1-2v) z R3
+ 3z3 } R5
= Pz f 1 (x,y,z) say
where x = x. - x. etc
R2 =x2 +y2 +z2
Thus the coupled, opposing point forces constituting
the dipole QZ induce a az stress component at i given by
of az = 2Pzdz. azl
af Q' Z az
Q z
{(1-2v) 1 + 6(1+v) z5 15z 4 }
887(1-v) R
3
R5 R7
Expressions for other stress and displacement
components due to the point dipole are derived from the
respective expressions for these quantities induced by a point
load.
z a
139
In the two-dimensional analysis involving line
dipoles described in Chapter 5, it was found that lateral
thrusts associated with the line dipole polarised in the
vertical direction could be suppressed by superimposing
a horizontally polarised dipole, of intensity (TIT) times
the vertical dipole intensity. Therefore it is inferred
that for the three-dimensional case, shown in Figure 6.3(b),
the lateral thrusts associated with the dipole QZ may be
suppressed by dipoles QX and Qy each with intensities
(lv) that of Q. An X-directed point force, of magnitude
Px, applied at the co-ordinate origin induces a az stress
component given by
• 87(1-v) { - (1 -2y) + 3xz2 }
Therefore, opposing point forces applied near the co-
ordinate origin and separated by the distance 26x, induce
a az stress component given by
a z
• 8
7(1-v) ax { (1-2v47 + 31,c5 2 }
• 8-a (0-v) {- (1-2v)R3 + R5 2 3-
(-(1-2v)
+ R5z2
_— )}
Similarly, the horizontally polarised dipole Q,
induces a az stress component given by
2 a =
Qy {- (1-2v) 1 + 3z2 -? (- (1- 2v) + 5z )}
z 87r(1-v) R3 R5 R5 R2
Using dipoles QX , Qy each of strengths (TIT) ) Q , superposition of the dipoles Q X, Q,, QZ produces the
hexapole of intensity Qz, and this induces a az stress
component given by
(1-2v) 1 (1 + 6z2 15z4
az = QZ 8Tr(1-v)2 R3 R2 R4
Q' z P
X
Q' z E 2Sz. Pz Q 1
I(xi,yi,z.) Y
X
Q = Q' + Q' + Q' z z x y
140
The hexapole singularity Qz is a centre of compress-
ion in the medium without lateral thrust. Expressions
for the other components of stress and displacement due
to a hexapole singularity are given in Appendix V.
z z
(a) (b)
FIGURE 6.3: CONSTRUCTION OF A COMPRESSIVE DIPOLE (a), AND A COMPRESSIVE HEXAPOLE FROM THREE DIPOLE SINGULARITIES (b)
Singularities for controlling the tractions tx and
ty on the surface representing the excavation boundary
have been developed from opposing pairs of point forces
with parallel, non-coincident lines of action. Figure
6.4(a) illustrates a pair of point forces of equal
magnitude and opposing lines of action operating in the
X-Z plane, which together constitute the shear dipole SX,
if the distance 26z. is infinitesimally small. The
rotational action associated with this dipole is suppressed
by the superposition of the dipole SZ of ecruai magnitude
but opposite sense of rotation, as shown in Figure 6.4(b).
The resultant singularity is a shear centre without
moment. An X-directed point load Px applied at the co-
ordinate origin induces TYz and TzX stress components
given by
Px 3xyz Tyz 8n(1-v) R
T zx 6n(1-v) {(1-2v)R3 + 3 ?}
The Tyz stress component induced by the SX dipole
is given by
213,6Z.dz~ T =
a { 3xvz } yz 8n (1—v) 3z —R
_ Sx { 31Y 1131Y_? ..f. } 8(1-v) R5 R7
and the TZx stress component is given by
SX a z 3x2z
Tzx - 8n (1-v) az { (1-2v) R3 + Rue}
= 811(lxv) {(1 2v) R3 + 3x2 2 2z 2 } 3 (1-2v) R 15 5 R7
The Tyz and TZx stress components induced by a
Z-directed point load Pz are given by
Pz } Tyz 811(1-v) { (1-2v) - + R
Pz 3xz 2 Tzx 811(1—v) f(1-2V
+ --7— }
Therefore the Tyz and TZx stress components induced
by the SZ dipole are given by
= SIz a 3vz 2
Tyz 811(1-v) ax {(1-2v)R3 + R5 }
r = 811(1-v) {-3(1 2v)3 159z 2
}
Sz a x 3xz 2 Tzx - 8Tr(1-v) āx {(1-2v) + -- }
Sz {(1-2v)13 + 3z2 - 3(1-2v)xs
15x 2 } 811(1-v) R R R R
141
P x
pX SX
2Sz. 1
z z (a) (b)
S' = 2Sz.P x j x
I(xi,y.,z.)
S = S' + S' x x z
Superposition of the SX and SZ dipoles produces
the Sx shear quadrupole, for which the induced stresses
Tyz and Tzx are given by
SX {(3v 15xyz 2
Tyz 47(1-v) R5 R
Sx 1 3vyl
15z2z2 TzX =
47(1-v) {(1+v) - R R.
FIGURE 6.4: CONSTRUCTION OF A SHEAR DIPOLE (a), AND A
SHEAR QUADRUPOLE FROM COUNTERACTING SHEAR
DIPOLES (b)
Expressions for other stress and displacement
components due to the shear singularity Sx are obtained
in the manner described above, and are given in Appendix
V. Expressions for stresses and displacements due to the
shear singularity Sy, polarised in the Y-Z plane, are
obtained from those due to the singularity Sx by cyclic
permutation. The Tyz and Tzx stress components are
given by
Sy 1 3v x 2 15y2 z2
Tyz = 4 Tr(1-v)
{(1+v) 1 - R'
142
143
T = Sy { 3vxy 15xyz 2 zx 47 (1-v) R5 R }
It is useful to note the forms taken by the
expressions for the az, Tyz, TZx stress components
induced by the compressive and shear singularities as
z tends to zero. The expressions for the limiting case
are as follows.
Compressive singularity Qz:
(1-2v) 1 az = Q z 8Tr (1-W r
T = 0 yz
T = 0 zx
Shear singularity Sx :
a z = 0
Sx 3vxy Tyz = 4Tr (1-v) r5
T = Sx { (l+v) r -3-YYL zx 4Tr (1-v )
Shear singularity Sy :
a z
= 0
v )Sy 1 3vx2
T __
yz 4Tr (1-v) { (1+ ~- - }
_ S 3vxy v Tzx = 4Tr (1-v ) r
where 2 2 2
r = x + y
The inference from these expressions is that,
when compressive and shear singularities are distributed
over excavation segments, there is coupling between
the stresses induced in the plane of the excavation by
the distributed shear singularities, but the compressive
and shear singularities are completely decoupled.
6.3 Imposed Distributions of Singularity Intensity on Excavation Segments
In modelling the excavation of a thin slot, the
requirement is to determine the intensities and distribu-
tions of singularities producing known tractions tx, ty,
tZ on surfaces in a continuum which match those induced
by excavation. If account is taken of the distribution
of intensity of a particular singularity required to
satisfy conditions around a single excavation of simple
shape, the overall distribution needed to simulate
mining of more complex shapes of excavations may be more
readily achieved by imposing modified forms of the ideal
distributions of singularities over particular excavation
segments. For example, the two-dimensional analysis
of stress distribution around a narrow opening in a
uniaxial stress field showed that an elliptical distribu-
tion of compressive singularity is required to generate
a traction-free final excavation surface. In three
dimensions, the excavation shape exhibiting the same
degree of symmetry as the narrow slit in two dimensions
is axially symmetric; i.e., circular in plan. It is
inferred that, in a uniaxial stress field normal to the
plane of excavation, a spheroidal distribution of
compressive singularities is required to generate traction-
free excavation surfaces. Similarly, considering the
square excavation shown in Figure 6.5, it may be inferred
that since the diagonals AC, BD and the bisectors EF,
GH of pairs of opposite sides are axes of symmetry,
compressive singularity intensity must vary elliptically
in these directions.
144
In the Boundary Element analysis, the excavation
is divided into a number of rectangular excavation
segments, and these are of three types: internal
segments, segments with one edge coinciding with an
excavation boundary, and segments with two edges
coinciding with excavation boundaries. The intention
is to impose different distributions of singularities
over the different types of elements so that the inferred
ideal variation of singularity intensity over the area
of a symmetric excavation can be achieved most readily.
B
G
C
145
E F
A
H
D
FIGURE 6.5: AXES OF SYMMETRY FOR A SQUARE EXCAVATION, ALONG WHICH ELLIPTICAL VARIATION OF SINGULARITY INTENSITY IS INFERRED
Consider the edge elements initially. The two-
dimensional analysis showed that the ideal elliptical dis-
tribution of compressive and shear singularities could be
approximated by a quasi-elliptical distribution consisting of
superimposed uniform and parabolic distributions. This
suggests that for an excavation segment adjacent to an abutment
or pillar, the intensity of a singularity will vary over the
segment according to the relationship
t. = to {O.188 + 0.812
(2x. xj x~ 2)} 0<x. xJ2 x.
J2
=tof 1 (x.)
146
where xj is the x co-ordinate of point j, relative to
element local axes X., Y.
tj is singularity intensity at point j
to is singularity intensity at point xj2
The distribution is illustrated in Figure 6.6(b). The stresses
and displacements induced by this singularity distribution
are obtained from the expressions for these quantities due to
a point singularity. Suppose a point singularity of intensity
T. induces a stress component a given by
a = T. f2 (x,y,z) (6.1)
The singularity distribution shown in Figure 6.6(b) induces
a stress component c at a point i with local co-ordinates
(xi, yi, zi) relative to the local co-ordinates of the
segment given by
a = to j2 ]2f (x.) f2 (x,y,z) dxjdyj
x jl 1'jl
(6.2)
where x = xi -x.
Y yi yj
x = z. -z.
The integration may be performed analytically, since the
functions obtained from the product of fl and f2 in equation
(6.2) are all tractable.
The distribution of intensity of a singularity most
appropriate for a corner segment may be inferred by noting
that the distribution should be compatible with the quasi-
elliptical distributions on the adjacent edge segments. An
elliptic paraboloidal distribution superimposed on a uniform
distribution provides this compatibility, and also leads to
quasi-elliptical variation of intensity along the diagonal of
the segment. The distribution is illustrated in Figure 6.6(c)
and is represented by the expression
Z (a)
147
...---ZJ'
z (b)
Z (c)
FIGURE 6.6: DISTRIBUTIONS OF SINGULARITY INTENSITY OVER INTERNAL, EDGE AND CORNER EXCAVATION SEGMENTS
148
ti = to (0. 188 + 0.812 (wlxj+w2yj+w3x~ 2+w4yi 2+w5xiyj ) )
= t o f 3 (x~ 1y~ ) (6.3)
where x, y are local co-ordinates for the segment
w1 etc. are constants determined by the relative
side lengths of the rectangular segment.
For the element geometry defined in Figure 6.6(c), the
constants in equation (6.3) describing the distribution
are given by
w = 1 (1 + cos26) 1 xj 2 2
w = 1 (1 cos26) 2 yj2 2
w3 = - x2 (1 + co~26)
J2
(6.4)
w = _ y2
cos26) (1 4
~2 2
W = 1 5 x~ 2y3 2
Y12 where 6 = arctan (x J 2
Stress and displacement components induced by this
distribution of a particular singularity are obtained from
the expressions for these quantities due to the point singularity, by integration over the area of the corner segment, in the
manner indicated by Equation (6.2) for an edge segment. Thus,
if a point singularity of intensity Tj induces a stress
component given by equation (6.1), the corner segment
singularity distribution induces a stress component given by
Q = to xj2 yj2 f 3 (x~,y3 ) f2 (x,y,z) dx.dy. l f jl y3
149
To illustrate the application of these singularity
distributions, expressions are given below for the az stress
component induced at point i (xi, yi, zi) by uniform, quasi-
elliptical and elliptic paraboloidal distributions of compressive
(hexapole) singularities, corresponding to interior, edge and
corner segments, shown in Figures 6.6(a), (b), (c). In all
cases, inferred ranges for the double integrals are xl, x2,
Y1, Y2 where
x1 = xi - xil etc
(a) uniform distribution:
Zz Z2 Z4 = (1-2v) 1 [[(1+ —) sin 2a - (2+ + ) sin2a
z z q 2 2 4 8Tr (1-v) 2 Z
R R R x Y 22 2 2 2
- ā (1 +) sin 4a] ] R2 x4 y
where a = arctan (RZ) 1 1
(b) quasi-elliptical distribution:
__ (1-2v) 1 r CC { (1+ !1) sin 2a - a q 1. 2 R2 z z 87(1-v)2
C1 R
Z2 Z4
Z2 2 (2 + R2 + ~) sin 2a - 4 (1 +
Z) sin 4a} R2
+ C {- ln(R+y) +
4 2z2 z (2R+y) 3 R(R+y) R3(R+y)2
- y ln(R+x) + 1E111 + z (1+ Z2)sin2a RU2 2 R2
xyz _ ( + 2R2 1 ) R3U2 U2
where
U2 = x2 + z2
C = 1.232x? 1 J 2
C2 = 0.232x 2 + 2x. 2 xi - xi2
C3 = 2(xi - x~ 2)
(c) elliptic paraboloidal distribution:
az = qz (1-2v) [[ci 87T(1-v) 2 0 0 1 1 2 2
+ C I + C I + C I ]] 3 3 - 4 4 5 5
where
Co = 0.188 + 0.812 (w x. + w y. + w x? + 1 1 2 1 3 1
C 1
C 2
C
=
=
=
w4y.2 + w5x.yi ) 1
0.812 (-w - 2w x. - w y.) 1 3 1 5 1
0.812 (-w - 2w y. - w x.) i 4 1 5 1
0.812 w 3 3
C = 0.812 w 4 4
C = 0.812 w 5 5
and w1 etc are defined by equations (6.3);
10 = 1 { ( 1 + 2 Z ) sin 2a - (2 + z2 4 + Z) sin 2a - z R2 R2 R4
2 2 - 1 (1 +
z ) sin 4a} R2
150
I = -ln (R+y) + 2z2 z`' (2R+y) 1 R(R+y) R3(R+y)2
151
I 2
= -ln (R+x) + 2z 2 z" (2R+x) R(R+x) R 3 (R+x) 2
I = y ln (R+x) { -2xyz + z - 1 (14Z ) sin 2a + XyZ 2R2 RU2
(1+ ) U2 3 R2 R3U2
I = x ln(R+y) { _ 2xyz + z 1 (1 + z2 ) xyz 2R2 ) } sin 2a + (1+ " RV R2 R3 V2 V2
I s
= - R + --- - Z 4
R3
where V2 = y2 + z2
6.4 Three-Dimensional Boundary Element Solution Procedure
The objective is to determine total stresses and
mining-induced displacements around a set of narrow, parallel-
sided openings of finite area in the plane of a tabular orebody.
The openings are divided into a total of k rectangular
excavation segments. To simulate excavation of the openings
it is necessary to determine the intensities of the compressive
and shear singularity distributions which, when applied over
all segments in a stress-free continuum, induce known tractions
tx, ty, tz on each boundary element. It is assumed that it is
sufficient to realize these tractions at the centre of each
element to satisfy the required conditions over the complete
element. The expressions for stress and displacement components
due to the different types of point singularities, given in
Appendix V, can be integrated analytically to give expressions
for these components due to uniform distributions of these
singularities, and due to the distributions illustrated in
Figure 6.6(b), (c). In general, the induced stress and dis-
placement components at a point i due to a distribution of
shear singularities polarised in the X-Z plane, of intensity
sxj, on segment j can be written
(asxij sxJ + asy.. sYJ
[ asxiJ asyij agzij
+ agzij
sxJ
s Y J
qzj
6xi k
j=1
k E
j=1 7
zj)
6xi = asxij sxJ
ayi • bsx1J sxJ
xj azi = csxiJ sxJ
Tx = dsx. SxJ
xj Tyzi
asxij sxj (6.5)
Tzxi fsxlJ sxJ
ux j = Ux j S xi xi xj
uxj = Uxjs yi yi xj
ux j = xj s zi Uzi x j
Similar equations can be written for the stress and dis-
placement components at i due to the distributions of the s
and qz singularities on segment j; i.e.,
YJ cr xi = asy1J s 1J
zj Q xl = agz1J qZJ
etc.
etc.
If the excavations are represented by k excavation segments,
induced stress components at the point i are obtained by
superposition of the components induced by the various dis-
tributions of singularities on the various segments; i.e.,
152
= [At] [q]
where [Ai]is a row vector, of order 3k, of influence
coefficients,
[q] is a column vector, of order 3k, of segment
singularity intensities.
Thus all induced stresses and displacements are given by :
153
a xi
a yi
a Zi
TXy i
= [Ai] [q]
= [Bi] [q]
= [Ci] Eq]
= [Di] [q]
Tyzi = [Ei] [q] (6.6)
TZX1 = [Ft] [q]
= [Uxi] [q]
= [q]
= [UziJ [q]
Equations(6.6) allow stress and displacement
components to be determined at any point i in the medium
if the segment loads are known. The magnitudes of these loads
are determined from the known values of induced traction at
the centre of each boundary element. To model excavation of
openings, induced stress components Tzx' Tyz' az at the
centre of any element i must be equal to the known induced
tractions tx, ty, tz on the element; i.e.,
TZXl = tXl
T = t yzi yi
c= t zi zi
uxi
u i Y
uzi
154
As noted earlier, the Tyz and Tzx stress components
induced at a point by the compressive singularity tend to
zero as the point approaches the plane in which the
compressive singularity operates. Also, the stress
components induced at a point by the shear singularities
tend to zero as the point approaches the plane in which
the shear singularities operate. Due to this de-coupling,
the magnitudes of the unknown segment loads are obtained
by solution of the two sets of simultaneous equations
[TZ] [qz] = [tx]
(6.7) [Ts] [s] = [ts]
where the square matrix [Ta], of order k, consists
of rows of[ .Ci] vectors , determined for each element in turn;
[qz] is the column vector of compressive
singularity intensities;
[tz] is the column vector of known induced
tractions tz.;
the square matrix [Ts], of order 2k, consists
of rows of [Ei] and EF;] vectors, determined
for each element in turn;
[s] is the column vector of shear singularity
intensities;
[ts] is the column vector of known induced
tractions tyi' txi'
In equations (6.7), both the [Ti] and [TS] matrices are leading diagonal dominant. The sets of simultaneous
equations can therefore be solved readily by Gauss-Siedel
iteration. No over-relaxation is required.
6.5 Validation of Boundary Element Program
The requirement is to demonstrate that, for selected
problems, the three-dimensional Boundary Element program
produces results which are consistent with results from
155
independent analysis of the same problem. One closed form
solution exists which is suitable for program validation.
Sneddon (1946) has determined expressions for stresses and
displacements around a penny-shaped crack subject to uniform
internal pressure, from which stresses and displacement
around a crack in a uniaxial field are obtained directly
by superposition. For a crack of radius c in a uniaxial
field pz directed normalto the plane of the crack, stresses
and displacements at points in the plane of the crack, defined
by the ratio P = r/c, , where r is the radial distance from the crack centre, are given by:
251. a z = 0
= ar = - (v+~) Pz ce
— 2p (1-v) uz = z 7Gc (1-p2)2
(6.8)
p>1
6z = pz {1 + {(p2 -1) - sin-1 (p)}}
6r = 2p {(p2-1)-1/2 - (v+ z) sin -1 (p)}
(6.9)
a0
= 2pz 1 { 2v (p 2-1) - (v+Z) sin (p) } TI
The expression for uz in equations (6.8) implies
that displacement varies elliptically over the crack. The
existence of the terms (•p 2 - 1)1 in equations (6.9) is
accounted for when one notes that the penny-shaped crack can
be considered as a degenerate spheroid. The same term occurs
in expressions for stresses in the plane of a long crack,
which in cross-section is a degenerate ellipse. It is inferred
that the term sin-1 (p) takes account of the curvature of
the boundary of the penny-shaped crack. It is noted that
for both ar and a0 , the curvature term also involves
Poisson's Ratio, and that in the formal analysis, Gr
and a0
are coupled directly.
156
Figure 6.7(a) compares stresses in the plane of a
penny-shaped crack, determined using the Boundary Element
program, and calculated usingSneddon's solution. As the
Boundary Element program can only accept rectangular elements,
it was possible to make only a fairly coarse representation
of the circular plan area of the crack, even when 121 elements
were disposed over the area. It is observed that the dis-
tribution of the az stress component is practically
identical for the different methods of analysis. The Boundary
Element analysis slightly overestimates the circumferential
stress component, ae , and underestimates the radial stress
component, ar , when compared with the distributions obtained
from the analytical solution. It has been found that(Gr + ae)
at any point is virtually identical for the numerical and
analytical solutions, as indicated in Table 6.1. It is
suggested that the minor- discrepency in the results of the Boundary Element analysis arises from inability to represent
boundary curvature properly, resulting in inadequate
resolution of (ar + ae) into its components. Confirmation
of this was provided by analysis of a long slot, with length/
width ratio 11:1 . In this case, all stress components in
the plane of the slot agreed satisfactorily with the
analytical solution for a long crack.
Sneddon (1946) has tabulated values of are ae , az for points on the central normal to a penny-shaped crack.
Figure 6.7(b) shows the variation of these stress components
along the central normal to a crack, determined by Boundary
Element analysis and provided by the analytical solution.
Excellent agreement is indicated between the results from
the two methods of analysis.
Figure 6.7(c) compares displacement over the crack
determined numerically and calculated from the expression
given in equations (6.8). The agreement between the dis-
placement distributions is seen to be satisfactory.
Figure 6.7 (d) shows the variation of compressive
singularity intensity with distance from the crack centre.
The intensity varies elliptically in the radial direction,
157
Table 6.1 Analytical and Numerical Solutions for (or + 6e) for Points in the Plane of a Penny Shaped Crack in a Uniaxial Compressive Stress Field
p
Analytical
a crr + e
Numerical
1.2 0.499 0.504
1.4 0.215 0.219
1.6 0.119 0.122
1.8 0.075 0.077
2.0 0.052 0.053
p = r/c, where r = radial distance to point from
crack centre
c = crack radius
158
0.6
0.4
0.2
a/PZ 0
-0.2
-0.4
-0.6
-0.8
1.4 • o B.E. Solution C.F. Solution
1 .2
1.0 az/pz
(on z=0.0)
0.8
e
a/ pz z
0.6
0.4
0.2
0. 0
-0.1 R/c
FIGURE 6.7(a)
2.0 1.0 1.2
ar/pz (on z=0.0)
(on z=0.0) 0
0 0
FIGURE 6.7(b)
1.0 0.2 0.0 0.6 0.8 0.4 R/c
3.0
Elliptical Distribution
Segment Loads 2.0
qz
1.0
0.2
0.4
0.6
0.8 1.0
R/c
FIGURE 6.7(c)
FIGURE 6.7(d)
FIGURE 6.7: STRESSES IN THE PLANE OF, AND PERPENDICULAR TO THE PLANE OF A PENNY SHAPED CRACK IN A UNIAXIAL FIELD (a), (b), DISPLACEMENT DISTRIBUTION OVER THE CRACK (c), AND SINGULARITY DISTRIBUTION WHICH MODELS CRACK FORMATION (d)
159
160
reflecting the uz displacement distribution. Thus the
compressive singularity intensity follows aspheroidal dis-
tribution over the area of the crack, as was inferred previously
from consideration of the two-dimensional crack problem and
the axial symmetry of the three-dimensional problem.
The conclusion from this stage of program validation
is that the three-dimensional Boundary Element program
provides a satisfactory method for determining stresses and
displacements around narrow parallel-sided openings of finite
plan area in a compressive stress field directed normal to the
plane of excavation.
Further interests in program evaluation were to
assess the performance of the program in modelling openings
in shear stress fields, and to determine the range of
excavation span/height ratios for which the narrow, parallel-
sided slot remained a satisfactory geometrical approximation.
The Boundary Integral (B.I.) program described by Lachat and
Watson (1976) has been used in these phases of program
evaluation.
Figure 6.8(a) illustrates a square slot in an infinite
medium, oriented with its major faces parallel to the X-Y
plane and its centre located at the co-ordinate origin. The
pre-mining stress field is a unit shear stress pzx = 1.0.
Two different excavation geometries were analysed with
Watson's program. These were slots with span/height ratios of
10 and 5. The variation of TzX along the local X axis for the excavation is shown in Figure 6.8(a), for these two
excavation geometries, and for the infinitely narrow slot.
The similarity between the stress distributions from the B.I.
analysis of the 10:1 slot and the Boundary Element analysis
is taken to indicate satisfactory performance of the program
components concerned with analysis of shear stress. The
downwards concave section of the stress distribution curve
for the 5:1 slot is associated with the requirement that
T = 0 on the vertical walls of the excavation. zx
t I t t t 1 .0 1 .0
t
1.2 1.6 1.8 2.0
t t
1.7
1.6
1.5
1.4 T zx pzx
1.3
1.2
p Span/Ht = 5 V Span/Ht = 10 O B.E. Soln
1 .0
0.8
0.6
Tzx/pzx
0.4
0.2
0.0 0.4 Az/HS (b)
FIGURE 6.8: DISTRIBUTION OF SHEAR STRESS AROUND SQUARE OPENINGS WITH VARIOUS SPAN/HEIGHT RATIOS IN A UNIT SHEAR FIELD
0.0 0.2 0.6 0.8 1.0
HS i 162 a
Spall 'Ht = 5
Span,'Ht = 10
O
0.4
0.2
0 .0 0.8 1.0 Az/HS
-0.2
-0.4
-0.6
-0.8 (b)
7.0
6.0
_ 5.0 -U x 10
HS 4.0
3.0
2.0'
1.0
0 .0 0.2 0.4(C) 0.6 0.8 1.0
FIGURE 6.9: STRESS AND DISPLACEMENT DISTRIBUTIONS AROUND SQUARE OPENINGS WITH VARIOUS SPAN/HEIGHT RATIOS IN A UNIAXIAL COMPRESSIVE FIELD
x/HS
163
The variation of Tzx along the central normal to
the excavation is shown in Figure 6.8(b). The position of
the reference points is measured relative to the lower surface
of the excavation. The close correspondence between the
results for the various problem geometries reflects the more
uniform stress gradients in this area.
Displacements over the excavation are not presented,
due to similarity with results discussed below for excavations
in a compressive stress field.
Distributions of stress and displacement around
square slots, excavated in a uniaxial field pz = 1.0,
with span/height ratios of 10:1 and 5:1, are compared
with those for the infinitely narrow slot in Figure 6.9.
Figure 6.9(a) shows good correspondence in the abutment
area between the distributions of 6 and az along the
local X axis of the excavation, for the different excava-
tion geometries. There is a discrepancy in the slot
abutment area between the ax distribution for the 5:1
slot and those for the other problem geometries. This is
caused by the proximity of the vertical side of the
excavation, and the requirement that ax = 0 at that surface.
The distributions of stress along the Z axis for the
excavation shown in Figure 6.9(b) indicate the virtual
identity of results obtained in that area. The distribu-
tion of the uZ displacement component over the excavated
area is shown in Figure 6.9(c) for the case where pZ/G =
0.01. For the narrow slot, the displacement varies
elliptically over the excavated span, and the displacement
distributions for increasing excavation height parallel
that for the narrow slot exactly.
The inference drawn from these results is that slots
with span/height ratios greater than 5 may be represented
adequately as infinitely narrow slots, if one is prepared to
ignore stresses in the region immediately adjacent to the
vertical walls of the excavation. In this region the narrow
slot model predicts unrealistically high stress levels which
are a direct consequence of the modelling procedure.
164
A series of problems was analysed to assess the
performance of the Boundary Element Program in determining the
state of stress in pillars isolated by the mining of adjacent
stopes. A mined area with a single remnant pillar is re-
presented in Figure 6.10(a), one-quarter of the problem region
being illustrated. Two problem geometries, with stope span/
height ratios of 10:1 and 5:1 were analysed with Watson's
B.I. program, for comparison with the results of Boundary
Element Analysis of a narrow slot. The field stress was uniaxial,
directed perpendicular to the plane of excavation.
Figures 6.10(a) , (b), (c) show stresses in the pillar
and abutment area of the excavation,calculated at the mid-
height of the excavation. The axial stresses in the pillar
are practically identical for the narrow stope and the
10:1 stope, as illustrated in Figure 6.10(a). The
Boundary Element analysis underestimates the axial stress
in the body of the pillar for the 5:1 stope by less than
10%. The Qy and ax stress components at various points
along the local X-axis for the problem region, shown in
Figures 6.10(b), (c), indicate that the confining stresses
which are generated in the body of the pillar are determined
adequately using the Boundary Element Program. The dis-
placement distributions over the excavation lower surface,
along the local X-axis for the problem, are shown in
Figure 6.10(d). For each of the three problem geometries
the distributions are skewed elliptical, towards the
central pillar, in agreement with the notion that the
pillar is softer in compression than the surrounding
abutments.
6.6 Assessment of Slot Modelling Procedure
The three-dimensional Boundary Element program
described above was developed with the objective of providing
a method of estimating the states of stress which are
generated in pillars created during the mining of tabular
or lenticular orebodies.The results of the validation problems
analysed with the program suggest the method should be useful
for design of mining layouts where stope spans are greater
2.6 165
1.2
1.4
0.4 a
Pz 0.3
0 .7
0.6
0.5
0.2
0.1
0.0
O Span/Ht = 5
V Span/Ht = 10
O B.E. 2.4
2.2
2.0
1.8
az Pz
1.6
1.0 0.0 0.2 0.4 0.6 0.8 1.0 3.0 3.2 3.4 3.6 3.8 4.0
Distance from Pillar Centre (x/HS)
FIGURE 6.10(a)
0.0 0.2 0.4 0.6 0.8 1.0 3.0 3.2 3.4 3.6 3.8 4.0 Distance from Pillar Centre (x/HS)
FIGURE 6.10(b)
1
1.0
0.8
0.6 -uz/HS
x10-2 0.4
166
0.0 0.2
0.4 0.6 0.8 1.0 3.0 3.2 3.4 3.6
3.8
4.0 Distance from Pillar Centre (x/HS)
(c)
1.0 2.0 3.0
Distance from Pillar Centre (x/HS)
(d)
FIGURE 6.10: STRESS AND DISPLACEMENT DISTRIBUTIONS AROUND A SQUARE ROOM WITH A CENTRAL SQUARE PILLAR IN A UNIAXIAL STRESS FIELD
167
than about five times the mining height ororebody strati-
graphic, thickness. Typically this would apply in cut-and-fill
stoping in inclined orebodies,where crown or floor pillars
may be left between successive stopes, or in the design of
permanent pillars, such as regional support pillars, in flat-
lying orebodies.
The method allows adequate determination of the state
of stress in the body of a pillar. The main disadvantage
of the method is that it is not possible to estimate the
state of stress at pillar boundaries. This is a restriction
on its application to design of cut-and-fill stopes, when
knowledge of the state of stress in the immediate working area
is of some concern.
The studies using the B.I. program have confirmed
the general validity of the approach used in the analysis,
that narrow, parallel-sided openings may be represented by
infinitely narrow openings, and that singularities may be
developed specifically to handle the modelling of these
openings.
CHAPTER 7
CHAPTER 7 : DIRECT FORMULATION OF THE BOUNDARY ELEMENT METHOD FOR COMPLETE PLANE STRAIN
7.1 Objectives in Development of the Direct Formulation
In the indirect formulations of the Boundary
Element Method described earlier, distributions of
singularities are applied over excavation boundary
elements, and the intensities of the distributions are
adjusted to achieve known values of traction or displace-
ment on all boundary elements representing excavation .
surfaces. The most suitable form of singularity must
be determined by trial and error, and the distributions
of singularities which satisfy the imposed boundary
conditions have no direct physical significance. For
complete plane strain problems, it has been found that
when the ratio of cross-sectional area to surface area
of excavations is high, uniformly distributed strip loads
can be used to realize the known boundary values. When
the ratio is low, e.g. for long, narrow, parallel-sided
slits, distributions of higher order singularities must
be used, and the openings must be modelled as infinitely
thin slits.
In a direct formulation of the Boundary Element
Method, the solution for unknown boundary values is
obtained by solution of an equation which relates
excavation-induced tractions and displacements on the
excavation surfaces. This boundary constraint equation
is established using expressions which are themselves
fundamental solutions of the governing differential
equations of elastostatics, and by application of the
Reciprocal Work Theorem. Direct formulations of the
Boundary Element Method have been described for the
strictly two-dimensional problem by Rizzo (1967), for a
homogeneous medium. The direct formulation for three-
dimensional analysis described by La chat and Watson (1976)
168
also takes account of non-homogeneity of the medium.
The singularities used in these formulations were
either unit line loads, for the two-dimensional case, or
unit point loads, for the three-dimensional case.
In this work, a direct formulation is developed for
non-homogeneous media and complete plane strain, the
aim being to allow design analyses for orebodies with
elastic properties different from those of the country
rock. A second objective was to determine if higher
order singularities, such as quadrupoles of various
types, could be used in direct formulations, in place of
line load singularities. -she interest in this area arose
from the possibility of modell'ng crack generation and
propagation around mine openings, or slip on geological
features such as faults, and was prompted by the success-
ful application of these singularities in indirect
formulations. Finally, a basic objective was to assess
the relative merits of direct and indirect formulations.
In the discussion that follows, attention is confined
initially to the complete plane strain, direct formulation
of the Boundary Element Method for a homogeneous medium.
This is subsequently extended to include non-homogeneous
media.
7.2 Establishment of Boundary Constraint Equations
The nature of the complete plane strain problem has
been described in detail in Chapter 3, and is illustrated
again, for the sake of clarity, in Figure 7.1. A slice
of an excavation, whose surface is S* and which is
arbitrarily oriented in a triaxial stress field, is
illustrated in Figure 7.1(a). At any location on the
boundary, the final boundary conditions may be defined
in terms of imposed final boundary tractions txf, tyf, tzf, or imposed displacements ux, uy, uZ . The problem
is to determine unknown surface values of traction or
169
(a)
(b)
170
X
Projection of S on X-Z plane
tzj I / l/
txj ./ -ux .
— Normal to u21 Element of
Surface at point j
J
1 z
(c) (d)
FIGURE 7.1: SLICE OF THE SURFACE OF AN OPENING IN A MEDIUM SUBJECT TO TRIAXIAL STRESS, AND PROBLEM SPECIFICATION FOR COMPLETE PLANE STRAIN ANALYSIS
171
displacement, and to determine total stresses and
excavation-induced displacements at selected internal
points in the medium. The analysis is effected in terms
of the tractions tx, ty, tz and displacements ux, uy, uz
induced on the surface S, geometrically identical to S*,
in an otherwise non-loaded continuum, illustrated in
Figure 7.1(c). The solution is obtained by considering
separately the completely uncoupled plane and antiplane
problems, the former involving tx , tz , ux , u z , the latter involving t uy.
The requirement is to establish and solve boundary
constraint equations for the plane and antiplane problems.
Consider the problem in the X-Z plane initially, with
tractions tx, tz assumed known at all points on the
surface S. Figure 7.2(a) shows the trace of the surface
S projected on the X-Z plane. At point i on S, the
outward normal to the boundary makes an angle Bi with the
Z axis. Local axes X', Z', oriented as shown, are
established at i. At another point j on S, the normal
N. to the surface makes an angle Sji with the local Z'
axis for point i. Relative to these axes, tractions and
displacements at j are t'., t'., u'., u'.. Considering x3 z3 x3 z3
all points j on S, the surface can be considered to be
loaded by tractions t'. t'., producing displacements x3 z3
u'., u'.. Call this Load Case 1. x3 z3
In Figure 7.2(b), a line load, of unit intensity/
unit length parallel to the Y axis, is applied at i
parallel to the local X' axis. Suppose tractions and
displacements induced at j (expressed relative to the X',
Z' axes) by the unit line load at i directed parallel
to the local X' axis at i are T i xis T, xi, U, xis U, xi where
xj z~ xj zj
T'X~ = a'X~ sine.. + ,xi COss.
T'Z~ = TIzx. sins,. + iXi cos$.
J
172
x
z (a)
x
z (b)
FIGURE 7.2: LOAD CASES FOR ESTABLISHING BOUNDARY INTEGRAL EQUATION
173
and 6'X~ etc. are stress components induced by the unit
line load, expressed relative to the local axes at i.
The sixrface S can be considered to be loaded by
tractions T' xi , T' xi producing displacements U' xi u' xi
xj zj x3 zj Call this Load Case 2.
The Reciprocal Work Theorem may be applied to the
systems of forces and displacements acting on the surface
S; i.e. the total work done by the forces of Load Case
2 acting through the displacements of Load Case 1 is
identically equal to the total work done by the forces
of Load Case 1 acting through the displacements of Load
Case 2. Noting that the components of force on an
element of surface dS are given by
f' . - t' . dS etc.
x3 x3
applying the Reciprocal Work Theorem yields the equation
f {T''xl u' + T'xl u' } dS S {t' U` xi + t' U' xl}dS xj xj zj zj x3 xj zj zj S (7.1)
The surface S is divided into n rectangular (strip)
elements, and, in Load Case 1, it is assumed that tractions
and displacements are uniform over each element. Equation
(7.1) becomes
n n j= 1 {T'xj uxj + T'xi uzj} dSj = ., {txj U'xj +
S S . . i
t' U' xi } dSj (7.2) zj
where the integrations are performed over the length S.
of each element j. When the line load singularity occurs
in the range of the integration, the Cauchy Principal
Value of the integral is taken.
Putting JTI X1 dSj = F'X~
JutX~ dSj = UIt etc. equation (7.2) may be written
174
= n E j=1
,xi [UI xj , UI xjxi ] t xj t , (7.3) E C
,xi , uxj F xj F Zjxi ] J., uzj
Similarly, applying a unit line load at point i in
the direction of the Z' axis at point i, and proceeding in
the same way as indicated above, yields the equation
E [F,zi F,zi1 xj = E [uitzi UI'zixj j=1 xj zj J u' . j=1 t' xj zj ] zj
(7.4)
Combining equations (7.3) and (7.4):
F, xi F,xi_ xj zj
n 2
j=1
U' X3 n
= E j=1
UI' xi
UI' xi
xj zj t'.
F'zi F,zi xj zj
u' z j UI' zi UI' zi xj zj ti.
(7.5)
Equation (7.5) is derived from an identity that must
be invariant under a co-ordinate transformation. Transforming
to the excavation (X,Y,Z) axes from the element local axes
yields the equation
_ xi xi F F u n x3 z3 x3 n
j=1 FX. F . U. j=1
UIxi UIxi x3 z3
t X3
Ulzi zi Ul
xj zj
tz j
(7.6)
Taking the point i as the centre of each boundary
element in turn, and proceeding in the manner described
above, yields n equations similar to equation (7.6). These
equations may be combined into a single set of simultaneous
equations expressed in matrix notation by
txl UIxi UI
xi Ulxl Ulxl xl zl x2 z2 uxl
F,xl Fxl Fxl F,xl Xl Zi x2 z2
zl zl zl zl UIxl Ulzl UIx2 Uiz2 Fzl zl zl zl xl Fzl Fx2 Fz2 uzi tzl
x2 tx 2 UI x2 UIx2 UI x2 UIx2 xl zl x2 z2 F
x2 x2 x2 x2 xl Fzl Fx2 Fz2
tz2 UI z2 UIz2 UIz2 UIz2 xl zl x2 z2 uz 2 F
Z2 Z2 Z2 z2 xl Fzl Fx2 Fz2
175
[F] [u] = [UI] [t] (7.7)
For example, considering the simplest case where the
surface is divided into two elements, the relationship
between tractions and displacements at the centre of
each element j is given by
In general, the boundary constraint equation for
the plane problem (i.e. equation (7.7)) represents, for
the case of n boundary elements, a set of 2n simultaneous
equations in 2n unknowns.
In the solution of the boundary constraint equation
the requirement is to group all known quantities on the
R.H.S. of equation (7.7). When mixed boundary conditions
exist, for elements on which displacements are imposed
the appropriate terms are interchanged between the[u]and[t]
vectors, and corresponding columns are interchanged, with
change of sign, between [F] and [UI] .
The boundary constraint equation for the antiplane
problem is formulated in the same way as for the plane
problem. Line loads acting parallel to the Y axis, of
unit intensity/unit length, are applied at the centre of
each boundary element in turn. Application of the Reciprocal
Work Theorem in each case yields a set of n simultaneous
equations in n unknowns represented by
[F y] [uy] = [UIy] [t y] (7.8)
176
The boundary constraint equations (7.7) and (7.8)
are established from the discretized problem geometry, the
imposed excavation boundary conditions and the solutions for
stresses and displacements induced by unit line loads
applied in the three co-ordinate directions in an infinite
medium. Expressions for these quantities are given in
Appendix II. It is noted that although equations (7.7) and
(7.8) suggest that sufficient information exists to determine
unknown boundary values uniquely, this is not so when
excavation boundary conditions are specified in terms of
tractions only. In this case, any arbitrary rigid body
displacements may be imposed on particular solutions of
equations (7.7) and (7.8) and still satisfy the field
equations of elasticity. In this situation, one is free to
employ any convenient device to obtain a solution for
displacements which suits the requirements of the particular
problem being analysed. In the current work, all displace-
ments are determined relative to an arbitrarily selected
datum point remote from the area of excavation.
7.3 Solution of Boundary Constraint Equations
The requirement is to set up and solve equations (7.7)
and (7.8) for the unknown surface values for the plane and
antiplane problems, taking account of conditions imposed
at excavation boundaries, the pre-mining stress field and
the geometry of the excavation boundaries. Once the
excavation boundaries have been divided into the required
number of discrete elements, the vectors [t] and [ty]
of known surface values can be constructed in a straight-
forward way. The terms of the D1, DUI], [Fy] and [UIy] matrices are obtained by integrating tractions and
displacements induced by unit line loads, in the various
co-ordinate directions at the centre of each element,
over the range of each element. Although quite simple
expressions for the integrals can be obtained analytically,
177
it has been found more efficient in the Boundary Element
program to evaluate the integrals using Gaussian quadrature.
Four-point quadrature with a weight factor of unity has
been found to provide sufficiently accurate determination
of the integrals.
The systems of simultaneous equations represented by
equations (7.7) and (7.8) are not necessarily well-
conditioned. To improve conditioning, in assembling the
[F] and [UI] matrices for equation (7.7), all matrix coefficients of the form UIx etc. are multiplied by 2G xj and divided by Rmax , where G is the Modulus of Rigidity
of the material and Rmax is the maximum distance between
elements. In the vector [t] of known surface values,
all known surface tractions are divided by 2G, and all known
surface displacements are divided by max'
The set of equations is solved by Gaussian elimina-
tion. In the program developed in this work, the block
solver described by Lachat and Watson (1976) has been
employed. After solution of the equations, all displace-
ments are multiplied by max' and all tractions are
multiplied by 2G. A similar procedure is followed to
improve the conditioning of the constituent equations of
equation (7.8), except that G is used instead of 2G as the
multiplier and divisor of the appropriate terms.
7.4 Boundary Stresses
After solution of the boundary constraint equations,
the values of the excavation-induced tractions (tx, ty, tZ)
and excavation-induced displacements (ux, uy, uz),
expressed relative to the excavation (X,Y,Z) axes, are
known at the centre of each boundary element. The require-
ment is to find the induced stresses at each element,
relative to the local axes for the element, and superimpose
the field stresses, expressed relative to these axes, to
obtain the total stresses.
178
Figure 7.3 represents part of the boundary of an
excavation. The global axes are X,Y,Z, with the X,Z axes
as shown and the Y axis directed out of the plane of the
paper. For the element i, the local axes are L,Y,N,
with the N axis directed into the solid. Consider the plane
problem initially. Induced tractions tx and tZ transformed
to the local L,N axes for element i yield tractions tQ, tn.
