l ,

\

ABSTRACT

It is the purpose of this stuqy to present an analysis of some

of the patterns of stresses which may.be associated-with Room-and-~11ar

~aning Systems. Three series of tests were performed, using photoe1astic

techniques, to simulate simp1ified "Room-and-Pi11aru Hining Systems.

Stress patterns were' obtained for two series of mode1s in which the ratio

of opening width to pillar width were 1:1 and 3:1 respective1y.. A final

series of tests involved the extraction of the pi11ars. . .'

Previous studies have been reviewed and summari?ed in order to

clarify the objectives of this study.

The theor,y of photoelasticity, the experimental procedures that

were used~ and the techniques for a numerical solution of the 1aboratory

data have been exp1ained in previous works20,30, and are only summarized

in this thesis.

The experimental results have been il1ustrated b.Y a series of

figures and photographs; significant trends have been discussed.

Comparisons of. the experimental results with those from previous

works have been made and justification of the use of certain empirical

methods to approximate the maximum stress concentrations has been made.

Possible applications of the .results obtained to field conditions

are brieflydiscussed.

i

STRESSES IN TWO-DIMENSIONAL MODELS

OF ROOM-AND-PILLAR MINING SYSTEMS

By

Hyun-Ha Lee

A thesis submitted to the Faculty of

Graduate Studies and Research in

partial fulfi1ment of the

requirements for the degree of

Master of Engineering

Department of Mining Engineering

and Applied Geophysics

McGill University

Montreal, Canada

@) Hyun-Ha Lee 1969

April, 1969

ACKNOWDEDGEMENTS

The author wishes to express bis gratitude to Professor John

E. Udd, Director of tbis research, whose guidance, advice, interest

and encouragement were invaluable in carrying out this work.

The author also wishes to thank Mr. Van Thuy Ho, a student in

the DepartIDent of Mining Engineering and Applied Geophysics, for bi.s

assistance, and other colleagues for their various suggestions during

the progress of the research.

Messrs. H. Tidy, J. Karounis and D. Leishman, Technicians in

the Department of Mining Engineering and Appli.ed Geophysics, provided

much assistance in preparing templates for the models studied, and

the writer would like to express bis sincere appreciation.

ii

CONTENTS

SYMBOLS AND CONVENTIONS •••••••••••••••••••••••••••••••••••••••••• 1

~C~HAP __ ~TE~R~O_NE ______ I_NT~~R_O~D_U_C~T_I_O __ N •••••••••••••••••••••••••••••••••••••••

1.

2.

Review of Previous Work .................. ~ ........ . Summary of Previous Work •••••••••••••••••••••••••••

Nature of the study ••••••••••••••••••••••••••••••••

CHAPTER '!'WO - THE PHOTOELASTIC TECHNIQUE AND EXPERIMENTAL

PROCEDURES •••••••••••••••••••••••••••••••••••••••••

2

4

6

7

8

1. Theor,y of Photoelast1city •••••••••••••••••••••••••• 9

2. Treatment of Photoelastic Data

Assumptions made in the Study

• ••••••• G ••••••••••••

••••••••••••••••••••••

il

il

Separation of Principal stresses ••••••••••••••••••• 12

Determination of (p + Q) from Laplace's Equation •••• 12

). Model Material for Photoelastic Studies •••••••••••• 18

Models ••••••••••••••••••••••••••••••••••••••••••••• 19

4.

5.

The Photoelastic Bench

Experimental Procedure

•••••••••••••••••••••••••••••

• ••••••••••••••••••••••••••••

19

20

CHAPTER THREE - PRESENTATION OF RESULTS •••••••••••••••••••••••••• 22

CHAPTER FOUR - INTERPRETATION OF EXPERIMENTAL FINDINGS, SUMMARY

1.

AND BECOMMENDATIONS •••••••••••••••••••••••••••••••

First Series of Tests (Wo/Wp=l)

Vertical Loading (Sx=O, Sy:/:O)

Horizontal Loading (Sy=O, Sx~O)

•••••••••••••••••••••

•••••••••••••••••••

89

89

90

92

111

2. Second Series of Tests (Wo/Wp=3) •••••••••••••••• 92

Vertical Loading (Sx=O, SY10) ••••••••••••••••••• 93

Horizontal Loading (Sy=O, Sx:/:O) ••••••••••••••••• 93

3. Third Series of Tests ••••••••••••••••••••••••••• 95

Vertical Loading (Sx=O, SY10) ••••••••••••••••••• 97

Horizontal Loading (Sy=O, Sx:/:O) ••••••••••••••••• 97

4. Calculation of the l~ stress Concentratio~ •• 98

5. Summary of the Results •••••••••••••••••••••••••• 99

6. Recommendations for Further Study •••••••••••••• 101

BIBLIOGRAPHY •••••••••••••••••••••••••••••••••••••••••• c ••••••• 103

APPENDIX

Computer Programme

iv

\

SYMBOLS:

x, y, z

Sx

Sy

P

Q

Wo

Wp

A

R

K

SYMBOLS AND CONVENTIONS

Carte sian coordinates

Horizontal applied stress

Vertical applied stress

Major principal stress

Minor principal stress

Width of opening

Width of pillar

Distance from the edge of the plate to the center

of the outside opening

One half of the width of the outside opening

Maximum stress concentration in pillars

other s.ymbols used in the thesis are referred-to in the texte

CONVENTIONS:

Compressive stresses are denoted as positive; tensile stresses

are denoted as negative.