Then induced stresses an, Tra at i are given by
an t n n
T = t
To obtain the induced stress component az at i, it is
necessary to determine the directional derivative of the
tangential displacement numerically. Transformation of
ux, uZ displacements from the global axes to the local axes
for element i, for element i and its adjacent elements
i - 1 and i + 1 yields tangential displacements uQ-1,
uQ , uQ+l. Distances At and Ak between element centres
are as shown in Figure 7.3.
FIGURE 7.3: GEOMETRIC PARAMETERS FOR DETERMINATION OF DIRECTIONAL DERIVATIVES OF DISPLACEMENT AT EXCAVATION BOUNDARY
The state of strain at the centre of element i is
taken as the average strain between elements i-1 and i+1. Thus, at i,
8u EQ =- ā x
i i-i i+1 - i 1 uQ - uQ uQ uQ -
2 ( ~Q1 + AQ2 )
Now, for plane strain
Et (1-v2) (c Q )
Q E Q (1-v) n
E v or a - ( EQ + (1-v) 6ri
Also a = v(un + a)
Thus, for the plane problem, the induced boundary
stresses 6Q, a , a , TnQ can be obtained directly fromnythe induced boundary tractions and displacements.
For the antiplane problem, the induced stress
component Tyn at the centre of element i is given by
Also, Tyn ty 8u au
'ky = - sat + ay )
8u ā Q
ui - ui-1 ui+1 _ ui _ 1 v Y v v
2 ( AQ1 + AQ2 Y.
Then
TQy = Gy9y
179
Thus all induced boundary stress components can be determined
from the surface values of the induced tractions and
180
displacements. The total stress components at the centre
of any element are obtained by transformation of the field
stresses to the local axes for the element, and super-
imposing the induced stresses.
7.5 Displacements and Stresses at Internal Points
Once-the boundary tractions and displacements are
known, displacements at internal points can be obtained
by further application of the Reciprocal Work Theorem.
Figure 7.4(a) illustrates a point i (xi,zi) in the medium,
at which it is required to determine excavation induced
displacements u1, uy, u1. Suppose a unit line load is
applied at i in the X-direction, as shown in Figure 7.4(b).
Applying the Reciprocal Work Theorem, and integrating
around the singularity at i and the boundary of the excava-
tion yields the equation
xi ux {t x3 Ux] + tzj UX.
}dS - {TXT ux
. + TZ~ uz3}dS
S S
= Efft xt UX~ + tZJ UZG}dS~ - E {TXi u
Xi + TZ~ uz.}dS~
j=1 S. J 1 S.
J J (7.9)
X
Z
x ®mai
/ Tx% i l XĪ
xi
/ rZ~i I
— ~ xi UZ1
\
(a) lb)
FIGURE 7.4: LOAD CASES FOR DETERMINING DISPLACEMENTS AT INTERNAL POINTS IN THE MEDIUM
181
Similarly, excavation induced displacements in the
Z and Y directions at i are obtained by applying unit line
loads at i in the respective directions, i.e.
n n
ui = E ( {t Uzi + t uzi }ds - E {T i u + Tzi u j}dS z
j=1Jxjxj zj zj j xjxj
j=1 zjz j
Si Si (7.10)
n
uy E pt
yj Uyj TYi
uyj}dSj (7.11)
j 1 Si
Induced stress components at internal points in the
medium are determined by calculating induced strain components,
and applying the appropriate stress-strain relationships.
Expressions for the strain components at any point in the
medium are obtained by partial differentiation of the
expressions for the displacement components. For example,
the ex strain component at the internal point i is obtained
from equation (7.9) :
Dux = Clx axi
n 3Txi aTxi x ?1 E J { ax uxj + ax.
uzj}asj - j-1 s i 1
i
n auxi auxi
'=1J {txj axi + t
i
axi}dSj s.
aTx au
i xi ] Expressions for aX, , -ā etc. are found by differ-
s 1
entiation of expressions for traction and displacement due
to unit line loads. Similar expressions can be established
for the strain components ez, yxy' yyz, yzx' Also, from the
plane strain assumption, ey = 0.
u , x
i
u x
Element j Element j'
Induced stress components are then calculated from
the equations
ai = A i + 2G e x x
az = AAi + 2G EZ
a = v (ax + az )
Txy = G Yxy etc.
where Ai is the volumetric strain.
It is noted that the. determination of both displace-
ments and stresses at internal points involves integration
of expressions for tractions and displacements due to unit
line loads, or integration of partial derivatives of these
expressions, over the range of each boundary element. The
integrations are performed conveniently and efficiently in
the program using Gaussian quadrature.
7.6 Symmetry Code
Figure 7.5 shows the cross section of an excavation which
is symmetric about the Z axis, and a representative boundary
element j of the surface together with its reflection in the
Z axis, element j'.
182
V 1 V zJ zJ z
FIGURE 7.5: PROBLEM SPECIFICATION FOR AN OPENING WHICH IS SYMMETRIC ABOUT THE Z-AXIS
ux1
uzl
ux j
u z3
u xj'
uZj'
uxn
u zn
txl
tzl
t x j
tzj
t ' xj
tZj ,
txn
(7.12)
183
Considering the plane problem initially, the boundary
constraint equation may be written
Fxl Fzl
., Fxl Fxl _. Fxl Fxl , Fxl F,xl xl zl x j zj xj' zj' xn zn
Fzi Fzl .. Fzl Fzl .. Fzl zl zl zl xl zi • xj zj • xj' Fzj' . Fxn Fzn
Fxi Fxi Fxi Fxi .. Fxi Fxi xi xi
xl zl • xj zj xj' zj' Fxn Fzn
Fzi Fzi • . Fzi Fzi .. Fzi Fzi zi zi xl zl xj zj x3' zj' .. Fxn Fzn
Fxi' zl xj xi Fxi' .. Fxi' Fxi' .. Fxi ' Fxi ' xi' xi'
zj xj' zj' Fxn Fzn
Fzi' F zl1 ' .. FZi F:13:1 .. FZ1 , Fzi' . Fzi' FZ1 ' xl z xj xr zj' • xn zn
Fxn Fxn Fxn Fxn Fxn Fxn xn xn xl zl xj zj xj' zj' Fxn Fzn
Fzn Fzn Fzn Fzn .. Fzn Fzn _ Fzn Fzn xl zi xj zj xj' zj' xn zn
Ulxl UI ..UI xi xi Ul xl .. Ulxl xl xl xl xl zl xj zj xj' UIzj, " UIxn UIzn
Ulzl Ulzl .UIzl UI •• Ulzl zl . zl zl xl zl xj zj xj ' UIZJ ' .. UIxn UIzn
UIxi UIxi • • UIxi UIxi UIxi UIxi UIxi UIxi xl zl xj zj xj' zj' xn zn
UIZi UIZi .. UIZi UIZi .. UI J, UI UI zi zi xl zl xj zj xj' zj' UIxn UI zn
UIxi' UIxi ' .. UIxi ' UIxi' .. UIxi' xi' xi' xi' xl zl xj zj x3' UI l zj' • • x3' zj' UI zn
UI xi ' UIz1' • • UIZi' UIZi' • • [Jlzi ' UIzi' zi' zi ' xl zl xj zj xj' z3' UI UI xn zn
UIxn UIxn ..UIxn UIxn Ulxn UIxn xn xn xl zl xj zj xj' zj' • UI UI xn zn
UIzn UIzn UI zn UIzn UI UI zn zn xl zl xj zj x3' zj' •' UIxn zn tzn
If the problem is symmetric with respect to load
conditions as well as being geometrically symmetric about
the Z axis,
txj , = - txj tz., = t
2.
and
uz j, u zj
184
Symmetry with respect to load conditions implies
that the field stress component pzX is zero. When both
load and geometric symmetry requirements are met, equation
(7.12) may be written
(Fxl) (Fxl) .. (Fxl) (Fxl) ..
xl zl xj zj
(Fxl) (Fzl) .. (FXV) (Fz~) ..
(Fxl) (Fzl) .. (FXV) (Fz~) ..
(Fxl ) (Fz1) .. (FXV) (FZ~) ..
(UIX1) (UIxl) .. (UIX~) (UIzj) txl
(UIXi) (UIZi) .. (UIXl) (UI11) .. tzl
(UIXi) (UIzi) .. (UI' ) (Ulx ) .. tx]
(UIxi) (UIZi) .. (UIg) (UI2 ) .. tzj
uxl
uzl
u xj
u zj
(7.13)
where (FXV) = FX-. - FXV,
(FZ~) = Fzj + Fz~, etc.
185
Exploitation of one degree of symmetry halves the
order of the system of simultaneous equations and the
number of matrix coefficients to be calculated, and produces
a coefficient matrix [F] one-quarter the size of that
generated if symmetry is not exploited. For symmetry of
load and geometry with respect to both X and Z axes, the
order of the system of equations is reduced to one-
quarter of the system generated if symmetry is not exploited,
and the size of the coefficient matrix [F] is one-
sixteenth that of the original matrix. Computational
efficiency is therefore improved by reduction in matrix
generation time, and by reduction in time taken for the
solution of the boundary constraint equation.
The same principles are followed in implementation
of symmetry code for the antiplane problem. In this case
the requirement of symmetry with respect to loading as
well as geometry requires that the field stress component
pxy be zero.
7.7 Validation of Boundary Element Program
The performance of the program has been evaluated
by taking as trial problems a long hole of circular cross
section in a triaxial stress field, and a long narrow
slit in various stress fields. Figures 7.6 and 7.7 show
stresses and displacements respectively for the circular
hole problem. The pre-mining stress components for this
problem were taken as px = 0.397p, p = 0.429p, pz = 0.924p,
pxy = 0.116p, pyz = 0.208p, pzX =-0.042p, where p = 0.01G,
and G is the Modulus of Rigidity of the material. The value
of Poisson's Ratio used was v = 0.25. Thirty-five boundary
elements were disposed around the complete circumference of
the opening. Figure 7.6 shows that the principal stress
magnitudes calculated using the Boundary Element program
are practically identical with those determined from the
25
iyO2/P(on Z=0.0)
— ANALYTICAL SOLUTION o COMPUTED BY
B.E. METHOD
2.0 X PX = 0.379P
Py =0.429P PZ =0.924P PXy =0.116P PyZ = 0 208P PzX = 0 042P
61IP (on Z =0.0) 1 .0
186
b)
-2.0 3.0
Distance from centre of hole (Z/R)
-4.0
000
°3/P (on Z =0.0) t 1
2.0 3.0 Distance from centre of hole (X/R)
(a)
4.0
FIGURE 7.6: STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A TRIAXIAL STRESS FIELD
4.0 1.0 2.0 3.0 U/R o Distance from centre of hole (XI R)
x 1R x 10 -3(on Z= 0.0) -0.5
-1.0 y%R x10-3 (on Z= 0.0)
-12 (a)
6.0
Uz /R x 10-3 (on X: 0.0)
0
Ux /Rx10 3 (on X=0-0)
(b)
4.0
U/R
2.0
0
-2.0
0.5 Uz /R x10-3 (a►.) Z = 0.0)
FIGURE 7.7: DISPLACEMENT DISTRIBUTION AROUND A CIRCULAR HOLE IN A TRIAXIAL STRESS FIELD. (PROBLEM PARAMETERS ARE AS DEFINED IN FIGURE 7.6)
187
188
analytical solution for the problem, both at the boundary
of the opening and at internal points in the medium. Figure
7.7 indicates that displacement components determined with
the Boundary Element program agree very closely with the
values calculated from the closed form solutions, for both
the excavation boundary and internal points, except for
the minor discrepancy in the computed values of ux along the
X axis for the excavation. The discrepancy seems to be of
little consequence, since it is not reflected in any of
the calculated stress components. Similar minor discrepancies
were noted in the computed values of displacements using
the indirect formulation described in Chapter 4.
For the thin slit problem, it has been found that for
excavations with width/height ratios of 10:1 and 20:1, the
variation of displacements around excavation boundaries and
the stress distribution in the medium approach those
calculated from the closed form solution for an infinitely
thin slit. Figure 7.8 shows stress and displacement
distributions around a 20:1 slit in a uniaxial stress field
pz, of magnitude 0.01G. In Figure 7.8(a), good correspondence
is observed between the u2 displacement component over the
excavation calculated using the Boundary Element program and
the elliptical distribution of uZ corresponding to the
closed form solution. Figures 7.8(b) and 7.8(c) show
convergence of the analytical and computed distributions
of the ax and az stress components in the plane of the mid-
height of the opening, and along the central normal to the
opening. Figure 7.9 shows displacement and stress distribu-
tions around a 20:1 slit in a longitudinal shear field,
with pyZ = 0.01G. The correspondence between the Boundary
Element and analytical distributions is regarded as acceptable.
7.8 Use of Higher Order Singularities in the Boundary
Element Algorithm
The bases of the direct formulation of the Boundary
Element Method are the Reciprocal Work Theorem, and
0.4
-u C
x 10- 2 (on Z=0.05C)
0.2
0
1 .0 a/
PZ
189
1 1.—______ 2C ,..I 0.75 ■ :~ O.iC - T 1 1
T w X
0.6
O B.E.Solution
Closed Form Solution
0.2 0.4 0.6 0.8 10
Distance from Centre of Excavation (x/C)
FIGURE 7.8(a)
2.0
1.5
0.5
0.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.7
3.0
Distance from Centre of Excavation (x/C)
FIGURE 7 .8 (b)
a/ Pz
O
0.6 0.2 0.4
1.0
0.8
0.6
0.4
0.2
1.0 1.2 1.4 1.6 1.8 2.0
Distance from Centre of Excavation (z/C)
0.0
-0.2
-0.4
x/pz (on x=0.0) a
-0.6
-0.8
FIGURE 7.8(c) -1.0
190
FIGURE 7.8: DISPLACEMENT AND STRESS DISTRIBUTIONS AROUND A NARROW EXCAVATION IN A UNIAXIAL STRESS FIELD
2C _t
0.1C I ` X T
Z
0 B.E.Solution
Closed Form Solution
0
Tyz/pyz (on Z=0.0)
1.6
1.4
1.2
Tyz/p 1.0
yz 0.8
0.6
Tyz/pyz (on x=0.0) 0. 4
0.2
0.0
2.0
1.8
191
1.2
1 .0
0.8
-u 0.6 x
10_2
(on Z=0.05C)0.4
0.2
0.2 0.4 0.6 0.8 1.0
Distance from Centre of Excavation (x/C)
(a)
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Distance from Excavation Surface (Ax/C, Az/C)
(b)
FIGURE 7.9: DISPLACEMENT AND STRESS DISTRIBUTIONS AROUND A NARROW EXCAVATION IN A LONGITUDINAL SHEAR STRESS FIELD
1 .8 2.0
192
knowledge of particular (singular) solutions to the field
equations of elastostatics. The choice of the type of
singularity used to provide the perturbations of traction
and displacement to establish boundary contraint equations,
through application of the Reciprocal Work Theorem, is
perfectly arbitrary. The development of particular higher
order singularities, which performed satisfactorily in
indirect Boundary Element formulations for complete plane
strain, thin-slit problems, has been described in Chapter
5, and expressions for stresses and displacements induced
by these line quadrupoles and dipoles are given in
Appendix IV. The main property of these singularities that
might be exploited in a direct formulation of the Boundary
Element Method is the highly concentrated disturbance of
stress and displacement which they induce in the medium.
Higher order singularities have been used to set up
the boundary constraint equations in exactly the same way
as described in Section 7.2 using line load singularities.
For the plane problem, unit shear and normal quadrupoles
are applied at the centre of each boundary element, in
place of the unit line loads applied parallel to the local
X' and Z' axes for the element. For the antiplane problem,
a unit longitudinal shear dipole is applied at the centre
of the element, in place of the unit longitudinal line
load. All other aspects of the solution procedure are
identical to that described previously.
The performance of the program employing these higher
order singularities was assessed by comparing computed
boundary stresses around simple excavation shapes with
those determined from the closed form solution for the
problem. Table 7.1 compares boundary stresses around a
circular hole in a uniaxial field, calculated from the
analytical solution, and the Boundary Element solutions
using line load singularities and the higher order
singularities. Twenty-five elements were used to define
193
the excavation boundary for each Boundary Element solution.
The results indicate that the boundary stresses were
determined slightly less satisfactorily with the higher
order singularities than with the line load singularities,
particularly where the boundary stress gradients are high,
as in the excavation sidewalls. Increasing the number of
elements did not improve the agreement between the analytical
solution and the Boundary Element solution using higher
order singularities, which contrasts with the case when
line load singularities are used in the Boundary Element
algorithm.
Similar slightly inferior performance of the Boundary
Element program utilizing the higher order singularities
was observed in the solution of the antiplane problem
involving a circular hole in a unit longitudinal shear
field, and of the plane and antiplane problems for narrow
slits. The inference from these results is that a Boundary
Element algorithm employing the higher order singularities
may require better representation of functional variation
(i.e. of t and u) with respect to element intrinsic co-
ordinates than does one using the simple singularities. It
is probable that this is a direct result of the highly
concentrated disturbances in stress and displacement
associated with these singularities. In spite of these
shortcomings, a significant conclusion from the work was
confirmation of the principle that singularities of types
other than those used conventionally in direct formulations
might be used profitably for particular applications.
1 I 1
l f 1 I pz = 1.0
194
Table 7.1 : Comparison of Boundary Stresses around a
Circular Hole in a Uniaxial Field, Determined
from Closed Form Solution, and Boundary Element Program with Simple (S) and Higher Order (HO)
Singularities.
0(deg) 60 C.F.S. BEM(S) BEM(HO)
0.0 -1.000 -0.979 -1.029
14.4 -0.752 -0.722 -0.773 28.8 -0.071 -0.042 -0.066
43.2 0.875 0.902 0.916
57.6 1.851 1.878 1.930
72.0 2.617 2.644 2.725
86.4 2.984 3.009 3.105
100.8 2.860 2.885 2.976
115.2 2.276 2.301 2.369
129.6 1.375 1.401 1.435
144.0 0.381 0.411 0.405
158.4 -0.458 -0.438 -0.467
172.8 -0.937 -0.917 -0.964
7.9 Non-Homogeneous Media
7.91 Development of Boundary Constraint Equations
Figure 7.10(a) represents a cross-section through a
set of long, parallel openings excavated in non-homogeneous rock. The medium consists of the primary (infinite)
domain (denoted I) and inclusions (II and III). Each
region, i.e. the infinite domain and the inclusions, is
assumed to be homogeneous, isotropic and linearly elastic,
and the interfaces between the infinite domain and the
inclusions are welded. For the sake of simplicity, the
infinite domain and the inclusions are considered to be
at the same initial stress state, but this choice is
arbitrary . Any state of initial stress in each region,
defined, for example, by direct measurement, can be
accommodated in the analysis with equal facility.
Consider the antiplane problem initially, with the
final boundaries of excavations subject to imposed final
tractions t,,. The problem may be resolved into four
components, consisting of the uniformly stressed medium,
and three homogeneous infinite continua, stress free at
infinity, corresponding to the primary domain and the
inclusions, illustrated in Figures 7.10(b),(c),(d). In
the real problem (Figure 7.10(a)), the surface of an
excavation in the primary region is denoted Si, while the
195
Material Type I
/
/ ~
T`
r52
/ Tx
Tz
~' ?
:mir J
Liz UZ
196
X
i
(0 )
(b)
(c )
(d)
FIGURE 7.10: PROBLEM SPECIFICATION FOR A NON-HOMOGENEOUS MEDIUM
197
interfacial surfaces are denoted S2 and S. In Figure 7.10(b), surfaces S1, S2, S3 which are geometrically
identical to these real surfaces, are shown inscribed in
a continuum of material type I. The real surface of an
excavation in the inclusion of material type II is
denoted S4 in Figure 7.10(a) and its trace inscribed in a
continuum of material type II, shown in Figure 7.10(c),
denoted S. The prime is used to indicate that the normal to this surface at any point is directed in the opposite
sense to that of the surface S2 in Figure 7.10(b). In a.
similar way, the inclusion of material type III and its
excavation are represented in Figure 7.10(d) by the surfaces
S3 and S5 in a continuum of this material.
The aim is to establish a single boundary constraint
equation for the combination of the three problems illus-
trated in Figures 7.10(b),(c),(d). For simplicity in the
development, each surface is represented by a single
element (numbered in the same way as the surfaces),
although in practice a number of discrete boundary elements
would be disposed around each surface. It is assumed that
induced traction and displacement components ty, uy are
uniform over each boundary element. By applying unit
line loads to each element in a region and by successive
application of the Reciprocal Work Theorem, the boundary
constraint equation can be constructed for each region.
For the problem
7.10(b)), the equation
-
FYI FYI FYI
Y1 Y2 Y3
FY2 FY2 FY2
Y1 Y2 Y3
FY 3 FY 3 FY3
Y1 Y2 Y3
in
- u -
Y1
u Y2
u Y3
the infinite
is
=
domain,
UI y1
UIY2 yl
UIY3 Y1
Region I (Figure
UI Y1 Y1 YI
Y2 UIY3
UIT UIY2
Y2 Y3
UIY 3 UIY 3
Y2 Y3
tY1
t Y2
t Y3
(7.14)
198
7.10(c),
For the problem
the boundary
FY2, FY 21 Y2' y4
FY4 FY4 Y2 y4
An equation similar
in Region
constraint
uY2,
u Y4
to
II,
equation
equation
defined by Figure
is
UIY2 , UI1'21 Y2 y4
UIY4, UI1'4 Y2 Y4
(7.15) can be
tY2 ~
tY4
(7.15)
established for the surfaces S 3' and S5 in Region III,
defined by the problem illustrated in Figure 7.10(d).
At the interfaces between the infinite domain and
the inclusions, neither induced displacements uy2, uy3,nor
induced tractions ty2 ty3,are defined. The conditions to
be satisfied on the interfacial surfaces are that continuity
of displacement be maintained across the interface, and
that the net induced traction at the interface is zero;
i.e. for the interface elements
U Y2
u Y2
ty2, _ - tY2
etc.
Taking account of these conditions for all interface
elements, equations (7.14), (7.15) and the boundary constraint
equation for Region III can be combined and recast to give
the boundary constraint equation for the complete antiplane
problem:
UI171 71
UI172 71
UIy2 71
0
0 0
o 0
UIy2 0 74
0
(7.16)
t71
0
0
0
0
t 74
t ys
UIY4 74
0
199
FYI -UIYI FYI -UIY'
FYI 0 0 uyl Y Y Y Y Y
Fyi —UIY2 FY2
—UIy2 Fy2
0 0 ty2 71 Y Y Y Y
FY3 —UI y2 FY3 —UIY3 FY3 0 0 u Y2 71 y Y Y y3
0 UIy2IF721 0 0 F74 0 t173
0 UIy2,Fy2, 0 0 Fy4 0 u73
o
F75 uys 0
0 0 UIy3,Fy3, 0
0 0 UIY3,F73, 0
F731 u
74
Equation (7.16) may be written
[Fy] [uy] = [illy] 5:y ] (7.17)
200
Inspection of equation (7.16) shows that the [Fy]
and [Uiy] matrices are horizontally banded, corresponding
to blocks of equations describing the problem in each
region. It is seen that two constraint equations are
written for each element defining an interface between
the infinite region and an inclusion, and this significantly
increases the order of the system of equations to be solved
when dealing with non-homogeneous media.
The boundary constraint equation for the plane problem
is constructed in the same way as for the antiplane problem,
and is similar to equation (7.16). The order of the system
of equations is twice that for the antiplane problem, and
the complete boundary constraint equation may be written
[F] [u] = [UI] [t ] (7.18)
7.92 Computational Procedures
The procedure followed for the solution of problems
involving a non-homogeneous medium is similar to that
followed for a homogeneous medium. Using the known
geometry of the excavation and inclusion boundaries, discrete
linear elements are generated to represent the surfaces
S1 - S5 in Figures 7.10(b),(c),(d). The vectors [t] and
[ty] of known boundary values are then constructed, taking
account of the imposed final boundary conditions, the pre-
mining stress field and the orientation of the boundary
elements. The terms of the [F], [UI], [F ] and [UIy] matrices are obtained by integrating tractions and dis-
placements induced by unit line loads, applied at the centre
of each element in the various co-ordinate directions and
in each problem region, over the range of each element.
The matrices [F], [UI], [F ], [UI ] are constructed in
blocks, region by region, as suggested by the horizontally
banded structure of the matrices apparent in equation
(7.16). Considering the plane problem as an example, for
201
excavation boundary elements the terms of the type FXV
UIX~ are disposed in the [F] or [UI] matrices, with a
negative sign introduced if necessary, depending on
whether surface tractions or displacements are specified
on element j. For interface boundary elements, all
terms FXV, UIx are disposed in the [F] matrix, due account xj
being taken of the sense of the normal to the element and
the problem region being examined in calculating the
coefficient. The terms of the [F] matrix are then written
row by row to a random access (mass storage) file, with
blocks of zero terms being incorporated as required. The
rows of the [UI] matrix are multiplied immediately after
they are generated by the [t] vector, to establish the
known RHS vector of the set of equations. The set of
equations is then solved by Gaussian elimination.
After solution of the boundary constraint equations,
the values of the excavation-induced tractions (tx, ty, tz)
and the excavation-induced displacements (ux, uy, uz)
are known at the centre of each excavation boundary element,
and each element defining the interface between the
infinite domain and the inclusions. Using the techniques
described previously for a homogeneous medium, stresses at
excavation boundaries and at the interfaces between
inclusions and the infinite domain are determined from
these quantities, taking due account of the elastic properties
of the region being considered.
Displacements and stresses at internal points in a
particular region are determined from the displacements
and tractions on elements defining excavation boundaries
in the region, and interface elements defining the region.
Thus the induced displacement at a point i in a region is
obtained from equation (7.9), where the integrals are
evaluated over the range Sj of each element j, and n is
the total number of elements defining the interfaces and
excavation boundaries for the region. Induced stresses
at a point in a region are calculated from the induced
202
strains, determined in the same way as for the homogeneous
medium, using the appropriate elastic constants for the
region.
7.93 Validation of Solution Procedure for Non-
Homogeneous Media.
The performance of the program in the analysis of
problems involving.non-homogeneous media has been evaluated by
considering as trial problems firstly, a solid cylindrical
inclusion in an infinite medium subject to combined biaxial
(plane) loading and antiplane loading, and secondly a
circular opening in a circular inclusion in an infinite
medium subject to biaxial loading.
For the solid inclusion, the solution to the plane
problem is quoted by Jaeger and Cook (1976). Within the
inclusion, the analytical solution indicates that the state
of stress is homogeneous, while outside the inclusion, the
stress components vary according to modified forms of the
Kirsch Equations. The solution to the antiplane problem
was not available in the literature, and has been obtained
by exploiting the analogy between displacement and stress
for the elastostatic problem and potential and velocity
for the hydrodynamic problem. Bray (1977) has examined the
problem of fluid flow through an isotropic infinite medium
containing an isotropic circular inclusion with permeability
different from that of the infinite medium. Using the
solution to this problem, it has been found that for field
stress components pxy and pyZ in the infinite medium,
stresses in and around the inclusion are given by the
following expressions:
(a) within the inclusion:
2G o
Txy G+Go Pxy
2G o p Tyz
G+G yz
203
(b) outside the inclusion:
Txy = pxy (1 - t a2 cos26) - pyz t a sin20 r2 r2
2
Tyz = - pxy t a2 sin20 + pyz (1 + t
a cos20)
r r2
where G o = Modulus of Rigidity of the inclusion
G = Modulus of Rigidity of the infinite medium
G - Go t =
G + Go
a, r, 0 are as defined in Figure 7.11.
Figure 7.11 compares the distributions of stresses
within the solid circular inclusion, and in the medium
surrounding the inclusion, calculated using the Boundary
Element program and determined from the closed form
solutions. The parameters for this test problem were:
Modulus of Rigidity for the infinite domain, G, 100.0 pz;
Modulus of Rigidity of the inclusion, Go, 20.0 pz;
Poisson's Ratio for both domains 0.25; ratio of field
stress components Px :pz :Pyz = 0.6:1.0:0.8, and other
stress components zero. Figure 7.11(a) shows the variation
of stresses along the X axis for the problem. It is noted
that within the inclusion, the calculated stresses are
effectively homogeneous (except near the interface), and
close to the values predicted from the closed form solution.
The discontinuities in the az and Tyz stress components
required at the boundary by the closed form solutions are fairly wellmatched by the results from the B.E. analysis. The main discrepancy between the predicted and calculated
values may be due in part to the rather crude method used
for calculating the tangential component of strain at a
boundary. Similar conclusions apply to the variation
of stresses along the Z axis for the problem, as shown in
Figure 7.11(b); i.e. the state of stress in the body of
the inclusion is effectively homogeneous, and the
1 4
12
1
1 6
08-
CrX (on Z- 0.0) 0 3 c 06-
04Q-° 0
ŌZ (on Z=0.0)
0" 0 0 0 fl • "Tāz(on Z-0.0)
0.0 10 20 30 Distance from centre of inclusion (Xla)
(a1 10-
08-
4.0
G-z (on X- DO)
06-
0
0.4 Q v• v 0 0 Yy (on X_0.0)
02 d(on X:0 0)
8 8 8
_ 0 0
o B.E. Solution
— C.F. Solution
0 (on Z- 0 0)
a-
0.0 10, 20 30 Distance from centre of inclusion (Z/a)
(b)
FIGURE 7.11: STRESS DISTRIBUTION IN AND AROUND A SOLID CYLINDRICAL INCLUSION IN A TRIAXIAL STRESS FIELD
40
205
calculated variation of stress components in the infinite
domain matches fairly well with that predicted from the
closed form solutions.
For the second validation problem, Savin (1961)
provides a closed form solution for the problem of an
infinite medium subject to uniaxial stress containing a
circular inclusion, in which there exists a concentric
circular hole. The material properties used for this test
problem were G = 100.0 px, Go = 20.0 px, Poisson's Ratio
0.25 for both regions, and apart from the defined value of
px, the other five stress components were zero. Figure
7.12(a) compares the calculated stress variation along the
X axis with the variation predicted from the closed form
solution, and Figure 7.12(b) shows the comparison along the
Z axis. In both cases the agreement between the calculated
and predicted stress distributions is regarded as satisfactory.
Continuity of stress in the radial direction and the dis-
continuity in tangential stress between the inclusion and
the infinite domain are reproduced adequately by the
Boundary Element analysis, for both reference directions.
7.10 Appraisal of Boundary Element Direct Formulation.
The validation problems analysed with the Boundary
Element program, described in Section 7.7, indicate that
the direct formulation can handle a wide range of problem
geometries with equal facility.
This consideration alone suggests that for the
particular assumptions made in this work in the development
of the different solution procedures, the direct formula-
tion is preferable to the indirect formulation, in which
different singularities must be introduced explicitly to
allow analysis of narrow, parallel-sided slits. It is
possible that better performance by the indirect formula-
tion,in the analysis of extreme problem geometries,could
be achieved by allowing variation of fictitious load
0 51- 206
z/Px (on Z=00 — Closed form solution o Boundary element solution -04
-06-
x px (on Z=0.0)
R2 2 R1 X
Distance from centre of hole (X/R, ) 3.0
20
06
04
lpx •0
-02
(a) -08-
25
2.0
1•0 crxiPx (on X:0-0
0-z/Px (on X =0 0)
0
10
20 30
4 0 Distance from centre of hole (Z/R, )
(b)
FIGURE 7.12: STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A CIRCULAR INCLUSION IN A MEDIUM SUBJECT TO PLANE STRAIN
207
intensity with respect to element intrinsic co-ordinates.
However, the author's opinion is that, since in the direct
formulation a problem is solved in terms of the real problem
parameters of tractions and displacements on boundary
elements, there should be less risk with this formulation
of obtaining spurious solutions to problems. In spite
of this, there is still a degree of arbitrariness in the
direct formulation. In the current work, unit line loads
have been used to provide the second load case for
application of the Reciprocal Work Theorem, for the
establishment of boundary constraint equations as quadrupole
singularities have been found to perform somewhat less
satisfactorily. It is possible that for some types of
problems, and for imposed higher order variation of traction
and displacement with respect to element co-ordinates,
these singularities may provide solutions superior to
those obtained using the conventional singularities.
Gaussian quadrature has been used throughout the
program for the direct formulation, for the evaluation of
integrals of tractions and displacements induced over
boundary elements by unit line loads. This resulted in
significant improvement in execution time for the program,
compared with the case where integrals were determined from
analytical expressions, with no apparent loss of accuracy
in solution. The reason for improved efficiency is that
the expressions for the integrals obtained analytically
all involve trigonometric quantities, the computation of
which is a relatively slow machine operation. The use of
Gaussian quadrature, and the different equation solvers
used in the programs for the direct and indirect formulations,
precluded direct comparison of the relative computational
efficiencies of the two methods. It is noted, however,
that the direct formulation requires more central memory
than the indirect formulation. This is associated with
the large block of memory required by the Gaussian
elimination equation solver used in the direct formulation.
CHAPTER 8
CHAPTER 8 : MINE DESIGN APPLICATIONS OF THE BOUNDARY ELEMENT METHOD
8.1 Preliminary Considerations
The preceding chapters have been concerned with the
development and validation of various formulations of the
Boundary Element Method. Methods of analysis have been
established and programs written to allow determination of
stress and displacement distributions in supported mine
structures in both tabular and massive orebodies.
In this chapter, the significance of the antiplane
component of complete plane strain analysis is assessed, a
method for estimating the stability of mine pillars is des-
cribed, and an analysis is undertaken of a stoping block at
the Mount Isa Mine, Australia, to demonstrate design appli-
cations of Boundary Element methods. The direct formulation
of the method is used for these studies, due to the versa-
tility provided by its capacity to handle a wide range of
excavation shapes.
8.2 Design Problems Requiring Complete Plane Strain Analysis
It has been noted earlier that a basic requirement in
conventional plane strain analysis, that a principal stress
direction be parallel to the long axis of the excavation,
will not be satisfied generally. However, it is clear that
for pre-mining stress fields that are hydrostatic, or
approach this state of stress, antiplane stress components
either vanish or are insignificant, and conventional plane
strain analysis is completely adequate for determination of
the stress distribution around openings with any orientation
in the field. These considerations suggested that it was
worthwhile to establish the conditions under which conven-tional plane strain analysis is inadequate, and complete
plane strain analysis required, to determine the stress
distribution around an opening.
208
209
Factors to be considered in determining the significance
of the antiplane problem in plane strain analyses are the
field principal stress ratios, the orientation of the long
axis of the excavation relative to the principal stress
axes, and the shape of the opening. Several hundred analyses
have been conducted, varying these parameters in turn. The
differences between the plane strain and complete plane
strain solutions to a particular problem have been assessed
by comparing excavation boundary stresses determined using
the two methods. The procedure followed was to select par-
ticular pre-mining principal stress ratios and excavation
geometry, and to vary the orientation of the axis of the
opening in one of the principal planes. The situation is
illustrated in Figure 8.1. This introduced only one anti-
plane component when the pre-mining stress field was trans-
formed to the excavation local axes. It allowed the state
of stress in the sidewall of the excavation to be used for
direct evaluation of differences between the results of com-
plete plane strain and conventional plane strain analyses.
A circular hole and an elliptical hole with major axis/minor
axis ratio of five were used as the trial excavations.
X CO
Z
(a)
(b)
FIGURE 8.1: PROBLEM GEOMETRY FOR ASSESSING THE SIGNIFICANCE OF THE ANTIPLANE COMPONENT OF COMPLETE PLANE STRAIN
Table 8.1 compares principal boundary stresses in the
sidewalls of these openings, for a range of field principal
stress ratios, when the long axis of the opening is parallel
210
or sub-parallel to the intermediate or minor field principal
stress. The significance of the antiplane problem for any
analysis is indicated in the table by the ratio of the anti-
plane stress component to the average of the plane prin-
cipal stresses. Inspection of the results for the cir-
cular hole indicates that differences exceeding 5per cent exist
between the results of plane strain and complete plane
strain analyses when the antiplane stress is greater than
about 0.35 times the average plane principal stress. For
the range of field principal stress ratios which might be
encountered in practice, this occurs when the axis of the
excavation is inclined at an angle greater than 20° to a
field principal stress direction. Examination of the
results for the elliptical hole given in Table 8.1 shows
that the high stresses associated with the plane problem
tend to mask the significance of the antiplane problem.
However, it is suggested that when the antiplane/plane
stress ratio is greater than about 0.35, the differences
between the conventional plane strain and complete plane
strain results are sufficiently great not to be ignored.
Finally, it is observed that when the ratio of the field
principal stresses, p, : PZ„ is 0.75, the differences between the plane strain and complete plane strain results
are insignificant. This suggests that if both the inter-
mediate and minor field principal stresses are greater
than 0.75 times the major field principal stress, the
antiplane problem may be ignored, for any excavation geo-
metry.
In Table 8.2, the results of conventional plane
strain and complete plane strain analyses are compared for
the case where the excavation axis is sub-parallel to the
major principal field stress. The results are in general
agreement with the conclusions stated above, but also
reveal an inconsistency for one particular problem. For
the circular hole, field principal stress ratios 0.6: 1.0:
0.5, and hole axis inclined at 20° to the major principal
field stress, the conventional plane strain analysis sug-
gests that the maximum boundary stress occurs in the crown
2.41
2.37
2.24
2.04
1.79
2.41
2.38
2.30
2.17
2.00
0.00
0.11
0.21
0.30
0.36
2.41
2.39
2.32
2.22
2.10
2.41
2.39
2.34
2.26
2.15
(°A) p (°A) c
10.56
10.32
9.59
8.48
7.11
10.56
10.39
9.87
9.04
7.95
Table 8.1 Sidewall Boundary Stresses for Circular and Elliptical Holes in a
Triaxial Stress Field, determined by Conventional Plane Strain and
Complete Plane Strain Analyses. Hole Axis Sub-Parallel to Intermediate
or Minor Principal Stress Direction.
Px1 : Pyz pz'
0.6 : 0.0 1.0
Px' . Py' Pz'
0.6 . 0.25 : 1.0
Px' . Py' • Pz'
0.6 . 0.5 : 1.0
Px' • Py' Pz'
0.6 . 0.75 1.0
CIRCULAR HOLE
Pyz
(Px+Pz)
00 10°
20°
30°
40°
0.00
0.16
0.32
0.45
0.57
Pyz (0A) p (0A) c 'i (Px+Pz)
2.41 2.41 0.000
2.32 2.37 0.22
2.06 2.27 0.43
1.66 2.09 0.64
1.17 1.85 0.82
(aA ) p (QA) c
2.41
2.41
2.34
2.38
2.15
2.28
1.85
2.12
1.48
1.90
(°A)p (°A)c Tpx P ) (°A)p (°A)c Pyz
-~TPZ)
0.00
'0.05
0.10
0.14
0.16
ELLIPTICAL HOLE
(°A) p (°A) c Pyz
'2(Px+Pz)
10.5E 10.56 0.00
10.23 10.36 0.22
9.26 9.76 0.43
7.78 8.77 0.64
5.96 7.47 0.82
Pyz
"Px+Pz)
0.00
0.16
0.32
0.45
0.57
(cA) p (°A) c Pyx
"Px+Pz)
10.56 10.56 0.00
10.40 10.44 0.11
9.92 10.04 0.21
9.17 9.42 0.30
8.26 8.62 0.36
(°A)p (c°A)c Pyz
'1 (Px+Pz)
10.56 10.56 0.00
10.48 10.49 0.05
10.24 10.27 0.10
9.87
9.93
0.14
9.42
9.50
0.16
00
10°
200
30°
40°
Note: (a) Axes x', y', z', x,y,z and the orientation of the hole axis, 1P, are defined in Figure 8.1
(b) (OA)p is sidewall stress at position A, defined in Figure 8.1 determined by conventional plane strain analysis. (c) (OA)c is sidewall stress determined by complete plane strain analysis.
(d) The major axis of the elliptical hole is parallel to the x-axis. N
IJ
0° 10° 20° 30° 40°
Table 8.2 Boundary Stresses for Circular and Elliptical Holes in a Triaxial Stress Field, determined by Conventional Plane Strain and Complete Plane Strain Analysis. Hole Axis Sub-Parallel to the Major Principal Stress Direction.
Px': Py': Pz' 0.6: 1.0: 0.0
Px': Py': Pz' 0.6: 1.0: 0.25
Px': pyI : pz.
0.6: 1.0: 0.5 Px': Py': Pz' z 0.6: 1.0: 0.75
CIRCULAR HOLE
(aA) n (a )c B a
11) (aA) p (aA) c aB Pyz
':(Px+Pz) Pyz
A 1/2(Px+Pz) Pyz
(aA)p (°A)c aB 11(px+pz) (aA) p Pyz (GA) c GB 1/2 (Px+Pz)
0°
10°
20°
30°
40°
-0.59 -0.59 1.80 0.00
-0.50 -0.59 1.77 -0.54
-0.24 -0.58 1.68 -0.89
0.16 -0.52 1.55 -1.01
0.50 -0.41 1.39 -0.97
0.83 0.83 1.56 0.00
0.82 0.91 1.53 -0.30
0.78 1.12 1.47 -0.51
0.73 1.38 1.37 -0.62
0.67 1.65 1.25 -0.64
0.95 0.95 1.31 0.00
0.96 1.12 1.29 -0.16
1.08 1.34 1.25 - -0.28
1.29 1.56 1.19 -0.36
1.53 1.79 1.11 -0.38
1.66 1.66 1.06 .0.00
1.68 1.70 1.06 -0.06
1.75 1.78 1.03 -0.12
1.85 1.90 1.00 -0.16
1.97 2.03 0.96 -0.17
(°A)p (aA)
-0.59 -0.59
-0.25 -0.90
0.88 2.75
2.20 4.32
4.02 5.95
Pyz il(Px+Pz)
0.00
-0.54
-0.89
-1.01
-0.97
( GA) p (aA ) c
2.20
2.20
2.46
2.86
3.18
4.02
4.29
5.33
5.66
6.68
ELLIPTICAL HOLE
Pyz 12(Px+Pz)
0.00
-0.30
-0.51
-0.62
-0.64
(aA) p (aA) C
4.99 4.99
5.16 5.24
5.64 5.89
6.39 6.76
7.30 7.70
Pyz 11 (Px+Pz )
0.00
-0.16
-0.28
-0.36
-0.38
(aA ) p (aA) c
7.78
7.78
7.86
7.88
8.11
8.15
8.48
8.55
8.93
9.02
Pyz '1(Px+Pz)
0.00
-0.06
-0.12
-0.16
-0.17
Note (a) Principal stress axes, hole local axes and other geometric parameters are as defined in Table 8.1 (b) aB is the boundary stress at position B, defined in Figure 8.1 n~i
of the opening. The complete plane strain analysis indi-
cates that the maximum boundary stress occurs in the side-
wall of the excavation. The antiplane/plane stress ratio
for the problem was 0.28.
The generalization drawn from these results is that
if the absolute value of either antiplane/plane stress
ratio (i.e. for the reference axes used here -=i p z k(p +p )
or ~ (p
y+pz) is greater than 0.35, a complete plane strain analysis is required to ensure satisfactory determination
of the stress distribution around an opening in the medium.
In general, the converse is not true. For most cases it
appears that conventional plane strain analysis may be
adequate when the antiplane/plane absolute stress ratio is
less than 0.35. Particular situations, determined by exca-
vation geometry, field principal stress ratios and excava-
tion orientation, may occur which satisfy this condition
but for which failure to take account of the antiplane
problem will lead to a misleading determination of the
stress distribution around the opening.
8.3 Study of Pillar Stability
8.31 Objectives and Scope of Study
The criterion for stability of a pillar under mining
conditions in which the strength of the rock is exceeded
in the body of the pillar has been discussed in Chapter 1.
The aim in this section is to establish and assess a
method of estimation of pillar stability, based on the
Boundary Element Method of analysis.
A mine may be regarded conveniently, if somewhat
simplistically, as an assembly of structural elements con-
sisting of pillars, abutments and the country rock. The
requirements are to demonstrate that the statically indeter-
minate structure can be resolved into these elements,
213
214
implying that continuity of traction and displacement can
be satisfied at interfaces between the elements, and to
establish the load - convergence performance characteristics
of the elements. Figure 8.2(a) illustrates an idealised
cross section of part of a stoping block. It is resolved
into its component elements in Figure 8.2(b). Assuming
that a pre-mining stress is normal to the orebody, and
ignoring any transverse tractions which are induced over the
interfaces between support elements and the country rock,
the performance characteristics of the pillar, abutment, and
the country rock at these locations are illustrated in
Figure 8.2(c). It is noted that the effective width of the
abutment, Wa , is undefined, as are the distributions of
load and convergence at the support positions. These
issues are resolved below.
The parameters required for the assessment of pillar
stability are the stiffness of the pillar in the failing
regime, and the mine local stiffness. It is proposed to
estimate post-peak pillar stiffness (X') from pillar stiff-
ness ( A) in the elastic regime. This is obtained from the
slope of the load-convergence characteristic of the pillar;
i.e.
_ AP AS
Mine local stiffness (k1) is defined by the slope of
the load-convergence characteristic for the country rock at
the pillar position; i.e.
k1 AP _ AS
Mine local stiffness is, therefore, positive by
definition.
It is sometimes convenient to describe pillar and
Pillar Performance Characteristic Pillar Load
P
Mine Abut-
local ment
.C. Load P a
Abutment Performance Characteristic
Mine Local P.C.
Pillar
215
(a)
W -~ a
63:132,
(b)
Convergence at Pillar Position S
(c)
Abutment Convergence Sa
FIGURE 8.2: REPRESENTATION OF INTERACTION BETWEEN COUNTRY ROCK AND PILLAR AND COUNTRY ROCK AND ABUTMENT IN A SUPPORTED MINE STRUCTURE
mine stiffness properties in terms of deformation moduli.
Considering a long pillar of width W and height H, and
taking a slice of unit thickness through the pillar,
pillar deformation moduli in the elastic and post-peak
regimes and the mine local modulus are defined by:
E = A W
E P
H K1 = kl W
Comparison of mine local stiffnesses at different
pillar positions for different pillar widths may be made
using the normalised mine local stiffnesskla ,,where
k1
kla W
In order to demonstrate the determination and appli-
cation of these quantities, the extraction has been
examined of an 8m. thick tabular orebody using long rooms
and rib pillars. The orebody and country rock have the
same elastic properties, with Young's Modulus 50GP a and
Poisson's Ratio 0.25. The pre-mining principal stresses
were px = 9MPa, p = 6MPa, pz = l2MPa, where the Zaxis is
perpendicular to the plane of the orebody, and the Y axis
parallel to the long axis of excavation. These conditions
were chosen to simulate excavation of an orebody in hard
rock at a depth of approximately 450 m. below ground surface.
8.32 Methods of Estimation of Mine Local Stiffness and Pillar Stiffness
Figure 8.3(a) shows the 8m. thick orebody, in which two stopes, each of span Ss, have been excavated to generate
a 12m. wide central pillar. To estimate the mine local stiffness, it is necessary to determine convergence of the
216
country rock at the pillar position for various magnitudes
of the distributed pillar load.
This has been done by applying uniformly distributed
loads of various magnitudes P at the pillar position, as
shown in Figure 8.3(b) and calculating the displacement
distributions under the loaded strips using the Boundary
Element program. Convergence at the pillar position due
to the strip loads may be taken as the average convergence
over the pillar width, or the convergence at the vertical
centre line through the pillar position.
217
S S •T. 12m
=1 T 8m
1
(a)
It t t t a
II4 4 441 (b)
FIGURE 8.3: APPLICATION OF UNIFORMLY DISTRIBUTED LOAD AT A PILLAR POSITION TO DETERMINE MINE LOCAL STIFFNESS
In determining the load-convergence characteristic for the
pillar, the pillar axial load P for particular adjacent
stope spans has been obtained by calculating the stress
distribution across the midheight of the pillar, and
integrating the az stresscomponent across the width of
the pillar using the trapezoidal rule. Pillar convergence
may be determined from the mining induced uz displacements
across the ends of the pillar. The representative value
of the pillar convergence may be taken as the average
convergence across the pillar, or the convergence across
the vertical centre line of the pillar. Repetition of
this procedure for a number of stope spans provides the
information to establish the pillar performance character-
istic.
The appropriate convergences to be used in
establishing the performance characteristics for the
pillar and the country rock may be decided by taking
account of the requirement for displacement continuity
at the pillar - country rock interface. Figure 8.4
shows plots of load versus convergence of the country
rock at the centre of the loaded strip for 15m, 20m and
30m stope spans, and pillar load versus central
convergence across the pillar. It is observed that
the intersections of the country rock performance
characteristics,for the various stope spans, with the
pillar performance characteristic do not correspond to
the actual load -convergence equilibrium positions for
the pillar. This means that these methods of estimating
the performance characteristics of the pillar and the
country rock do not result in proper coupling at the
pillar/country rock interface.
In Figure 8.5, the pillar performance characteristic
has been obtained by plotting pillar load versus convergence
across the centre line of the pillar, while the country
rock performance characteristics for the various stope
spans have been obtained by plotting the applied strip
load magnitude versus average convergence under the loaded
strip. The figure indicates that the intersections of
the country rock performance characteristics with the pillar
performance characteristic are virtually coincident with
the calculated load - convergence equilibrium positions
for the pillar, as is required by the continuity condition.
A physical explanation of why the continuity relationship
is satisfied by establishing performance characteristics in
218
5 10 15 20 25
400
Load-Convergence Characteristics for Country Rock
Stope Spans A - 15m B - 20m C - 30m
Pillar Load-Convergence Characteristic
17
300
200
Load P (MN)
100
219
Convergence S(mm)
FIGURE 8.4: PILLAR AND MINE PERFORMANCE CHARACTERISTICS BASED ON CONVERGENCES AT THE CENTRE LINE OF THE PILLAR f'.
400
220
p Load - Convergence Characteristics
for Country Rock
Stope Spans A - 15m
B - 20m C - 30m
p Pillar Load - Convergence
Characteristic
300
200
Load P
(MN)
5
10 15
20
25
Convergence S(mm)
100
FIGURE 8.5: PILLAR AND MINE PERFORMANCE CHARACTERISTICS BASED ON CONVERGENCE AT THE CENTRE LINE OF THE PILLAR AND AVERAGE CONVERGENCE OVER THE LOADED STRIPS AT THE PILLAR POSITION
the way described is that it is the core of the
pillar which controls convergence of the country rock.
On the other hand, the work which would be done if the
pillar load, approximated here by a uniformly distributed
load, were gradually reduced to zero, involves
contributions from the displacements of all the elements
used to represent the loaded strip.
Mine and pillar stiffnesses estimated from the
performance characteristics in Figure 8.5 are given in
Table 8.3.
An alternative estimate of pillar stiffness can
be made by assuming that mining-induced stress and
strain in a pillar are homogeneous. For plane strain
conditions, pillar convergence is given by
S = H (1_v) . { (1+v) PV + px }
where Pi = mining-induced axial load
Pillar stiffness is therefore given by
_ E W (8.1)
(1-v2) H
For the case considered here, the calculated
pillar stiffness is 80GN/m , and the deformation
modulus 53.3 GPa. The fact that the pillar stiffness
and modulus determined from the Potindaty Element analysis
are higher than these values is taken to be indicative
of the confinement which develops in the body of the
squat pillar.
8.33 Stiffness Properties of the Abutment Area
In a properly designed mine pillar, the horizontal
stress component increases from zero at the pillar lines
to a maximum at the pillar core, as shown in Figure 8.6(a).
It is the effective confining stress in the pillar core
which results in the higher ultimate load capacity of a
221
222
TABLE 8.3 Mine Local Stiffness and Pillar Stiffness, for 12m Wide Pillar in 8m Thick Orebody
Stope Span (m) kl (GN/m) kla (GPa/m) K1 (GPa)
15.0 15.0 1.25 10.0
20.0 14.0 1.17 9.36
30.0 12.7 1.06 8.48
Pillar Properties
Pillar stiffness A = 99.9GN/m
Pillar deformation modulus E = 66.6GPa P
TABLE 8.4 Abutment Width and Mine Local Stiffness in Abutment Area, for Various Stope Spans (Figure 8.3(a) refers)
Stope Span (m) Wa (m) kl (GN/m) kla (GPa/m) K1 (GPa)
15.0 13.0 19.7 1.52 12.2
20.0 14.0 19.0 1.36 10.9
30.0 14.0 18.1 1.29 10.3
Abutment Properties
Stiffness A = 183.1GN/m
Deformation Modulus E = 107GPa P
TABLE 8.5 Pillar and Mine Stiffness Properties in Stoping Blocks with Various Pillar Widths and Width/Height Ratios, at Constant Extraction Ratio of 75% (Figure 8.8 refers)
W (m) W/H Ss (m) a (GN/m) EP (GPa) k1(GN/m) kla (GPa/m) K1(GPa)
4.0 0.5 12.0 27.3 54.6 11.7 2.93 23.4
8.0 1.0 24.0 59.0 59.0 11.7 1.46 11.7
12.0 1.5 36.0 96.5 64.5 11.8 0.98 7.9
16.0 2.0 48.0 142.5 71.2 11.9 0.74 5.95
pillar with high width/height ratio, when compared with
a pillar of the same width, but lower width/height ratio.
Crushing of the pillar, and consequent generation of a
potentially unstable state, requires the destruction of
the pillar core.
The existence of the peak in the distribution
of the ax stress component suggests a possible method
of defining the minūmum effective mine abutment. At
the end of a narrow stope excavated in an orebody with
the major field stress component directed.normal to the
plane of the orebody, the distribution of the induced
a stress component is as shown in Figure 8.6(b). By
analogy with an isolated pillar, if the distance from
the stope limit to the peak in the ax distribution is D,
the width of the effective abutment Wa may be taken
as 2D. The effective abutment may then be treated as
a pillar confined on one side.
Distribution of ox at Distribution of
Mid-Height of Pillar ox in Stope Abutment
223
X ln\ /r11
/A`
(a)
FIGURE 8.6: METHOD OF ESTIMATION OF THE EFFECTIVE ABUTMENT WIDTH
Effective abutment widths have been determined
for the simple stoping block illustrated in Figure 8.3(a).
The procedure described for an isolated pillar has
been used to determine the mine local stiffness in the
abutment area, and the abutment stiffness. Load-
convergence curves for the abutments associated with
the various stope spans are shown in Figure 8.7. The
Load P
(MN)
350
300
200
100
2 4 6
o Performance Characteristics for Country Rock in Abutment Area
Stope Spans A-15m B-20m C-30m
~ Abutment Performance Characteristic
8 10 12 14
Convergence Sa (mm)
224
FIGURE 8.7: ABUTMENT PERFOru~NCE CHARACTERISTIC AND MINE PERFORMANCE CHARACTERISTICS IN THE ABUTMENT AREA
16
1
,f/ mA/ 1 2m , —1E /7-1
1 (a)
a- 8m
I - 24m--- J
18m
I (b)
12 2m_ 36m —.1
I1 1 3 1 1 (Im (c)
i [..- 1 6m -+--- 48m 'i
1 I
I 1 I 1 1I8m (d )
IL FIGURE 8.8: STOPE AND PILLAR LAYOUTS IN A TABULAR OREBODY TO ACHIEVE AN
EXTRACTION RATIO OF 0.75
226
plots of abutment performance characteristic and
country rock performance characteristics indicate
correct coupling of the abutment and country rock.
Abutment and local stiffness properties
determined from Figure 8.7 are given in Table 8.4.
From Tables 8.3 and 8.4 it is noted that
normalised local stiffness in the abutment area
is approximately 20% higher,for any stope span,
than that at the location of the isolated pillar.
The abutment stiffness is approximately 60% higher
than the stiffness of the isolated pillar. These
results properly reflect the increased restraint
imposed in the abutment area by the adjacent country
rock.
8.34 Evaluation of Pillar Stability
In designing a stope and pillar layout, the
aim is to achieve a high extraction ratio while ensuring that unstable collapse of pillars cannot occur.
Pillar stability is assured if the mine local
stiffness (k1) at the pillar position and the post-
peak stiffness of the pillar (A') satisfy the
relationship k1 + X'>0 •
In terms of pillar and mine moduli, the criterion
for stability is K1 + Ep'>0
In order to assess how this criterion may be
satisfied in practice, a series of stoping layouts
was designed to achieve 75% extraction from the 8m
thick orebodydescribed earlier. Each stoping block
consists of six stopes and five pillars, as shown in
Figure 8.8. The width/height ratio of pillars varies
from 0.5 to 2.0, and stope spans vary from 12m to 48m
to provide the required extraction ratio.
Mine local stiffness and pillar stiffness in
the elastic range have been determined for the
central pillar in each block, using the methods
established in Section 8.32. Pillar and country
rock performance characteristics are shown in
Figure 8.9, and pillar and mine stiffness properties
are given in Table 8.5. For a pillar width/height
ratio of 0.5, the pillar deformation modulus is close
to the value of 53.3 GPa obtained by assuming a pillar
operates in homogeneous, uniaxial compression. The
increase in pillar deformation modulus with increasing
width/height ratio is the result of constraint imposed
on the pillar ends by the country rock. Mine
modulus is inversely proportional to the span supported
by the pillar.
Salamon (1970) has published a set of stiffness
coefficients, based on the assumption of uniaxial
compression of pillars, for a stope block similar
to that examined here. These can be used to assess
the validity of the methods used here for determining
mine local stiffness and pillar stiffness. The
relationship between mine local stiffness, pillar
stiffness and the stiffness coefficients is given by
equation (1.13) in Chapter 1. For the central pillar
in the block with a stiffness of 27.3 GN/m, the
calculated local stiffness is 11.9 GN/m, compared with
the value of 11.7 GN/m determined from the Boundary
Element analysis. These results imply that the methods
described here for the determination of both mine and
pillar stiffnesses are satisfactory.
In assessing the stability of pillars in the
stoping blocks illustrated in Figure 8.8, it is
necessary to estimate the post-peak stiffness of
227
100 A
..? B
0 54 6o 140 20 10 50 30
Pillar Widths
A 4m
B 8m
C 12m
D 16m
700
600
500
400
Load P
(MN)
300
200
I 1 0 a
1
228
O Pillar Performance Characteristic
Mine Local Performance Characteristic
Convergence S(mm)
FIGURE 8.9: CENTRAL PILLAR AND CORRESPONDING MINE PERFORMANCE CHARACTERISTICS FOR MINING LAYOUTS SHOWN IN FIGURE 8.8
pillars from experimental measurements of elastic/
post-peak stiffness ratios. Published data on
elastic/post-peak stiffness ratios is given in
Figure 8.10. The majority of the data has been
obtained by large scale tests on specimens of South
African coal, with side lengths of specimens up to
2m. The system of loading designed by Cook et al.
(1971) and used by Wagner (1974) involved cutting a
jacking slot at the mid-height of the coal pillar.
This was designed to maintain the natural boundary
conditions on the specimen ends. Van Heerden (1975)
end-loaded specimens through a concrete slab.
Wawersik's results (1972) were obtained by laboratory
tests on Tennessee marble. The data reported by
Brown and Hudson (1972) was obtained by laboratory
tests on block-jointed and unjointed specimens of a
rock-like material.
The data presented in Figure 8.10 may be
separated into three domains by the lines A, B, C.
Line A is defined by the upper limit of average
elastic/post-peak stiffness ratios determined by
Wagner. It delineates the upper limit of pillar
stiffness ratios determined by laboratory or field
tests on intact specimens, and represents the most
optimistic estimate of the post-peak stiffness of
a pillar in unjointed rock. Line B represents the
upper bound of Van Heerden's and Wawersik's data.
The domain between lines B and C includes all of
Wagner's data for the ratio of -X/X' calculated
using the maximum slope of the post-peak loading
curve, some of Wagner's data using the average slope
of the post-peak curve, and Brown and Hudson's test
on an intact, rock-like specimen. Line C represents
a lower bound for all data.
229
The lines A, B, C defining the bounds of the grouped
3.0 3.4 1.0 2.0 Specimen Width/Height Ratio
FIGURE 8.10: ELASTIC/POST—PEAK STIFFNESS RATIOS DETERMINED IN FIELD AND LABORATORY TESTS ON ROCK SPECIMENS
Starfield and Wawersik (1972) Van Heerden (1975) Wagner (1974)- using peak a' Wagner (1974)- using average X' Brown and Hudson (1972)
0 0 V A
0
0 0
(H60) *
(SO) *
230
9.0
8.0
7.0
6.0
3.0
2.0
1.0
5.0
-1/X'
4.0
data in Figure 8.10 have been used, in association
with the stiffness data in Table 8.5, to estimate
values of the pillar stability index kl+a' for the central pillars in the various hypothetical stoping
blocks. The results are plotted in Figure 8.11.
Although the estimate of kl+X' based on the line A
in Figure 8.10 approaches the condition for stability,
the estimate based on line B, which is probably more
realistic, suggests pillar failure will result in
instability. For constant pillar width, mine local
stiffness increases relatively slowly with decrease
in stope span, as indicated by the data in Table 8.3.
The suggestion is that for massive orēbody rock with
the same elastic properties as the country rock, any
pillar failures will result in instability, whatever
the extraction ratio or pillar dimensions.
Considering a pillar with width/height ratio
of 0.5, the stiffness ratios X/X' for jointed and
unjointed specimens may be used to estimate the
stability index for various patterns of joint
development in a pillar. Using Brown's nomenclature,
H60 and SO joint patterns give values of the stability
index of 5.3 and 4.5, indicating the pillar will fail
in a stable manner. For H30 jointing and an initially
intact specimen, the stability index values are -24.7
and -15.6, indicating unstable failure.
8.35 Discussion of Procedure for Pillar Stability Analysis
The assumptions that have been made in the
stability analysis procedure are that the post-peak
performance of a pillar can be estimated from its
elastic performance and width/height ratio, and
that the data presented in Figure 8.10 is applicable
to the mining situation considered in the study.
231
232
120
100
80
-(kl+X' )
60
40
20
0.5 1.0 1.5 2.0
Pillar Width/Height Ratio
FIGURE 8.11: VARIATION OF PILLAR STABILITY INDEX (NEGATIVE VALUE) WITH PILLAR WIDTH/HEIGHT RATIO
kl+Al estimated from Line A of Fig 8.10 0 B O
C 0
Wagner (1974) indicates that the design
strategy employed in South African coal mining
is to create yielding panel pillars between stable
barrier pillars, implying that the former assumption
is adequate for that situation. The extensive
testing of model coal pillars which has been
conducted in South Africa also presents a sound
data base for design.
Apart from differences in rock type, the
main question concerning the applicability of the
available data to the problem considered here is
the difference in pillar geometry between the test
specimens and the long rib pillars analysed in the
design. With regard to pillar strength, Holland
and Gaddy (1957) state that the minimum lateral
dimension determines the effective width of a pillar.
Wagner (1974) suggests that the effective width of
a long, narrow pillar is twice the apparent width.
This proposal is based on the proportionally greater
effective pillar core area which exists in long pillars,
compared with pillars of limited length. In the
case of post-peak stiffness of a pillar, it is
clear that the confining stress acting parallel to
the long axis of a rib pillar will affect the pattern
of crack development, and thus influence the post-
peak performance of the pillar. The suggestion is
therefore that the available experimental data may
not allow realistic assessment of the stiffness of
pillars in the failing regime, for plane strain
conditions.
The absence of data on the post-peak behaviour
of pillars in hard rock presents an obstacle to progress
in pillar design. Due to the loads involved, the
only practical way of obtaining this data is to
instrument pillars, and mine adjacent rock to induce
233
234
pillar failure. Such large scale experiments
would be difficult and expensive to conduct,
involving a considerable amount of reliable, sophis-
ticated instrumentation to measure stresses and
displacements throughout a large volume of rock.
It is noted that the evaluation of pillar
stability in terms of pillar post-peak and mine
stiffnesses represents a considerable simplification.
The basic criterion for stability of a pillar
involves consideration of the energy available
locally in the country rock, and the energy required
to destroy the pillar. The former may be estimated
using Boundary Element Methods, simply by calculating
the energy release increment when a pillar is mined.
Calculation of the energy required to destroy a
pillar requires modelling of the development and
propagation of fracture and yield in rock under the
mining-induced loads, and therefore presents
significant difficulties.
Finally, it has been shown that the orientation
and continuity of jointing in a pillar has a dominant
effect on its post-peak performance. In particular,
the results suggest that continuous jointing oriented
parallel to the pillar axis, and discontinuous
jointing oriented to favour slip, promote stable
pillar deformation in the failing regime.
8.4 The Mount Isa Lead Orebodies
8.41 Introduction
The Mount Isa Mine, at Mount Isa, Queensland,
Australia, is a major producer of copper, lead, zinc
and silver, obtained by underground mining methods.
Mine production is 26 000 tonnes per day, of which
10 000 tonnes per day is lead - zinc - silver ore.
Mineralization in the mine area is confined
to the Urquhart Shale, which is a sequence of dolomitic,
pyritic and tuffaceous shales. The shale sequence is
conformable with other members of the Mount Isa Group
of sediments, which are up to 4 km wide. Bedding in
the group dips west at 65°. The detailed geology of the
mine area has been described by Bennett (1965).
Lead - zinc - silver mineralization occurs as bands of galena and sphalerite concentrated within
the shale beds. The orebodies range in stratigraphic
thickness from 5m to 50m, lie mainly in the northern
part of the mine, and are arranged en echelon, with the
western (hangingwall) orebodies extending further to the
north. A representative cross-section through the northern part of the mine is shown in Figure 8.12.
The narrow orebodies are mined by cut-and-fill stoping. The mining sequence is indicated in the
cross-section shown in Figure 8.13. Cut-and-fill
(MICAF) stopes have been mined from 13B sublevel to 11
level, generating crown pillars in each orebody with
the previously mined stopes above 11 level.
235
'" E 8 E
-
236
•
•
---+--------------------:'/~~.._---_+-----------3300mR.L.-
/
/ /
/
•
-, I
I .- / . I
/ . " I
I / "-'
•
/ /
i I i
i i i i ;
-
ria COPPER ORE
a LEAD ORE
EJ "SILICA-DOLOMITE-
S GREENSTONE
[Z) CARBONACEOUS MYLONITE
~ STOPED AREA
~ -"-~ FILLED STOPE
~~~------+------------2700mRL-
Scal. of .... tr.s 10 0 10 .., HBB
FIGURE 8.12: GENERAL CROSS-SECTION (LOOKING NORTH) THROUGH THE NORTHERN PART OF THE MOUNT ISA MINE
E 0 0
E 0 0 m
237
2 9 0 0 r o R .-
28C.:~ 4_
FIGURE 8.13: CROSS-SECTION THROUGH NARROW LEAD OREBODIES SHOWING CROWN PILLARS GENERATED BY CUT AND FILL STOPING
The wide orebodies are mined by sub-level open
stoping. Pillars may be recovered in the open
stope blocks, in which case extensive use is made of
cemented and uncemented backfill in the primary stope
voids. Mining and associated Rock Mechanics practice
at Mount Isa have been discussed by Davies (1967),
and Mathews and Edwards (1969).
Figure 8.14 shows the mining layout for the
extraction of blocks of stopes in two orebodies,
located between No. 13 and 15 levels, which are 611m
and 728m below ground surface. Mining of 6/7 Orebody
south of the 7000m mine northing, and 8 Orebody
south of the 6840 northing, is by sublevel open
stoping. North of these co-ordinates the orebodies
are to be mined by cut-and-fill stoping. The stope
extraction sequence is such that open stoping and
pillar recovery in 6/7 Orebody north of 6840
E 0 0
~o
E 0 0 ID ID
8 OREBODY
M I.C.A F. _ (II/L-I3/LZ 3/L _
I • C _ 2800 L674
U73/74 rD r
4 r4.
L678 075/76
a
0 m ~D J
•/L_ IS Co--.
6 and 7 OREBODY
I4C_ 2800
14 IL_
f
les
'S/L.
2700 E 0
E 0 0 r iD
E 0 0 m
14 LEVEL PLAN
6,7 and 8 OREBODIES
Scale: I 2500
8 OREBODY
6 and 7 OREBODY
1600 E
• a
1 L687I , I L692 78/79 UBO/81
a
Modified M.I.C. A.F.
ID J
0 Dl D
/Y /'
0 D1 ID -J
f
L698 U62
MQ 7l•
M.I.C.A.F. (II/L- 13/L)
Modified M.I.C.A.F.
4
FIGURE 8.14: MINING LAYOUT FOR EXTRACTION OF ADJACENT THICK SECTIONS OF LEAD OREBODIES
will be well advanced before significant cut-
and-fill stoping takes place in 8 Orebody on the
immediate footwall side. The sequence of mining
in the open stope blocks is designed to integrate
primary stoping and pillar recovery, and is
illustrated by the mining of the stope and pillar set
M674-M676-M678 in 8 Orebody. M678 stope is mined
and filled with cemented sandfill, M674 stope mined,
and M676 pillar extracted by blasting into the M674
stope void. After extraction of the ore, M674-676
void is backfilled.
Similar triplet stope blocks are used
throughout the open stoping area. Permanent pillars,
such as M660, M671 and M680 in 8Orebody, are generated
by this extraction scheme. Maintenance of the integrity
and stability of M671 pillar is critical, since it
provides access and services to operations on the
hangingwall side of 6/7 Orebody.
8.42 Site Conditions in Mining Area
The orebody and country rocks in the area of
interest have similar material properties, with a
uniaxial compressive strength of approximately 170MPa,
measured both parallel and perpendicular to bedding,
Young's Modulus in the range 70-80 GP a, and Poisson's
Ratio 0.23. Analysis of the results of a trial
stoping programme (Brady (1977)) in which a pillar
was mined to failure suggested the strength of the
rock mass is described by the relationship
a1 = 9.34 a30.75 + 94.0
where al, a3 are the principal stresses at failure.
Three sets of structural features occur in the
area, with approximately mutually orthogonal orientations.
These are:
239
(a) Bedding planes, with dip 65°, dip
direction 270°. They are the most frequently
developed and most continuous planes of weakness
in the rock mass.
(b) A joint set dipping east at 20° - 25°.
Members of this set rarely continue more than 1 - 2 m.
before being offset.
(c) A joint set which trends E - W, and is
near vertical. These also have limited continuity.
The frequency and continuity of joints is
such that the rock mass would be classified as
only moderately jointed.
The pre-mining state of stress in the mine
area has been measured by Hoskins (1967) and Brady
et al. (1976). Based on these measurements, the
estimated pre-mining stresses at the mid-height of
the stope block (14 Level) are: a1 = 19.8 MPa, dipping E 25° ;
a2 = 13.6 MPa, dipping w 65° ;
a3 = 7.9 MPa, directed N-S horizontal.
8.43 Analysis of Cut-and-Fill Stoping
Figure 8.15(a) shows a cross-section (looking
north) through an isolated cut-and-fill stope. This
situation occurs in the northern extension of 6/7
Orebody. The main Rock Mechanics concerns are ground
conditions in the immediate working area, particularly
the stope back, during up-dip advance of mining, and
the behaviour of the final crown pillar created by
the mined stope and the mined and filled stope above.
Ground conditions in the working area may be
assessed from the boundary stresses in the stope back,
and the rate of energy release, Ç. These parameters
240
have been calculated for various stages of up-dip
advance, and are shown in Figure 8.15(b). It has
been assumed that the fill may be neglected. The
boundary stress at the centre of the stope back
exceeds 80% of the rock mass strength at a stope
height of 97 m. At a stope height of 109 m,
general failure in the stope back is predicted. The
indication is therefore that ground conditions will
deteriorate significantly at stope heights greater
than 97 m. Established mining practice is to install
long, fully-grouted tendons in advance of mining
to achieve control of any local instability of the
stope back.
Figure 8.15(b) indicates a rapid increase in
the rate of energy release at a stope height of about 100m.
This suggests that at that stage, up-dip advance of
the stope is equivalent to stripping the crown pillar
from the underside. The energy release rate for
mining near the crown pillar level is a factor of 100
below the rate of 30MJ/m3 which is associated with
slight to moderate damage to rock in South African
gold mines. However, the energy release rate may
not be insignificant. Harries (1977) indicates that
in typical blasting practice with ANFO explosive,
approximately 6% of the available explosive energy
is transferred to the rock as dynamic strain energy,
and that the specific energy of ANFO is 3.81 MJ/Kg.
Assuming a typical powder factor of 0.9 Kg/m3, the
strain energy released in rock by blasting is 200 KJ/m3.
Energy release rates exceeding this are achieved as
the final crown pillar level is approached. The
suggestion is therefore that the energy released
by excavation, and the dynamic stresses thereby induced,
may be of some consequence under conditions where the
static state of stress approaches that necessary to
cause failure of the rock mass.
241
The extent of zones of overstressed rock in the
crown pillar was assessed from the rock mass failure
criterion and the calculated states of stress in the
body of the pillar. The predicted overstressed zones
are plotted in Figure 8.16(a). To assess the
significance of this overstressing, similar analysis
was conducted on a stope geometry with a modified
crown pillar shape, which resulted in the zones of
overstressing shown in Figure 8.16(b). The sensitivity
of the zones of overstressing to the change in pillar
geometry suggests that the originally designed pillar
will fail as a whole.
It was not possible to assess the mode of
failure of the pillar, for the reasons discussed in
Section 8.3. The mine local stiffness at the pillar
position was 13.3 GN/m, and the pillar stiffness
107.6 GN/m. If the pillar were to fail in an
unstable way, the energy release rate would be 721
KJ/m3. This would produce vibration levels remote
from the pillar site in excess of those that would
be experienced if a volume of rock comparable to the
pillar volume were blasted.
8.44 Rock Performance in Open Stope Block
The layout of stopes and pillars in the open
stope block for extraction of the southern sections
of 6/7 and 8 Orebodies is shown in Figure 8.14. Issues
to consider in the mining of stopes and pillars in
the block are (i) the integrity and stability of
the 13 Level crown pillars generated by the stopes
between 15B sublevel and 13 Level and the previously
mined cut-and-fill stopes above 13 Level; (ii) the
performance of the permanent pillars generated at the
end of the extraction programme at 6585N, 6700N,
6805N and 6905N; (iii) the sequence of extraction of
stope and pillar units in the area between 6712N and
6895N where 6/7 and 8 Orebodies are immediately adjacent.
242
- 300
o Boundary Stress in Stope Back
o Energy Release Rate
. 200
- WR(KJ/m3)
100
20 40 60 80 100 110
Up-dip Advance (m)
(b)
110
100
80
Final
Crown
Pillar
T 15m — 13/L
Q
(MPa)
60
40
20
- 15/B
// /
/ /
/ /
Up-dip /~/
Advance
of Stope
Sill level
110m
(a)
FIGURE 8.15: BOUNDARY STRESSES AT THE CENTRE OF THE STOPE BACK AND INCREMENTAL RATE OF ENERGY RELEASE DURING THE UP-DIP ADVANCE OF CUT-AND-FILL STOPING
(a)
Zones of Overstressed Rock
244
Zones of Overstressed Rock
■
(b)
FIGURE 8.16: ZONES OF OVERSTRESSED ROCK GENERATED IN THE FINAL CROWN PILLAR OF THE CUT—AND—FILL STOPE SHOWN IN FIGURE 8.15
The average down-dip dimension of the crown
pillar between the 15B-13 Level open stopes and
the filled stopes above 13 Level is llm, compared
with the 15m pillar analysed in Section 8.43, while
the stratigraphic width of the stopes in the open
stope block is always greater than the 10m stope
width considered previously. The results obtained
in Section 8.43 indicate that when reasonable strike
spans are generated in the open stope block, crown
pillar failure will occur. The longest strike span
generated is 85m, following mining of the M662-M667
stope and pillar unit in 8 Orebody. Failure of the
crown pillar occurred following extraction of the
M665 pillar. The extent of the failure has been
reported by Fabjanczyk (1978) and is shown in
Figure 8.17(b). Cemented fill in M667 stope has
controlled the collapse of failed ground, as indicated
in Figure 8.17(b). Penetration of fill from the 8
Orebody stopes above 13 level was prevented by the
cemented sandfill plugs placed at the bases of
these stopes. Violent failure of the crown pillar
was not reported. Clearly, similar crown pillar
failures are to be expected as other stope and pillar
units are extracted.
The probable performance of permanent pillars
in the stoping block, and potential problems relating
to extraction sequence, have been assessed by plane
strain analysis of sections through the stoping block,
using cutting planes dipping E25° and passing through
the 1600E mine co-ordinate on 14 Level.
Figure 8.18(a) shows the section representing
the completion of mining in the block. The analysis
indicates that a zone of tensile stress develops
245
Sandfill \ \
Cemented Sandfill
fz Horizontal Cracking
Area of Crown Pillar Collapse
M662-5 Stope Designed Stope Limit
(a)
13 Level
M667 Stope (Fill)
Broken ore /in Stope
/////////////
/0/Hi/MHZ
(b)
Horizontal Cracking
Scale: 1:500
246 80/B 80/B 9 0/B
Filled Stope
Air\ i l / 04(/‘ N'A`°
FIGURE 8.17: EXTENT OF ZONES OF FAILURE IN A CROWN PILLAR GENERATED BY OPEN STOPING (FROM FABJANCZYK (1978))
70/B
lī,
rn m
(T.
z
1600E
L674-6-8 iN•
N. \ \ \ N \ I L683-5-8
M674-6-8 'I\ NI M683 j\/
• a Lb92-5-8
L690 Pillar
)----
M671 Pillar Assumed Zone
of destressing Scale: 1:2000 \`
V \~
M660 Pillar M671 Pillar
(a)
M680 Pillar
Scale: 1:2000
rn rn rn rn m v CO
0 0 O O O
O O O
1600E
M651-4-7
M660 Pillar
(b)
FIGURE 8.18: ZONE OF TENSILE STRESS INDICATED BY ELASTIC ANALYSIS OF MINING LAYOUT AND ASSUMED ZONE OF DESTRESSING FOR SUBSEQUENT ANALYSIS N
248
throughout the barren remnant between 6/7 and 8 Orebodies.
It is clear that significant stress re-distribution must
occur throughout and adjacent to the area in which the
elastic analysis indicates the development of tensile
stress. There are two major implications of this zone
of stress re-distribution. Firstly, because the zone
continues across the section of the barren remnant which
abuts L680 and M680 pillars, it suggests that these
pillars may not be as effective as support elements at
the end of extraction as assumed in the elastic analysis.
Secondly, destressing of rock in the hangingwall of 8
Orebody stopes, and associated instability under gravity
loading, indicate that the M674-M678 block and M683
stope should be mined and filled, before mining commences
between 6710N and 6845N in 6/7 Orebody. Apart from the
premature mining of L683 stope, it appears that this
sequence is being followed. Strength/stress ratios,
which represent virtual factors of safety obtained from
the elastic analysis and the rock mass failure criterion,
are shown in Figures 8.19(a), (b), for various points
in M671 and L690 pillars. The values suggest that, if
the pillars at 6800N performed elastically, there would
be no risk of failure in compression in the permanent
pillars.
To make some assessment on the effect of de-
stressing in the barren remnant between 6/7 and 8
Orebodies on the possible performance of M671 and
L690 pillars, the assumption has been made that rock
in the area de-stresses completely. The state of stress
in the pillars was determined for the zone of de-
stressing illustrated in Figure 8.18(b). Factors of
Safety against failure at internal points in M671 and
L690 pillars are indicated by the bracketed numbers
in Figures 8.19(a), (b). Although significant reductions
in the Factors of Safety occur throughout both pillars,
the observation is that both pillars can sustain the
states of stress imposed by any de-stressing in the
area. It is also noted that the proportionate
249
1.35 1.66 1.99 2.32 2.34 2.42 x x x x x x
(1.11) (1.35) (1.59)(1.69) (1.71) (1.68)
2.41 1.92
x x (1.65) (1.55)
x 1.91 (1.46)
x 1 .70 (1.30)
x 1.58 (1.24)
1.57 1.47
x x (1.18)(1.18)(1.24)(1.30)(1.33) (1.33)(1.34) (1.35)
x 1.70 (1.46)
x 1.96 (1.73)
x 2.35 (2.12)
1.52 2.31 2.71 2.85 2.82 2.61 2.18 1.57 x x x x x x x x
(1 .39) (2.17) (2.52) (2.61) (2.53) (2.3o)(1.89) (1.35)
M671 Pillar
(a)
1.38 1.43 1.51 1.58 1.61 1.60 x x x x x x
Unbracketed: 6800N Pillar Operating Bracketed : 6800N Pillar Destressed
Scale : 1:250
1.45 1.67 1.96 2.18 2.13 1.18 x x x x x x
(1.31) (1 .42) (1.66) (1 .71) (1 .63)(1 .01 ) 1.60 1.67 1.89 1.92 1.49 1.27 x x x x x x
(1.12) (1 .45) (1 .75) (1 .77) (1 .34) (0.97)
1.62 1.74 2.19 2.37 2.14 1.36 x x x x x x
(1.46)(1.59)(1.98)(2.13)(1.92)(1.08)
L690 Pillar (b)
FIGURE 8.19: STRENGTH/STRESS RATIOS IN M671 AND L690 PILLARS
reduction in Factors of Safety is lower in L690
pillar than in M671 pillar. This reflects the
better proportions of the latter pillar.
The stiffness of M671 pillar, and mine local
stiffness at the pillar position with the M680 pillar
and the surrounding rock destressed, were determined
in the manner described previously. Pillar modulus
was 69.8 GPa, and normalized mine local stiffness
was 0.44 GPa/m. These values may be compared with
those obtained for the crown pillar in cut-and-fill
stoping of 71.7 GPa and 0.89 GPa/m. The lower
value of the normalised mine stiffness at the central
transverse pillar position in the open stope block
indicates the increased deformability of the structure
associated with the longer adjacent stope spans in this
layout.
8.45 Discussion of Case Study
The study of the extraction of the complex
mining layout described here indicates the relative
ease with which the Boundary Element Method may be
used to assess the significance of potential mining
problems. It is clear that, for the layout examined
the possibility of conducting three-dimensional analyses
is limited, and it is doubtful if such analyses would
yield results of much more significance that the two-
dimensional elastic analysis. Two-dimensional methods
of analysis which allow for non-linear behaviour of the
rock mass probably have greater scope for application.
However, there remains in this case the significant
problem of modelling the excavation sequence precisely,
due to the stress-path dependence of the solution.
The author's view is that elastic analysis of mining
250
layouts at different stages of extraction, taking
due account of the existence of destressed or
failed zones, provides a useful first assessment
of the scale and seriousness of identified mining
problems.
251
CHAFER 9
252
CHAPTER 9: SUMMARY AND CONCLUSIONS
Several versions of the Boundary Element Method,
suitable for the analysis of different types of mining
layouts, have been developed, and their performances assessed.
The principle followed in the development of these methods
of analysis was that singularities could be designed and
solution procedures employed which could exploit the geometric
properties of a particular mining system. Priority in
program development has been given to complete plane strain
methods of analysis, for two reasons. Firstly, a typical
supported mine structure consists of a greater number of
openings than one could reasonably expect to model in a proper
three-dimensional analysis, and only two-dimensional analysis
is generally feasible. Secondly, the application of the
complete plane strain concept removes the limitation on
conventional plane strain methods of analysis, that the long
axis of openings be parallel to a pre-mining principal stress
direction.
It has been demonstrated that complete plane strain
problems may be analysed in terms of subsidiary, decoupled
problems. Conventional plane strain analysis solves the plane
component of the problem. The antiplane component deals with
induced tractions and displacements acting parallel to the long
axes of excavations.
The indirect formulation of the Boundary Element
Method for complete plane strain analysis, using uniform strip
load singularities, is a direct extension of an existing
formulation for conventional plane strain. The treatment
of the antiplane component of the complete plane strain problem
followed that for the plane problem, and involved the develop-
ment of solutions for stresses and displacements induced by
an infinite antiplane line load. The program has been shown
to provide an efficient method of analysis, but difficulties
arise when narrow, parallel-sided slits are examined.
253
A method of analysis has been described for problems
involving long, narrow, parallel-sided openings, such as
occur in tabular orebody extraction using long rooms and rib
pillars. Singularities have been specifically designed to
take account of the proximity of the surfaces of the
openings, which are modelled as infinitely thin slits. For
the solution of the plane component of the complete plane
strain problem, suitably coupled sets of four line loads
have been used to generate singularities which are centres
of normal compression and pure shear. A pair of opposing
line loads with non-coincident lines of action has been
coupled to generate the shear singularity for control of the
antiplane stress component. In the Boundary Element solution
procedure, distributions of the singularities over the
excavation segments have been used which effectively deal
with the high stress gradients near the ends of this type of
opening.
The methods established for the construction of
singularities to handle the two-dimensional slit problem have
been employed in the development of centres of compression
and shear required for the three-dimensional Boundary Element
analysis of tabular orebody extraction. In this case excavations
have been modelled asinfinitely thin slots. The centre of
compression consists of three pairs of coupled, directly
opposing point loads, while the shear centre consists of two
pairs of counteracting couples. The distributions of singularity
intensity over excavation segments have been chosen to be
compatible with those used for the two-dimensional problem.
This has also required the development of a distribution
function for the segments which represent the corners of
excavations. Stresses and displacements in a simple mine
structure, consisting of a room with a central square pillar,
have been determined with the Boundary Element Program, and
with a three-dimensional Boundary Integral Program. Comparison
of the results obtained from the two methods has suggested
that mining layouts generating openings with a width/height
ratio greater than 5 may be analysed with the Boundary Element
Program.
254
A direct formulation of the Boundary Element Method
has been developed for the complete plane strain analysis of
structures in non-homogeneous media. Gaussian quadrature has
been used in integration routines throughout the program,
and this appears to improve efficiency without significantly
affecting accuracy. Narrow parallel-sided slits, and other
openings with greater area/perimeter - ratios, maybe handled with equal facility with the method. The flexibility this provides
in the analysis of irregular mining layouts represents the
main advantage of the method.
An assessment has been made, using the direct
formulation, of the probable performance of rock in and around
a pair of close-spaced orebodies being mined, in their wider sections, by sub-level open stoping, and by cut-and-fill
stoping in the narrow sections. The energy release rate
during cut-and-fill stoping has been found to be well below
the levels associated with roof falls and .unstable failure
in South African longwall mining of gold reefs, but comparable
to the strain energy released in rock during blasting. The
design of permanent pillars in the open stope block appeared to be sound, even after allowing for the complete de-stressing
of one of the pillars.
A simple technique has been established for the
estimation of pillar and mine local stiffnesses, and an
attempt has been made, using these stiffnesses, to assess the
stability of pillars in a series of hypothetical stoping
blocks in a tabular orebody. The study was somewhat inconclusive due to the lack of suitable information on the post-peak behaviour of hard rock masses. However, the inference that
has been drawn from the results is that persistent jointing
parallel to the pillar axis, or discontinuous jointing
oriented to favour slip, will result in pillar yield. When
jointing is not well developed, and the orebody rock has the
same elastic propertiesas the country rock, the results have suggested that the risk arises of unstable pillar failure
when the rock mass strength is exceeded.
255
Three issues have been identified which require
further research activity if pillar design practice and
underground mining extraction strategies are to be improved.
The initial requirement is to establish a method of
analysis which will handle adequately the strain - softening
behaviour of over-stressed pillars. A suitable approach
might be to model the country rock as an elastic continuum,
thereby exploiting the efficiency of the Boundary Element
Method, and to treat pillars as inclusions within which more
complex constitutive equations are obeyed. A finite
difference technique, for example, might be used to model
the behaviour of the material within the inclusion.
The lack of field data on the post-peak behaviour of
rock masses other than coal deserves attention. The high
load capacities required of jacking systems, and uncert-
ainties about the appropriate boundary conditions imposed
by such systems of loading, suggest that model and full
scale pillars in hard rock mines should be instrumented,
and adjacent excavations mined to induce pillar failure.