The principal strass difference, (P - Q), is alw~s positive.

1

CHAPTER .QM!

INTRODUCTION

The room-and-pillar method, in general use in the mining of

minerals, consists of alternately spaced rooms (or areas of extraction)

and pillars. The method is usually association with fIat or gently­

dipping bedded deposi ts. In planning this method i t is important to

de termine local stress concentrations induced qy' openings which are

adjacent to each other. A better knowledge of stress distribution in

mines is a prerequisite for improved mine design.

Many mining problems are directly concerned with concentrations

of stresses which may cause failures near mine openings as extraction

proceeds. In establishing sare stoping spans for room-and-pillar

methods there is, as yet, no substitute for experience. There are

advantages with wide spans, but, traditionally, spans have decreased

with depth largely because, with depth, conditions appear to demand a

more positive control. Certain spans could have been established as

acceptable through mining practice26•

There is much about pillar loading and pillar strength that we

do not know. If tr..:: loading of pillars could be estimated, much un­

certainty could be avoided, more ore could possibly be made immediately

available, and safety would be improved. DeterIllining the interna1ly

stressed condition of the pillars, however, is extremely difIicult.

The problem of accurately determining the stresses which exist in rocks

underground has long been of interest to Mining Engineers. An immediate

objective of stress analysis in relation to the analysis of underground

2

mine structures is an application of the results of these analyses to

achieve more economical mining operations4•

Within récent years photoelasticity has become an important tool

for mechanical and structural engineers, particularly with reference to

the design of complicated structures and parts in which high stress

concentrations are likely to cause failure. Mathematical treatments

of Many stress distribution problems are ver.y laborious, if not im­

possible, and the experimental technique has proved to be of great

value in these instances.

In this study, for the purpose of photoelastic stress determina­

tions, models of the prototypes were made from CR-39 plastic plates.

These models were placed in the optical path of a polariscope and

examined while being subjected to stresses thought to be similar to those

applied to the prototypes. The stress concentrations which were obtained

were then examined on a comparative basis.

It should be made clear that these model studies have been based

upon the assumption that the rock surrounding mine openings is "elastic"

and "isotropic" in character. None of the methods for solutions of

underground stresses explained all of the stress phenomena observed

because of the lack of accurate knowledge of the physical properties of

rock under field conditions and the great complexity of these con­

ditions due to inhomogeneity of the rock, geologic discontinuities, and

other factors.

In the case of simple ideal problems, a theoretical approach is

perhaps the most satisfactory since it provides an exact solution. In

most instances, however, underground openings do not have simple

boundaries. In most cases, a large number of openings is involved and

these are arranged in a manner which is difficul t to analyse mathematically.

It is in the approximate solution of these more difficult problems that

photoelasticity has its best applicationl •

In the early development of the technique, photoelastic analysis

was used for the determination of the distributions of stresses around

tunnels or shatts, or in pillars or arches. There are malliY' cases in

underground practice in which openings are located sufficiently close

to each other that the introduction of one opening affects the stress

concentration around the others. This condition is of primar,y importance

since stress concentrations are increased when two or more openings are

in close proximity. In the case of multiple openings, interest is

centered upon the point of maximum stress concentration in addition

to the stress distribution in pillars formed by th~ :Jmùtipl(? openings,

and the relationship between stress concentrations and size and shape

of pillars.

To date, there have been only a few solutions for problems

involving stress distributions around multiple openings. Solving

these problems by the theor,y of elasticity involves ver,y complex

equations (even for the Most simple geometl'ic shapes) thus~ only a few

have been made. The photoelastic method, however, lends itseif re~

to the solution of such complex problems.

1. Review of Previous Work

The first problem to have been considered was that of two

circular openings and the pillar between them. Ling1,17 attempted,

and solved this problem analytically and determined the critical

compressive stresses within"'the pillar (on horizontal diameters at

4

the boundar.y o~ the holes). He defined that the di~ferenoe between the

critical stress "st" (Sx=O) associated with a sma1l p/h ratio (pillar

width to opening height) and that associated with a large p/h ratio is

less than 1.

Duvall7,8,24,25 solved a similar problem, in whioh piates con-

taining two, three and rive circu1ar, and oval openings were subjected

to uniaxial stress fields applied perpendicular to the lins defined by

the centers of the series of openings. The stress conoentrations at

significant points were determined using photoelastic techniques.

Capper also studied the problem of three circular holes in a

plate subjected to a stress applied perpendicular to their line of

centersl • His solutions were obtained by photoelastic methods.