Since failure may occur in an unstable way, high speed data-
logging of stress and displacement within the pillar would
be required. Cable connections between transducers in the
pillar and the logging unit would be unacceptable. Both
the instrumentation and the tests would therefore be complex
and expensive. The results from such tests would be used to
validate or modify the numerical model discussed above. It
would also provide data for evaluation of the simpler methods
of pillar stability analysis, and allow better assessment
of the role of rock structure in pillar stability.
Finally, practical methods are required for modifica-
tion of the post-peak properties of a pillar which has
been found by analysis to be potentially unstable. The
objective would be to generate sets of suitable oriented
fractures which would allow local energy dissipation in
the pillar, while maintaining its support capacity.
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APPENDICES
T xy
APPENDIX I Stresses and Displacements induced by a Point
Load in an Infinite, Isotropic, Elastic
Continuum (Kelvin Equations)
For load P applied at the co-ordinate origin, in the
Z-direction:
ax
_ Qy
6z
z 2 3 (1
- (1-2v)
+ (1-2v)
-2v) }
}
I
871- (1-v) R3
P ?
{
1/2_
{ R2
2
8Tr (1-v) R3
8 (1-v) R3 {3 Tr
P 3xvz 8Tr (1-v) Rs
Tyz 817(1-v) { 3 T -TT + (1-2v) }
2
Tzx 811(1-v) R3 {-TT + (1-2v) }
P xz ux = 16ffG (1-V) R3
uy 1671G (1-v) R
2
zuz 167 (G(1-v) {
+ (3-4v) R}
266
where R2 = x2 + y2 z2
267
APPENDIX II Stresses and Displacements induced by Infinite
Line Loads in an Infinite, Isotropic, Elastic
Continuum
(a) For line load, of intensity pz/unit length, applied
at the co-ordinate origin, in the Z-direction
= pz z 2x2 ax 47 (1-v) r2 {— (1-2v)}
pz 2v z 6y 47r (1-v) r2
pz z 2z2 az - 4rr (1-v) r2 {r2 + (1-2v) }
TXy = Tyz = 0
pZ X 2z 2 Tzx 47(1-v) r2
{ -- + (1-2v) } rr
xz ux - 87G(1-v) r2
u = 0 Y
uz
•
8PG (1-v) {r2 - (3-4v) ln r }
(b) For antiplane line load, of intensity py/unit length,
applied at the co-ordinate origin, in the Y-direction
a = a = 6 = T = 0 X y Z zx
= X
Txy 27 r2
pz
z Tyz = 2Tr Fr
ux z = u = 0
P uy = 2G ln r
268
where r 2 = x2 + z 2
APPENDIX III Stresses and Displacements due to Infinite
Strip Loads
(a) Transverse Load qx (Figure 4.2(a) refers)
1
a - qx [(3-2v) ln r2 + cos28 ]2 x 87(1-v)
qx 2 ay - 87r (1-v) 2v[ln r ]2
1
az =
8~ (qXV) [(1-2v) ln r 2 - cos26 ]2
Txy = Tyz = 0
T zx g~ ((v) [4(1-v)e - sin20 ]2
q
ux 8~rG(1-v) [4(1-v) (x-zi6) - (3-4v)x in r]2
u = 0 y
q 1 x
uz 81rG (1-v) zi In r
2
(b) Longitudinal Load qy (Figure 4.2(b) refers)
Q = a = a z = T = 0 x y zx
269
1
1
1
1
2
q
T xy = 27r [mr]1 2
_ [e] 27r
u = u = 0 x z
1
uy = - [xinr-x+z.Ol2 2TrG
(c) Normal Load qz (Figure 4.2(c) refers)
q 1 z
ax 8ir (1-v) [4v0 - sin20]
2
1
Q = qz 4v [o]
y 87r (1-v ) 2
c1 1
az = 8n (l-
[41_o + sin20]2
Txy = T = 0
1 qz
Tzx 8~rG (1-v) [(1_2'.) ln r2 - cos20 ]2
qz ux 87rG(1-v)
zi [ln r]2
u = 0 y
270
1
1 q
uz = 8~rG (1-v) [(3_4) (x-x In r) - 2 (1-2v) zie ]
2
where r2 = (x. - x.)2 + z,2 a = 1,2
271
x.71, x
72 are X co-ordinates of element ends
xis zi are co-ordinates of the point of interest
e l ,e 2
are as defined in Figure 4.2(d)
APPENDIX IV Stresses and Displacements due to Infinite
Line Quadrupoles and Dipoles
(a) Centre of Compression qz (Figure 5.3(d) refers)
__ (1-2v) 1 8z2 + 8z" ax qz 4~rr (1-v) 2 (rT r ~— )
(1-2v) 1 2z2 ay __ qz 4rr (1-v) 2 (~ T.7—,
(1-2v) 1 4z2 8z" az __
qz 47(1-v)2 (TT + Tr—)
Txy = Tyz = 0
(1-2v) xz 4xz3 __ Tzx qz 4Tr (1-v) 2 (T7 rr 6
u = q (1-2v)2 {(1-2v)x 2xz2
X Z 8G(1-v) r }
u = 0 Y
_ (1-2v) z 2z 3 __ uz qz 8'rrG(1-v) 2
{(1-2v)--7 + r" }
(b) Antiplane Shear Dipole sy (Figure 5.5 refers)
a = a = a = T = 0 X y z zx
272
Xz Txy
• r r"
T = (1 2z2)
yz 2ir r2 r"
u = u = 0 x z
uy 27G r 2 sy z
(c) Shear Centre sx (Figure 5.4(b) refers)
sx 6xz 8xz3 cx
= 27
(1-v) r" + r6 )
sx 4vxz a 27 (1-v) r4
X 2xz 8Xz3 6z = 27r (1-v) ( rT r6
Txy = Tyz = 0
x 1 $z2 + $z4 )
T zX • 27r(1-v) (r 2 r" I~
ux 4TrG(1-v) {- (3-2v) r2 + rfi-- }
sX 2
uz • 47G (1-v) {- (1-2v) r 2r4 }
273
=
z
Qz
z
a = Q (1-2v) 1 {2(1-v)
{ 2 (1-v)
6z2
87(1-v)2 R3
87r (1-v)
(1-2v)
R3
1 87r (1-v) 2 R3
{ + —
x
Qy
Q z
- 3 (1-2v) ,2 15x2 z2 }
R4
- 3 (1-2v) - 1515-5y2 z }
15z4 } ~
APPENDIX V Stresses and Displacements due to a Point
Hexapole and a Point Shear Quadrupole
(a) Centre of Compression Qz (Figure 6.3 refers)
274
T = Q (1-2v) 3_._Y { (1-2v) 5z2
z 87r (1-v) 2 R5 xy R2
__ (1-2v) 3yz 5z2 Tyz Qz 87r 87(1-v)2 R5 { 1 R2
}
Tzx = Qz 87(1-v)2 R5 { 1 - R2 }
u = Q (1-2v) x (1-2v 3z2 x z 167rG (1-v) 2 R3 R2
u Q (1-2v) y~ (1-2v 322
fi— ) y z 167rG (1-v) 2 R R =
(1-2) 2
uz Qz 167rG (1-v) 2 R3 {-(1-2v) R2 }
(1-2v) 3xz 5z2
ux =
uy =
uz =
1 z 2
Sx 8TrG(1-v) R3 {- (1-2v) R— }
1 x 2
Sx 8nG(1-v) R 3 {- (1-2v) R2 }
1 3xyz - Sx 8TrG (1-v) R5
275
(b) Shear Centre Sx (Figure 6.4 refers)
1 3xz 5x2 ax = Sx 4'n (1-v) R5 (1 - R2 )
1 3xz ay = Sx 4n (1-v) R5
5z2 R2 )
1 3xz 5z2 az = Sx 411- (1-v) R5 (1 R2 )
Txy g 1 3yz (v
5x2
xy x 4Tr (1-v) R5 ( R2 )
_ 1 3xy 5z2 Tyz S 4Tr (1-v) R5 (v R2 )
Tzx Sx 4Tr(1-v) R3 (1+v - 3vyl - 15x2z2) R2 R4
APPENDIX VI USER INFORMATION AND INPUT SPECIFICATIONS FOR
BOUNDARY ELEMENT PROGRAMS
A. PROGRAM BEM11
1. The program is the algorithm for the indirect form-
ulation for complete plane strain described in Chapter 4.
The rock mass is assumed to be homogeneous, isotropic
and linear elastic. The variable names and other notation
used in the input specification are now defined.
2. Mine axes are X (North) , Y(East) , Z (down) . The
local axes for the excavation are x,y,z, as shown below.
The y-axis is parallel to the long axis of the excavations,
and this is specified by its dip (ALF) and bearing (BET)
relative to the Mine axes, as shown below. The x-axis lies
in the horizontal plane.
x
Long axis of opening
Z
3. Magnitudes of the field principal stresses are FP1,
FP2, FP3, and their dips and bearings are ALF1, BET1;
ALF2, BET2; ALF3, BET3 respectively.
4. In representing the boundaries of excavation or
the position of points in the medium, position co-ordinates
are specified relative to the excavation local (x,y,z) axes.
5. The boundary of an excavation is defined by dividing
the boundary into a number of SEGMENTS. Segments may be of
three types:
(a) straight lines (b) circular arcs (c) elliptical arcs.
The range of a segment is defined by an initial point
and a final point. The convention used is that when the
boundary segment is traced from its initial point to its
final point, and one faces the direction of travel, the
solid material lies on the right hand side.
276
Straight line Segments
X
(XL,ZL)
(XO, ZO ) z Circular Segments
X
B Elliptical Segments
xo,zo = co-ords. of initial point
xl,zl = co-ords. of final point
xc,zc = co-ords. of centre of circle
RDS = radius of circle
THET 1= polar angle of initial point (deg)
THET 2= polar angle of final point (deg)
Line CB is drawn from the centre C in the direction of the +z axis.
The polar angles are measured in a counter-clockwise direction from CB.
xc,zc = co-ords. of centre C
SEMIAX= length of one semiaxis
RATIO = b/a, where b = length of other semiaxis
PSI = polar angle of axis a
THET 1= polar angle of initial point
THET 2= polar angle of final point.
277
C (XC,ZC)
z
B Segments may be combined to form a boundary of virtually any arbitrary shape.
e.g.
circular
straight line
elliptical
NSEG = the total number of segments used in defining all the boundary surfaces
of a problem.
6. Each segment is divided into a number of elements,
NELR. In the case of straight line and circular segments,
the elements are all of equal length. For elliptical
segments, elements are small when the curvature is small,
and conversely.
7. The equation solver in the program is a modified
form of Gauss-Siedel iteration. For problem geometries
which do not involve parallel-sided slits, 20 cycles
(NCYC=20) are usually sufficient to obtain a satisfactory
solution for the unknown element loads.
8. The calculation of excavation-induced displacements
requires specification of the Young's Modulus of the rock
mass, EMOD. The units of EMOD must be the same as those
used for the Principal Stresses; i.e. if FP1 , FP 2 , FP3 are input in MPa, so must EMOD.
9. Stresses are determined at interior points in the
278
medium defined by the nodal points of a grid. The boundaries
of the grid are specified by lines parallel to the x and
'z axes. Grid lines are parallel to the boundary lines
as shown below. There are NLX and NLZ grid lines parallel
to the x and z axes. XW1 XW2
----------------~~X I I I I
ZW1------I' lIiilll'}NLX
ZW2------~~~~~~, y
Z NLZ 10. Input Data Format for Job Execution
The input data deck structure is shown on the
following sheet.
Solution for the unknown element loads can be a time
consuming process. In some cases it is more satisfactory
to determine the unknown element loads and boundary
CARD
TYPE Al
A2
A3
A4
A5
A6
Si
S2
S3
G
INPUT FOR BEM11
1 10 20 30 40 50 60 70 80
NOPEN INCLOSE
NPROB L N S E.G
' I: F P 1
- N C YCl
- --
, AL F 1
ALF2
- RN U[
; 1 ∎
EM 0 D`
; 1
I
I
BE T,1
1
[I.
1
I: L1 1 I 1 ;
1 1, .FP 2 1 1 I !BE T 2 1' I. 1 1 i 1
F P 3 ,, AL F 3 i BET 3 1 I I1 I 1 1
1 A L F . 1 , . •B E T . 1 i , ; 1; 1 ;
1 1 1, IZ L i 1 1 ;
N E L R ;; ,!; X 0 I I _Z 0 ' 1 X L 1 I I I; i I
NELR X C i I ; Z C , T H E T 1 , I , IT H_ET 2
I , 'T H E T2
• , R D S ; i I:. I
P S
_l_____
1 1 NELR 1 X C 1 Z C T H ET 1 S EM I A,X ∎ I 'R A T I 0
1 X W 1 1 , X W 2 : 1 1 ,ZW 1 Z W 2 . ,NLXXI 1 N L Z 1 1 1 1 •
i 11
1
I
r
r--
•
stresses and displacements in an initial run, and to
determine stresses and displacements at interior points
in a subsequent run. This is accomplished by storing on
permanent file all element and other required problem
data in the initial run, and retrieving this data from
permanent file in any subsequent runs. There are thus
two distinct types of jobs, indicated by the control
parameters NOPEN, NCLOSE, whose use is described below.
Local file TAPE8 is used for problem data storage and
retrieval.
Initial Run
Card A 1 : run type identifiers.
(2I5)
Cols 1-5 NOPEN
NOPEN = 0 initial run
NOPEN = 1 restart run
Cols 6-10 NCLOSE
NCLOSE = 0 no problem data written on TAPE8
NCLOSE = 1 problem data written on TAPE8
for filing and subsequent restart
Card A 2 : problem identifier and job execution data
(3I5,2F10.0)
Cols
Cols
1-5
6-10
NPROB
NSEG
problem identification number
total number of boundary segments
Cols 11-15 NCYC number of iterative cycles
Cols 16-25 RNU Poisson's Ratio for the rock mass
Cols 26-35 EMOD Young's Modulus for the rock mass
280
281
Card A 3 : magnitude and direction of major principal
(3F10.0) field stress
Cols 1-10 FP1
Cols 11-20 ALF1
Cols 21-30 BET1
Card A 4 : magnitude and direction of intermediate
(3F10.0) principal field stress
Cols 1-10 FP2
Cols 11-20 ALF2
Cols 21-30 BET2
Card A 5 : magnitude and direction of minor principal
(3F10.0) field stress
Cols 1-10 FP3
Cols 11-20 ALF3
Cols 21-30 BET3
Card A 6 : dip and bearing of long axis of excavation
(2F10.0)
Cols 1-10 ALF
Cols 11-20 BET
Card A 7 : used on restart run only
(I10)
Cols 1-10 MAXJ total number of boundary elements
282
Cards S1
Segment cards - linear segments (I10,4F10.0)
Cols 1-10 NELR
Cols 11-20 XO
Cols 21-30 ZO
Cols 31-40 XL
Cols 41-50 ZL
Cards S2
(I10,5F10.0)
Segments cards - circular arc. segments
Cols 1-10 NELR
Cols 11-20 XC
Cols 21-30 ZC
Cols 31-40 THET1
Cols 41-50 THET2
Cols 51-60 RADIUS
Cards S3
(I10,7F10.0)
Segments cards - elliptical arc segments
Cols 1-10 NELR
Cols 11-20 XC
Cols 21-30 ZC
Cols 31-40 THET1
Cols 41-50 THET2
Cols 51-60 SEMIAX (a)
Cols 61-70 RATIO (b/a)
Cols 71-80 PSI (of semi-axis a)
Card G (4F10.0,215)
Grid specification
Cols 1-10 XW1
Cols 11-20 XW2
Cols 21-30 ZW1
Cols 31-40 ZW2
Cols 41-45 NLX
Cols 46-50 NLZ
11. Output from Job Execution
(a) Input Data
(b) Stresses and displacements at the centres of
boundary elements
(c) If a grid is specified, stresses and displace-
ments at the grid nodes.
283
xtOMOECK GEN LOM'BlN/CEN/CX(10O) .CZ(100) .EX1 (100) .EZ1 (1003 ,EX2 (100) •EZ2 (100) .
1 PL(100).PM(100).PN(100).PLM(100).PMH(l0O).PNL(100). 2 AL (100) .BL (1003 .CL (100) .FL (100) .DM(10O3 .EM(100) • 3 AN(100).BN(100).GN(100).FN(l00).OL(1OO).OM(I00). 4 ON (100) .SIG (3.100) .DALF (3.100) .DBET(3.100) .0V (100) . 5 OGAM(3.100).SIGX(100).5IGY(100).SI(iZ(100).04(100). 6 TAUXY(100),TAUYZ(100).7AUZX(100).SINB(100).CDSB(100) . 7 I.J.MAXI•MAXJ.PY.NN.NCYC•NCLOSE.FX.FY,F2,FXY.FYZ.F2X• 8 NPROB.TOLIRNU.RNUl.RNU2oRNU3oCOEF.PY43.FAC.DX0L(100) , 9 D2OL(100).DY0M(100).0X0N(100).DZON(100).DU(100).G.R4U34
xDECK MAIN PROGRAM BEM11(INPUT.pUTPUT.TAPE1=INPUT,TAPE7=0UTPUT.TAPE2•TAPE6 ,
1 TAPES) XCALL GEN
READ(1.5) NOPEN.NCLOSE 5 FORMAT(2I5)
READ(1.10) NPROB.NSEG.NCYC.RNU,EMID IF(NOPEN.GT.0) GO TO 500
10 FORMAT(3I5.2F10.0) LRITE(7.15) NPROB
15 FORMi7(I41///,7X.35HBOUNOARY ELEMENT ANALYSIS. METHOD 9///. 1 7X.12HPROBLEM NO. ,I3.7X.32HCOMPLETE PLANE STRAIN CONDITIONS ) 4RITE(7.20) NSEG.NCYC.RNU.EMOD
20 FORMAT(1H //.7X.24HNSEG. NCYC. RNU. E MOD =•2I4.F5.2.F10.1) READ(1.25) FP1•ALFI,BETI READ(1.25) FP2.ALF2.BET2 READ(1.25) FP3,ALF3.BE73
25 FORMAT(3F10.0) 4RITE(7.30) FPI,ALFI.BETS.FP2,ALF2.BET2.FP3 , ALF3 ,BET3
30 FORMAT(1H //,7)044HPRINCIPAL STRESS MIGNITUDES AND ORIENTATIONS/. 1 14 .16X.14H MAGN DIP MG/. 2 14 .11X.34FP3.F5.2.F5.1.F6.1/. 3 14 .11X,3RFP2.F6.2.F5.]•F6.1/. 4 14 •11X,3HFP3.F6.2.F5.1.F6.1) READ(1.35) ALF.BET
35 P0RNAT(2F10.0) 4RITE(7.40) ALF.BET
40 FORNaT(lH /.7X.26HLONG AXIS OF OPENINGS DIPS.F5.1.164 DEGREES IOWA 1RDS.F6.1.0k DEGREES)
NN=O 1=0 NSEGG=D PY=ATAN(1.0)*4.0 TOL=1.E-4 RNU1=1.0-RNU COEF=8.0*PYxRNU1 RNU2=1.0-2.0xRNU RNU3=3.0-2.0xRNU FAC=PY/180.0 PY43=4. DxPY/3.0 G=E113D/2.0/(1.0+RNU) RNU34=3.0_4. DXRNU ALF1=ALF1xFAC BETS=bETIZFAC ALF2=ALF2*FAC BET2=BET2xFAC ALF3=ALF3xFAC BET3=BET3xEAC ALF=ALFxF AC BET=BET'FAC U1=C05(ALF1)x;CO5(BETI) U2=COSiALF2)=COS(BET2) U3=C05(ALF3)=COS(BET33
V1=COS(ALF1)*5IN(BET1) V2 =COS (ALF2) x5IN (BET2) v3=CO5 (ALF3) x5IN (BET3) U1=5Ik(ALF1) 142=51k (AL F2) 44=SIN (ALF3) FU=Ulxu 1XFP1+U2xU2xFP2+U3xU32(FP3 fv=V1xvIxTPI+V2xV2xFP2+V3xV3xFP3 FW=U 1 xSl1 xFP 1+(2xW2WP2 iW xWOIFP3 FUV=U1xVIZFPI+U2xV2xFP2A03xv3xFP3 FVi.V1ZJ1xFP1+V2xu2xFP2+V3XI43 P3 FW=U1xU 1 WP1+Dxu2XFP2+WxU3xFP3 )0U=SIN(BET) )0d= -COS(BET) X1=0.0 YU=COS (ALF) xtOS (BET) YV=COS(ALF)XSINIBET) Y{.GSIN (ALF) ZU=-SIN (ALF) xtOS (BET) 2V=-SIN (ALF) *5Ik (BET) ZI. COS (ALF) Fx=XUx)(UxFU+XWNV KFV+XWX141F14+2.01(XUxXVXFUV+>NxX41xFVU+XUxXUxFW) FY=YUxYUrFU+YV1YVxEV+Y4P:Y •P;F14+2.0x(YUxYVXEUV+YVxYUxFVU+YUxYUxFW) FZ=ZU12UxFU+2Vx2VxFV+Zytx11AF14+2.0x(ZUxZVxFUV+ZvxZNxFVN+ZHxZUxFW) FXY=XUxYUxFU+XVXYVxFV+XWXYWWW4. (XU*YV+XVYYU)**UV+(XV1YL1+XUxYV)xFVU
1 +(XLWxYU+XUxTW17xFW FYZ=rilxZUxt-U+YVx2vxFV+mp7143(rN+(YUrZV+YvxzL)xFUV+(YVx2u+YUx2v)XEVu
1 +(114ZU+YUx214)xFW FZX=2ux uXFU+ZVxXVXFV+ZI.XXW PW+(Z)J*XV+ZVX*J) XgUV+(ZVxXU+ZUx)v) ZFVU
1 +(Z1RX()+ZUXXLA xFW 4RITE(7.41) FX.FY.F2.FXY.FYZ.FZX
41 FORMAT(IH //.7X.45HFIELD STRESS COt ONENTS REL TO HOLE LOCL AXES/. 1 14 •10)63HFPX.F7.3/. 2 14 .10X.34FPY.F7.3/. 3 14 .10X.3HFPZ.F7.3/. 4 14 .10X.4HFPXY.F7.3/. 5 14 .10X.4HFPYZ.F7.3/. 6 IN •10X.4HFPZX.F7.3)
45 IF(NSEGG.EO.NSEG) GO TO 90 NSEGG=NSEGG+1 NELG=0 READ(1.50) NELR.XO.ZO.XL.ZL.RDS.RATIO.PSI
50 FORMAT(I10.7F10.0) RNELR=NELR IF(RDS.LT.TOL) GO TO 70 IF(RATIO.LT.TOL) RATIO=1.0 WRITE (7,55)
55 FORNaT(14 //.4X1P8HELEr- UTS.1X,64CENT )(114X•6HCENT Z.5X.5HTHET1.5X. 1 5HTHET2.5X.64RA0IU5.4X.5HRATIO.5X.34PSI) I. ITE(7.60) NELR.X0.Z0 .XL.Zl.R0S.RATI0.P57
60 FORNAT(1H0.6X.I3.7F10.3) SINPSI=SIN(PSI'PY/180.0) COSPS I=COS (PS PPY/180.0) GD=RDS/10000.0 GA=RATIOrCOS((XL-PSI)XPY/180.0) IF(ABS(GA).LT.GD) GA=GD GB=RATIOxtOS((a-ASI)xPY/180.0) IF (ABS (GB) .LT.GD) GB=GO CHII=ATAN2(SIN((XL-PSI)ZPY/180.0),GA) CHI2=ATAN2 (SIN ((ZL-PSI) XPY/180.0) .GB) DCH I = (CHI2-CH I S) /RNEL R IF(ABS(DCkI).LT.GD) GO TO 61 GC=0C41/ABS(DCHI) GO TO 62
61 62 65
GC=-1.0 DCHI=DCHI+:(ZL-XL)/ABSCZL-XL)-GC)XPY/RNELR 1=1+1 RNELG=NELG CHI=CHI1+RNELGxDCHI
510 100
WRITE:7.151 NPROB WR I TE (7 .510) FORMAT(1H ///.7X.41HTHIS RUN USED ELEMENT DATA FROM PERM FILE) CONTINUE READ(10105) X1l10(W2.Z1d1141m2.NLX.NLZ
EX1(I)=RDSXiCOS(CHI)=SINPSI+SIN(CHI)xCOSPSIXRAT10)+XO 105 FOR11 T(4F10.0.215) EZ1(I)=RDSX(COS(CHI)=COSPSI-SIN(CHI)XSINPSIXRATIO)+20 WRITE(7.110) XW10(412.ZN1.ZIJ2 CHI=CHI+DCHI 110 FORMRT(1H //.7X.27HX.Z BOUNDARIES OF PROB AREA/. EX2(I)=RDSX(COS(CHI)xSINPSI+SIN(CHI)XCOSPSIXRATIO)+XO 1 8)04F7.2) EZ2(I)=RDSX(COS(CHI)*COSPSI-SIN(CHI)XSINPSIXRATIO)+.O WR1TE(7.115) NLX.NLZ CX(I)=0.5X(EX1(I)+EX2(I)) 115 FORMAT(1H //.7X.19HGRID OVER PROS AREA/. CZ(I) =0.5X (EZ1 ( I) +EZ2 (I) ) 1 1H .7X.I4.26H LINES PARALLEL TO X-AXIS /. DX=EX2(I) {X1 (I) 2 1H 7)614.26H LINES PARALLEL TO 2-AXIS ) DZ=EZ2 (I) -EZI ( I) 120 DO 140 1=1.MAXI SINE (I) =-0Z/SORT COXXDX+OZXDZ) IF(NN.GT.0) GO TO 125 COSB(I)= DX/SORT(OXXOX+OZ*02) COSBI=COSB(I) NELG=NELG+1 SINBI=SINB(I) IF(NELG.LT.NELR) GO TO 65 125 CXI=CX(I) GO TO 45 CZI=CZ(I)
70 WRITE (7.75) DO 130 J=1.MAXJ 75 FORMAT(1H //.4X.8HELEMENTS.IX06HFIRSTX.4X.6HFIRSTZ.6X.SHLASTX.SX. COSBJ=COSB(J)
1 5HLASTZ) SINBJ=SINB(.1) WRITE(7.80) NELR.XO.ZO.XL.ZL RN=(CZI-CZ1(J))XCOSBJ+(CXI-EX1(J))XSINBJ
80 FORM1T(1H0.6X.I3.4E10.3) IF (RBS (RN) .LT.TOL) RN=TOL DX=(XL-X0)/RNELR L L =10X (I-J) +1000 Q1N 02=(ZL-20)/RNELR IF(LL.E0.0) RN=TOL DS=SORT (DXXOX+OZXOZ) RL1=(CXI-CX1(J))XCOS8J-(CZI-EZ1(J))'SINBJ
85 I=I+1 RL2=(CXI-EX2(JI)xC058J-(CZI-C22(J))XSINBJ SINS (I) =-02/DS RNSO=RNXRN COSB(I)=DX/DS R501=RLI RLI+RNSO RNELG=NELG R502=RL2xRL2+RNS0 EXI(I)=XO+RNELGXOX T1=ATAN (RL 1/RN) -ATAN (RL2/RN) EZI(I)=ZO+RNELGXDZ T2=2.0*RNx(RLI/R501-RL2/RS02) CX (I) =EXI (1) +0.5XDX T3=(PNSO-RL1xRL1)/RS01-“RNSO-RL2XOL2)/RS02 CZ(I) =EZ1 (I) +0.5XDZ T4=ALOG(RS01/R502) EX2(I)=EXI(I)+OX COSD=COSBIXCOSBJ+SINBIXSINSJ EZ2(I)=E21(1)+OZ SIND=SINBIXCOSBJ-COSBIXSINBJ NELG=NELG+1 CL'S2D=2.0XCOSDXCOSO-1.0 IF:NELG.LT.NELR) GO TO 85 SIN2D=2.0XSINDXCOSD GO TO 45 TL=RNU3XT4+T3
90 M$XI=1 T11 2.0XRNUXT4 MAXJ=I TN=-RNU2XT4-T3 DO 95 1=1.MAXI TNL=4.0XRNUIXT1-T2 C0S2BI=2.0XCOSB (I) XCOSB (I) -1.0 AL(J)=0.5X (TN+TL)-0.5x(TN-71.)XCOS2D-TNLXSIN2D SIN28I=2.0XSINB(I)XCOSB(I) BL (J) =TM PLC!) =0.5X (FZ+F)0 -0.5X (FZ-FX) XCOS2BI{ZXX5IN28I CL (J)=0.5x(TN+TL)+0.5*(TN-TL)*COS2D+TNLx5I1120 P11(I) =FY FL(J)=TNLXZ0520-0.5X(TN-TL)XSIN20 PN(I)=0.5X(FZ+FX)+0.5*(FZ-FX)xCOS213I+FZXXSIN2BI TL11=2.0XRNU1XT4 PLM (I) =fYZXSINB (I) +FXYXCOSB (I) TMN=4.0XRNU1XT1 PMY (I) =FYZX COSB (I) +FXYXS INB (I ) DM(J)=-TMMXSIND+TLMXCOSD PNL (I) =FZXXCOS2B I-0. 5x (FZ-FX) X5IN2B I EM(J)=TMNxCOSD+TLMXSIND 0L(I)=0.0 TL=4.0XRNUxT1-T2 011(1)=0.0 TM=4.0xRNUXT1 ON(I)=0.0 TN=4.0xRNU1XT1+T2 95 CONTINUE TNL=RNU2XT4-T3 GO TO 100 AN (J)=0.5X(TN+TL)-0.5X(TN-TL)XCOS2D-TNLXSIN2D
500 READ(10505) MAXJ BM(J)=TM 505 F0RNA7t110) CN(J)=0.5X(TN+TL)+0.5X(TN-TL)XCOS2D+TNLXSIN2D READ :8; NPROB. Pr: TOL.RNU.RNUI.COEF.RNU2.RNU3.FAC.PY43.G.RNU34.FX.
1 FY.FZ.FXY.FYZ.FZX. (EX1 (K) .EZ1 (K) .EX2 (K) .EZ2 (K) .CXCK) CZ (K) . , FN (Ji =TNL XCOS2D-0.5* (TN-TL) XSIN20 TNET1=ATAN(RL1/RN) 2 COSB .K) .51116(K) .PL (K) .PM(K) .PN (K) .PLM(K3 .PMN (K) .PNL (K) .OL (K) THET2=ATAN(RL2/RN)
3 011(K) *ONO() .K=1.111XJ) MAYI=MIXJ NN=O
TTI=RNX(THETI-THET2) TT2=RL 1-0L2 TT3=0.5X(RLIXALOG(R501)-RL2XALOG(R502))
N CO U,
TT4=0.5ERN •ALOG (RS01/RS02) TUL=4.0xRNUlx(TT2-TT1) -RNU34XTT3 TLL =TT4 TVM=-4.0xRNU1X(TT3-TT2+TT1) TUN=TWL T4N=RNU34X(TT2-TT3)-2.0=RNU2=TTl OXOL (J) =TULICOSBJ+TILXSINBJ DZOL (J) =-TULxSINBJ+TI.LXCOSBJ DYOM(J) =TVM DXON(J)=TUNICOSBJ+TLN*SINBJ 020N(J)=-TUNxSINBJ+T44XCOSBJ IF(NN.GT.0) GO TO 130 IF(J.NE.I) GO TO 130 IF(NOPEN.E0.1) GO TO 130 DENOM=CL (J) * N (J) -CN (J) XFL (J) OL (J) =-COEFX (PN (J) IFN (J) -ANL (J) XCN (J)) /DENOM OM(J) =-COEFXPhN (J) /EM(J) ON (J) =-COEFX (PNL (J) XCL (J) -PN (J) XFL (J)) /DENOM
130 CONTINUE IR I TE (2) (AL (J) . BL (J) . CL (J) . FL (J) . DM (J) . EM (J) . AN (J) . BN (J) . CN (J) .
1 FN(J).J=1.MIX1) I.RITE C6) (0X0L (J) .DZOL (J) .DYOM(J) .DXON (J) ,DZON (J) .J=1.MAXJ)
140 CONTINUE REWIND 2 REWIND 6 IF(NN.GT.0) GO TO 145 IF(NOPEN.E0.1) GO TO 145 CALL SOLVER
145 DO 150 I=1.MAXI CALL STRESS
150 CONTINUE Cxxxxxxxxsxxxxxxxxxxxxxcxxzxxx
DO 600 I=1,I1 XI DU(I)=0.0 DV(I)=0.0 DW(!) =0.0 READ (6) (DXOL (J) .DZOL (J) ,DYOM(J) .DXON (J) .DZON (J) .J=1.MAXJ) DO 610 J=1.MAXJ DU (I) =DU (I) +OL (J) XOXOL (J) +ON (J) XOXON lJ) Dv (I) =DV (I) +OM (J) Xa'ON(J) DW (I) =DW (I) +OL (J) xDZOL (J) +ON (J) *DZON (J)
610 CONTINUE DU(I) =DU(I)/COEF/G*1000.0 DV(I)=DV(I)/COEP/G*1000.0 Du(I)=OW(I)/COEF/GX1000.0
600 CONTINUE Cxzxzsx-:--.-----zszzzzxxzzxx
IF(NN.GT.0) GO TO 170 WRITE(7.155) NN
155 FORMATC1H /41.I4,19X.43HSTRESS COMPONENTS REL TO ELEMENT LOCAL AXE 1S.26X.23HDISPLACEMENT COMPONENTS//.5X.IHI,8X.2HCX.8X,2HCZ.6X. 1 4H5IGL.6X.4HSIGM,6X.4HSIGN.5X.5HTAULM.5X.5HTAUM1.5X.5HTAUNL.IX. 2 61-1 U 1.15.5X,5HV h►5.4X.6H W
GO TO 160 170 WRITE(7.171) NN 171 FORMAT(1H ///.I4.20X,40HSTRESS COMPONENTS REL TO HOLE LOCAL AXES.
1 28X.23HDISPLACECENT COMPONENTS//.5X.IHI.8X,2HCX.8X.2HCZ.6X. 2 4HSIGX.6)(.IWSIGY.6X.4HSIGZ.5X.5HTAUXY.5X.5WTAUYZ,5X.5HTAUZX,5X, 3 5HU M5.5X,5HV MM5.5X.5HW MS')
160 CONTINUE WRITE(76165) (I.CX(I).CZ(I).SIGX(I).SIGY(I).SIGZ(I),TAUXY(I),
I TAUYZ CI) .TAUZXCI) ,DU CI) .DV (I) .DWCI) .I=1. -I XI) 165 FORMAT(1H .15.8F10.3,3F10.2)
WRITE C7.161)
181 FORMAT(1H //.7X,1OHPRINCIPAL STRESSES) WRITE(70178)
:76 FORMAT(1H0.3X,2H I.8X.2HCX ,aX.214C12 .6X.4HSIG1 . 1 X.17HALPHA BETA GAM 1NA.6X.4NSIG2. 1 X,37HALPNA BETA GAMq.6X,4HSIG3.1X.17HALPHA BETA G 2AMSi/) WRITE(7.179)(I.CX(I).CZ(I).SIG(1,I).DALF(1.1).DBET(1.I).DGAM(I.I),
1 SIG (2, I) .DALF (2, I) .DBET (2. I) .DGAMC2. D ,SIG (3. I) .DALF (3. I) . 2 DBET (3.I) .DGAM(3.1) . I=1.5 XI)
179 FORM/17(1H .I5.2F10.3,F10.3.3F6.1.F10.3.3F6.1.F10.3.3F6.1) IF(NM.EO.NL)0 GO TO 999 NN=NN+1 IF(NN.GT.1) GO TO 186 COSBI=1.0 SINBI=0.0 MgXI=NLZ DIV=NLZ-1 DELX=(XW2-X41)'CIV DIV=NLX-1 DELZ=(2142-2H1),DIV CX(1)=XW1 DO 180 I=1.MAXI C2(I)=ZW1
180 CONTINUE DO 185 I=2.PWXI CX(I)=CX(I-1)+0ELX
185 CONTINUE 186 CONTINUE
IF(NN.E0.1) GO TO 195 CZIi=C2 (1) +OELZ DO 190 I=101AXI C2(I)=CZN
190 CONTINUE 195 CONTINUE
REWIND 2 REWIND 6 GO TO 120
999 STOP END
VECK SOLVER SUBROUTINE SOLVER
xCALL GEN M=0
5 DO 20 I=1.MAXI OL I=-ONL (I) CCOEF OMI=-PhM (I) XCOEF ONI=-PN (I) aCOEF READ(2) CAL (J) .BL (J) .CL (J) .FL (J) .0M(J) . EM (J) . AN (J) . BN (J) . CN (J) .
1 FN(J).J=1.I1XJ) DO 10 J=1.MAX1 IF(I.EO.J) GO TO 10 ONI=ONI-CL (J) X0L (J) -CM (J) XON (J) OL I =OL I-FL (J) X0L (J) -fM (J) *ON (J) OMI=OMI-EM(J) XOM(J)
10 CONTINUE DENOM=CL (I) XFN (1) - N (I) XFL (I) OL (I) _ (ON I *FN (D -OL IXCN (I)) /DENOM QM(I)
M
= OM I /EM( I)
_ (OL IXCL (I ) -ONIXFL (I)) /DENOM
20 CONTINUE
REWIND 2 IF(M.LT.NCYC) GO TO 5 IF(NCLOSE.LT.1) GO TO 25 WRITE(8) NPROB.PY.TOL.RNU,RNUI.COEF.RNU2,RNU3,FAC.PY43.G,RNU34.FX.
1 FY.FZ.FXY.FYZ.FZX.(EX1(K).EZ1cK).EX24K1 •E22 .1).CX4K).CZ•K). 2 COSB (K..SINB (K) .PL (K) .PM(K) .PN (K) .PL (K) .PMV K. .PNL (K) .OL (K) . 3 OM(K) 'ON 4K).K=1.MaXJ)
25 RE TURN END
')ECK STRESS SUBROUTINE STRESS
2ALL GEN READ (2) (AL (J) .BL (J) .CL (J) .FL (J) .DM(J) EM (.1) .AN (J) .BN (J) .CN (J) .
1 FN(J).J=1.M9XJ) SLGLI=0.0 SIGM1=0.0 SIGNI=0.0 TAULMI=0.0 TAUMYI=0.0 TAUNL 1=0.0 DO 5 J=1.1'AXJ SIGL I=S IGL I+AL (J) *OL (J) +AN (J) *ON (J) SIGMI=SIGMI+el (J) *GL (J) +BM (J) *ON (J) SIGNI=SIGNI+CL(J)t0L (J)+CN(J) *ON (J) TAULMI=TAUL111+OM(J)*011(J) TAUMYI=TAUMYI+EM(J) *OM (J) TAUNL I=TAUNL I+FL (J) *OL (J) +F 1 (J) *OM (J)
5 CONTINUE IF (NN.GT.0) GO TO 20 SIGX ( I) =SIGLI/COEF+PL (I) SIGY CI) =SIGMI/COEF+PM(I) SIGZ (I) =SIGN I/COEF+PN (I) TAUXY(I)=TAULMI/COEF+PLM(I) TAUYZ ( I) =TAUIII I/COEF+PrTI (I) TAUZX (I)=TAUNLI/COEF+PNL (I) GO TO 25
20 SIGX(I)=SIGLI/COEF+FX SIGY(I)=SIGMI/COEF+FY SIGZ(I)=SIGNI/COEF+FZ TAUXY (I) =TAULMI/COEF+FXY TAUYZ (I) =TAUR1I/C0EP+FYZ TAUZX(I)=TAUNLI/COEF+FZX
25 CONTINUE RJ1=SIGX CI) +SIGY(I) +SIGZ (I) RJ2=SIGX (I) *SIGY (I) +SIGY (I) *SIGZ (I) +S !CZ (I) *SIGX CI) -
1 (TAUXY(I)*TAUXY(I)+TAUYZ(I)*TAUYZ(I)+TAUZX(I)*TAUZX(I)) RJ3=SIGX(!)*SIGY(I)*SIGZ(I)+2.0*TAUXY(I)*TAUYZ(I)*TAUZX(I)-
1 (SIGX (I) *TAUYZ ( I) *TAUYZ(I)+SIGY CI) *TAUZX(1)*TAUZX(I)+ 2 S IGZ( I)*TAUXY(I)*TRUXY(I))
TRJ4=RJ1**2-3.0*RJ2 IF (TRJ4.LE.0.0) TRJ4=TOL RJ4=SORT (TRJ4) TC=(27.0*RJ3+2.0ri1J1**3-9.0*RJ1xRJ2)/(2.0*RJ4**3) IF (TC.L T.-1.0) TC=-1.0 !F (TC.GT.1.0) TC=1.0 THET=ACOS (TC) /3.0 DO 35 K=1.3 GO TO (26.27.28) K
26 ANG=THET GO TO 29
27 ANG=PY43+THET GO TO 29
28 ANG=PY43—THET 29 CONTINUE
SIG (K. I) _ (RJI+2.0*RJ4*COS (ANG) ) '3.0 TA= (SIGY (I) —S IG (K . I)) * (SIGZ (I) —SIG (K . I)) —TAUYZ (I) *TAUYZ (I) TB=TAUYZ (I) *TAUZX (I) —TAUXY (I) * (SIGZ (I) —SIG (K .1) ) TC= TAUXY (1) *TAUYZ (I) —TAUZX (I) X (SIGY (I) —SIG (K . I)
STS=SORT(TA*TA+TB*TB+TC" ) DCX=TA/STS DCY=TB/STS DCZ=TC/STS DALE (K. I) =ACOS (DC)G /CAC DBET(K.I) =ACOS (OCT)/FAG DGRM(K. I) =ACOS (DCZ) /FAG
35 CONTINUE RETURN END
288
B. PROGRAM TAB4
1. The program is the algorithm for the indirect
formulation for the complete plane strain analysis of
tabular orebody extraction, described in Chapter 5.
Openings in the plane of the orebody are modelled as long,
narrow, parallel-sided slits. The rock mass is assumed
to be homogeneous, isotropic and linear elastic.
2. Mine axes (X,Y,Z) are as specified for BEM11.
3. The local axes for excavations are x,y,z, where
the y-axis is parallel to the long axis of excavations,
and the x-axis lies in the plane of the orebody. The
orientation of axes is specified by the angles ALF, BET, ROT. ALF, BET are the dip and bearing of the dip vector
for the orebody, and ROT is the angle, measured in the
plane of the orebody, between the dip vector and the long
(y) axis of the excavations.
4. Magnitudes, and orientations of the field principal
stresses relative to the Mine axes, are FP1, ALF1, BET1
etc, as for BEM11.
5. The real thickness of the orebody, TH, and the z
co-ordinate of the midplane of the orebody in the x-z
plane, are defined in the figure below.
6. The number of mined excavations in the plane of the
orebody is NSTOPE. The span of each stope is defined by
the x co-ordinates XO, XL of the stope limits. Each
stope is divided into a set of segments equal to the number
of elements NELR defining one surface, e.g. the footwall side, of the narrow excavation.
289
*-X zp
xp
xwz
zwz. f
nix
ntz
z
7. The elastic properties of the rock mass are defined
by its Young's Modulus EMOD and Poisson's Ratio RNU.
8. The number of iterative cycles to solve for the
segment loads is specified by NCYC. Typically 10 cycles
are sufficient to guarantee convergence to a satisfactory solution.
9. Stresses and displacements at internal points in
the medium are calculated at the nodes of the grid
illustrated above. The boundaries of the grid are XW1,
XW2, ZW1, ZW2, and the grid consists of NLX lines parallel
to the x-axis, and NLZ lines parallel to the z-axis.
10. Input Data Format
The structure of the data input deck and the format
of data is illustrated on the following sheet.
On Card Al, NPROB is the problem identifier.
There are NSTOPE cards of type S.
Other cards have been described previously for
BEM11.
11. Output Data
(a) Input data
(b) Stresses and displacements of the centres
of the elements defining the footwall sides
of excavations
(c) Stresses and displacements at the nodes of the
grid defining the problem area.
290
CARD
TYPE
Al
A2
A3
A4
A5
S
G
INPUT FOR TAB4
1 10 20 30 40 50 60 70 80
N PR OBIN STO-PE
, I .F .P 1
N C
,
1
YCI
I A. L F 1 _
RNU l
I I • B E T, 1,
E MO DI `i L
i
T Hl , 1 Z PI , 1
1
1 .
1 ; 1 ,
1! 1 I' 1 I ∎ ,'
! , .FP2 I ALF2 1; 1; BETZ , I. 11 I l l
! r ;
l; 1 1
1 ; 1I 1
1 I
1 I'
' I
I, 1
I l
I, 1 , :FP 3 1 , 1AL F 3 1 1 1 1 1 ,8 E T 3 I, 1 ALF 1 BET I R O T 1 '.
N E L R , 1 1 1 X 0 • I I 1) X L ', 1 , 1 1 i 1; XW1 . •XW 2 D I ZW 1 I Z W 2 NLXI
1 1 I
1 .NL Z
I I r 1
1 1 I i t
I I. I, , 1 I I' f 1 I I 1 1 1 1! I
r r I I I
1 I I ' I ; '
i I I 1 1
i 1 ! 1 I 1
I ' I I I , I I '
'. i I
I I
I I 1 I
I 1 I I
I I i
1 1 i i I I I
,,! I I I
I I ,
I' I l '~
I
I 11 1 I I I I
; ,
I I I I
1
I , ,
1
I I I, I I I I
I I 1 t I
MCOMDECK GEN COPFUN/GEN/CX(50).CZ(50).EX1(50).EZ1(50).EX2(50).E22(50).
1 COFI.COF2.COF3.00F4.BAS.NCYC.RNU.RNU1.RNU2.RNU3.FAC.PY.PY23.TOL. 2 PX.PPY.PZ.PXY.PYZ.PZX.OZ(50).SX(50).SY(50).A02(50).B0Z(50). 3 COZ(50) .FOZ(50) .DXOZ(50) .0202(50) .ASX(50) .CSXC5C:.FSX(50) . 4 DXSX(50).DZSX(50).DSY(50).ESY(50).DYSY(50),IT(50).SIGX(50). 5 SIGY(50).SIG2(50).TAUXY(50).TAUYZ(50).TAUZX(50).SIG(3.50). 6 OALF(3.50).DBET(3.50).DGAM(3.50).DU(50).DV(50).0U(50) , I.J. 7 MXI.MAXJ.CXI.PI(8).X1(2).BSX(50).RCZ(50).00F5.COF6.Z
MDECK MAIN PROGRAM TAB4(INPUT.OUTPUT.TAPE1=INPUT.TAPE7=OUTPUT.TAPE2.TAPE6.
1 TAPE8=1002) *CALL GEN
REAO(1.10) NPROB.NSTOPE.NCYC.RNU.EMOO.TH.ZP 10 FORMT(3I5.4F10.0)
LRITE(7.15) NPROB 15 FORMT(1H1///.7X.51HTABULAR EXCAVATION ANALYSIS (COMPLETE PLANE ST
1RAIN)///.7X.12HPROBLEM NO. .I3) IRITE(7.20) NSTOPE.NCYC.RNU.EM70
20 FORMT(1H //.7)(126HSTOPES. NCYC. RNU. E POD =.2I4.F5.2.F10.1) WRITE (7.21) TH
21 FORMT(1H0.6)(1117HOREBODY THICKNESS.F6.2) READ(1.25) FPI.ALF1.8ET1 READ(1.25) FP2.ALF2.BET2 REA0(1.25) FP3.ALF3.BET3
25 FORMT(3F10.0) 4RITE(7.30) FP1.ALFI.BETI.FP2.ALF2.BET2.FP3.ALF3.BET3
30 FORMT(1H //.7X.44HPRINCIPAL STRESS MAGNITUDES AND ORIENTATIONS/. 1 1140.16X.14HPAGN DIP BRG/. 2 1HO.11X.3HFP1.F6.2.F5.1.F6.1/. 3 1H0.11X.3HFP2.F6.2.F5.1.F6.1/. 4 1H0.11X.3HFP3.F6.2.F5.1.F6.1) REAO(1.35) ALF.BET.ROT
35 FORMT(3F10.0) 4RITE(7.40) ALF.BET.ROT
40 FORMAT(1)4 /.7X.12HOREBODY DIPS.F5.1.16H DEGREES TOWAROS.F6.1.8H DE 2GREES/. 2 1HO.6X.26HLONG AXIS OF EXCAVATION IS.F8.1.241.1 DEGREES FROM DIP VE 3CTOR)
NN=O ML=O 1=0 NSEGG=O PY=ATAN(1.0)'4.0 TOL=1.E-4 G=EM3D/2.0/(1.0+ANU) RNU1=1.0-RNU RNU2=1.0-2.0M2MU RNU3=3.0-2.0'RNU FAC=PY/180.0 PY23=2.0=AY/3.0 COF1=RNU2/4.0/PY/RNU1/RNUI COF2=COF1/2.0/G COF3=1.0/2.0/PY COF4=COF3/0 C0F5=1.0/2.0/PY/RNU1 COF6=COF5/2.0/0 BAS=0.232 ALF1=ALF1aFAC BET1=BEfD TAC ALF2=ALF2'FAC BE T2=BE T2AC ALF3=ALF3+FAC 8ET3=8ET3ZFAC
ALF=ALF=FAC BET=BET=FAC U1=COS(ALFI) OS(BET1) U2=COS (ALF2) tOS (BET2) U3=C0S (ALF3) MCOS (BET3) V1=COS (ALF1) =SIN (BET)) V2=COS(ALF2)'SIN(BET2) V3=C0S (ALF3) =SIN (BET3) H1=SINCALF)) 12=SIN (ALF2) W3=S IM (ALF3) FU=U1=J1MF P1+U2=U2WP2+U31113XFP3 FV= V 1=V 1=FP 1+V2'2=FP2+V3N3WP3 FI.H1 J1=FP1+4Q'.Q P24C0=W=FP3 FUV=U1=V1MFP1+U2=V2MFP2+U3=V3aFP3 FVIV1s411MFP1+V2fii2XFP2+V34.13 FP3 FW=H1'UI P1+42=U2XFP244 'U3MFP3 XU=N (BET)
D XV=O5 IBE X41=0
SI{.0
YU=COS (ALF) MCOS (BET) YV=COS CALF) =SIN (BET) 't% SIN CALF) ZU=-S IN CALF) =ZOS (BET) ZV=-SIN CALF) =SIN (BET) 2WCOS CALF) FX=XLMU=FU+4(V=XVWV+41wW FN+2.0=(XUACVSFUV+XV2KXWMFVU+XLI=XUWW) FY=YU=YUxFU+YVZYV V+YLPCY4AF4i+2.0= (TU=YV=FU V+YV=YL:iF V4)+Y1RYU'F W> F2=2U=2ll=FU+2VXZVWV.10444AFI+2.0=(ZU34V=FUV+2Vz21 VU421)52UaFW) FXY=XU=YUzFU+XVXYVYFV+)0.1xYL1F41+(XU=YV+XV=YU) =FUV+(XVxV14+44xYV)'FVH
1 +(X10YU+XU=Y111XFW FYZ=YU=ZUW-l1+YVSZVSFV+Y4F44AFW+CYU=ZV+YVa2U) =FUV+(ri=Z11+f. V) aFVH
1 +(YLDQU+YU 1.0 WWU FZX=2L1xXU=FU+2V=XVaFV421 4NCZU=XV+ZV=)Q1) 7'FUV+(ZVX 4+ 1#)M 1FV4
1 +(2(4041+2U=)2.0 1JJ ROT=ROTWAC R0T2=2.0'ROT PX=0.5=CFY+FX)-0.5=(FY-FX)=COS(P0T21-FXY=SIN(R0T2) PPY=0.5=(FY+FX3 +0.5=CFY-FX)'COS (R0T2) +FXY=SIN cR0T2) P2=FZ PXY=FXY=COS CR0T2) -0.5= (FY-FX) =SIN (R0T2) PYZ=FZX=S IN CROT) 4FYZaCOS CROT) PZX=FZXaCOS CROT) -FYZ=SIN (ROT) 4RITE(7.41) PX.PPY.PZ.PXY.PYZ.PZX
41 FORMT(1H //.7X.58HFIELD STRESS COPPONENTS RELATIVE TO EXCAVATION 1LOCAL AXES /. 2 1H0.1OX.2HPX.F7.3/. 3 1H0.10X.2HPY.F7.3/. 4 1l!0.10X.2HPZ.F7.3/. 5 IMO.10X.3HPXY.F7.3/. 8 1H0.10X.31.1PYZ.F7.3/. 7 1140.10X.3HPZX.F7.3)
DUP2=TCL LRITE (7.55)
55 FORMT(1H0.18X.11HEXCAVATIONS//.4X.8HELENENTS.5X.5H5IDEl.5X•5HSiDE 12.8X.2HZP)
49 CONTINUE IF(NSEGG.EO.NSTDPE) GO TO 70 NSEGG=NSEGG+1 NELG=0 READ(1.50) NELR.XO.XL
50 FORMT(I10.2F10.0) IRITE(7.60) NELR.XO.XL.ZP
60 FORMT<IH /.7X.15.3E10.3)
RNELR=NELR DEL X= CXL-XD) /RNELR SDX=0.5:0ELX 02=0.5XTH
65 1=1+1 NELG=NELG+1 RNELG=NELG IT(1)=1 7F(NELG.E0.1) IT(I)=2 IF(NELG.EO.NELR) ITCI)=3 CXCI)=XO+RNELGXDELX-SDX EX1(I)=CX(I>-SDX EX2 C I) =CX C I) +SDX E21(I)=ZP EZ2(I)=ZP CZ(I)=ZP+02 RCZ (I) =2P+DUME 02(1)=0.0 SX(I)=0.0 SYCI)=0.0 IFCNELG.LT.NELR) GO TO 65 GO TO 49
70 CONTINUE MAXI=" MAXJ=I REAO(1.71) X1.11.%L2.2N1.il2.MlX.MLZ
71 FORMAT(4F10.0.2I5) LRITEC7.72) XW110442.ZW1.ZI.2.NLX.NLZ
72 FORMAT(1H //.7X.27HX.Z BOUNDARIES OF PROB AREA//.6X.4F7.2//. 1 1W /.7X.19HGRID OVER PROS AREA//.7X.I4.26H LIMES PARALLEL TO X-AX 2I5 //. 3 7X.I4.25W LIMES PARALLEL TO Z-AXIS )
C 75 CONTINUE
DD 95 I=1.MAxI CALL COEFFS WRITE(2)(A0ZCJ).BOZ(J).COZ(J).FOZCJ).ASX(J).SSXCJ).CSX(J).FSX(J).
1 DSY(J).ESYCJ>.J=1.MAXJ) LRITE(6) (DXQZ(J) .DZOZCJ) .DXSX(J) .DZSX(J) .DYSYIJ> .J=1.I79XJ) IF(NN.GT.0) GO TO 94 WRITEC8) CCOZ(J).FSX(J).ESY(J).J=i.MAXJ) IF<I.LT.MAXI) GO TO 94 REWIND 8
94 CONTINUE 95 CONTINUE
REWIND 2 REWIND 6
C IFCNN.GT.0) GO TO 100 CALL SOLVER WRITE(7.1000) IOZCI),I=1.MAXI)
1000 FORMAT<1H //7X.2E13.5) C
100 CONTINUE DO 105 I=1.MAXI CALL STRESS
105 CONTINUE LRITEC7.110) NM
110 FORMAT(1W ///.I4.20X.40HSTRESS COMPONENTS REL TO EXCAVATION AXES. 1 26X.23HDISPLACEMENT COMPONENTS//.5X.1HI.8X.2WCX.BX.2HCZ.6X. 2 4HSIGX.6X.4HSIGY.6X.4H5IGZ.5X.5HTAUXY.5X.5HTAUY2.5X.5HTAUZX•NX. 3 1WU.l1X.114V.11X.lHWM
WRITE (7.115) cI.CX(I) .CZ (I) .SIGX(I) •SIGYCD .SIGZ(I) .TAUXY(I) 1 TAUYZ(I).TAUZX CZ) @DU CI) *DV CI) .DWCI).1=1.MAXI>
115 FORMAT(1H .I5.8F10.3.3E12.44 WRITE(7.120)
120 FORIrtIT(1N //.15X.18WPRINCIPAL STRESSES) WRITE(7.125)
125 FDRr T(1)40.4X.1NI.0X.21+CX.8X.2HCZ.6X.4HSIG1.1X.17WALPHR BETA GAMM IA.6X.4H5IG2.1X.17WALPHA BETA GAlr .6X.4HSIG3.1X.17HALPHA BETA GA 2Mr1an
LRITE(7.127) (I.CX(I) .CZ(I) .5IO(1.1) .DALF(l.I) .DBET(1.I) .DGAM(1.I) . 1 SIG(2.I).DALF(2.I).OBET(2.I).DGiW(2.I).SIG(3.I)•DALF(3.I). 2 DBE7C3.I) .OGAM(3.I) .I=l.MAXI)
127 FORMAT(1HO.I5.3F10.3.3F6.1.F10.3.3F6.1.F10.3.3F6.1) MN=MN+1 IFCML.EO.NL)O GO TO 999 ML=ML+1 IF(NL.GT.1) GO TO 176 MAXI=NLZ DIV=MLZ-1 DEL X= (X42-XW1) /D I V DSV=NLX-1 DEL Z= (212-I1a1) /D I V CX(1)=X411 D0 170 I=1.MAXI CZ(I) =Z1aS
170 CONTINUE CZU=ZP-OZ CZL=ZP+OZ IFCZWI.GE.CZU.AND.ZWI.LE.CZL) GO TO 171 IF (21a1. LT. CZL) RCZI=ZW1+OZ IF (ZNS . GT. CZL) RCZI=2111-DZ CO TO 172
171 CONTINUE RCZI=ZP+TOL
172 CONTINUE 00 173 I=1014XI RCZ(I)=RCZI
173 CONTINUE DO 175 I=2.1AXI CX(I)=CX(I-1)+OELX
175 CONTINUE 176 CONTINUE
IFOIL.E0.1) GO TO 191 CZM=CZ (1) +DEL Z DO 180 I=1.MAXI CZCI)=CZN
180 CONTINUE IF(C7I1.GE.CZU.AHD.CZf(.LE.CIL) GO TO 186 IFCCZN.LT.CZU) RCZI=C2N+OZ IF(CZN.GT.CZU) RCZI=CZM-DZ GO TO 187
186 CONTINUE RCZI=ZP+TOW
187 CONTINUE DO 190 I=101AXI RCZ(I)=RCZI
190 CONTINUE 191 CONTINUE
REWIND 2 REWIND 6 GO TO 75
999 STOP END
*DECK SOLVER SUBROUTINE SOLVER
CALL GEN
EST (J)=COF3IPI(2) DYSY(J) =COF412xPI (1) GO TO (998.25.30) IT(J)
25 CK1=XJ(1)*XJ(1) -2.0XXJ(1)=XJ(2)-CXDCXI+2.0'XJ(2)=CXI CK2=2.0x(CXI-XJ(2)) GO TO 35
30 CK1=XJ(2)KXJ(2)-2.O'XJ(1)'X.)(2)-CXI'CXI+2.O'XJ(1)KCXI CK2=2.0x(CXI-XJ(1))
35 CONTINUE OXJ2=(XJ(1) -XJ (2))' (KJ (1) -XJ (2)) 23=Z212 24=Z33ZZ 25=Z412 DOXJ=1.OIOXJ2 AOZJ=COF IKDOXix (CKI' (2.0'Z21PI (3) -PI (2)140(2x (PI (6) +4.Ox221PI (4) -
1 2 .0124'PI(5))+4.0122,4,1C1)-8.01421PI(2)+2 .0' 4'P1(3)-PI (7) ) 60ZJ=COF1x2.01RNIJ1O0XJ'( -DC 11m1(2)+CK2' (PI (6)+Z2 win (4))+2.0K22=PI(
11)- Z2WPI(2) -PI (7)) CO2.J=COF1 ZDOXJ' (-CK1' (PI (2) +2.0Y42IPI (3) ) +CK2' (PI (6)-2.07122=7r (4) +
1 2.0*24*P1 (5)) +3.0x2,2117 I (2) -2.07441PI (3) -PI (7) ) FOZJ=COF1xDOXJx(C(1'(2.0=Z3xPI CS) -'PI(4))+CK2'(2.0,Z3 *Pr (3)-
1 2.0)12KPI(2))-2.04121PI(8)-4.04123WI(4)+2.01Z5Y471(5)) OX12ZJ=COF2=DOXJ'(CK1'(RNU211/P1(6)+Z21P1(4))+CK2'(RNU2xPI(7)-
1 2. 01RNU I x223117 (1) +72113I (2)) -0.5171NU2'P I (6) +RNU3K22KP I (6) + 2 Z4xP1(4)) 0ZOZJ= -COF2100XJ'(CK1'(2.0=R1(U1xZ1PI(1)+2xPI(2))+CK2x(RNU212=PI(
16) -Z3=2I (4)) I(7)-2.01RNUx231PI (1)+Z31P1(2)) ASXJ=COF5xDOXJX(CKIX(3.012x17I(4)-2.0123'P1(5))+CK2'(4.012113I(2)-
1 2.0321131(1) -2.0123113I (3)) +6.01CZWP1(6) +7.01231PI (4) -2.0145xPI (5) ) BSXJ=COF5x00XJx4.0xRNU' (CA 110.S1ZxpI (4) +CK2'(0.5'ZxPI(2)-0.5=Z1 'I(
11)) +21PI(6)+0.51231P1 (4) ) CSXJ=C0F5100XJ* (CAP (-VIP I (4) +2. 0123/43 1(5)) +CK2' (-2. 0EL'P I (2) +
1 2.0Z231P1(3)) -2.0'Z 1 (6) -5.0'Z3xPI (4) +2.01:131PI (5) ) FSXJ=COF5'O0XJ*(CK1'(2.0x721PI(3) -PI (2))+CK2x (PI (6) +4.0=22113I (4)-
1 2.Ox24>PI(5)) -PI (7) -5.01(72xPI (2)+4.0742131(1)+2.0124xP1(3)) DXSXJ=C0F61DOXJ'(CK1'C-2.0102NU1'Z=PI(1)+ZxP1(2))+CK2'(-RNU3K21PI(6
1) -23131(4)) +RNU312' (PI (7)-22KPI(1))+Z31PI(2)-23xP1(1)) DZSXJ=C0F6xDOXJ' (Mix (- NU2WI(6)h221PI(4))+CK2'(-81U2xPI(7)-
1 2.0=RNUKZ2W1(1)+17.2W1 (2))+0.5'RMU2KPI(6)-04U2zZ2/PI(6)+2.0*221P I 2(5) +Z41PI(4)) OSYJ=C0F3=WXJ' ({K 1121PI (4) +C1C212* (-PI (2) +P1 Cl)) -2.012'P1 (6) -
1 Z3*PI (4)) ESYJ=COF3KOOXJ' (CK 1 xP I (2) +C1(2' (-PI (6) -,22142 I (4)) +P 1(7) -
1 2.0xI2WI (1) +721PI (2) ) DYSYJ=C0F4100XJ' (CK 112x17I (1) +0(2KCZY42I (6) -2211321 (7) +Z3112 I (1) ) A02(J)=AOZ (.1)=BAS+AOZJ BOZ (J) =6OZ (J) 12)AS+6OZJ COZ(J)=COZ(J)' AS+COZJ FOZ(J)=FOZ (J)x6AS4FOZJ DXOZ (J)'07322 (J)16A54DX9ZJ ova (J)=DZ0Z (.1)'6A5+OZ0Z.1 ASX(J)=ASX(J)1BA5+A5XJ 65X(J)=65X(J)1ōA5+6SXJ CSX (J) =CSX (J) KBAS+CSXJ FSX(J)=FSX(J)x6AS+F5XJ DXSX(J)'DXSX(J)*BAS+OXSXJ DZSX(J)=DZSX(J)'5A5+0ZSXJ OSY(J) =DST (..1) KBAS+0SYJ EST (J) =EST (J)x)AS+ESYJ DYSY(J)=DYSY(J)121AS+DYSYJ
998 CONTINUE RETURN END
KOECK STRESS
M=0 5 DO 20 I=1.MAXI
PZI=-P2 PZXI=-PZX PYZ I =-PYZ READ(8) (COZ(J).FSX(J).ESY(J).J=1.MAXJ) DO 10 J=1.r1 XJ IF(J.EO.I) GO TO 10 PZI=PZI-CO2(J)x1Z(J) PYZI=PYZI-ESY(-I)*5Y(J) PZXI=PZXI-FSX (J) X5X (J)
10 CONTINUE OZ (I) =PZI,COZ (I) SY (I) =PYZIASY (I) SX (I) =PZXI/FSX (I)
20 CONTINUE M=M+1 REWIND 8 IF(M.LT.NCYC) GO TO 5 RETURN END
*DECK COEFFS SUBROUTINE COEFFS
=CALL GEN DO 998 J=I.MAXJ DO 5 L=1.8 PI(L)=0.0
5 CONTINUE XJ (1) =EX1 (J) XJ (2) =EX2 (J) CXI=CXCI) CT=1.0 2=RCZ (I) -EZI (J) IFCZ.E0.0.0) Z=TOL DO 15 K=1.2 IF(K.EC.2) CT=-1.0 X=CXI-X.1(K) R2=XXX+ZX2 R4=R21322 P1(1) =P1(1)+CT*ATAN (X/Z) PI(2) =PI (2)+CT*XXR2 PI(3)=PI(3)+CTKX/R4 PI(4) =PI (4)+CT,R2 PI(5) =PI (5)+CT,R4 PI(6) =PI (6)+CT*0.5XALOG(R2) PI(7) .PI C7)+CT'X PI(8) =PI (0)+CTXX'X
15 CONTINUE PI(1)=PI(1) Z Z2=Z12 Z3 Z212 AOZ (J)=COF1X(2.0=721PI(3) -PI (2)) BOZ(J)=-COFIX2.0*RNUKPI(2) COZ (J)=-COF1X(PI(2)+2.0'72X11 (3)) F0Z(J)=COF12(2.0123=PI(5) -ZxP1(4)) DXOZ (J) =COF2x (RNU2IP I (6) +221P! (4) ) DZOZ(J)=-COF2X(2.0*RNU1*ZKAI(I)+Z1PI(2)) ASX(J)=COF5X(3.0=Z1PI(I)-2.OXZ3=PI(5)) BSX(J)=C0F5X2.01RNUSZ'PI(4) CSX(J)=COF5X(-ZSPI (4)+2.0XZ31PI(5)) FSX(J)=C0F5X (2.0x22xPI(3) -PI (2)) DXSX Li) =COF5X(-2•0xBNU1xZWI(1)+Zx17I(2)) DZSXCJ)=COF5X(-RNU2XPI(6)+Z21?1(4)) DSY(J)=-COF3=2x91(4)
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C. PROGRAM TAB5
1. The program is the algorithm for the elastic analysis
of tabular orebody extraction based on the indirect
formulation described in Chapter 6. Mined openings in the
plane of the orebody are of finite plan area, and are
modelled as infinitely thin slots. The rock mass is
assumed to be homogeneous, isotropic and linear elastic.
2. Mine axes (X,Y,Z) are as described for BEM11.
3. Mined openings or parts of openings are laid out on
a rectangular grid, and the edges of excavation segments
are parallel to the local x,y axes which lie in the plane
of the orebody. The orientation of the y-axis is defined
by the angle ROT between the dip vector for the orebody,
described by its dip and bearing ALF, BET, relative to the Mine axes, and the y-axis.
4. Magnitudes, and orientations of the field principal
stresses relative to the Mine axes, are FP1, ALF1, BET1 etc, as for BEM11.
5. The real thickness of the orebody is TH, and the
depth of the midplane of the orebody below the x-y plane
is ZP.
6. The mined area is represented by a total of NSEG
primary excavation segments. These are subdivided in the
program into secondary segments, the faces of which are
boundary elements for the complete excavated area. The
extent of a primary segment is defined by the x and y co-
ordinates of its corners, XO, YO, XL, YL, as illustrated in figure (a) below. The number of secondary segments into
which a primary segment is divided is described by the product of NELX and NELY, counted parallel to the x and y
axes respectively.
X 297
(Xo,yo)
nely
(xl,yl)
netx (a) y
Within the program, each secondary segment I is
assigned a type code, IT(I), depending on whether the
segment abuts unmined ground or a mined area. The
code numbers assigned to the different types of segments
are indicated in figure (b).
The secondary segment code numbers are assigned by
the program from data supplied by the user defining the
types of boundaries for the primary segments.
If both NELX and NELY are greater than unity, the
code for defining primary segment edges (ITYP(L), L=1,4)
is given in figure (b), and the input format is 4I2.
X 410- X X
© y 20 Edge Code Numbers ITYP(L) For Primary Segments (b)
6 5 9
2 1 4
7 3 8
Secondary Segment Code Numbers ITO)
If both NELX and NELY are unity, or either is unity,
the input code for edge types is that for secondary
segments, shown in figure (b). If both are unity, the
input format for ITYP(L), L=1,4 is 6X,I2. If either
is unity, demonstrated by the cases shown in figure (C),
the input format is 2X, 3I2.
Nelx=1
I 1 ( K N
ITYP(2)=1T(I )
ITYP(3)= IT(J)
(TYP(4)=1T1J)
Nely=1
J
K
N
298
X
(C )
7. The elastic properties of the rock mass are defined
by its Young's Modulus, EMOD, and Poisson's Ratio, RNU.
8. The number of iterative cycles to solve for the
segment loads is specified by NCYC. Typically less than
10 cycles are sufficient to achieve satisfactory convergence.
9. Stresses and displacements at internal points in the
medium are calculated at the nodes of a planar grid. The
grid plane may be parallel to the xy, yz or zx planes,
defined by ITREG = 1,2,3 respectively. The grid is
specified by boundary lines parallel to the co-ordinate
axes, given by BL1, BL2, BUl, BU2, and these are NELl
and NEL2 grid lines. The distance of the grid plane from
the reference plane is specified by ELP.
10. Input Data Format
The structure of the data input deck and the format
of data is illustrated on the following sheet.
Card Al
problem title
(8A10)
up to 80 alpha-numeric characters describing
the problem
CARD
TYPE
Al
A2
A3
A4
A5
A6
A7
S
G
INPUT FOR TABS
10 20 30 40 50 60 70 80
„ ____ ,T_1 TLE_,,
NSTR TIN FNSHNCASE
1 '. F P 1
L.L_! , ! N S E G
t I A L F 1
1 _ '..
NCYC
!I ,
1 i
N,REG
' I, I
I' I I, I. ! I I I I I I III I I , I ,
-
.---- - -
'--- -
._ •- .
' I 1 I • RNU I ! EMOD 1 1; 1 , 1 THS '_. . ,,, I 1 B E 7 1 1 1 ! I It i 1. 1 1 III 1 1 1 1 1 I ,
! I! I I ;F P 2 ',! I! A L F 2 '1; 1 1 ,,B E T 2 I! 'I IIII
1 I I'
I
I i
' I I
J
1 IIII
111111111
1 i l l l 1 1 1 1 1
I I! I! I II I
II II , 1 1
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1:
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I I I H A X J . I ! I I I 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1! I I I l i I 1 I 1 1 1 1 1 1 1' 1 I
N E L XI N E L Y ' I• ' 1 X 0 1 I Y O 1 1 ' 'X,L J 1 1 1 1;
BU ll
1 1 1 1 1 t
I YL
1
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1 1 1 1 ! 1 1
I'
1
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BL11 ' 1 1
! 1 1 1 1 1 1 1 1
BL2]
I I
!I
1 1 1 1 !; i I 1 I 1 I 1 1 1 1 1 1 1 1
1 1 1 1! ! I 1 I 1 1 I, ! i l l 1 1 1 ! 1 1 1! 1 11 1 1 1 11 11 1 I I !
! I r! ! 1 1.I I I
I I i: 1 1 1 „ 1
III '1 1 • 1 1 1 1:1 ill r .--
I .
• I i ,
I I
1 11 ' I 1 I I•
1, 1
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1 I I !
1 1 1 , 1 I ' I 1 I I I I I I I
300
Card A2 control information and rock mass properties
(4I5,10X,
2I5,3F10.0) Cols 1-5 NSTRT = 0 for initial run
= 1 for restart run (read
data from TAPE8)
Cols 6-10 NFNSH = 0 no data written to
TAPE8
1 data written to TAPE8
for filing and restart
job identification number
number of excavation
primary segments
number of iterative cycles
in equation solutions
number of problem regions
or grids specified
Poisson's Ratio for rock
mass
Young's Modulus for rock
mass
Orebody thickness
Cards A3,A4,A5 principal stress magnitudes, dips and bearings
(each 3F10.0)
Card A6
orients y-axis in plane of the orebody, as
(3F10.0)
described in Paragraph 3.
Card A7 required only when NSTRT=1, in which case
(I10) the data deck consists of cards A1,A2,A7
and G only
Cols 1-10 MAXJ total number of excavation
secondary segments
=
Cols 11 -15 NCASE
Cols 16-20 NSEG
Cols 31 -35 NCYC
Cols 36-40 NREG
Cols 41 -50 RNU
Cols 51 -60 EMOD
Cols 61 -70 TH
301
Cards S there are NSEG cards S, each defining an
(2I5,5F10.0, excavation primary segment, as described
4I2) in Paragraph 6.
Cards G there are NREG cards G, each defining a
(3I5,5F10.0) grid, as described in Paragraph 9.
11. Output Data
(a) Input data
(b) Stresses and displacements at the centres of
elements defining the footwall sides of
excavations.
(c) Stresses and displacements at the nodes of
the specified grids.
2?
.271
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WRITE(7.42) PX.PPY.PZ.PXY.PYZ.PZX 42 FORMAT(114 //.7X.58HFIELD STRESS COMPONENTS RELATIVE TO EXCAVATION
1LOCAL AXES Z. 2 1H0.10X.2HPX.F7.3/. 3 1640.10X.2HPY.F7.3/. 4 1N0.10X02HPZ.F7.3/. 5 1140.10X.3HPXY.F7.3/. 6 1H0.10X.3NPYZ.F7.3/. 7 1H0.10X.3HPZX.F7.3) TZI=-PZ,COF1 MI=-0YZ,COF3 T2XI=-PZX/COF3 I=0 NSEGG=O WRITE (7.30)
30 FORMAT(1140/.4X.6HSEG NO.10)4 NELX NELY11eX.2u)0.6X.2HY0.8X.2HXL.8X. 1 214YL.8X.2wEP.2X.8HS7OPE HT.X.10HEDGE TYPES)
700 IF(NSEGG.EO.NSEG) GO TO 100 NSEGG=NSEGG+1 NELG=0 REA0(1.25) NELX.NELY.)(O.YO.XL.YL.ZP. (ITYP(L) .L=1.4)
25 FORP T(2I5.5F10.0.4f2) IRITE(7.33) NSEGG.NELX.NELY.X0.YO.XL.YL.ZP.TH.(ITYP(L).1=1.4)
33 FORMiT(1H0.6X.I3.215.6F10.3.2X.4I2) RNELX=NELX DX=(XL-X0)/RNELX RNELY=NELY DY=(YL-VO)/RNELY DO 111 J=1.NELX DO 111 K=1.NELY I=I+1 SOX(I)=0.5=OX SOY(I)=0.5XDY RJ=J CX(I) =XO+(RJ-0.5) ZDX EX(I) =CX(I) RK=K CV(I) =YO+ (RK-0.5) 30Y EY(I) =CY(I) SOZ=0.5=TH CZI(I)=ZP+SDZ CZ2 (I) =ZP+TOL EZ (I) =EP IT(I) =5 IF(ITYP(1).LT.1) GO TO 112 IF(J.E0.1) GO TO 170 IF(J.EO.NELX) GO TO 180 IF(K.E0.1) GO TO 190 IF(K.EO.NELY) GO TO 200 GO TO 110
170 IF(K.E0.1) GO TO 171 IF(K.EO.NELY) GO TO 172 IT(I) =ITYP (1) GO TO 110
171 IF(ITYP(1).E0.5.AND.ITYP(4).E0.5) GO TO 110 IF (ITYP (1) .LT.S.AND. ITYP (4) .E0.5) GO TO 173 IF(ITYP(1).E0.5.ANO.ITYP(4).LT.5) GO TO 174 IT(I) =6 GO TO 110
173 IT(I) =ITYP (1) GO TO 110
174 IT(I) =ITYP (4) GO TO 110
172 IF(ITYP(1).E0.5.AND.ITYP(2).E0.5) GO TO 110
IF(ITYP(1).LT.5.RND.ITYp(2).E0.5) GO TO 175 IF(ITYP(1).E0.5.ANO.ITYP(2).LT.5) GO TO 176 IT(I) =7 GO TO 110
175 IT(I) =ITYP (1) GO TO 110
176 IT (I) =ITYP (2) GO TO 110
180 IF(K.E0.1) GO TO 181 IF (K.EO.NELY) GO TO 182 IT (I) =ITYP (3) GO TO 110
181 IF (ITYW (4) .E0.5.AND. ITYP (3) .E0.5) GO TO 110 IF (ITYP (4) .LT.5.AND.ITYP(3) .E0.5) GO TO 183 IF(ITYP(4).E0.5.ANO.ITYP(3).LT.5) GO TO 184 IT(I)=0 GO TO 110
183 IT(I) =ITYP (4) GO TO 110
184 IT (I) =ITYP (3) GO TO 110
182 IF(ITYP(2).E0.5.AND.ITYP(3).E0.5) GO TO 110 IF(I7YP(2).L7.5.AND.ITYP(3).E0.5) GO TO 185 IFCITYP(2).E0.5.ANO.ITYP(3).LT.5) GO TO 186 IT(I)=8 GO TO 110
185 IT(1) =ITYP (2) GO TO 110
186 IT(I)=ITYP(3) GO TO 110
190 IT (I) =ITYP (4) GO TO 110
200 IT(I)=ITYP(2) GO TO 110
112 CONTINUE IF(ITYP(2).GT.0) GO TO 113 IT(I) =ITYP (4) GO TO 115
113 CONTINUE IFCNELY.E0.1) GO TO 114 ITCI) =ITYP (3) IF(K.E0.1) IT(I)=ITYP(2) IF(K.EO.NELY) IT(I)=ITYP(4) GO TO 115
114 IT (I) =ITYP (3) IF(J.E0.1) IT(I)=ITYP(2) IF (J. EO. NEL)0 IT (I) =ITYP (4) GO TO 115
110 CONTINUE ITI=IT(I) IF(ITI.E0.5) IT(I)=1 IF(ITI.LT.5) IT(I)=ITI+1
115 CONTINUE OZ(I)=0.0 SX(I)=0.0 SY(I)=0.0
111 CONTINUE GO TO 700
100 CONTINUE M9XI=f MiXJ=I
800 CONTINUE C C CALCULATE COEFFICIENTS
DO 505 J=1.NEL1 C JJ-J-1
DO 120 I=1.M9XI RJ=JJ CALL COEFFS DO 505 K=l.NEL2
120 CONTINUE KK=K-1 REWIND 2 RK=KK
1=I+1 C EQUATION SOLVER A02CI>=8L1+RJxO1 C B02(I)=BL2+RK202
IF(NN.GT.0) GO TO 404 COZ(I)=ELP REWIND 6 505 CONTINUE CALL SOLVER MAXI=I IF(NFNSH.LT.1) GO TO 404 GO TO (510.515.520) ITREG WRITE(8)(EX(J).EYCJI.EZCJ)•SDX(J).SDY(J).SX(J).SY(J).91(J)•IT(J). 510 00 511 I=1.MAXI
1 J=1.IAXJ).PY23.PX.PPY.PZ.PXY.PYZ.PZX.TOL.COFI.00F2.COF3.COF4.FAC• CXCI)=AOZCI) 2 SNU.RNUI2.RNUP2.RNU4.RNUM4.RNU32.RNUP1.RNU2.RNU1.RNU.BAS.BASE. CY(I)=B0Z(I) 3 SDZ.ZP CZ1CI)=COZCI)
404 CONTINUE 511 CONTINUE C WRITE(7.512) NELI.NEL2.BLI.BL2.BU1.6U2.ELP C CALCULATE STRESS COPPONENTS.OISPLACE1ENT COMPONENTS 512 FORD T(1H ///.7X.12MNELX. NELY = .2I6//. C 17X.20HX0. Y0. XL. YL. ZP = .5F10.3)
DO 500 I=1.PgXI GO TO 525 CALL STRESS 515 DO 516 I=1.MRXI
500 CONTINUE CX(I)=CGZ(I) WRITE (7 .35) CY C P =AOZ (I )
35 FORFAT(1H ///.3)02H I.7X.2WCX.11X.2HCY.8X.2HCZ.6X.414SIGX.6X.4HSIGY. C21(1)=6.02(1) 1 6X.4HSIG2.5X.5HTAUXY.5X.5MTAUYZ.5X.5HTAUZX.11X.1HU.11X.1NV.11X. 516 CONTINUE 2 11414/) WRITE(7.517) NELS.NEL2.8l1.Bl2.BU3.BU2.ELP WRITE(7.40) (I.CX(I).CY(I).CZ1(I).SIGX(I).SICY(I)rSIGZ(I).7AUXY(I) 517 FORMAT(1H ///.7X.IPHNELY. NELZ a .216//.
1. TAUYZ(I).TAUZXCI)rU(I).V(I)•N(I),I=1.MAXI) 1 7X.20HY13. Z0. 1'L. ZL. XP = .5F10.3) 40 FORMATC1H .I4.9F10.3.3E12.4) GO TO 525
WRITE(7.550) 520 DO 521 I=1.11 XI 550 FORPATCIN /i.7X.18MPRSNCiPAC STRESSES) CX(I)=692(I)
WRITE(7.179) CY(I)=COZ(I) 176 FORMaT(1140.3X.214 I.B HS X.2HCXrBX.2HCY.BX.2NC2.6X.4IGS.X.I7HALPHA B CZICI)=AQZCI)
ZETA GAFT.NX.4HSIC2.1X.17MALPNq BETA GAH1ii.6X.4HS2C3.X.17HALPHA 521 CONTINUE 2BETA GAMMA) WRITE (7.522) HELI.NEL2.BLI.BL2.BUI.BU2.ELP 141ITE(7.179)(I.CX(I).CYCI).C21(I).SIG(I.1).DALF(1.I).DBET(1eI). 522 FORMAT(1N ///.7X.12HNELZ. NELX = .2I6//.
1 OGAM(l.I).5IGC2.I).DALF(2.I).D6ET(2.I).DGAM(2.I) .SIG (3.I). 1 7X.20HZ0. X3. ZL. XL. YP = .5,10.3) 2 DALF(3.11.DBET(3.D.DGAM(3.II.I=1.VIAXI) 525 CONTINUE
179 FORPRTC1H0.15.4F10.3.3F6.1.F10.3.3F6.1.F10.3.3F6.1) CZU=ZP-SDZ IF(NN.EQ.NREG) GO TO 503 CZL=ZP+SDZ NN=NN+1 DO 530 1 1.PRXI
C C21I=CZ1(I) C ITREG=1 NOMINATES PROBLEM AREA IN X-Y PLANE IF(CZII.GE.CZU.AND.C2II.LE.CZL) GO TO 535 C ITREG=2 NOMINATES PROBLEM AREA IN Y-Z PLANE IF(CZII.LT.CZU) CZ2I=CZII+SDZ C ITREG=3 NOMINATES PROBLEM AREA IN Z-X PLANE IF(CZII.GT.CZL) CZ2I=CZII-SDZ C GO TD 540
GO TO 796 535 CONTINUE 795 CONTINUE C22I=ZP+TOL
NN=1 540 CZ2(I)=C22I READ(1.797) PAXJ 530 CONTINUE
797 FORPAT(I5) REWIND 2 READ(8) (EX(J).EY(J).EZ(J).SDX(J)•SDY(J).SX(J).SY(J).02(J),IT(J). GO TO 600
1 J=1.MAXJ).PY23.PX.PPY.PZ.PXY.PYZ.P2X.TOL.COFI.COF2.COF3.00F4.FAC. 503 CONTINUE 2 SNU.RNUI2.RNUM2.RNU4.RNUM4.RNU32.RNUPI.RNU2.RNUI.RNU.BAS.BASE. STOP 3 SDZ.ZP END
796 CONTINUE ;DECK SOLVER READ(1.26) ITREG.NELS.NEL2.BL1.RL2.BUI.BU2.ELP SUBROUTINE SOLVER
26 FORPAT(3I5.5F10.0) 3tALL GEM 1=0 M=0 RNELI=NEL1-1 5 DO 10 I=1.PPXI 01=01U1-6L1)/RNELI PZI=TZI RNEL2=NEL2-1 PYZI=TYZI 02=(BU2-6L2)/RNEL2
PZXI=TZXI READ (6)(COZ(J).EOZ(J).FOZ(J).CSX(J).ESX(J)IFSX(J).CSY(J).ESY(J)•
1 FSYCJ).J=1.MaXJ) DO 20 J=1.111XJ IF(I.EO.J) GO TO 20 PZI=PZI-COZ(J)10Z(J) PYZI=PYZI-ESX(J)xSX(J)-ESY(.1)XSY(J) PZXI=PZXI-FSX (J) XSX (J) -FSY (J) XSY (J)
20 CONTINUE OZ(I)=PZI/C0Z(I) SY(I)=(PYZI-.ESX(I)TSX(I))iESY(I) SX(I) = (PZXI-SY(I) 7SY(I)) /FSXCI)
10 CONTINUE 11=M+1 REWIND 6 IFCM.LT.MCYC) GO TO 5 RETURN END
XDECK STRESS SUBROUTINE STRESS
*CALL GEN SIGXI=PX SIGYI=PPY SIGZI=P2 TAUXZI=PXY TAUYZI=PYZ TAUZXI=PZX UI=0.0 VI=0.0 NI=0.0 READ (2)(AOZ(J).BOZ(J).COZ Li) .DOZ(J).EOZ(J).FOZ(J).UOZ (J).VOZ(J).
1 1422 (J) .ASX(J) .BSX(J) .CSX Li) .DSX(J) .ES% (J) .FSX(J) . USX (J) .VSX(J) . 2 1.15X (J) •ASY(J) .BSY(J) .CSY(J) . DST (J) .ESY(J) .FSY(J) .USY(J) .VSY(J) . 3 NSY(J).J=1.MAXJ)
DO 10 J=1.MAXJ SIGXI=SIGXI4A0Z(J)X02(J)■COF1+(ASX(J)'SXCJ)+ASY(J)X5Y(J))XCOF3 SIGYI=SIGYI+90Z(J)XOZ(J)xCOF1+(BSXCJ)'SXCJ)+6SY(J)XSYCJ)) 3C0F3 SIGZI=SIGXI+COZ(J)IOZCJ)1COF1+(CSXCJ)*5X(J)+CSY(J)X5Y(J))xCOF3 TAUXZI=TAUXYI+OGZ(J) )Z (J)1C0F1+(DSX(J)*SX(J)+DSYCJ)XSYCJ))ZCOF3 TAUYZI=TAUYZI+ēOZ(J)=02 (J)xCOF1+(ESX(J)'SX Ci)+ESY CJ)x5Y(J)) xCOF3 TAUZXI=TAUZXI+FOZ(J)XOZ(J) 11COF1+CFSX(J)*5XCJ)+FSY(J)x5Y(J))xCOF3 UI=UI+U0Z(J)XOZ(J)XCOF2+CUSX(J)XSX(J)+USY(J)XSY(J)) COF4 V I=V I+VOZ (J) XOZ CJ) XC0F2+ (VSX CJ) XSX CJ) +VSY (J) XSY CJ)) *COF4 NI=NI+C.GZ CJ) X0Z (J) mcCOF2+(N5X (J) x5X (J) +CŌY (J) XSY (J)) xCOF4
10 CONTINUE SIGX(I)=SIGXI SIGYI)=SIGYI SIGZ(I)=SIGZI TAUXY(I)=TAUXZI TAUYZ(I)=TAUYZI TAUZX(I)=TAUZXI U (I) =UI V (I) =VI W(1)=WI RJI=SIGX(I)+S2GY(D+SIGZ(I) RJ2=SIGX(I) =SIGY(I)+SIGY(I) XSIGZ(I)+SIGZ(I) XSIGX(I) -
1 CTAUXY(I) XTAUXY(I)+TAUYZ(I) XTAUYZ(I)+TAUZX(1) XTAUZX(I)> RJ3=5IGX (I) XS IGY (I) XS IGZ ( I) +2. 0zTAUXY ( I) XTAU YZ (I) *TAUZX (I) -
1 (SIGX(I>XTAUYZ(I)XTAUYZ(I)+SIGY(I)XTAUZX(I)XTAUZX(I)+ 2 SIGZCI>XTAUXY(I)ZTAUXY(I))
RJ4=SORT(RJ1xBJ1-3.07RJ2) TCs (27. 07RJ3+2. 0xRJ 1 =3-9. 0=RJ 1)4J2) / (2. OXRJ4xx3) IF(TC.GT.1.0> TC=1.0 IFCTC.LT.-1.0) TC=-1.0
THET=ACOS (TC) /3.0 00 35 K=1.3 GO TO (26 27.26) K
25 ANG=THET GO TO 29
27 ANG=PY23-THET GO TO 29
28 ANG=PY23+THET 29 CONTINUE
5IG(K.I)=CRJ1+2.0=4RJA=COS(AMG)) /3.0 TA= (SIGY (:) -SIG (K. I)) X CS IGZ (I) -SIG (K. I)) -TAUYZ (I) XTAUYZ (I) TB=TAUYZ (I) ZTAUZX ( I) -TAUXY (I) x (S IGZ (I) -510 (K . I) ) TC=TAUXY(I)*TAUYZCI>-TAUZX(I)x(SIGY(I) -SIG(K.I)) STS=SORT(TRZTA+TBX7H+TCXTC) DCX=TA,STS DCY=TB/STS DCZ=TC/STS DALF (K. I) =ACOS (DCX) /FAC OBET (K. I) =ACOS (DCY) /FAC DGAM(K. I) =ACOS (DCZ) /FAC
35 CONTINUE RETURN END
=DECK COEFFS SUBROUTINE COEFFS
HALL GEM CZI=CZ2 (I) DO 600 J=1.MAXJ DZ=CZI-EZ(J) IF(DZ.E0.0.0) DZ=TOL Z2=0Z0,0Z Z3=Z2XDZ Z4=Z2'Z2 ZIMV=1.0/DZ 1. 20 IF(IT(J).GT.1) M=47 IF(IT(J).GT.5) M=57 DO 605 L=101 PI(L)=0.O
605 CONTINUE IFCIT(J).E0.1) GO TO 635 ITJ=IT (J) -1
C C TRANSFORM TO LOCAL CO-OROS FOR EDGE ELEMENTS C
GO TO (610.615.620.625.610.615.620,625) ITJ 610 EXJ=EX(J)
SDXJ=SDX(J) EYJ=EY (J) SOYJ=SDY(J) CXI=CX(I) CYI=CY(I) SIMR=0.0 COSR-1.0 COS2R=1.0 GO TO 630
615 EXJ=-EY(J) SDXJ=SDY(J) EYJ=EX(J) SDYJ=SDX(J) CXI=-CY (I) CYI=CX(I) SINR=1.0 COSR=0.0 O
U,
C052R=-1.0 GO TO 630
620 EXJ=-EX(J) SDXJ=SDX(J) EYJ=-EY (J) SDYJ=SDY(J) CXI=-cx(I) CYI=-CY(I) SINR=0.0 COSR=-1.0 C0S2R=1.0 GO TO 630
625 EXJ=EY(J) SDXJ=SDY(J) EYJ=-EX (J) 5DYJ=SDX(J) CX1=CY(I) CVI=-CX (I) SINR=-1.0 COSR=0.0 C052R=-1.0
630 CONTINUE X11=ExJ-SDXJ XJ(1)=0.0 XJ (2) =2.0=SDX1 YJ1=EYJ-SDrJ YJ(1)=0.0 YJ(2)=2.0=SDYJ cxl=cxI-X11 CYI=CYI-YJ1 GO T1 650
635 CXI=CX(I) CYI=CY(I) XJ (1) =EX (J) -SDX (J) XJ (2) =EX (J) +SOX (J) YJ(1)=EY(J)-SDY(J) V.1(2) =EY (J) +SOV (J)
650 CONTINUE D0 660 KI=1.2 DO 660 KJ=1.2 CON=-1.0 IF(KI.EO.KJ) CON=1.0 DX=CXI-X1(K I) DY=CYI-YJ (KJ) DX2=0XXOX DY2=0Y=DY R2=0X2+OY2+22 R=50RT(R2) ALF=ATAN (DX=DY.R/0Z) RPY=R+DY RLRPY=ALOG(RPV) XY2=DX=DY=DZ V2=0Y2+22 RV2=R=V2 SIN2A=SIN(2.0=ALF) ZOR2=Z2/R2 ZOR2P=1.0+20R2 U2=0X2+22 RU2=R=U2 X2Y=DX2=OY RPX=R+DX RLRPX=ALOG(RP)0 XY2=DX=DY2 R3=R2=R
SIN44=5IN(4.0=ALF) ZOR4=ZOR2=ZOR2
PI(1)=PI(1)+CON=XYZ/RV2 PI(2) =PI (2)+CON=0.5=7.0R2P=5IN2A PI(3) =PI (3)+CON=XY /RU2=20R2=(1.0+2.0=22/U2) PI(4)=PI (4) +CON=XYZ/RU2 PI(5) =PI (5) +CON=XY7/RV2=ZOR2=(1.0+2.0=R2/v2) PI C6) =PI (6)+CON=(2.0+20R2+Z0R4) XSIN2A PI (7) =PI (7) +CON/4.0=20R2P=CZOR2P=S IN4A PI(6) =PI (8)+CON/R PI (9) =PI (9) +CON/R3 PI(10) =PI (10)+CON/R/RPX PI(11) =PI (11)4CONX(R+RPX)/R3/RPX/RPX PI(12) =PI (12)+CON/R/RPY PI(13) =PI (13)+CON=(R+RPY)/R3/RPY/RPY PI(14) =PI (14)+CON=RLRPY PI(15) =PI (15)+CON=RLRPX PI (16) =PI (16) +CON=ALF PI(17) =PI (17)+CON=(X2Y/RU2=(1.0/R2+2.0/U2)-2.0/R/RPY) PI(16) =PI (18)+CONXOY/R3 PI(19)=PI(19)+CON=DX/R3 PI(20) =PI (20)+CON=(XY2/RV2=(1.0/R2+2.0/V2)-2.0/R/RPX) IF- (IT CJ) .EG.1) GO TO 660 YORR=OY/R XY=DX=DY YORM=1.0-YORR D)0=DX2=DX XORM=1.0-DX/R DY3=DY2=DY 20v2=22/V2 PI(21)=PI(21)+CON=VORR PI (22) =PI (22) +CON=OY=RLRPX P1(23) =PI (23) +CON=xY/R PI (24) =P1(24) +CON=OX3/U2=YORN PI (25) =PI (25) +CON=OY3/v2=XORM PI(26) =PI (25)+CON=OY/V2=XORM PI(27) =PI (27)4CON=DY3/V2/V2=)CORM PI(28) =PI (28)+CON=XY/R3=(1.0+20V2) P1(29) =PI (29)+CON=(2.0+3.0=7802-Z0R4)=SIN2A PI(30)=P1(30)+CON=X2Y/RU2 PI(31) =PI (31)+CON=DY=(0.5+ZOV2) PI(32) =Pi (32)+CONXXY/R=(1.0+20V2) PI(33) =PI (33)+CDN=XV/R7 PI (34) =PI (34) +CON=X2Y=LOX/U2/R3 PI(35) =PI (35)+CON=DX3/U2/U2=YORM PI (36) =PI (36) +CON=XY2=0Y/V2/R3 PI(37)=P1(37)4CON=DX/R PI(38) =PI (38)+CON=Pt=(1.0+V2/R2) P1(30)=PI(39)4CDN/RX(3.0-V2/R2) PI(40) =PI C40)+CONXRXDY Pr (41) =P I (41) 4C3N=R PI(42) =PI (42)+CONXXY2/R3 PI(43)=P1(43)4CDN=h2Y/R3 PI(44)=P1(44)+CON=DX=V2/R3 PI (45) =PI (45) +COM/R= (3.0-U2/92) PI(46)=P1(46)+CON=RXOX PI (47) =PI (47)+CON=V2=RLRPX IF(IT(J) .LT.6) GO TO 660 20U2=Z2/V2 PI(48)=PI(46)+CON=XY2/RV2 PI(49)=PI(49)+CON=DX=RLRPY P1(50)=PI(50)+CON=Ox=213U2 PI(51)=P1(51)+CONXXY/R=(1.0+20U2)
ESYJ)=ESYJ FSY(J)=FSYJ U5Y(J)=USYJ VSY(J)=USYJ ISY (J) =USYJ
600 CONTINUE I.RITE(2) (A02(J).BOZ(J).COZ(J).DOZ CJ) .EOZ(J).F02(J).UO2(J).VOZ(J) .
1 402(J) .ASX(J) .BSX(J) .CSX(J) .DSX(J) .ESX(J) .FSXCJ) .US%(J) .VSX(J) 2 1SX(J).A5Y(J) .BSY(J).CSY(J).05Y(J) .ESY(J) .FSYCJ) .USY(J).VSY(J), 3 ISY(J) .J=1.MXJ)
IF(NN.GT.0) GO TO 999 1.RITE(6) CCOZ(J) .EOZ(J) .FOZ(J) .CSX(J) .ESX(J) .FSXCJ) .CSY(J) .ESY(J)
1 FSY CJ) .J=1.NRXJ) 999 CONTINUE
RETURN END
*DECK EDCOR SUBROUTINE EDCOR
*CALE GEN IF(IT(J).GT.5) GO TO 665 CK 1=XJ (2) xXJ (2) CK2=2.0xXJ (2) xCXI-CXIxCXI CK3=2.0*(CXI-XJ(2)) CK4=CK1) ASE4CK2 CK01=1.0/CK1 5TPI1=PI (23) -PI (24) -0I (25) +OZ1TI (2) STPI2=PI (33) +PI (34) -2.07421 (35) 4PI (36) -2.0)4,1 (27) STP13=3.0191(25)+2.0421(27) 401(28) AOZJ=CKO1x(CK4*AQZJ+CK3*(-PI(14)-RNU21PI(21)+22xP1(17))-RNU1 VIP I (2
12) +4.0xVZWPI(15)+111.1U2/1.01KSTPI1+02xPl(7)-0Z1P1(29)-72xSTPI3> BOL=CKOlx (CK 4x607J+CK3x(-RNUN21PI(14)+RNU21PIC30)-221PIC18) -
1 Z2xPI(12))+RNU4IIPI(22)+RNUPMxOZx?I(16)4 NU2xPI(31)-RNU2xPI(32)+ 2 RNU21azIPI(2)+22/3.OxSTPI2+DZ/3.Ox (PI (7) -PI (29)))
COL= CK01x(CK4xCOZ.HCK3x(-PI (14)+2.0x22xPI (12) -Z4IIP1(13) ) -0I (22) + 1 2.0x0ZxPI(4)+DZAQI(2)-0Z1 '1(3)) 00ZJ=CKOlx(C1C4*0Q2.J+CK3x(RNU2xPI(37)-RNU2xP1(15) -72xP I(19)-Z2xPI(1
10)) +RNU2WPI(38)+22xPI(39)) EOZJ=CKOlx (CK4xEQZJ4CIC3x (DZx9I (B) -Z31P1 (9)) -0Zx (PI (37) -0I (15)) +
1 Z3x(P1(19)+P1(10))) FOL=CKOI*(CK4IPOZJ+CK3x(-0I(4) -PI (2)+PI(3))+OZxPIC30)+2.0xDZ*PI (1
24) -23xPI (17) ) UOL=CKOIx(CK4xUO2J+CK3x(RNU21PI(22) -am 12xaZxPI(16)+OZxQI(4))-
1 RNU2xPI(40)-22x(RNU32) TIC14)401(30))) voZJ.GJ(01x (CK4ZVO2J40c3x (-RNU2aP1(41) -227431 <6)) 4oU2/2, 0* (PI (47) -
1 PI (46)) 422x (PI (37) -P1(15)) ) I.0ZJ=CKO1x(CK41140ZJ+CK3*(RNU211)ZxP1(14)-23IPI(12))+RNU23 DZxP1(22)+
1 RNUrQ122xPI (16) -22xPI (4) ) ASXJ=CKOIx(CK4#1SXJ+CK3*(DZx5TPI3-2.O'PI(16) -PI (4) -PI (7)+PI(29))+
1 DZx(-4.0PPIC30)-8.0xPI(14) -PI (21) -PI (43)+221P1(12)+72WAI (17))) BSXJ=CK01x(CK41i)SXJ4CK3x ORMUZ* (PI(15) -PI (4))-(OZ'STPI2+3.O*Q1(16)
2 +PI(7)-P1(29))/3.0)+02x(RNU2xPI(30)-RNUM4)0131(14)+PI(21) 4P1(43) -22 31PI(12))) CSXJ=CK01x(CK4xCSX .CK3*(-PI(4)-0I(2)+PI(3))4OZx(PI(30)+2.0xPI(14)
1) -,23x?I(17)) 05XJ=CK01 x (CK4xCSXJ+CK3x (RNUIITZ*P 1(8) -02x91(39)) -0Zx (RNU1(P 1(37) -
1 PI(15))-4.019I(37)+3.0x9I(15)+PI(44))) ESXJ=CKOlx(CK4xESXJ4CK3x(RNUx (PI (37) -PI (15))-22x (PI C19)4PI(10)))+
1 RNU1P1 (38) +22IIPI (39) ) FSXJ=CK01x<CK41FSXJ+CK3x(-01(14)-RNUxPI(21)+22xPI(17))_RNUPlx (PI (2
12) -021P1(16))+RNU/3.0x(STPI1-3.0x0ZW1 (I6))-Z2XSTP13+02*(3.0xP1C1 26) +PI (7) -PI (29) ) )
USXJ=CKO1x(CK4xUSXJ4CK3x(RNU32147Zx9I(14)+(MAPI(30))+02*(RNU2x (PI (2 12)-0Z*I(16))+3.01P1(22)+PI(31) -PI (32)-3.0x02xPI(16)+02=P1<2)))
VSXJ=CK01x (CK4xVSX1-CK3xVZx (PI (37) -0I (15)) -02xPI (38) )
PI(52)=PI(52) +CON*Rx (1.O+U2/R2) PI (53) =PI (53) +CONx0X/U2KYORN PI (54) =PI (54) 4CONx0X3/U2/U2xYURN PI (55) =PI (55) +CONXXY/R3x(1.0+23112) PI (56) =PI (56) +CONxu21KL RP PI (57) =PI (57) +CONx0YxU2.423
660 CONTINUE C
A01.J ZINVx(RNU2xP1(1)-0I(2)+PI<3)) BOZJ=Z INV* (RNU2xPI (4) -Pt (2)+PI(5)) COZJ=ZINVx(2.0xP1(2) -PI (6) -PI (7)) D02J=RNU2 01 (8) -Z2xPI (9) EOZJ=DZKP:(10)-23xPIC11) FOL=02'PI(12)-Z3=PI(13) UO2J=-RNU2IFI (14) -22IPI C12) VOZJ=-0NU2ZP7(15)-221PIC10) 1-02J=-2.03RNU1xP1(15)-P1(2) ASXJ=DZx CPI (12) +PI (17) ) 8SXJ=DZx(-2.0*ANUzP1(12) -PI (18)) CSXJ=DZYPI(12)-2.3KFI(13) DSXJ=DZXC-RNUIXPI (10) -PI (19) ) ESXI=RNUaPI(8) -J2xPI (9) FSXJ=ZINVx (RNUNPI (1) -PI (2) +01 (3) ) USXJ=-2.0KRNUI KPI(16)+PI (4) V5XJ=-0ZAP1 (8) ISXJ=RNU2xPI (14) -22W1 (12) ASYJ=0Zx(-2.0xRNU1AI(10) -PI (19)) BSYJ=DZx CPI (10)+PI(20)) CSYJ=DZ1PI(10)-'Z3*PI(11) DSYJ=DZx(-RNU1ZPI(12)-PI(18)) ESYJ=ZINV*(RNUW1(4) -PI (2)+PI(5)) FSYJ=RNUIPI(8) -224PI(8) USYJ=-OZIPI(8) vSYJ=-2.0xKNU11PI(16)+PI(1) ISYJ=RNU2xAI(15)-22AP1(10) IF (IT (J). CO. 1) GO TO 680
C C CORNERS. EDGES C
CALL EDCOR GO TO 600
680 CONTINUE AOZ U) =AOZJ BIM (J)=BOZJ COZ CJ) =COZJ DOZ CJ) =1302J EOZ(J)=E02J FOZ (I) =FOZJ UOZ CJ) =UOL WIZ (J) =%OL 1.402 (J) =1.GL ASX(J)=ASXJ 65X(J)=BSXJ CSX (J) =CS XI DSX(J)=05XJ ESX(J)=ESXJ FSX(J)=FSXJ USX (J) =USXJ VSX(J)=VSXJ ISX (J) =ISXJ ASY(J)=ASYJ BST (J) =BSYJ CSY (J)=CSYJ DSY(J)=05YJ
46XJ=CK01X(CK4X46XJ+CK3X(-RNU2X CPI (22) -0Z421 (16)> -0ZX (P1(16) -PI(4) 1)) +RNU2X (pi (40)+Z21431(14))-22zcPI(30) +2.0=P1(14))>
ASYJ=CK01X(CK4XASYJ+CK3X0ZZCRN)2XPI (ED -PI (39))-0ZX(RNU2X (PI (37)-1 PI(15))-4.019I(37)+3.0=FI(15)+PI(44)>) BSYJ=CK01X (CK4385YJ+CK310Z1(PI (8) -PI (45)) -02X cPI (15) -PI (42) +22XP1
110) ) ) CSYJ=CK011(CK4ICSTJ+CK3XDZX(PI(8)-Z2XP1(9))-0ZZ CPI (37)-PI(15))+
1 23Z (pi (13) +PI(lO)) ) OSYJ=CKOIX(CK4XDSYJ+CK3X(RNUX(PI(16)-P1(4))-0Z 3.0X5TPI2-(3.OzPI(1
16) +PI(7) -PI (29))/3.0)+OZX(RNUXCPI(30)+2.0( 1(14))+PIC21)+P!(43)- 2 2.0XP1(14)-Z2XPI(12)) )
ESYJ=CK011(CK4zESTJ+CK3XC_iNUP1XPI(14)+RNUX (PI (30)+2.0ZPI(14))- 2= 1 (PI (10)+PI(12)))-RNUPIX(PI (22)-0ZVI cis) )+RNUX(3.OXPI(22)+PI(31)- 2 PI(32)-3.0 DZXP1(16)+OZXPI(2))+22/3.0XSTP12+02/3.0X(3.0=PI(16)+ 3 PI(7) -PI (29))) F5YJ=CK011(CK4ZFSYJ+CK31(RNUX (PI (37) -PI (15)) -221(P I (19) +p1 (10) ) ) +
1 RNUZPI (38) +Z2XPI (39) ) USTJ=CK01X (CK41USYJ-CK3ZOZ1(PI (37) -PI (15)) -0ZsPI (38) ) v5YJ=CKOlX(CK4XySYJ+CK31DZX(2.OXRNUIzPI(14) -PI (21))+RNU2*ZX (PI (22
1) -OZIPI(16))+0Z/3.0z (STP 11-3.01DZIPI(16))) 46YJ=CK01Z fCK4z,SYJ+CK3z (RNU219I (41) -Z2XPI (8)) +RNU2/2.0X (PI (47) -
1 PI(46))+22X (PI (37) -PI (15))) GO TO 670
665 CONTINUE 5XC=2.0ZS0XJ 5YC=2.0ZS0YJ THET=ATAN (SYC/SXC) C052=COS(2.0XTHET) SLN2=SIN(2.OXTHET) SX2=5XCZSXC TANT=SYC/SXC U1=(1.04C052/2.0)/5XC 1.42=(2.0-C052)/2.0/5XC/TANT W =- (1. 0+C052/2.0) /5X2 U4=-(2.0-C052)/2.0/7ANT/TRNT,3x2 W5=1.0/5XC/SYC 0<1=-441-2.OICXI-CY1 Xlh CK2=-12-2.017C1I34.14 CXIXU5 CK3=40 CK4=N4 CK5=15 CK=BAS+1.111tX744.4ICYI444=CX1ZCX1+414=EYI ICYI •N 51CX11CYl 1PLF=DZWI (16) STP I1=P1(23) -PI (24) -P1 (25) -3.0Z7ALF+OZ112I (2) STP12=3.0XPI(26)+2.0XPI(27)+PI (28) STP13=3.01PI (16) +P1 (7) -PI (29) 571214=3.0191(49)+PI(50) -PI (51)-3.017ALF+OZZPI(2) 57P15=P1(33)+PI(34) -2.0XPIC35)+PI(36)-2.0191(27) STP16=3.0ZPI(22)+PI(31) -PI (32)-3.0XZALF+OZXPI(2) STPI7=3.OW I (53) +2.0XPI (54) +PI (55) R11=CK1X(-PI(14)-RNU2191(21)+Z2XPI(17)) RI2=CK2Z(-RNUr12XPI(15)+RNU2XPI(46)-22z (PI (19)+P1(10))) R 13=CJ(3Z (RNU 121(P 1(22) -ZAL F) - RNU2/3. 015TP I1+7215TP I2-0Z15TP 13) RI4=CK4Z(RNU121 (PI (49)-7ALF)-RNU2XSTP14-22/3.0XSTP15-0Z/3.0XSTPI3) RI5=CK5X(-RNU12XPI(41)+RNU2IPI(52) -UWI (39)) A0L=CKX4402J+R 11 +R I249I3+R 1449I5 R I 1=CK 1 X (-RNUC2XP I (14) +RNU219I (30) -724 (PIC 16) +P I (12)) ) RI2=CKZ.I (-PI (15) -QNU219I (37) +Z21P I (20) ) RI31CK3*CRNOI2'(P1(22)-2ALF) QNU2X5TPI6_22/3.0ZSTP15-O2./3.OZSTPI3) RI4=CK4X(RNU12Z(PI (49)-7ALF)-RK'12/3.0XSTP11+72XSTPI7-02Z5TP13) RI5=CK5*(-RNU121P1(41)+RNU219:(36)-.2XP1(45)) BOZJ=CKI80L+RI14912+4/13+R144915 RI1=CK11C-er (14) +2.034.2x131 (12)-04191 (13) ) RI2=CK2X(-01(15)+2.01P2431(10)-Z4XPI(11))
RI3=CK3X (PI (22) +02X (-2. 0XP I (4) -PI (2) +P I (3)) ) R 14=CK4X (P! (49) +02X (-2.0ZP I (1) -01 (2) +P I (5)) ) RI5=CK51(-0I(41)+2.0=22XP1 ce) -Z4XPI(9)) COZJ=CK=COZ1+R11+RI2+R13+RI4+RI5 R I1=CK1X (RNU21(P I (37) -P1 (15)) --Z2X (PI (19) +PI (10)) ) R I2=CK2X:RNU2X (PI (21) -PI (14))-Z2X (PI (18)+PI(12))) RI3=CK3X( -Rita 19I(38)-221PI(39)) RI4=CK4X(-RNU21PI(52)-22XPI(45)) R15=CK5/..OZ(RNU2X5TP11-721STP15-0ZX5TPL3) D0ZJ=CK1DOZJ+RI1+4412+R13+914+RI5 R I 1=CK 1 X (DZXP I (8) -231P 1 (9) ) R12=CK21(-P1(1) -PI (2)+PI(5)) RI3=CK31(DV((PI(37) -P I (15))-231 (PI (19)+PI(10))) RI4=CK4:OZ1(-PI(48)-2.OIPI(15)4 21PI(20)) R15=CK51DZ1 (PI (21) -PI (14)-72l (PI (18)+PI<12))) COL =CKzEOZJ+RI1+R12+R13+RI4+RIS RI1=CK1Z(-PI(4) -PI (2)+PI(3)) RI2=CK21021 (PI (8)-227i (9)) RI3=CK311)ZI(-PI(30)-2.01,I(14)+121PI(17)) RI4=CK410ZI (PI (21) -PI (14) -121 (PI (16)+PI(12))) RI5=CK51D21 (Pi (37) -PI (15)-721 CPI (19)+PI(10))) FOZJ=CKIF02J+RII+RI2+RI3+R14+RI5 RI1=CK11(RNU2IPI(22)-RNU12)1ZALF+OZIPI(4)) RI2=CK21(-RNU21P I (41) -721 1(6) ) RI3=CK3X(RNU2IPI(40)4 21(RNU321P1(14)+PIC30))) R 14=CK4X (RNU2/2. 0z CPI (SB) -PI (40)) -2.21( (PI (21) -PI (14)) ) R15lCK5I(RNU2/2.01 (PI (47) -PI (45))-Z2X (PI C37) -PI (15))) UOZJ=CK KJ02J+R I 14912+9I3+R 14+R IS RII=CKSIC-tNU21P1(41)-121PI(6)) RI2=CX.2I(RNU2I91(49)-2NUI2ZZAEF+OZ1P CI) ) R I3=CK31(RNU2/2.01CPI (47) -PI (46)) -221(P I (37) -PI (15)) ) RI4=CK4X(RNU21P1(46)+Z2*(RNO321121(15)+9I own )) RI5=CK5* (RNU2/2.01CPI (56) -0I (40)) -Z21(P I (21) -PI (14)) ) vOZJ=C.KIVOZJ+I1+R I2+R13+R 144915 RII =CK I Ia21(RNU21121(14) -22W.! (12) ) RI2=CK210Z1(RNU21p1( IS) -12191(10)) RI3=CK3Z<-RNU210Z1(P1(22)-2ALF)-221(PI(16)-PI(4))) R14=CK4X(-0NU210Z1(PI(49)-2ALF)-12*(P1(16)-P1(1))) RI5=CK51DZI (RNU21P1(41) -22191(0) ) 1 ZJ=CKX4OL49I1491249 OW414+4115 RII=CXII (PI (16) -PI (4)+OZISTPi2 -SIP I3) RI2=CK2z0ZI (PI ce) -PI(39)) R 13=-0K3101* (-4.01P I (30) -6. 01P I (14) -0i (21) -P I (43) +72191 (12) +
1 12191(17)) RI4=CK410Z1 CPI (14)-P1(43)+121,I(12)) RI5=CK510ZI (-3.01PI (37) +2.019I (15) +PI (44) ) ASXJ=CKZASXJ+R1149I2+913+RI44915 RY1=CK11DZI (RNU2IPI (0) -P1(39) ) RT2=CK21(RNU21 (PI (16) -PI (1))-(DZZSTPIS+STP13)/3.0) RY3=CK31DZ1(RNU21 (PI (37) -PI (15))-4.01P1(37)+3.01,1(15)+PI (44)) RY4=-0k410Z1(RNU21(PI (48) +2.01PI (15)) +P1(37) +Pi (42) -2.0191(15) -
1 121PI(10)) DPII=PI(21) -PI (14) RY5=C1(510Z1(RNU21OPI1-P1(21) -PI (43) +2.01PI (14) +22431 (12) ) AS TJ=C K ZASYJ+R Y 1 +R Y249 Y349 Y449 Y5 R11=CK1X(RNU2X(PI(16)-P1(4))-(DZISTPI5+STPI3)/3.0) RI2=CK2ZOZ1(RNU21PI(6) -PI (45)) R 13=CK3lOaz (RNU2KP I (30) +2.01P1 (14) ) 49I (21) +P I (43) -2.01P I (14) -
1 22191(12)) RI4=CK41aZX(RNU21 (PI (21) -PI (14))-4.01P1(21)+3.01?1(14)+PI(57)) RI5=CK51DZZ (RNU21(PI (37) -PI (15)) -PI (37) -PI (42) +2.01PI (15) +2219I (10
1)) 65XJ=CK185XJ491149I2+R 13+9144915 RY1=CKIZ0ZX(PI ce) -PI (45))
~Y2=C"2" (PI (lS) ~I (II +OZ"STPI7-STPIJ) ~Y"J=CI(J"OZ" (P I OS) ~I C"I2) +iZ2%PI (10l) RY"I=<II;'I'SOZ" (--1.0%PI C"I8)-6.0llPI (lS) ~I (37) ~I C~) +iZ2%PI OOl +iZ2 xPI (
120l) RYS=CII;S"OZX(-3.0%PIC21) +2. o lIP I (1'1) +PI (S7» aSY4=CII;*aSY4+RYl+RY2+RY3+RY4+RYS RIl=CII;IZ(PIC3)~I('I)~I(2» RI2=Co;2ZOZ"CPI (B)-z2%PI (9» RI3=CI(3ZOZ"(~I(JO)-2.0llPI(I'I)+l2l1PI(17» RI'I=CI('I'SOZ"cPI (21) ~I C1'1)-z2"(PI CIS) +PI (12») RIS=CI(SZOZ"CPI C37> ~I US)-z2"(PI US) +PI (10») CSXJ=CI(ZCSXJ+RII+RI2+RI3+RI4+RIS RY1 =CI(IZOZ" (PI CB)-z2l1PI (9» RY2=Co;2"(PI(S)~ICI)~IC2»
RY3=CI(3XRIS-1:I(S RY"I:CI('I'SOZ"C~I ("18) -2.0llPI (15) +iZ2l1PI (20)) RYS=CI(SXRI'I-1:I('I CSY4=CI(ZCSY4+RYI+RY2+RY3+RY4+RYS RIl=CI(IZOZ" (RHUlIPl (e)-PI(39» RI2=CI(2"(RHU" (PI (16)-PI(I»-(DZXSTPI5+STPI3);3.0) RI3=CI(JZOZ" (RHU'" (PI (37) -PI <1S» --1. OllPl (37) +3.0llPI US) +PI C .... » RI"I:CI('I'SOZ" (-RHU" (PI ("18) +2.0llPI (IS»-PI (37)-PI ('12) +2.0llPI(lS) +
1 Z2%PI (10» RIS=CI(S~,"(RI'IU'" (PI (21) -PI <1'1» -PI (21) -PI ("13) +2.0llPI U'I) +iZ2l1PI (2)
1)
DSXJ=CI(lIOSXJ+RI I+RI2+R IJ+RI4+RIS RYl=CI(I'"(RHU" (PI (16)-PI ('I»-(OZXSTPIS+STPI3);3.0) RY2=CI(2l1OZ'" (RHUlIP I (e) -P I ("15) )
OPI2=PI (30) +2.0llPl (1'1) RY3=~3l1OZ'"(RHU:o:OP12+PI (211 +PI ("13) -2.0llPl (1'1) -z2l1PI (12» RY"I:CI("IZOZ'" CRtru:o:OP Il--1.0llP 1(21) +3. OllPI <1'1) +PI (S7) ) OPI3=PI (37) -PI (15) RYS=CI(SlIOZ" (RHU:o:OP13-PI (37) -PI (~) +2.0llPl <1S) +iZ2l1PI UO» OSY4=Cl(lIOSYJ+RYI+RY2+RY3+RY4+RYS RI1=CI(I'" (RHU" (PI (37) -PI (15» -z2,,(PI (19) +PI (10») RI2=CI(2" (RHU"'(PI (21) -PI (1'1» -z2" (PI (le) +PI (12) » RI3=CI(3'" (-RHUlIPI (38) ~lIPI (39» RI"I:CI("""(-RHUlIPI (S2) ~lIPI ("15» RIS:CI(S;3. 0'" (RHUXSTPI l-z2XSTP IS-oZXSTPI3) ESXJ=CI(*[SXJ+RII+RI2+R13+RI4+RIS RY1=CI(I" (-RHUPI lIP I (1'1) +RHU:o:OP12~'" (PI <1e) +PI (12») RY2=CI(2" (-RMUPlllPI US) -RHUlIOPI3+Z2l1PI (20» RY3=CI(3" (RHUPI " (PI (22)-zAlF") -RHUXSTPI6-(Z2"5TPI5+OZ"5TPI3);3 .0) RY"I:CI(""" (RHUPI'" (PI ("I9)-zAlF")-RHU;3.0"5TPIl+Z2"5TPI7-oZXSTPI3) RYS=CI(S"(-RHUPllIPl ('Ill +RHUlIPI (38) ~lIPI ("15» CSYJ=CI(lIESYJ+RY1+RY2+RY3+RY4+RYS RIl:Cl(I,,(-PI (1'1) -RHU~I (21) +Z2l1PI (17» RI2=CIt2" (-RHUP I lIP I (IS) +RHU'" (PI ("18) +2 • OllP I <1S»~" (PI US) +PI (l0») RI3=CI(3"'(Rl'lUPI"(PI(22)-zAlF")-RHU;3.0"5TPI1+Z2"5TPI2-oZ"5TPI3) RI"l=CI("""(RHUPI"(PI("I9)-zAlF")-RHU"5TPI4-Z2;3.0"5TPIS-oZ/J.0"5TPI3) RIS=CI(S"'(-QHUPllIPI ('II) +RHUlIPl (S2) ~lIPI (39» F"SXJ=CI(~SXJ+RI1+RI2+R13+RI4+RIS RYI=CI(I"'(RHUxoPI3-z2,,(PI(19)+PICIO») RY2=CI(2" (RHU:o:OPIl-z2'" (PI ue) +PI (12») RY3=~3" (RHUlIP I (38) +iZ2l1P I (3S) ) RY"I:~""" (RHU%P I (S2) +iZ2l1P I C"I5) ) RYS=CI(S;3.0" (RHUXSTP I I-z2XSTPIS-oZ"5TPI3) F"SYJ=CI(ZFSYJ+RYl+RY2<RY3+RY4+RYS . RII=CI(IZOZ" (RHU2 lIP I (1'1) +PI (30) +2.0llPl (1'1» RI2=CI(2ZOZ"'(RHUI2l1PI(IS)-PI(37» RIJ=CI(3ZOZ'" (-RHU2'" (PI (22)-lAlF")-STPI6) RI"I=CI("IZOZ" (-RHU2'" (PI ("I9)-zALF")-STPII;3.0) RIS=CI(5ZOZ"CRHU2l1PI('II)+PI(38» USXJ=CI(-USXJ+RII+RI2+R13+RI4+R15
C
RYl=ClI;l"OZX(RHU2%PI(I"I)-oPIl) RY2=CI(2"OZX(RHU2%PI(IS) +PI ("18) +2.0%PI CIS» RY"J=<I(JZOZ*(RHU2"(P! (22)-zALf) +STPII;3.0) ~Y"I=<I('I'SOZ" (RHU2X(PI ("I9)-zALF"l +STPI'I) ~YS=CI(SlIOZZ(RNU2l1PIC'Il)+P1(52»
VSYJ=CII;XVSY4+RYl+RY2+RY3+RY4+RYS RIl=<l(lZOZ"(PI(37)~I(lS» RI2=<o;2ZOZ" (PI (21) ~l <1'1» RIJ=CI(3ZOZ%PI(3S) RI'I=CI(~ZlIPI (S2) RIS=<I(SZOZ/3.0XSTPII VSXJ=CI(XVSXJ+RII+RI2+R13+RI4+RIS USYJ=CI(ZUSYJ+RI1+R12+RIJ+RI4+RIS RII=CI(I"C-RHU2*(PI (22)-zALF")-oZ"(PI (16)-PI ("I») RI2=CI(2*(RHU2l1Pl('I1)-z2l1PI(S» RIJ=CI(3" (-RHU2" (PI ("10) +Z2l1PI (1'1»+Z2" CPI (JO)+2.0%PI CI'I») RI"I=CI("""(-RHU2/2.0" (PI (S6) -PI C"IO» -z2"(PI (211-PI (1"1») RIS=CI(S"C-RHU2/2.0"(PI('I7)-PI("I6»~"(PIC37)-PI(15») ~XJ=CI(~XJ+Rll+RI2+RIJ+RI4+RI5 RYI=CI(I"(RHU2l1Pl('II)~lIPl(e» RY2=<"'2" (RHU2'" (PI ("19) -lAlF"l +OZ"'(PI (16) -PI (1») RY3=CI(3XRIS.-1XS RY"I=CI(""" (-RHU2" (PI C"I6) +l2l1Pl (5» +Z2,,(PI ("18) +2. o lIP I US») RYS=CI( S XR I'I.-1X'I ~Y4=CI(~YJ+RY1+RY2+RY3+RY4+RYS
C 'mAtlSF"ORM BACI( TO Ela:AV AllES C
670 COttTIHUE )PY=O. S'" (AQZ-J+&GIZ-J) )CMY=O. S'" (AQZ.J-60z.J) AQZCJ)=)PY+)CMYZCOS2R BDZ(J)=)PY-)CMYZCOS2R CQZ(J)=CQZJ OOZ (J) =OQz.JZCOS2R EQZ CJ) =-f"QZ-JXSIHR+EQz.JZCOSR F"QZ(4)=F"Qz.Jzr~R+EQZ-JXSIHR UQZ CJ) =UQz.JZCOSR+IIOZJXS INR IIOZCJ)=-UQz.JXSIHR+IIQZ.JZCOSR ~Z(J)=~z.J )PY=O. S'" (ASXJ+8SXJ) XMY=O.S"(ASXJ-BSXJ) ASXCJ)=)PY+)CMYZC0S2R BSX(J)=)PY-)CMYZCOS2A CSX CJ) =CSXJ OSXCJ)=OSXJZC0S2R ESX(J)=-fSXJr.sINR+ESXJZCOSR F"SX(J)=F"SXJ*COSR+ESXJXSIHR USX(J)=USXJ*COSR+VSXJXSIHR VSXCJ)=-USXJ"5IHR+VSXJ*COSR ~X(J)=~XJ )PY=O.S'" (ASYJ+8SYJ) )CMY=O.S"'(ASYJ-BSYJ) ASY(J)=)PY+)CMYZCOS2R aSY(J)=)PY-)CMYZCOS2R CSY(J)=CSYJ OSYCJ)=OSYJZCOS2R ESY(J)=-fSYJXSIHR+ESYJZCOSR F"SY(J)=F"SYJZCOSR+ESYJXSIHR USY(J)=USYJZCOSR+VSYJXSINR VSY(J)=-USYJXSINR+VSYJ*COSR ~Y(J)=~YJ REltJRH END
w o 1.0
310
D. PROGRAM BIM3
1. The program is the algorithm for the direct
formulation for complete plane strain analysis described
in Chapter 7. The rock mass consists of an infinite,
isotropic, elastic medium in which are embedded isotropic,
elastic inclusions. Openings may be excavated in the
infinite region or in the inclusions.
2. Mine axes (X,Y,Z) are as described for BEM11.
3. Local axes (x,y,z) for inclusions and openings, in
terms of which the problem geometry and boundary conditions
are specified, are as described for BEM11, with the y-axis
parallel to the long axis of excavations.
4. The pre-mining state of stress is assumed to be the
same in the infinite medium and the inclusions. Field
principal stress magnitudes are FP1, FP2, FP3. The
orientations of the principal stress axes are defined by
dips and bearings ALF1, BET1 etc. relative to the mine
axes.
5. The order of presentation of data is illustrated
on the format sheet which follows. The method of presenta-
tion is as follows.
Data Set A general problem data
Data Set B
Data Set C
data for the infinite region, speci-
fying elastic properties and
excavations in this region
data for inclusions in the infinite
region. There is a Data Set C for
each inclusion. This set defines
elastic properties of the inclusion,
the boundaries of the inclusion, and
the, boundaries and boundary conditions
on excavations in the inclusion.
CARD
TYPE
Al
A2
A3
A4
A5
A6
81
B2
B3
B4
Cl
C2
C3
C4
C5
D1
INPUT FOR BIM3
10 20 30 40 50 60 70 80
T I
NPROB]
T L E , I I , ,,,,I, 1 1
NPARLJ STRJFI
, I 1 1 i •BET 1
I I ,
NI
, ___L_! 1_; 1 1 1 1 I I l I l t
PS
PSI
- P SI_
----
NP
- - -
NREG KSXL KSZ
: : 1' ,A L F1
L' I j I ; ; 1 I I I 1
1 1 1F.P 1 t I ,! 1 ! 1 l 1 I , 1
I! I'
1 1 1 ,
11 I 1 . 1! 1 I I
I I I 1 1'
1 ,
1 I F P 2 • 1 ; 1 1 ,A L F 2 ,, I I I IB E T 2, I 1 1 1 l I I l I i
1 1 F P 3 , A L R3 I ,BET 3 1 1 1 1 1 1 {{ I I
1 1 1 1
1'{!
i 1 1 ALFF BET . . 1 I I I
1 I II 1 I 1 R, E G 1 1 NXCVS ' AN U I 1! 1E.MOD
1 1, ; 1 1! i
1 1 I I I l I i
1 1„! I
I I I 1 1 Z L
1 I I 1 I
1 1 I
',•
1 ,,
1
RDS
I l 1 I
1 1
I I 1 R A T 10 - 1 1 1 1 1 ;J X C 1 1 N S E G 1, 1 1, 1 1
,' N E L R 1 1 1 1 IX0 1 1 1 1, Z O • IIXL
BCONI I 1 TEM1 ''I,TEM2 I T,EM3 111; ,1, , ,
1 ' '
'
,
I 1
RDS
I1I 1 1 I I 1 1 ,
; RATIO
REG I, NXCVS ; R N U 1 EMO D I I NSEG
NELR X 0 I Z 0 . I XL ZL
1 JXC NSEG I 11 1 I ! NELR 1 .X0 , 1 Z 0 t ; I !XL i: ; Z L 1 1 i RDS '' RAT 10
BCONI IPARI
1 T EMI I TEM2 _ , TEM3 I !
IREG
I 1
.'XG 1' ZG GAMl I GAM LEN LEN NP 1 I 1 1 1 I • 1 1
r, I I! i 1 1 1 1 1 1 1 I 1, 1 1 I 1 I I i I' 1 I ,
1 , i , • I I I
• , I I I ---
I
I I I i ,
Cols 1-5 NPROB
Cols 6-10 NREG
NREG=1
NREG=5
Cols 11 -15
Cols 16 -20
Cols 21 -25
Cols 26 -30
KSX
KSX=0
KSX=1
KSZ
KSZ=O
KSZ=1
NPAR
JSTR
Data Set D data for defining grids over problem
areas of interest.
DATA SET A
Card Al
(8A10)
title of problem, up to 80 alphanumeric
characters
Card A2
problem control information.
(7I5)
312
problem identification
number
number of regions
infinite region only
infinite region + 4
inclusions
symmetry code for x-axis
no symmetry about x-axis
symmetry about x-axis
symmetry code for z-axis
no symmetry about z-axis
symmetry about z-axis
no.of problem areas (grids)
which will be specified by
cards of type D
code for indicating job
start conditions
JSTR=0 initial run
JSTR=1 restart run, problem data
is read from TAPE8, and
only cards A1,A2,D are
required
Cols 31-35 JFIN code for indicating job
termination of initial run
JFIN=O no subsequent runs to follow
JFIN=1 problem data written to
TAPE8 for filing for
restart run
Cards A3,A4,A5 magnitude, dip and bearing of each field
(each 3F10.0) principal stress
Card A6
dip and bearing of the long axis of
(2F10.0)
excavation
313
314
DATA SET B
1-10 IREG No.which identifies infinite
region
Card Bl
Cols
Cols 11-20 NXCVS no.of excavations in the
infinite region
Cols 21-30 RNU Poisson's Ratio for infinite
region
Cols 31-40 EMOD Young's Modulus for infinite
region
Card B2
Cols 1-10 JXC no.which identifies excavation
in infinite region
Cols 11-20 NSEG no.of segments defining excavation
boundary
Card B3 Segment card defining all or part of excavation
boundary
Cols 1-10 NELR no.of elements in this segment
of boundary
Cols 11-80 XO,ZO,XL,ZL,RDS,RATIO,PSI are the same
as used for defining segments in the indirect
formulation, BEM11.
Card B4 boundary conditions imposed on elements of
segment
Cols 1-10 BCON
if BCON=TRACS, then boundary values
of traction on the segment are
specified, and TEM1=TXF,TEM2=TYF,
TEM3=TZF. If the surface is traction-
free, TXF=TYF=TZF=0.0; if BCON=DISPS,
then boundary values of displacement
on the segment are specified, and
TEM1=ux,TEM2=uy ,TEM3=uz.
Card Cl
Cols 1-10
Cols 11 -20
Cols 21 -30
Cols 31 -40
Cols 41 -50
315
Cols 11-40
imposed boundary values, defined
above, of traction or displacement
DATA SET C
If NREG = 1 there will be no Data Set C.
IREG no.which identifies inclusion
NXCVS no.of excavations in inclusion
RNU Poisson's Ratio for inclusion
EMOD Young's Modulus for inclusion
NSEG no.of segments defining the
interface between the inclusion
and the infinite region.
Cards C2 segment cards for defining inclusion interface
with infinite region
there are NSEG cards C2
variables NELR etc. have the same significance as
defined for excavation boundaries
the boundary of the inclusion must be described
such that the infinite region lies on the R.H.S.
as the boundary is traced.
Cards C3,C4,C5 define excavations, excavation segments
and boundary conditions for excavations in the
inclusion. Inputformat is the same as for cards
B2,B3,B4.
Note: there are (NREG-l) sets of C Data cards
DATA SET D
Card D1 defines geometry of grid shown in figure below
Cols 1-5 IPAR problem area identifying no.
Cols 6-10 IREG no. which identifies region
in which the problem area
IPAR exists
) 3 XG,ZG
GAM1
GAM2
LEN1
LEN2
NP1
Cols 76-80 NP2
Cols 11 -20
Cols 21 -30
Cols 31 -40
Cols 41 -50
Cols 51 -60
Cols 61 -70
Cols 71 -75
corner of oblique grid
inclination of upper arm of
grid
inclination of lower arm of
grid
length of upper arm of grid
length of lower arm of grid
no. of grid points along
upper arm
no. of grid points along
lower arm
316
Note: there are NPAR cards of type Dl
X (xg,zg)
Lent
t z
/IL Le n1
6. Output from Program
(a) Input data
(b) Boundary stresses and displacements around
openings in the infinite region
(c) Stresses and displacements at the centres
of elements defining the interfaces between
the infinite region and an inclusion
(d) Boundary stresses and displacements around
excavations in the inclusion
(e) The energy released by excavation
(f) Stresses and displacements at the nodal
points of the specified grids.
C C C
*COtOECK GEM COPtOMiGEN1itX(50) .C2(50) .EX1 (50) .E21 (50) .EX2(50) .E22(50) .
1 SING (50) .COSB (50) .05 (50) .ITB (50) . TXF (50) . TYF (50) . TZF (50) . 2 OU(50) . D V (50) . DW (50) . TX (50) . TY (50) . T2 (50) . 6 VX2 (200) . 3 BVY ()0).5TX2(200).BTY(100).INDEXl(5O),IN0EX2(5O)•SIGX(5O). 4 SIGY(50).SIG2(50).TAUXY(50).TAUYZ(50).TAU2X(50).SIG(3.50). 6 DALF(3.50).DBET(3.50).DGAM(3.50).UX(50).UY(50).U2(50). 6 XGP (50.4) .ZGP (50.4)
C@T'ON/GEN2/GAF(4).RWF(4).RMAX.GM00(5).GNO02(5).P1(2O).P723.PY. 1 PY2.PYM.TOL.COF1(5).FX.FY.FZ.FXY.FY2.E2X..NREC.FAC.TRACS.0ISPS. 2 TITELI(6).TITEL2(6).TITELD(6).RNU34(5).RHUI2(5).9NU32(5)0NER2. 3 MER6.RNU(5).E,OQ(5)0GMDD2P.CM;lDP,RLAM(5).KXT.Kt7.RMU11(5). 4 IREG(5).NXCvS(5).IF1RD(5).ILASO(5),IXCv(5.20),IFIREL(5.20). 5 ILASEL(5.20).HZTEL(5).(IIFELP.NIFELT.MELEX(5.20).NELEXT(5).NELOX. 6 NLIBX2.MLIBY.MELREG(5).RMUIPI(5).RNU14(S).RMU1P2(5).NPAR.IL.IRP. 7 IPT.r,iXI
*DECK FAIN PROGRAM 6IM3(INPUT=1316.OUTPUT*1316.TAPEIaINPUT.TAPE7*OUTPUT.
2 TqPE2,TAPE6,7APE8) *CALL GEN
DIMENSION TITLE(B) carrot' DUM (4600) COPPON/FIXE/NL180.NCC.LUM.LDLM.N6LM.JMAT.Pt T COMM/CHANGE/GT(300) DATA TRACS0DISPS/54TRACS.5N0ISPS/ DATA GAF" -0.8611363116.-0.3399010436.Q.339fl6I0436.O.8611363116/ DATA RKF/Q.3470546451.D.6521451549.O.6 52 1 45 1 549.0.347854p451/ DATA TITEL1/56HIrPOSED BOUNDARY TRACTIONS TX TY
1 TZ/ DATA TITEL2/56)1IMP05E0 BOUNDARY DISPLACEPENTS W DV
1 OW READ(1.5) (TITLE (I),I=1al)
5 FORh1T(8A10) READ(1.10) NPROB.NREC.KBX.KSZ.HPAq.JSTR.JFIN
10 FORr]T(715) WRITE (7.15) NPROB. (TITLE (1) .1=1 .8)
15 F0RFAT(1H ///45X.4514BOUNDARY ELEMENT ANALYSIS DIRECT FORMULATION 1 /45X.45H ///52X. 2 31HINFINITE REGION WITH INCLUSIONS//52X.32NC0rPLETE PLANE STRAIN 3CONDITIONS///7X.12HPROBLEM NO. .I3.4X.8A10/7X.15H )
WRITE(7.20) NREG.KSX.KSZ 20 FORMAT(IH /n)615HNO. OF DOMAINS .I2//7X.20NSYMPETRY CODES KSX.
1 I2//24X. 3FN(52. I2) IF(JFIN.E0.0) GO TO 22 WRITE(7.21)
21 FORP T(1H //7X.39BELEPENT AHD PROS DATA FILED FOR RESTART) 22 CONTINUE
IFCJS7R.E0.3I GO TO 500 READ(I.25) FPI.ALF3.BET1 READ(1.25) FP2.ALF2.BET2 READ(1.25) FP3.ALF3.8ET3
25 FORPAT(3F10.0) WRITE(7.30) FPI.ALF1.6ETI.FP2.ALF2.BET2.FP3.ALF3.OET3
30 FORPPT(IH //7X.44HPRINCIPAL STRESS MAGNITUDES AND ORIENTATIONS,/ 1 16X.14HMAGN DIP DRG//11)(.3HFP1.F6.2.F5.l.F6.1//11X.3HFP2.F6.2. 2 F5.1.F6.1/i1lX.3HFP3.F6.2.F5.l.F6.l) READ(1.35) ALF.BET
35 E0R,1T(2r10.0) WRITE(7.40) ALF.BET
40 FORPAT(IH //.7X.26HLONG AXIS OF OPENINGS DIPS.F5.1.1614 DEGREES TOW IAROS.F5.1.ON DEGREES) PY=ATAN(1.0)*4.0 PY23=2.0*PY/3.0 TOL=1.E-4
FAC=PY/160.0 C
PY2=2.0*>'Y PYhP-0Y KXT=1+2*KSX KZT=I+21.XSZ ALF!=ALF1xFAC BETI=6ET1xFAC ALF2=ALF2xFAC BET2=BET2*FAC ALF3=ALF3*FAC BET3=BET3xFAC ALF=ALFWAC BET=BETIFAC U1=COS(ALF1) 1C05(BETI) U2=COSCALF2) 1LOS(BET2) U3=COS(ALF3) 11COS(BET3) V1=COS(ALF1)ESIN(8ETI) V2=CO3 (ALF2) *SIN (8ET2) V3=COS(ALF3)*SIM(8ET3) W1=SIM(ALF1) W2=SIN(ALF2) 1.051N (ALF3) FU=U1*U1*FP1+U2*U2*VP2+U3*U3xFP3 Fv=v1*V1*FP1+V2*V2*FP2+V3*V3xFP3 FL=U14 1 xFP 144Q1.Q*WP2+4fi"W11FP3 FUV=U1*V111FPl+U2"w2xFP24U3xN3*FP3 FVwVl" 11*FP1+V27/WarP2+N,l1.OxFP3 FW=W 11U l aWP 1+1Q1121iP2+4O)413*FP3 XU=SIM(8ET) Xv=-COS (BET) X6=0.0 'rU=COS CALF) aCOS (8ET) Yv=COS CALF) *SIN (BET) n SINCALF) ZU=-SIN CALF)11COS (BET) ZV=-SIN (ALF) *SIN (BET) ZL=COS (ALF) FX=X1PLXIlaFU+XV*XV*FV+)34EXIMU+2.0x ()O"XVIFUV+XV*X(1pFvN+)1PF L11Fw) FY=YU*YUxFU+YVarry*Fv+YLAfAFW+2.0* (1'UxYVxFUV+YV*YLAFV11+Y40*YUxgw) F2=ZU*QIPIFU+2V*ZVIFV+210.7. FM+2.0*(ZU1¢V"FUv42V*71AFvU+21.a¢L1Fw) FXY=XU*1'U*FU+X(V*YWFV+ 041114 F 4,00JxYV+XV*;YU) 1FUV+(XV*Y)4+XLAYV) *FVW
1 +(71. YU+ J*YLO1FW FYZ=YUxZU1FU+YVa2V1FV+Y1+41o471+CYUx2V+YvJ) aFUV+(YVa2F+►YLA2v) 1FV(
1 +(YLA2IP+YUa2F0 1FW F2X=ZUWXU*FU+2VWXVaFV+17.E410341,144.(ZU*XV+2V*XQP) 1TUV+(ZV71704+2U*XV) WW1
1 +(ZLA)d1+ZU*76.D1FW 4RITE(7.44) FX.FY.FZ.FXY.FYZ.FZX
44 FORM T(1H //.7X.46HFIELD STRESS COMPONENTS REL TO HOLE LOCAL AXES/ 1/11X03HFPX.F7.3//11X.3HFPY.F7.3//11X03HFPZ.F7.3//lOX.4HFPx7.F7.3// 2 10X.4HFPYZ.F7.3//10X.4IFPZX.F7.3)
C C READ REGION DATA GENERATE ELEMENT DATA C
CALL INSEG
FORM COEFF MATRICES CREATE RANDOM ACCES FILES
CALL COEFFS C CSET UP AND SOLVE X2 PROB C
NL180=NLIBXZ NC-C=1
LUN=2 LBLrt600 NBL f1=1 ItilT=NER2 JMAT=P11ATA71L IBX2 DO 50 I=1.NL IBXZ ST CI) =BTXZ (I)
50 CONTINUE C
CALL 105E11 C
DO 55 I=1.NLISXZ BVXZ (I) =BT (I)
55 CONTINUE C C SET UP AND 50L VE Y PROS C
NL IBO=NL IBY L Url 6 t?:AT=NER6 JrsIT=PIEREOANL IBY DO 60 I=1 .NL toy era) =BTY( I) CONTINUE
CALL IDSOL
DO 65 I=1.NLIBY BVYCI) =BT CI) CONTINUE JL X=0 JL Y=0
DO 75 IR=1.NREG IF CNIFEL (IR) .E0.0) GO TO 75 INIT= wrap (IR) IFIN=IL AV) CIR) DO 70 I=INIT.IFIN GO TO (66.67.68) ITB CI)
66 CONTINUE JLX=JLX+1 DU CI) =B VXZ CJL X) JLX=JLX+1 Du CI) =BV)¢CJL)0 JLY=JLY+1 DV (I) =BVYCJLY) GO TO 70
67 CONTINUE JLX=JLX+1 TX C I) =BVXZ CJL)0 JLX=JLX+1 TZ CI) =SVI (JL )0 JLY=JLY+1 TY CI) =B VY LILY) GO TO 70
68 CONTINUE JL X=JL X+1 TX C I) =B VXZ CJL X) JLX=JLX+1 TZ(I) =BVXZCJLX) JLX=JLX+1 DU (I) =BVX2 CJL X) JLX=JLX+1 DN C l) =B VXZ CJL X)
60 C
C
65
C
JLY= TYCI)
JLY+1=BVY CJL Y)
JLY=JLY+1 DV CI) =BVY CJL Y)
70 CONTINUE 75 CONTINUE
IF(NREG.E0.1) GO TO 86 DO 85 IR=1.NREG IF(IR.E0.1) GO TO 85 IF CN)cVS C IR) .E0.0) GO TO 85 NXC=NXCVS (IR) 00 84 IX=1 .NXC IA= !FUEL (IR. IX) IB=ILASELCIR. 1)0 DO 80 I=IA. IB GO TO (82.03) 1TB CI)
82 CONTINUE JLX=JL X+1 DU(I) =BV) CJLX) JL X=JLX+1 DuCI)=eV)CZ(JLX) JL Y=JLY+1 Dv CI) =DVY CJL Y) GO TO BO
63 CONTINUE JLXJL TXCI) =5V)a
X+1 (JL )0
JLX=JLX+1 12 (I) =S V)CZ CJL IO JLYJL TICI)=eVY
Y+1 (JLh
00 CONTINUE 84 CONTINUE 85 CONTINUE 66 CONTINUE
DO 90 I=1 .NPXI DU CI) =DU (I) ,IRM IX DV CI) =DV (I) ■Rf1AX DWI) =DUCI) MAX TX(I) =TX CI) 1P1002P TY CI) =IT CI) •GMIDP TZ(I)=T2 CI) IITY, 302P
90 CONTINUE IF (JFIN.E0.0) GO TO 420
C C PROS DATA FOR RESTART CJIITTEN TO FILE C
CRITE(8) MAXI CRITE (8) (CX(I) .CZ (I) .EX1 (1) .E2: (I) .EX2 (I) .E22 (1) .SINS (I) .COSB (1) .
1 05(I) . ITB (I) .DU (1) .DV (I) .0u(I) .TX(I) .TY(I) .TZ(I) . 2 (XGP(I.J) .ZGP(I.J) .J=1.4) .I=1.MAXI)
WRITE(8) (GMVD(I).GP002(I).C3FI(I).RNU34(I).RNUI2(I).RNU32(I). 1 RNU(I).EtO0(I).RLAN(I).RNUII(I),IREG(I).NXCVS(I) , IFIRD(I). 2 ILASD(I).NIFEL(l).NELEXT(I).NELREG(I).RNUIPI(I).RNU14(1). ] RNUIP2(I),I=1.NREG)
DO 410 I=1.NREG NXC=NXCVS(I) CRITE(B) (IXCV(I.J),IFIREL(I.J),ILASEL(I.J).J=1.NXC)
410 CONTINUE WRITE(8) RP(AX.PY23.PY.PY2.PYN.TOL.FX.FY.FZ.FXY.FYZ.F2X.FAC.GPUD2P ,
1 GrUDP.KXT.KZT.NIFELP.NIFELT.NELDX 420 CONTINUE
C C CALC BOUNDARY STRESSES. PRINCIPAL STRESSES. OUTPUT
C CALL BSTRESS
C WREL=0.0 DO 155 I=1.MAXI LREL =LREL +DS (I) M CTX C I) *0U C I) +TY (I) *0 V (I) +TZ C I) X0 W (1)) *0.5
155 CONTINUE WRITE(7.1005) WREL
1005 FORMAT(1H //7X.21HTOTAL ENERGY RELEASED.3X•E15.7) C C READ PROS DATA ON RESTART RUN C
IF'JSTR.E0.0) GO TO 530 500 CONTINUE
WRITE (7.505) 505 FORMAT(1H //7X.29HPRO8 DATA RETRIEVED FROM FILE)
READ (0) MAXI READ(8> (CX(I).CZ(I).EX1(I).E21(I).EX2(I) •E22(I).SINBCI).COSB(I).
1 DS(I).ITB(I>,DUCI).DV(I>.DW(1).TX(I).TYCI>.TZ(I). 2 CXGP (I.J) .2GP (I.J) .J=1.4) . I=1. MAXI)
READ(8) (GM130(1) .GMt102(I) .COFI (I) .RNU34(I) .RNUI2CI) .RNU32(I) . 1 RNU(I)•EM3O(I).RLAMCI).RNU11(1),IREG(I).NXCVS(I)•IFIRDCI). 2 ILASO(I).NIFEL(I).NELEXTCI).NELREG(I)•RMUSP1(I).RNU14(I). 3 RNU1P2(I),I=1.NREG) DO 510 I=1.NREG NXC=NXCVS (I) READ(B) (IXCV(I.J).IFIREL(I.J).ZLASEL(I.J).J=1.NXC)
510 CONTINUE READ(0) RMAX.PY23,PY.PY2.PYM•TOL.FX.FY.F2.FXY.FY2.F2X.FRC,G(t)D2P.
1 GMODP.KXT.KZT.NIFELP.NIFELT•NELDX 530 CONTINUE
C CALL STRESSES DISPLS AT NODES Of GRIDS DEFINING AREAS OF INTEREST C
IF(NPAR.E0.0) GO TO 130 C
DO 125 JA=1.NPAR REAO(1.100) IPAR.JREG.XG,ZG.GAM1.GAM2.RLEN1.RLEN2.NP1.NP2
100 FORMAT(2I5.6F10.0.2I5) WRITE(7.105) IPAR.JREG.X0.2G.RLENI.GAMI.NPI.RLEN2.GAM2.NP2
105 FOR(1iT(1H ///7X.13HPROB AREA NO. .I2.14R IN REGION NO. •I2/7X. 1 31(114*).//7X. 14AGRID CORNER AT.2F10.3//7X•6HSIDE 1.3X.2F10.2. 2 I3//7X.6HSIDE 2.3X.2F10.2.I3)
DO 107 IR=1.NREG IF(IREG(IR).NE.JREG) GO TO 107 IRP=IR
107 CONTINUE MAX I=NP 1 RNPI=NP1 RNP2=NP2 DLI=RLEN1/(RNPI-1.0) DL2=RLEN2/(RNP2-1.0> CXC)>=XG C2C1)=2G GAMI=GAMla4'AC GAM2=GAM2=FAC DDXI=OL1=LOS(GAM0) DD21=DLI'SIN(GAMI> D0X2=DL2*0OS (GAM2) 0022=0L2*SINCGAM2) DO 110 J8=2.MAXI JOr JB-1 CX(JB)=CX(JBM)+0DX1 C2 (JB) =C2 (JBMO +0021
110 CONTINUE
IL=0 DO 120 JB=1.NP2 IL=IL+1 IF(JB.E0.1) GO TO 110 DO 115 JC=1.MAXI CX(JC)=CX(JC)+00X2 CZ(JC)=CZ(JC)40022
115 CONTINUE 118 CONTINUE
C CALL DISSTR
C 120 CONTINUE 125 CONTINUE
GO TO 150 130 CONTINUE
WRITE (7.145) 145 F5RMAT(1H ///7X.23HN0 PROBLEM AREA DEFIMED/7X.2314
1 ) 150 CONTINUE
STOP END
*DECK INSEG SUBROUTINE IMSEG
*CALL GEN 1=0 TEMI=0.0 TEM2=0.0 TEM3=0.0 DO 500 IR=I.NREG READ(1.5) IREG(IR).N%cVS(IR).QNUCIR).ENODtIR).NSEG
5 FORMAT(2I10.2F10.0.I10) ITBI=3 IFIRD CIR)'1+1 GM30 (IR) =E'Y D (IR) /2.0/ (1.0+PNU (ZR) ) GMOD2(IR)=2.0XG1730(IR) COF1CIR)=4.0=PYM(1.0-RNJ(ZR)) RNUIl(IR)=1.0-ANU(IR) RNUI2(IR)=1.0-2.0=RNU(IR) RNU34CIR)=3.0-4.0=RNU(IR) RNUIPI(IR)=1.0.ANUCIR) RNU14(IR)=1.0-4.0MRNU(IR) RNU1P2(IR)=1.0+2.0=RNUCIR) RNU32(IR)=3.0-2.01RNU(IR) RLAM(IR)=2.0=RNU(IR)SGM30 (IR)/(1.0-2.0' NU(IR))
C C IREG IDENTIFIES REGION NXCVS NO OF IN IREG NSEG NO OF SEGS C DEFINING BOUNDARY C
IF(IR.GT.1) GO TO 15 (RITE(7.10) IREG(IR) .RNU(IR) .EMID(IR) .NXCVS(IR)
20 FORMAT(1H //4X.29HINFINITE DOMAIN - REGION MO. .I1/4X.3014 //7X.1514POISSONS RATIO .F4.2//7X.15HYOUNGS MO
2OULUS .F10.0//7X.1914NO. OF EXCAVATIONS .12) GMO02P=GMO02(1) GMODP=GMI3D (1) GO TO 100
15 CONTINUE WRITE (7.30) IREG (IR) .NSEG.RNU (IR) .EMOD (IR) .NXCVS (IR)
30 FORMATC1H //4X.11HREGION NO. .I1/4X.1214 .//7X.16HNO. OF 1 SEGS DEFINING REGION BOUNDARY.I3//7X. 15HPOISSONS RATIO .F4.2//7X. 2 15HYOUNGS MODULUS .F10.1//7X.1914NO. OF EXCAVATIONS .12)
25 CONTINUE NSEGG=O
TYF :I) =D.0 TZF =0.0
68 CONTINUE NEL G=NELG+1 IF(NELG.LT.NELR) GO TO 65 IF(ITBI.LT.3) GO TO 170 GO TO 35
70 CONTINUE DX= (4.-)13) /RNELR DZ= (ZI. -20) /RNEL R OS 1=SORT (DXU2+02a4)
85 I=I+1 SINS (I)=-0Z/OSI C056 (I) =DX/05I DS(I)=D5I ITS(I)=ITOI IF(ITBI.EO.3) GO TO 88 GO TO (86.87) ITBI
86 TAT (I)=TEN► TYF(I)=T r2 TZF (I) =TEM3 DU (I) =0.0 DV(I)=0.0 01.1(I)=0.0 GO TO 88
87 DU (I) =TEN! Dv (I) =TEr2 DU(I) =TEM3 TXF CI) =0.0 TYF (1) =0.0 TZF(I)=0.0
88 CONTINUE RNELG=NELG EX1(I)=XO+RNELGal7X EZ1(I)=213+RNELG3aZ CX (I) =EX1 (I) +0.5=0X CZ(I)=EZI(I)+0.510Z EX2(I)=EX1 CZ) 40X EZ2 (I)=E21(I)+OZ NELG=NELG+1 IF (NELG.LT.NELR) GO TO 85 IF(ITBI.LT.9) GO TO 170 GO TO 35
90 CONTINUE IL ASO (IR) =I IF (NXCVS (1R) . EC) .0) GO TO 500 GO TO 110
100 CONTINUE IF (NXCVS(1).GT.0) GO TO 110 IFIRD(I)=0 ILA5D(1)=0 GO TO 500
110 CONTINUE C
IX=O 112 CONTINUE
1X=1X+1 READ(11115) JXC.NSEG
115 FORMAT (2110) C C JXC IDENTS EXCAV IN THIS SUBREG C
IXCV (IR. IX) =JXC IFIREL(IR.IX)=I+1
35 CONTINUE IF(NSZGG.E0.NSEG) GO TO DO NSEGG=NSEGG+1 NEL G.0 READ(1.40) NELR.X0.Z001.2L.RDS.RATIO.PSI
40 FORrfl7(I10.7F10.0) RNELR=NELR IF(R05.LT.TOL) GO TO 50 IRITE(7.45) NELR.X0.20.XL.2L.ROS.RATIO.PSI
45 FORMAT(IH //4X•8►ELEhEN7S. 1X.6►lCENT X.4X.6HCENT Z.5%.5NTNF7I 1SX. 1 5HT1ET2.5X.6HRROIUS.4X.SHRATI0.5X.7NPS1//7X.13.6F10.3.F5.3) GO TO 60
50 CONTINUE I. ITE(7.55) NFLR.X0.20.XL.ZL
55 FORrflT(1N //4X.8HELErENTS.IX.6HFIRSTX.4X.6HFIRSTZ.5X.50LASTX.5X. 1 5HLASTZ//7X.1] .1F10.3)
60 CONTINUE IF (MS .LT.TOL) GO TO 70 IF(RATIO.LT.TOL) RATIO=1.0 SINPSI=SIN (PS1zPY/180.0) COSP51=COS(PSI4'Y/180.0) G0=RD5/10000.0 GA=RA7IO3lOSC(XL -Ps I)a?Y/180.0) IF (ASS (GA) .LT.GO) GA =GO GB=RATIO COS((ZL-PSI) WY/180.0) IF (ASS(08).LT.GO) 138=G0 CHI1=ATF 12 (SIN(CXL-PSI)aPY/190.0).GA) CHI2=ATAN2 (SIN C CZL-PSI)=PY/180.0).G6) OCHI=CCHI2-CH11) /RNELR IF(A85(OCHI).LT.GO) GO TO 61 GC=OCHI/A6S(DCHI) GO TO 62
61 GC=-1.0 62 OCHI=DCHI+C(ZL-XL)/A65(ZL-XL) -0)aPY/RNELR 65 I=1+1
RNELG=NELG CHI =CH II+RNELGIOCNI EX1(I)=RDSZ (COS (CHI)=SINPSI+SIN (CH I)1COSPSI ATIO)+XO EZI CI) =RDS; (COS (CHI) alOSPS I-SIN (CHI) ■SINPS IR)ATIO) +20 CHI=CHI+OCHI DO (I) =R05x (COS (CNI) +SINPSI+SIN (CHI) =COSPSITRATIO) +XO EZ2 (I) =RDSX (COS (CHI)'COSPSI-SIN (CHI) as IMPS BCRAT1O) +Z11 CX (I) =0.5* (EXI CI) +EX2 (I)) CZ(I) =0.5x (EZI (1) +EZ2 (I) ) DX=EX2 Cl) -EX1 (1) DZ=EZ2 Cl) -EZI (I) DS I=SORT (0Xx0X+OZzO2) SINS (I) =-0Z/OSI COSB (I) =0X/OSI 05(1)=051 ITB (1) =ITBI IF(ITBI.EO.3) GO TO 68 GO TO (66.57) IT87
66 TXF (I) =TEN' TYF (I) =TEM2 TZF CI) =TEro DU(I)=0.0 DV(I)=0.0 01.1(1) =0.0 GO TO 68
67 DU(I)=TEMI Dv (I) =TEb2 Du( I) =TEm3 TXF (I) =0.0
505 CONTINUE LRITE(7.120) JXC.NSEG C
120 FORMAT<SH //7X.ISHEXCRVATION NO. .I2/7X.1714 .//7X. C SET UP GAUSS P0INTS, SCALED INDUCED TRACTIONS. DISPLS 1 25NNO. OF BOUNDARY SEGMENTS .I2) C
NSEGG=0 DO 520 1=1.MAX1 125 CONTINUE DSI2=D5(I) /2.0
IF(NSEGG.E0.NSEG) GO TO 175 SINBI=SINB(I) NSEGG=NSEGG+1 COSBI=COSB(I) NELG=O 00 515 J=).4 REAO(1.40) NELR,XO.ZO.XL.ZL.RDS.RATIO.PSI XGP(I.J)=CX(I)+0SI2XCOSBIXGPF(J) READ(1.130) BCON•TEMI.TEM2.TEPfl Zl;PfI.J) =C2CI>-0SI2*SINBI•TPFC.))
130 FORMATCA5.5X.3F10.0) 515 CONTINUE C IFCITSCD.GT.2) GO TO 516 C PUT ITBI=1 FOR TRACS SPECIFIED. 2 FOR DISPLS TX(I)=(TXFCI>-FXXSIN6IfzXxCOSBI)/GP0D2P C TY C I) = (TYF C l) _XTXSiNS LFYZXCD5S I) /GM3DP
1751=1 TZ C I) = CT2F (I) -FZX*S INS I_FZ4tOSS I) /GPt102P DO 135 K=1.6 OU (I) =0U (I) /RPiiX TITELD (K) =TITEL1 (K) DV CI) =DV (I) /RPf1X
135 CONTINUE DN(I>=OW(I)/RP7<iX IFCSCON.EO.TRACS) GO TO 145 GO TO 519 1791=2 518 CONTINUE 00 140 K=1.6 TX(I)=0.0 TITELOCK)=TITEL2(K) TYCI)=0.0
140 CONTINUE TZ(1>=0.0 145 CONTINUE DU(I)=0.0
RNELR=NELR DV(I)=0.0 IF(RDS.LT.TOL) GO TO 155 DMCI)=0.0 IFCRATIO.LT.T0L) RATIO=1.0 519 CONTINUE LRITE(7.150> TITELD.NELR.X0.Z0.XL.Zl.qDS.Rq7IO.PSI.TEM1.TEM2.TEM3 520 CONTINUE
150 FpRNA7C1H //4X.6HELEMENTS.1)06HCENT X.4X.6HCENT Z.5X.5MTHETI.5X. C 1 514TWET2.5X.6 R4OIUS .4X.5HRATIO.5X.3HPSI.X.6A10//7X.I3.6F10.3. C FARM VECTORS (ME PROS. Y PROS) OF KNOLL/ BOUNDARY VALUES 2 F6.3.27X.3F10.3) C
GO TO 155 NIFELT=O 155 CONTINUE DO 525 IR=1.NREG
LPITE(7.160> TITELD.NELR.X0.Z0.XE.ZL.TEM1.7EM2.TEP 3 NIF=ILASD(IR>-IFIROCIA)+1 160 FORMAT(1H //4X.BHELETENTS.IX.6NFIRSTX .4X.6MFIRSTZ.5X.514LASTX.5X. IF NIF.E0.1) NIF=O
1 5WLASTZ.X.6R10//7X.I3.4P10.3.27X•3F10.3) NIFEL(IR)=NIF 165 CONTINUE NELEXTT=O
GO TO 60 IF(r1XCV5CIR).E0.0) GO TO 523 170 c0(1TINUE NXC=NXCVS CIR)
GO TO 125 00 524 IX=I,NXC 175 CONTINUE NEL = ILASELCIR.IX) -IFIREL(IR.IX)+1
ILASELCIR.IX)=I NELE (IR.IX) =NEL IF(IX.LT.NXCVSCLR))G0 TO 112 NELEXTT=NELEXTTNIEL
495 CONTINUE 524 CONTINUE IFCIR.GT.1) GO TO 500 523 CONTINUE ILgSD CIR) =I NELEXTCIR)=NELEXTT
500 CONTINUE NIFELT=NIFELT+NIF MAXI=1 525 CONTINUE t XJ=I NIFELD=NIFELT-NIFEL Cl)
C NELDX=1 XI-NIFELT C DETERMINE RMX
NIFELP=NIFEL(1)
C IF(NIFEL (1) .E0.0) GO TO 542 RMAX=TOL IA=IFIRD(1) r1RXIMkMRX1-1 IB=ILASD C1) DO 505 1=1.MAXIM DO 540 I=IA.IB CXI=CX(I) ITBI=ITB(t) CZI=CZCI) K=I+1 K=J+1 DO 504 J=K.MAXI GO TO (530.535) 1TBI CXJ=CX<J) 530 CONTINUE CZI=CZ (J) 8V72(J) =TX (I ) R=SORT C CCXI-0XJ) X2+ (CZI-CZJ) xx2) BVXZ (K) =TZ ( I) RMAX=AMRX1(R.Rt )O BVYCI)=TT(I)
504 CONTINUE
GO TO 539 NLIBY=ICC+ICD
535 CONTINUE RETURN END BVXZ =DU (I) *DECK COEFFS BVX2
(K) (K) D V ( (I)
BW (I) =DV (I) SUBROUTINE COEFFS 539 CONTINUE CALL GEN
540 CONTINUE COMMON F.EC(o0D).UVEC1(200).U'EC2(200).FYVEC(0OO).VVEC(100).
542 CONTINUE 1 DCX (5U P.Z ) .D (50) .DXGP (50.4) .07GP (50.4) .DS IND (50) . DCOSB (50)
ICA=2TJIIFELP 2 D05(50).106(50)
ICB=O REAL lX.LZ.NX•N2.NU12.NU94
ICC=NLFELP C IC0=0 C SUB COEFFS FOR LINE LOAD SINGULARITIES
C IF cNREG.E0.1) GO TO 570 NAT 500 DO 550 IR=2.NREG IPXWBO0 NIER=NIFEL (IR) NER2=hqT1LNLIB7Q DO 545 IB=I.NIFR IC=ICA+ICB
NREC2=NLIDXZiMIER2+1
ID=IC+4;IB LEN2=NREC2+1 -0 IE=I0+1 NER6=MATWNLIBY
IG=7D+2 NREC6=NLIBY,NER6+1 LEN6rNREC6+1 IH=ID+3 JhrITX=NER2AILIBXZ B V ( ) =0.0 JP TY=NER6=71L IBY BVXZ(IEIE)=0.0
CALL OPEN('S (5 • INDEX2. LEN6.0) BV)Q (IH) =0.0 DO 2 J=1.NA)W IC=ICC+ICO FVEC(J)=0.0 IO=IC+2*IB-1 IE=ID+1 FYVEC(J)=0.0
BW(IO)=0.0 2 CONTINUE
B VY (IE) =0. 0 JXi=0
545 CONTINUE J72=NLi67Q
ICB=ICB+4=h(IFEL(IR) JY=O
ICO=ICD+2)1IFEL(IR) ICX=O IRX=0 550 CONTINUE IRY=O DO 565 IR=2.14REG IN02=0 IF (NXCVS (IR) .E0.0) GO TO 565 1NO2 0 NXC=NXCVS(ZR)
DO 560 IX=I.NXC CXR=1000.0*Rt X
INIT=IFIREL(IA.0 CZR=1000.OsRhgx I7 IFIN=ILASEL(IR.IX) NLOX=0
DO 555 I=INIT.IFIN NIOX=O
IC=ICA+ICB NTOX=2~4IELDX
ID=IC+2*(I—INIT)+1 NLOY=O N IG=ICC IOY=O NTOY=NELDX IG=L+ICD C
IH=IG+I—INIT+1 DO B00 IR=1.NREG GO TO (556.557) ITB (I) C 556 CONTINUE
SET UP TEPP ELEMENT PARAfETERS FOR EACH REGION Q 0h (ID) =TX (I) BVXZ (IE) =TZ (I) C B W (IH) =TY (I) IJ=O
GO TO 555 IF(IR.GT.1) GO TO 20 557 CONTINUE IRR=O
B CONTINUE 6V(ID)=DU(I) BVXZ(IE)=DN(I) IRR=IRR+1
BW(IH)=DV(I) IF(NIFEL(IRR).E0.0) GO TO 9
555 CONTINUE INIT=IFIRD(IRR)
NUhEL=IFIN—INIT+1 IFIN=ILASD(IRA)75U=1 ICB=ICB+2YJ(U1EL ISU ICD=ICD+NUFEL T5 TO 11 5 15
= 560 CONTINUE
9 CONTINUE 565 CONTINUE 570 CONTINUE IF(IRR.LT.NREG) GO TO 0
NLi87Q=lCRrICB GO TO 40
BVXZ(IC)=0.0 CALL OPENPS(2.INDEXI.LEN2.0)
45
50
C
C C TRANSFORM TO LOCAL AXES FOR EL I C
C
COFX=COFICIR) COFXG=COFX=GP iO2 CIR) COFY=PY2 COFYG=COFY=SteD(IR) ThDC=GPOD2P/COF)C0 THY=GrODP/COFYG NU12=RNUl2(IR) NU34=RNU34(IR)
DO 600 I=1.MAXID JRX=O JRY=O ICX=ICX+1 IF(ICX.E0.1) GO TO 42 JX1=JX1+NLIBX2 JX2=JX2+NLIBX2
42 CONTINUE JXI=JXI+NLOX JX2=JX2+NLOX JY=JY+NLOY CXI=OCX(I) CZI=DCZ(I) SINBI=DSINB(I) COSBI=DCOSB(I) LX=COSBI
IF(KXJ.E0.1.AND.KZJ.E0.1) GO TO 60 CONTINUE DO 50 JA=1.10 PI(JA)=0.0 CONTINUE SINBJ=SZJ=OSINS(J) COSE1J=SXJ‘DCOSB(J) SINBJI=SINBJ=COSBI-COSBJ=SINBI COSBJI=COS6J=i0SB1+5INBJ=SINBI
00 55 K=1.4 OXG=52J*DXGP (J.K) -CXI D2G=SXJ=02GP(4.K)-CZI
DX=DXG= 3S61-0ZG=SINBI 02=DX0=SIN6I+0ZG=COSBI DX2=0X 02 DZ2=02XX2 R2=DX2+0Z2 R4=R2==2 RLNR=0.5=ALOG(R2) RMFK=RHF(K)
C LZ=SINBI NX=-SINBI
15 CONTINUE NZ COSbI DO 19 I=INIT.IFIN C ID=ID+1 C REF VALUES OCX (IO) =CX (I) C DCZ(ID)=CZ(I) DXR=CXR-CXI USINB(ID)=TSC=SINB(I) DZR=CZR-C21 OCOSB(ID)=TSC=CASB(I) XRI=-0IR=SINSZ+0XR=COSBI DDS(ID)=DS(I) ZRI=DZR=COSBI+OXR=SINBI 106(10)=1TBCt) RREF2=XRI=Q+ZRI==2 00 16 K=1.4 RLNRF=0.5=ALOG(RREF2) DXGP(IO.K)=XGP(I.K) CXU I =XR I=12/RREF2-iK134xRLNRF DZGP(IO.K)=ZGP(I.K) CXNI=XRI=2RI/RREF2
16 CONTINUE CZUI=CXNI 19 CONTINUE C21aI=ZRI==Q/RREF2-1U344RUIRF
GO TO (9.25.30) ISU CVI=-RLNRF C C 20 CONTINUE DO 300 J=1.MAX.10
INIT=IFIRD(IR) FXJXI=0.0 IFIN=ILASO(IR) FZJXI=0.0 15U=2 URJXI=0.0 TSC=-1.0 LRJXI=0.0 GO TO 15 FZJZI=0.0
25 CONTINUE FZJZI=0.0 IF (NXCVS (IR) .E0.0) GO TO 40 URJZ1=0.0 IX=0 LRJZI=0.0
27 CONTINUE FYJYI=0.0 IX=IX+1 VR.'YI=0.0 INIT=IFIREL(IR.I)0 C IFIN=ILASEL(ZR.1)0 DO BO KXJ=1.KXT.2 ISU=7 SXJ=2-KXJ TSC=1.0 TYC=SXj GO TO 15 TZC=SXJ
30 CONTINUE C IF(IX.LT.NXCVS(IR)) GO TO 27 DO 100 KZJ=1.K2T.2
40 CONTINUE SZJ=2 -KZJ FI XID=ID TXC=SZJ MAYJD=IO IF(J.NE.1) GO TO 45
P1 (1) =PI (1) +RHFKM)X/R2 URJZI=TXC=(NXxLX=THRJ%1MIXAZzYWRJXI+NZ=EX=TURJZI+NZ=LZ=TIRJZI) / P1(2) =P1 (2) +RHFKa.DX2a4)X/R4 1 RMAX+URJZI P1(3) =PE (3) +RHFK DX=DZ2/R4 LRJZI=TZC=(NXxNX=TURJXI+NXxJ1Z=T(JtJXI+NZKIX=TURJZI+NZ=NZ=TLRJZI) / P1 (4) =P1 (4) +RHFK=DZ/R2 1 RMAX4 RJZI PI(5)=P1(5)+RHFK=DX2=02/R4 FYJYI=FYJYI+TYC=TFYJYI
VRJYI=VRJYI+TYC=T~dRJYI PI(7)=PI(7)+RHFKxOR2 P1(7) =PI (0) +RHFKZRLNRNR 100 CONTINUE PI (0) =P7 (0) +RHFKIIOXKDL/ti2 60 CONTINUE PI(9)=P!(9)+RHFK=aZXD22/R4 C PI(10)=Pi(10)+RHFKOZ2/R2 C IRITE COEFFS IN TEPP ARRAYS
55 CONTINUE C C IF(IR.GT.1) GO TO 130
DSJ=DOS(J) IF(NIFELP.EG.0) GO TO 130 DSJ2=DSJ/2.0 IF(I.GT.1) GO TO 105 TJX1=0SJ2XCNU124P7(1)+2.0xPI(2)) JINX=1 TZJXI=0SJ2*(-NU12xPI(1)+2.0ZPI(3)) JFINX=2Z71IFELP TZXJXI=05J2z(NU12xP1(4)+2.0ZPI(5)) JINY=1 TURJXI=DSJ2x(PI(6) —NU34a?I(7)>-05J=CXUI JFINY=NIFELP TWRJX1=0SJ2=PI(B)-0SJ=CXNI 105 CONTINUE TFXJXI=SINBJZ=TXJXI+COSBJIXT2XJXI TSC —1.0 TF2JX1=COSBJI=TZJX1+S1U5JI=TZXJXI GO TO (150.155.135) IDB(J)
C 130 CONTINUE TXJZI=DSJ2=(—MJ12xPI(4)+2.0*PI(5)) GO TO (140.140.134) IDB(J) TZJZI=DSJ2=(NU12>IPI(4)+2.0zPI(C1)> 134 TSC=1.0 TZXJZI=DSJ2=(NU12ZPI(1)+2.0xPI(3)) 135 CONTINUE 1URJZI=DSJ2ZPI(0)-05J=42U1 JX1=JX1+1 TH/JZI=DSJ2=(PI(10)—NU34ZPI(7))—OSJZC2U1 JX2=JX2+1 TFXJZ I=S INOJ I'TX1ZI+COS8J I xTZXJZI FVEC (JXI) =75Ca4JRJXlzīY0( TFZJZI=C0SBJ1ZTZJZI+SINBJI=TZXJ2I FVEC(JX2)=T1CZURJ21z7t0C
C JX1=JX1+1 TFYJYI=DSJ2W1(1) JX2=JX2+1 TYZJY7=DSJ2xPI (4) FVEC (JX1) zT5CZUKJXIxTTt( TVRJYI= (-05J2xP1 (7) -0SJ VI) /RMAX FEC(JX2) =TSCZURJZI=TMX TFYJYI=SINBJI=TXYJYI+COSBJI=TYZIYI JX1=JX1+1 GO TO 65 JX2=JX2+1
60 CONTINUE FVEC(JX1)=FXJX1/COFX DSI=DDS(1) FVEC(JX2)=FXJZI/C0FX RLNSI2=ALOG(OS1,2.0) JX1=JX1+1 TFXJXI=0.5=C1lFX JX2=JX2+1 TFZJXI=0.0 FVEC(JX1)=FZJXI/COFX TURJXI=OSIX(1.0—RNU34(IR)=(RLNSI2-1.0)—CXUI) FVEC(JX2)=FZJZI/COFX Ti.RJXI=—O5IxCX1I JY=JY+1 TFXJZI=0.0 FYVECCJM =TSCXVRJYIZTMY TFZJZI=0.5=COFX JY=JY+1 TURJZI=-0S1xTZU1 FYVEC(JY)=FYJY1/COFV TEKJZ 1=051= (—RNU34 (IR) x (RLNS I2-1.0) —M 7) INDD=0 TFYJYI=PY GO TO 295 TVRJY1=DSIZ(—RLNSI2+1.O—CvI)/RMAX 140 CONTINUE
65 CONTINUE IF(INDO.E0.1) GO TO 145 C INOD=1 C TRANSFORM FROM LOCAL AXES TO GLOBAL AXES JX1=JXI+44IOX C JX2=JX2+4410X
FXJXI=TXCx(LXxLX=TFXJXI+LXZLZzTFZJXI+l2ZLXxTFXJZI+lZ=LZxTFZJZI)+ JY=JY+NIOY 1 FXJXI JRX=NLIBXZ-NTOX-2a71ELEXT(19) FZJXI=TZCx(LXZNXxTFXJXI+LX=M2ZTFZJXI+LZZNXZTFXJZI+l.Z=TIZ&TFZJZI)+ JINX=JRX+1 1 FZJXI JFINX=JRX+2ZNELEXT(IR) FXJZI=TXCXCNXZLXx7FXJXI+NXZLZ=TFZJXI+NZ=LX=TFXJZI+NZxLZ*TFZJZI)+ JRY=NLI8Y—NTOY—MELEXT(IR) 1 FXJZI FZJXI=TZCx(NXa't(XZTFXJXI+NXZNZ=TFZJXI+NZ:71)(=TFXJZI+NZEN2xTFZJZI)+ JINY=JRY+1 1 FZJZI JFINY=JRY+NELEXT(IR) URJXI=TXCX(LMXxTURJXI+LX(L2xTLRJX1+LZ=L XZTURJZI+LZ=LZ=TLRJZI)/ 145 CONTINUE 1 RMfX+URJX1 GO TO (150.155) 105(J) W 6RJXI=TZCx(LXx (XZTURJXI+LXZT(ZxTFRJXI+l2xT(XxT1JRJZI+42xT(2xTLRJZI) / 150 CONTINUE N 1 RPAX+WRJXI JX1=JX1+1 .p
JX2=JX2+1 FEC (JX1)=FXJXI/COFX FvEC(Jx2)=FXJZI/COFX JX1=JX1+1 Jx2=JX2+1 -FVEC(JX1)=F2JXI/COFX FVEC(Jx2)=FZJZI/COFX JRX=JRX+1 UvEC1 (JRX) =URJXI=TTUC UvEC2 (JRx) =URJZI=TriX JRX=JRX+1 UVECI CJRX)=WRJXI=T t UVEC2 CJRXI =WRJZIXTMX JY=JY+1 FYVEC CJY) =FYJYI/COFY JRY=JRY+1 vvEC (JRY) =vRJY1xR1Y GO TO 295
155 CONTINUE JX1=JX1+1 Jx2=JX2+1 FVEC(JX1)=—URJXI=TTLC FvECCJX2)=—URJZI=TTVC JX1=JX1+1 JX2=Jx2+1 FvEC CJX1) =-4 RJXI=TTV[ FVEC(JX2)=-4RJZIKRIX JRX=JRX+1 UVEC1CJRX)=fXJXI/COFX UVEC2 (JRX) =—FXJZI/COFX JRx=JRx+1 UvECl (JR)0 =—FZJXI/COFX UvEC2 (JRX) =—FZJZI/COFX JY=JY+1 FYVECCJY)=—'RJYI=TMY JRY=JRY+1 V VEC (JRY) =—FYJYI/COFY
295 CONTINUE 300 CONTINUE
J%:1=JXI+NTOX JX2=JX2+NTOX JY=JY+NTOY
C C WRITE R/A FILES C
IF(JX2.GE.JMATX) GO TO 305 GO TO 320
305 CONTINUE IND2=IND2+1 ICX=0 IF(JX2.EO.JMAT7) GO 70 310 CALL WR ITNSC2.FVECC1).MATW.IN02) DO 307 JA=I.NLIBXZ FVEC(JA)=FVEC(JA+JMATX)
307 CONTINUE NLIBX=NLIBX2+1 00 308 JA=NLIBX.TWXW FVEC(JA)=0.0
308 CONTINUE JX1=NLIB%.7 JX2=JXI+NLIBXZ GO TO 320
310 CONTINUE CALL ITT5 c2.FVEC cl) .)RTW. IN02)
J)(2=NL IBX2 JX1=0 DO 312 J l.)RTW
312 G0NTTNU FvECLJA)=0.0
320 CONTINUE IF(JY.LT.JMATY) GO TO 325 IN06=IN06+1 JY=O CALL WRITMS(6.FYVECCl).MATW.IND6) DO 323 JA=I.JMATY FYVEC(JA)=0.0
323 CONTINUE 325 CONTINUE
C C RH5 C
TVAL1=0.0 TVAL2=0.0 TVAL3=0.0 DO 350 JA=JINX.JFINX TvAL 1=TVAL 1 WVECI (JA) RSVXZ CJA) TVAL2=TVRL2+UVFC2GJA)18VX2(JA)
350 CONTINUE DO 360 JA=JINY.JFINY TVAL3=TVRL1+VNFG(JA)NT(JA)
360 CONTINUE IRX=IRX+1 6TXZ(IRX)=TVAL1 IRX=IRX+1 6TXZCIRX)=TVAL2 IRY=IRY+1 8TY(IRY) =TVAL3
600 CONTINUE IF(IR.EO.NREG) GO TO 600 JW1=4 IF(IR.E0.1.AND.NIFELP.GT,0) JW1=2 NLOX=NLOX+JWI*N IFELCIR) NL OY=NC.OX/2
C NIOX=O IFtIR.E0.(NREG-1))GO TO 610 IR2=IR+2 DO 605 JA=IR2.NREG NIOX=NI0X+42NIFEL (JA)
605 CONTINUE IFCIR-EO.1) GO TO 620
610 CONTINUE IFCNREG.GT.2) GO TO 612 NIOX=0 GO TO 620
612 CONTINUE DO 515 JA=2.IR NIOX=NIOX+2=t+ELEXT CJA)
615 CONTINUE 620 CONTINUE
NIOY=NIOX/2 C C TRAILING ZEROS C
NTOX=NTOX-2xNELEXT(IR+I) NTOY=NTOX/2
B00 CONTINUE IND2=IN02+1
IN06=IND6+1 CALL LR2TF5(2.FNEC(1).MRTN.IND2) CALL EBIIMB(6.FYVEC(1).MATE(.IND6) RETURN END
*DECK BSTRESS SUBROUTINE BSTRESS
BALL GEN REAL LX.LZ.NX.NZ SIGP=0.5=(FZ+FX) SIGs 0.5=(FZ-FX) IST=O ISV=O IS1.0 IR=1 EDAS=EFDD (1) , (1.0-RNU (1) *K2) RNUO=RNU (1) iRllU 11(1)
C IF(NXCVS(1).E0.0) GO TO 310 NXC=NXCVS(1) 00 309 IX=I.NXC IST=1ST+1 IA=IFIREL (1. DO IB=ILASEL (1.IX) ISU=1 GO TO 400
305 CONTINUE 309 CONTINUE 310 CONTINUE
IF(NREG.E0.1) GO TO 500 DO 330 IR=2.NREG ISV=ISv+1 IA=IFIRO(IR) IB=ILASD(IR) ISU=2 GO TO 400
315 CONTINUE 330 CONTINUE
DO 350 IR=2.NREG IF(NXCVS(IR).E0.0) GO TO 350 EDAS=EFOD (IR) , (1.0-RNU (IR) =2) RNUD=RNUCIR),RNU11 (IR) NXC=NXCVS(ZR) DO 345 IX=1.NXC IA=IFIRELCIR.IX) IB=ILASEL(IR.IX) ISE.K I5N+1 ISU=3 GO TO 400
340 CONTINUE 345 CONTINUE 350 CONTINUE
GO TO 500 400 CONTINUE
DO 195 I=IA.IB SINBI=SINB(I) COSBI=COSB(I) NZ=COSBI NX=SINBI L2=-SINBI L X=COSB I SIN2BI=2.0=SINBI=COSBI COSBB I=2. 0*COSB I*COSB I-1. 0 PLI=SIGP-SIGFPCOS2BI-FZX=SIN261
PMI=FY PNI=S IGP+SIGPPtOS2BI+FZX=SIN2B1 PLMI=-FY2=SINBI+FXY OSSI PMNI=FYZ=COSB I+FXY=S INBI PNLI=FZX=COS2BI-SIGMZSIN261
C SINNI=NZ=7Z(I)+NX:TX(I) TAUNLI=LZ=T2(I)+LXXTX(I) CXI=CX(I) CZI=C2(I) DUI=DU(I) DVI=OV(I) DUI=DN(I) IN*I-1 IP=I+1 IF(I.E0.IA) GO TO 5 IF(I.EO.IB) GO TO 5
2 CONTINUE DU IP1=DU (IID OVIPI=DV (IIO DWIFKDW (IID DUIP=DU(IP) DVIP=DV(IP) DWIP=DU(IP) OXI=CXI-CX(IID DZ1=CzI-CZ(Iro DX2=CX(IP)-CXI DZ2=CZ(IP)-CZI GO TO 50
S CONTINUE IFCI.E0.IA) ITS=1 IF(I.E0.IS) ITS=2 TOX=ABS(EXI(IA)-EX2(IS)) TOZ=A5S(E21(IA)-EZ2(IS)) IF(TOX.LT.TOL.AND.TOZ.LT.TOL) GO TO 10 IF(TOX.LT.TOL) GO TO 15 IF(TOZ.LT.TOL) GO TO 20
C GO TO (25.30) ITS
25 CONTINUE DUIFK0.0 DVI(xDVI DUIr1 DNI DUIP=DU(IP) DVIP=DV(IP) DWIP=DU(IP) DXI=CXI DZI=CZI-EZ1(I) DX2=CX(IP)-CXI DZ2=CZ (IP) -CZI GO TO 50
30 CONTINUE DUIT1 DU(IID OVIFKDV(IID DUIPKDW(IID DUIP=DUI DVIP=0.0 DWIP=0.0 DXI=CXI-CX(IM) DZ1=CZ I-CZ (IPO DX2=EX2(I)-CXI DZ2=-CZI GO TO 50
10 CONTINUE
■
GO TO (11.12) ITS 11 CONTINUE
IM=IB IP=I+1 GO TO 2
12 CONTINUE IM=I-1 IP=IA GO TO 2
15 CONTINUE GO TO (16.17) ITS
16 CONTINUE GO TO 25
17 CONTINUE DUIM=DU(IM9 DVIM=DV(IM) DUIM=DW(IMD DUIP=-0U1 DVIP=DVI DVIP=DWI DX1=CXI-CX(IM) 0Z1=CZI-CZ(IM) DX2=-2.0=CXI DZ2=0.0 GO TO 50
20 CONTINUE GO TO (21.22) ITS
21 CONTINUE DUIM=DUI DVIM=-0VI DWIM=-DWI DUIP=OU(IP) DVIP=DVIP) DWIP=DW(IP) DX1=0.0 DZ1=2.0 CZI DX2=CX(IP)-CXI DZ2=CZ(IP)-CZI GO TO 50
22 CONTINUE GO TO 30
50 CONTINUE C
UL I=L Z=DW I+L X*DU I UL IM=LZ=OWIM+LXzDUIM UL IP=LZ;DWIP+LX=DUIP DL 1=LZzDZI+LX=DX1 DL2=LZzDZ2+LX DX2 IF(ABS(DL1).LT.TOL) DL1=TOL IF (ABS (OL2) . LT. TOL) DL2=TOL
C IRR=IR IF(ISU.E0.2) IRR=I EPL I=-0.5Z ((UL I-UL IM) /DL I+(UL IP-UL I) /DL2) SIGLI=EDAS;EPLI+RNUO=SIGNI SIGMI=RNU(IRR);(SIGNI+SIGLI) GAMLMI=-0.5ZC(DVI-OVIM)/DL1+(DVIP-0VI)/012) TAULMI=GMDO (IRR) Ye(,AMLMI TAUMIII=TY (I)
C C CALC TOTAL STRESSES ; USE SIGX ETC FOR SIGL ETC
IPT=I SIGX(I)=SIGLI+PLI
SIGY(I)=SIGMI+PMI SIGY(I)=SIGNI+PNI TAUXY(I)=TAUNMI+PLMI TAUYZ(I)=TAUMNI+PMMI TAUZX(I)=TAUNLI+PNLI
C C CALC PRINCIPAL STRESS MAGNS AND ORIENTATIONS C
CALL PRINSTR 195 CONTINUE
GO TO (200.220.240) ISU 200 CONTINUE
IF(IST.GT.1) GO TO 201 I.PITE(7.1005) IREG(1)
1005 FORMAT(1H ///7X.78HSTRESSES AND DISPLACEMENTS AROUND OPENINGS IN I SHE INFINITE DOMAIN - REGION NO. 2
201 CONTINUE WRITE(7.1000) IXCVCI.IX)
1000 FORMIATC1H // 7X.15HEXCAVATION NO. .I2/7X 18H //7 1X.43HSTRESS COMPONENTS REL TO ELEMENT LOCAL AXES.40X.23HDISPLACEME 2NT COMPONENTS//3X.214 I.7X.214CX.8X.2HCZ.6X.4HSIGL.BX.41451GM.6X. 3 4HSIGN.5X.5HTAULM.5X.SHTAUMN.SX.SHTAUNL•BX.IHU.11X.1HV.11X.IHW) GO TO 250
220 CONTINUE IF(ISV.GT.1) GO TO 225 WRITE(7.5000)
5000 FORMAT(1H ///7X.54HSTRESSES AND DISPLACEMENTS AROUND INCLUSION BOU INDARIES/7X.54H 2)
225 CONTINUE WRITE (7.5500) IREG (IR)
5500 FORMAT(1H //7X.lOHREGION NO. .I2/7X.12H //7X.43HSTRESS ICOMPONENTS REL TO ELEMENT LOCAL AXES.40X.23HDISPLACEMENT COMPONENT 2S//3X.2H I.7X.2HCX.8X.2HCZ.6X.4HSIGL.BX.4HSIGM.6X.4ISIGN•5X. 3 SHTAULM.5X.5HTAUMM.5X.5HTAUNL.8X.1HU.IIX.1HV.11X.1H141
GO TO 250 240 CONTINUE
IF(ISW.GT.1) GO TO 245 WRITE (7.6000)
6000 FORMAT(1H ///7X.59HSTRESSES AND DISPLACEMENTS AROUND EXCAVATIONS I IN INCLUSIONS/7X.59M 2
245 CONTINUE WRITE(7.1000) IXCV(IR.IX)
250 CONTINUE WRITE/7.2000) CI.CX(I).CZCI).SIGX(I).SIGY(I).SIGZ(I).TAUXY(1).
1 TAUYZ(I).TAUZX(I).DUCI).DV(I).DW(I),I=1A.I8) 2000 FORMAT(1H /.X.14.8F10.3.3E12.4)
WRITE (7.3000) 3000 FORMAT(1H //.7X.66HPRINCIPAL STRESSES AND ORIENTATIONS RELATIVE TO
1 ELEMENT LOCAL AXES//4X.2H I.BX.2HCX.BX.2HCZ.6X.4HSIG1.X.17HALPHA 2'BETA GMMA.6X.4HSIG2.1X.17HALPHA BETA GAMfA.6X.4HSIG3.1X.17HALPH OR BETA GAMMA)
WRITE(7.4000) (I.CX(I) .CZCI) .SIO(1.I) .DALF(1.I) .DBETCl•I) .DGAMCI• U 1. 5IG(2.I).DALE(2.I).DBET(2.I).DGAM(2.I).SIG(3.1)•DALF(3.I). 2 DBET(3.I) .DGMI(3• I) .I=IA.I8)
4000 FORMRT(1H /.X.I5.3F10.3.3F6.1.F10.3.3F6.1.F10.3.3F6.1) GO TO (305.315.340) ISU
500 CONTINUE RETURN END
iDECK PRINSTR SUBROUTINE PRINSTR
.I2/7X.80H XXX*ZU=_X
=CALL GEN I=IPT SIGXI=SIGX(1) SIGXI=SIGY(I) SIGZI=SIGZCl) TAUXXI=TAUXYCI) TAUYZI=TAUYZCI) TAUZXI=TAUZX(I) RJI=SIGXI+SIGYI+SIGZI RJ2=SIGXI;SIGYI+SIGYIKSIGZI+SIGZI*SIGXI-(TAUXYI=TAUXYI+
1 TAUYZI=TAUYZI+TAUZXI*TAUZXI) RJ3=SIGXI=SIGYI*SIGZI+2.0xTAUZXI=TAUYZI=TAUZXI-(SIGXIXTAUYZIXZ2+
1 SIGYI*TAUZXIX 2+SIGZIXTAUXYI K2) TRJ4=RJ1*RJ1-3.0=RJ2 IF(TRJ4.LE.0.0) TRJ4=TOL RJ4=S0RT (TRJ4) TC=(27.0=RJ3+2.0=RJ1a3-9.0*RJl* J2),(2.0*(RJ4ZX3) IF(TC.GT.1.0) TC=1.0 IF(TC.LT.-1.0) TC=-1.0 THET=ACOS(TC)/3.0 DO 25 K=1.3 GO TO (5.10.15) K
5 RNG=TNET GO TO 20
10 ANG=PY23-THET GO TO 20
15 ANG=PY23+THET 20 CONTINUE
SIG CK.I)=(RJ1+2.0=RJ4*COS(ANG))/3.0 TA= CS IUYI-S IG CK. I)) = (S IGZI-SIG (K. I)) -TAUYZIa2 TB=TAUYZIXTAUZXI-TAUXYlX(SIGZI-SIG(K.I)) TC=TAUXYIXTAUYZI-TPUZXI* (S IGYI-SIG (K. I) ) STS=TAxTA+TIPKTB+TCKTC IF(STS.E0.0.0) STS=TOL STS=SORT(STS) DCX=TA/STS DCY=TB/STS DCZ=TC/STS DALF (K . I) =ACOS (DCX) /FAC DBET(K.I)=ACOS(OCY)/FAC DGAMCK.I)=ACOS(DCZ)/FAC
25 CONTINUE RETURN END
=DECK DISSTR SUBROUTINE DISSTR
xCALL GEN COMMON FXJX(50).FZJX(50).FXJZ(50).FZJZ(50).URJX(50).
1 4RJX (50).URJZ150).WRJZ(50).FYJY(50).VRJY(50).DSINB(50). 2 DCOS6(5O).005(5O).DXGP(50.4).OZGP(50.4).DTX(50).DTY(50). 3 DTZ(5O).DDU(50).DOV(50).DDW(50).DXUJX(50).DXIIJX(50). 4 DXTXX(50).DXTZX(50).OZUJZ(50).DZ11U2(50).DZTXZ(50). 5 DZTZZ (50).OZUJX(50).DZUJX(50).DZTXX(50).DZTZX(50). 6 D)IJJZ(50) .DXIiJZ(50) .Dxt.(50) .DXTZZ'50) .DxTYYC5D) . 7 DXVJY(50).OZTYY(50).OZVJY(50)
REAL NU12.NUIPI.NUIP2.NU.NU14.NU34.NU32.NUII.LAM.M302.MODG C C TEMP PARAMETERS FOR PROS REGION C
ID=0 IFCIRP.GT.1) GO TO 20 00 10 IR=I.NREG IF(NIFEL (IR) .E0.0) GO TO 10 INIT=IFIRDIRR)
IFIN=ILASD(IR) I5U=1 TSC=1.0 GO TO 15
9 CONTINUE 10 CONTINUE
GO TO 40 C
15 CONTINUE DO 19 I=INIT.IFIN ID=70+1 OSINB (ID) =TSC=SINB (I) DCOSB CID) =TSC*C05B (I) DOS CID) =D5 CI) DTX (ID) =TSC=TX ( I) DTY(IO)=TSCXTY(I) DTZ(ID)=TSCXTZCI) ODU (ID) =DU (I) DOV(ID)=DV(I) DDN(I0)=DN(I) DO 16 K=1.4 DXGP (ID.K) =XGP (I. K) DZGP (ID. K) =ZGP (I . K)
16 CONTINUE 19 CONTINUE
GO TO (9.25.30) ISU C
20 CONTINUE INIT=IFIRD(IRP) IFIN=ILASO(IRP) ISU=2 TSC=-1.0 GO TO 15
25 CONTINUE IF (NXCVS (IRP) .E0.0) GO TO 40 NxL=NXCVS(IRP) DO 35 IX=I.NXC INIT=IFIRELCIRP.1X1 IFIN=ILASEL(IRP.IX) ISU=3 TSC=1.0 GO TO 15
30 CONTINUE 35 CONTINUE 40 CONTINUE
C MAXJO=ID COFX=COFICIRP) COFXG=COFXXGM3D2(IRP) COFY=PY2 COFYG=COFYX12D0(IRP) NUI2=RNU12(IRP) NUIPI=RNUIPI(IRP) NUIP2=RNUIP2(IRP) NU=RNU(IRP) NU14=RNU14(IRP) NU32=RNU32(IRP) NUII=RNUI1(IRP) NU34=RNU34(IRP) L APk RL AM (IRP) MU02=GM3D2 (IRP) MODG=GMOD(IRP)
C C REF VALUES
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C
DxUJXJ=DxUJXJ+TxC10XuJXI OXW XJ=0X/4 JXJ+TZCxDXW XI DXTXXJ=OXTXXJ+TXCxSDXTXX DXTZXJ=DXTZXJ+TZCx5DXTZX
DZXJZI= NUIP2xPI (1) 4.0x NUWPI (11) -8.0W! (4) XTXZJ=0.0 DZZJZI= NUI2WI (1)+4.0x NU1P1xPI (11) -8.0xP! (12) XTZZJ=0.0 OZzX11I=2.0x NUIP2TPI (5) -0.OWPI (13) XTYYJ=0.0 02UJZ2=PI (7) -2.074,7(14) XVJYJ=0.0 OZWZI=- NU14WI (9) -2.0xPI (15) ZTYYJ=0.0 SDZT)I=SINBJ=0ZXJZI+C05BJx0ZZXJZI
C ZVJYJ=0.0 SDZT!Z=COSBJx0ZZJZI+5INB.ROZZXJZI
DZUJZJ=DZU.ZJ+TXCxOZUJZI 0 315 KXJ=1.1(XT.2 D7.W2J=DZWZJ+TZCADZNJZI XJ=2-KXJ 07.TXZJ=DZTXZJ+TXCx50ZTXZ
TYC=SXJ OZTZZJ=DZTZZJ+TZGx50ZTZ2 TZC=5XJ C DSJ2=D05 (J) /2.0 OZXJ:'!= -NU12x2.0xPI (5) -5.01:A1M) COSBJ=5XJx0C05121(J)
C DO 310 KZJ=1.KZT.2 52J=2-$(ZJ 51N5J=5ZJxo5IN6 (J)
DZZJX1=2.0x NU32WI (5)-8.0WPI.,13) DZZXJXI= Nu72W1(1) -4.0* NUII*PI (11)-8.0W1(4) DZUJXI=-2.0=PI(10)- NU34W! (9) D21UXI=PI (7) -2.0W! (14) SOZTXX=52N5J=02XJXI+C055Jx0ZZXJXI
TXC=SZJ S0ZT2X=C05BJZOZZJXI+51NBJx0ZZXJXI D0 205 JR=1.15 OZUJXJ=DZUJXJ+TXCxDZUJXI PI(JR)=0.0 DZ)4JXJ=DZW XJ+T2CaDZJ.JJX1
205 CONTINUE 0ZTXXJ=OZTXXJ+TXCx5DZTb( C DZTZXJ=OZTZXJ+TZCx502.2X
DO 210 K=1.4 C DX=SZJ=DXGP(J.K){XI DXXJ21=2.0x NU32W1(5)-0.0xAtas) D2=5XJ=DZGPCJA)-CZI DXZJZI=-2.0x NUI2WI(5)-8.070AI(13) DX2=0XXX2 DXZXJZI= NU7223,I(1)-4.0* NU11xPI(2)-8.0xPI(4) DZ2=DZxx2 DXUJZI=PI (9) -2.0W! (10) R2=0X2+0Z2 DXKJZI=-2.0xPI (14) - NU34WI (7) DX4=DX2=2 SOXTXZ=SINBJ 0XXJZI+COSBJxDXZXJZI DZ4=DZ2=x2 SDXTZZ=C0513JZOX2JZI+5INLJxDX2.JZ! R4=R2 D)OJJZJ=OXUJZJ+TXGxOXUJZ7 R6=R4;:)2 OXW ZJ=OXW ZJ+TZGxOXJZI XTZ=DXSDZ DXTX2J=DXTXZJ+TXCx50XTX2 RHFK=RHF(K) DXTZZJ=DXTZZJ+TZCx50XT2Z PI(1)=PI(1)+RHFK/R2 C PI(2)=P1(2)+RHFK=DX2/R4 DXXYJYI=°I(1)-2.0WPI(2) P1(3) =PI (3)+RHFKxDX4/R6 DXYZJYI=-2.0xWI(5) PI(4)=PI(4)+RHFKx0X21022/R6 DXVJYI=-P1(7) P1(5) =P I (5) 4RHFKxXTZ/R4 50XTYY=5IN5J=DXXYJYI+COS5J7AOXYZJYI P1(6)=PI(6)+RHFK=DX2xXTZ/R6 OXTYYJ=DXTYYJ+TYCxSOXTYY P1(7)=P1(7)+RHFKxDX/R2 DXVJYJ=DXVJYJ+TYCxVXVJYI P1(9) =P1 (8) +RHFKxDX2x0X/R4 C PI(9)=PI(9)+RHFKx0Z/R2 DZXYJYI=-2.0xPI(5) PI(10)=P1(10)+RHFKx0X2x0Z/P4 DZYZJYI=PI(1)-2.041(11) PI(11)=PI(11)+RHFKx0Z2/44 DZVJYI=-PI (9) P1(12) =P I (12) +RHFK=DZ4/R6 5OZTYY=SO4DJ=0ZXYJTIACIIS5J=1)ZYZJYI PI(13)=PI(13)+RHFK*XTZ=022/R5 DZTYYJ=DZTYYJ+TYCx5DZTYY P1(14) =P1(14)+RHFKxDX=DZ2/R4 DZVJYJ=D2VJY.J4TYCx0ZVJYI PI(151=P!(15)+RHFKx0Z=0Z2/R4 310 CONTINUE
210 CONTINUE 315 CONTINUE C C
DXXJXI= NU12xWI(1)+4.0x NUIP1xPI(2)-8.0=P7(3) OXUJX(.1)=DXUJ)UxDSJ2 DXZJXI= NUIP2=WI(1)-4.0x NUxP1(2) 8.0xPI(4) DUX CJ) =DXWXJ=D5J2 OXZXJXI= NU1P2x2.0ZPI (5) -8.0W1 (6) DXTXX(J)=DXTXXJZDSJ2 DXUJX1=- NU14WI(7)-2.0=91(8) DXTZX (J) =OXTZXJ V5J2 DXWXI=PI (9) -2.0=PI (10) DZUJZ(JI=OZUJZJ=C5J2 50xTXx=5INBJAI0xxJx1+CosBJA10xZxJX1 DZWZ (J) =DZWZJxDSJ2 S0XTZX=COS6.! 0XZJXI+SINBJx0XZXJXI DZTXZ (J) =DZTXZJOSJ2
C
COMMON 1000 FORMAT(1H //7X.9HLINE NO. .I2/7X.9H //7X.40HSTRESS 1ENT5 REL TO EXCV LOCAL AXES.43X.23HD1SPLACEMENT COMPONENTS//3X. ZTZZ CJ) =DZTZZJ*05J2
ZUJX (J) =DZUJXJx0SJ2 2 2H 1.7X.2HCX.OX.2HCZ.BX.4HSIGX.6X.4HSIGY.6X.4HSIG2.5X.SHTAUXY.6X. ZWJX (J) =DZW JXJDOSJ2 3 SHTAUYZ.SX.SHTAUZX.OX.IHU.11X.1HV.11X.1HW) ZTXX (J) =DZTXXJzvSJ2 WRITE(7.2000) (IP.CX(IP).CZ(IP).SIGX(IP).SIGY(IP).SIGZ(IP). ZTZX (J> =OZTZXJXDSJ2 1 TAUXY(IP).TAUYZ(IP).TAUZXCIP).UX(IP).UY(IP).UZ(IP),IP=1.MAXI) XUJZ CJ) =DXUJZJ*OSJ2 2000 FORMAT(1H /.X.I4.0F10.3.3E12.4) XWJZ (J) =DXIIJZJXDSJ2 6RITE(7.3000) XTXZ(J)=OXTXZJ'OSJ2 3000 FORMATCIH //7X.63HPRINCIPAL STRESSES AND ORIENTATIONS RELATIVE TO XTZZ CJ) =DXTZZJ*VSJ2 1EXCV LOCAL AXES//4X.2H I.BX.2HCX.BX.2HCZ.6X.4HSIG1.X.I7HALPHA BET XTYYCJ)=DXTYYJzoSJ2 2A GAM1q.6X.4HSIG2.1X.17HALPHA BETA GAMMA.6X.4HSIG3.IX.17HALPHA B XVJY(J)=DXVJYJ=DSJ2 SETA GAMMA) ZTYY(J)=DZTYYJ=05J2 WRITE(7.4000)(IP.CX(IP).CZCIP).SIG(1.IP).OALF(1.IP).DBET(1.IP). ZVJYCJ)=OZVJYJ=OSJ2 1 OGAM(I.IP).SIG(2.IP).OALF(2.IP).DBET(2.IP).OGAM(2.IP).SIG(3.IP>.
350 ONTINUE 2 DALF(3.IP).D6ETC3.IP).DGAM(3.IP),IP=1.MAX1) 0 355 K=1.20 4000 FORt1 T(1H /X.15.3F10.3.3F5.1.F10.3.3F6.1.F10.3.3F6.1) I(K)=0.0 RETURN
355 CONTINUE ENO XOECK ID50L
DO 360 J=1.MAXJO SU9ROUTINE IDSOL nooaa
SOLVER P1(1> =P I (1) +OXUJX (JI MOTX CJ) C SOLVER
ALL LOAD CASES ARE TREATED SIMULTANEOUSLY P1(2) =PI C2) +OXWJX (J) ><OTZ CJ) C SOLVER P1(3) =PI C3) +OXTXX (J)'•DDU CJ) C CHOICE OF UNITS FOR ANALYSIS GIVES ALL-CONDITIONED MATRIX SOLVER P1(4) =P I (4) +OXTZX (J) LOW (J) C ML16D- ORDER OF THE SYSTEM SOLVER P1(S)=PI(5)+0ZUJZ(J)=0TX(J) C NCC - NUMBER OF LOAD CASES SOLVER PI(6)=PI(6)+OZWJZCJ)=OTZ(J) C LUM - FILE HOLDING MATRIX SOLVER P1(7) =PI (7I +OZTXZ(JI MDU (J) C LBLM - RECORD LENGTH OF LUM SOLVER P1(6)=PI(0)+OZTZZ(J)=0DW(J) C MUM - NUMBER OF RECORDS SET TO FAST BLOCK SOLVER PI(9)=PI(9)+OZUJX(J)zDTX(J) C JMAT - NUMBER OF COEFFICIENTS PER FAST BLOCK SOLVER P1(10) =P I (10) +OZH JX (J)'•OTZ (J) C MEAT - MAXIMUM NUMBER OF EQUATIONS PER FAST BLOCK SOLVER P1(11' =P I (11) +OZTXX (J) zODU (J) C Fl - FAST INPUT BUFFER SOLVER PI(12)=PI(12)+0ZTZX(J)=VDWCJ) C FO - FAST OUTPUT BUFFER SOLVER PI(13)=PI(13)+OXUJZ(J)x0TX(J) C A - SLOW BLOCK LENGTH - NBLO = LENGTH OF FAST BLOCK SOLVER P1(14) =PI (14) +DXWJZ (J) ■OTZ CJ) C AO - SLOW OUTPUT BUFFER SOLVER PIC15)=PI(15)+OXTX2(J)XD0U(J) C BT - RIGHT HAND SIDES SOLVER P1(16) =PI (16) +OXTZZ(J) xDOW (J) PI(17)=PI(17)+OXVJYCJ)x0TY(J) C
C LUO LINE POINTER SOLVER
SOLVER > COMMON A0(600).FI(600).FO(600).A(3000) P1(16) =P I (1B) ,0XTYY CJ) •ODV (J)
PIC19)=PI(19)+OZVJY(J)XOTY(J) P1(20) =PI (20) +OZTYY(J) woDV CJ) COMMON/FI)ER(LIED.NCC.LUM.LBLM.NBLM.JMAT.MMiT
COMMim/CHANGE/BT (300) SOLVER SOLVER
360 CONTINUE EQUIVALENCE (ICUE.A0(600)) EPXI=(P I(1)+0I(2))/COFXG-(PI(3)+PI(4))/COFX DATA NBLO/5/ EPZI=(PI(S)+PI(6))/COFX0-(PI(7)+PI(6))/COFX DATA LUO/7/ SOLVER GAM2XI=(P1(9)+PI(10))/COFXG-(PIC11)+PI(12))/COFX+ LBUF=NBLPP .BLM SOLVER 1 0PI(13)+PIC14))/COFXG-(PI(15)+PI(16)>/COFX C INCREMENTS AND SLOW BLOCK FILE PARAMETERS SOLVER GAMDCYI=PIC17)/COFYG-PI(10)/COFY JA=NLIBD SOLVER GAMYZI=PI(19)/COFYG-PI(20)/COFY JALO=JA-1 SOLVER DEL=EPXI+EPZI JB=JA+1 SOLVER SIGXI=LAM DEL+MO02=EPXI IRI=1 SOLVER S IGZ I=L AMP•OEL+Mi02*EPZ I IA0=1 SOLVER SIGYI=NU'(SIGXI+SIGZI) C EONS READ TO DATE AND SLOW OUTPUT BUFFER CONTROL SOLVER TAUXYI=MODG;GAMXYI LA=O SOLVER TAUYYI=MO00x0AMYZI LB=I SOLVER TAUZXI=MODGzGAMZXI 1 CONTINUE SOLVER
SIGXCI)=SIGXI+FX SIGY(I)=SIGYI+FY SIGZCI)=SIGZI+FZ TAUXYCI)=TAUXYI+FXY TAUYZ (I) =TAUYZI+FYZ TPUZX(1)=TAUZXI+FZX IPT=I
C READ SLOW BLOCK LD SOL BLOCKS LC=MINO (NBLOZM'AT.JA-LA) LCLO=LC-1 L D=L CLO/M1AT+1 JC=1 JD=LBLM DO 2 IA=1.LD JE=JC
SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER CALL PRINSTk
400 CONTINUE WRITEC7.1000) IL
JF=JO 00 3 IB=1.N6LM
SOLVER SOLVER
W LJ 1-+
IR=(JF-JE)+1 CALL REAOMS (LUM.A (JE) . IR. IRI)
SOLVER SOLVER
DO 15 1B=1.NBLM 15= (JG-JF) +1 CALL WRITM5(LUM.RO(JF).SS.IRO)
JE=JE+LBLM SOLVER JF=JF+LBLM JF=JF+LBLM SOLVER JG=JG+LBLM IAI=IRI+1 SOLVER IAO=IAO+1
3 CONTINUE SOLVER 15 CONTINUE JC=JC+JIIAT SOLVER LB=1 JD=JD+JMAT SOLVER 13 CONTINUE
2 CONTINUE SOLVER C TRANSFER ONE EQUATION C LONGEST EQUATION SLOW BLOCK. LEADING DIAGONAL INCREMENT SOLVER DO 16 IB=JD.JE
LE=JA-LA SOLVER AO(LB)=A(111) LED=JB-LA SOLVER LB=LB+1 IF(LCLO) 4.4.5 SOLVER 16 CONTINUE
C REDUCTION OF SLOW BLOCK SOLVER JC=JC-1 5 CONTINUE SOLVER JO=JD+LED
JC=1 SOLVER JE=JE+LE JIA=LA+1 SOLVER 12 CONTINUE DO 6 IA=1.LCLO SOLVER C EONS READ TO DATE
C ELIMINATE THE UNKNOWN JIA SOLVER LRF=LA+LC ELT=A (JC) SOLVER IF (LAF-JA) 17.18.18 IF(ABS(ELT) -1.0E-9) 999.999.8 C REDUCTION OF REST OF THE SYSTEM
8 CONTINUE SOLVER 17 CONTINUE (LT=1.0E0/ELT SOLVER C FAST BLOCK FILE PARAMETERS JO=JC+LE SOLVER IFI=IAI JIB=JIA+1 SOLVER IFO=IAO DO 9 IB=IA.LCLO SOLVER C FAST OUTPUT BUFFER CONTROL
C MODIFY THE EQUATION JIB SOLVER LBF=1 ELTA=A(JD)aELT SOLVER LBG=LC+1
C THE MATRIX SOLVER JCF=JALO-LAF JE=JD+1 SOLVER 19 CONTINUE JF=JC+1 SOLVER C READ FAST BLOCK = 1 SOL BLOCK DO 10 IC=JIA.JALO SOLVER L CF=MIND (MYAT. JA{AF) A(JE)=A(JE)-ELTAXA(JF) SOLVER JC=1 JE=JE+1 SOLVER JO=LBLM JF=JF+1 SOLVER DO 20 IA=1.NBLM
10 CONTINUE SOLVER IT= (JD-JC) +1 C THE SECOND MEMBER FOR EACH LOAD CASE SOLVER CALL READMS(LUM.FI(JC),IT.IFI) SOLVER
JE=JIB SOLVER JC=JC+LBLM SOLVER JF=JIA SOLVER JD=JO+L BL M SOLVER DO 11 IC=1.NCC SOLVER IFI=IFI+1 SOLVER BT(JE)=BT(JE)-ELTA48T(JF) SOLVER 20 CONTINUE SOLVER JE=JE+JA SOLVER JC=1 SOLVER JF=JF+JA SOLVER JIA=LA+1 SOLVER
11 CONTINUE SOLVER DO 21 IA=1.LC SOLVER JD=JD+LE SOLVER C ELIMINATE THE UNKNOWN JIA SOLVER JIB=JIB+1 SOLVER ELT=A(JC) SOLVER
9 CONTINUE SOLVER IF CABS (ELT) -1.0E-6) 999.999.41 JC=JC+LED SOLVER 41 ELT=1.0E0/ELT SOLVER JIA=JIA+1 SOLVER JO=IA SOLVER
6 CONTINUE SOLVER JIB=LAF+1 SOLVER G MOVE REDUCED EQUATIONS TO OUTPUT BUFFER. WRITE AS NECESSARY DO 22 IB=1.LCF SOLVER
4 CONTINUE SOLVER C
MODIFY THE EQUATION JIB SOLVER JC=LE-1 SOLVER ELTA=FI(JO)1ELT SOLVER J0=1 SOLVER C THE MATRIX SOLVER JE=L E SOLVER JE=JD+1 SOLVER DO 12 IA=1.LC JF=JC+1 SOLVER JF=LB+JC SOLVER DO 23 IC=JIA.JALO SOLVER IF (JF-LBUF) 13.14.14 SOLVER FI(JE)=FI(JE) ELTA=A(JF) SOLVER
C WRITE SLOW OUTPUT BUFFER SOLVER JE=JE+1 SOLVER C ICUE IS START OF LAST EON JF=JF+1 SOLVER
14 CONTINUE 23 CONTINUE SOLVER ICUE=LB-JC-2 C THE SECOND MEMBER FOR EACH LOAD CASE SOLVER JF=1 JE=JIB SOLVER JG=LBLM
GJ L.1
JF=JIA 00 24 IC=1.NCC BT(JE) =BT (JE) -EL TAKBT (JF) JE=JE+JA JF=JF+JA
24 CONTINUE JD=JD+LE JIB=JIB+1
22 CONTINUE JC=JC+LED JIA=JIA+1
21 CONTINUE C PLO£ REDUCED EQUATIONS TO OUTPUT BUFFER.LRITE AS NECESSARY
JD=LBG JE=LE DO 25 IA=1.LCF JF=LBF+JCF IF (JF-LBUF) 26.26.27
C WRITE FAST OUTPUT BUFFER 27 CONTINUE
JF=1 JG=LBLM DO 28 I15=1.NBLM IU= (JG-JF) +1 CALL LRITP5CLUM•FOCJF).1U.IF0) JF=JF+LBLM JG=JG+LBLM IF0=IF0+1
28 CONTINUE LBF=1
26 CONTINUE C TRANSFER ONE EQUATION
DO 29 IB=JD.JE FO (LBF) =FI (IB) LBF=LBF+1
29 CONTINUE JD=JD+LE JE=JE+LE
26 CONTINUE LAF=LAF+LCF IF (LAF-JA) 19.30.30
C EMPTY FAST BUFFER 30 CONTINUE
JC=1 JD=LBLM DO 31 IA=1.NBLM 7V = (JD-JC) +1 CALL LRITrS (LUM.FO (JG . IV.IFO) JC=JC+LBLM JO=JO+LBLM IF0=IF0+1
31 CONTINUE LA=LA+LC M'AT=L BL'F/ (JCF+1) JI1 T=MATZ (JCF+1) IAI=IAO GOTO 1
C SOLVE LAST EQUATION 18 CONTINUE
JC=LB-1 ELT=AO (JC) IF(ABSCELT; -1.0E-9) 999.999.33
33 CONTINUO ELT=1.0E0/ELT
JD=JA SOLVER SOLVER DO 34 IA=1.NCC SOLVER SOLVER BT(J0)=BT(JO)sELT SOLVER SOLVER JD=JO+JA SOLVER SOLVER 34 CONTINUE SOLVER SOLVER C.BACKWARD PASS SOLVER SOLVER J11=2 SOLVER SOLVER JD=JA SOLVER DO 35 IA=1.JALO SOLVER SOLVER JC=JC-JB SOLVER SOLVER J0=J0-1 SOLVER SOLVER IF (JC) 36.36.37 SOLVER SOLVER 36 CONTINUE SOLVER SOLVER IAO=IAO-NBLM SOLVER SOLVER IAI=IAO SOLVER SOLVER JE=1 SOLVER SOLVER JF=LBLM SOLVER SOLVER DO 38 IB=1.NBLM SOLVER SOLVER IW=CJF-JE)+1 SOLVER SOLVER CALL REAQHS(LUN.A0(JE),IW.IAI) SOLVER SOLVER JE=JE+LBLM SOLVER SOLVER JF=JF+LBLM SOLVER SOLVER IAI=IAI+1 SOLVER SOLVER 3O CONTINUE SOLVER SOLVER JC=ICUE SOLVER SOLVER 37 CONTINUE - SOLVER SOLVER JE=JC+1 SOLVER SOLVER JF=JD+1 SOLVER SOLVER C CALCULATE THE UNKNOWN JO SOLVER SOLVER ELT=1.0E0,A0(JC) SOLVER SOLVER JG=JD SOLVER SOLVER JH=JF SOLVER SOLVER 00 39 IB=1.NCC SOLVER SOLVER C SUM OVER UNKNOWNS ALREADY CALCULATED SOLVER SOLVER JI=JE SOLVER SOLVER JJ=JH SOLVER SOLVER SUrt0.0E0 SOLVER SOLVER DO 40 IC=1.IA SOLVER SOLVER SUMSUM+AO(JI)K6T(JJ) SOLVER SOLVER JI=JI+1 SOLVER SOLVER JJ=JJ+1 SOLVER SOLVER 40 CONTINUE SOLVER SOLVER BT(JG)=(ST(JO)-SUM)ZELT SOLVER SOLVER JG=JG+JA SOLVER SOLVER JH=JH+JA SOLVER SOLVER 39 CONTINUE SOLVER SOLVER JB=JB+1 SOLVER SOLVER 35 CONTINUE SOLVER SOLVER RETURN SOLVER SOLVER 999 WRITECLU0.100)JIA SOLVER 100 FORMAT( 3941 THE MATRIX IS SINGULAR - EQUATION .I3) SOLVER SOLVER STOP SOLVER SOLVER END SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER SOLVER
W SOLVER W SOLVER W