GreenlO solved similar problems using theoretical teohniques.

A theoretical stress distribution developed by Howland15 concel'!"..s

the case of an intinitely wide plate. containing an intinite row of

circular holes; the plate being subjected to a unitorm stress applied

perpendicular to the line defined by the centers of the holes.

Duvall's experimantal photoelastic ~sis of the stress

distribution for five circular holes in a plate was fourd to be in

close agreement with the results of Howlarid's theoretical study.

Panek27 performed an extensive series of experiments with plates

containing two rectangular openings wi th corner fillets of varying

radii. The ratio of the major to the minor axis, the ratio of the

radius of fillets to the minor axis, the angle of inclination of the

major axis with referenoe to the horizontal, and the ratio of the

pillar width to the minor axis, were all varied to determine stresses

around the openings. A photoelastic method of analysis was used.

In addition to the works mentioned, Obert2:3,24,25 am Merri1118,25

in the USA, Denkhaus5 and Hoek12,1:3 in South Africa, and Trumbachev and

Melinikov:31 in the ussa, have studied the subject.

Summarr of Previous Work

1) It has been shawn by investigators that the stress concentrations

induced by a number of openings are greater than those induced by

a single opening of a similargeometrical shape.

2) In general, stress concentrations increase with the addition of

each hole until there are about five holes in the plate. At that

stage, concentrations remain almost constant with the addition of

successive holes.

:3) It can be said that the stress conditions existing in the central

pillars approximate those existing in the pillars formed by an

infinite number of holes.

4) As the opening-to-pillar width ratio increases, the average stress

concentration in pillars increases at a fllster rate than does the

maximum stress concentration.

5) In most instances a tensile stress, of approximately the magnitude

of the applied stress, is produced in the roof and floor of the

model mine openings.

It is worthy of note, though, that few studies have been con­

ducted in which principal stress distributions around a series of

rectangular (or square) pillars and openings under uniaxial vertical

and horizontal loading conditions have been determined. Since rooms

6

'\

e being mined are generally rectangular in shape, the results obtained from

such a study will be of practical value (e.g. a determination of the

stresses acting upon pillars and openings).

2. Nature of the StudY

The objective of the study was to obtain information concerning

the stress distributions in pillars formed by two or more rectangular

mine openings subjected to uniaxial stress fields. Relationships

between maximum stress concentrations and 3ize and shape of openings

are developed. When appl1ed to mining, the results serve as a guide to

the development of the stress distributions in pillars, and the deter­

mination of factors causing high local stresses.

A two-dimensional photoelastic method was used to study the

stress distributions. Detailed descriptions of the method and equip­

ment used are given in two previous theses20,30•

7

CHAPTER 1l:lQ

THE PHOTOELASTIC TECHNIQUE AND EXPERIMENTAL PROCEDURES

The photoelastic method of model testing was used for the

following reasons32:

1 ) wi th this method i t is possible to obtain an entire

picture of the stress distribution throughout the body;

2) it is possible to determine the stresses with the sarne

order of accuracy in al1 shapes of models, even in

irregular ones;

3) the principal stresses and maximum stress can be l!!)eated

qualitatively.

The problem of determining stresses in the structural components

of a mine is complicated by the heterogeneity of the rock, the irregu­

larity of the boundaries of individual openings and the complexity of

the system of openings. These complications necessitate the following

simplifying assumptions to achieve a solution1,7,8:

1 ) The distance from the exterior boundary to the system of

openings must be large in comparison to the size of the

openillgS.

2) The rock is homogeneous, and isotropie (i.e. its elastic

properties are the sarne in all directions). This

assumption is justified when the openings are created

at a great depth in the earth and the mate rials are not

stratified.

3) The stress distribution along the length of the openings

8

1s uniform, and indepement of the length. This is true

when the length 1s great in oomparison to the height am

width of the opening. Henee, the st~ of a cross-seotion

of the openings gives us results with negligible error due

to negleeting the stress in third dimension.

4) The stresses along the length of the opening are assumed

to be uniforme

5) The oross-seetional shape of the openings may be repre­

sented by one of several geometric figures, sueh as a

oircle, ellipse.

1. Theoty of Photoelastioity

The theor,y of photoelastioity may be obtained from aqy standard

book on the subjeot3,6,9,1l,16. However, a ooncise summar,y of the

principles is:

a) If light is passed through a stressed plate made of an

isotropie, elastic and transparent material such as glass,

and in.direotion normal to the plane of stress, it is

polarized into two components in the plane of stress,

parallel and perperrlioular to the prinoipaL stress direc­

tions at eaeh point in the plate.

b) Each wave component is retlœded proportionally to the

magnitude of the stress aoting in that direction.

c) Retardation at aqy point is proportional to (p - Q), the

prinoipal stress differenoe. The directions of the prin­

oipal stresses at eaoh point and the direotion of the

optio axes are the sarna.

9

d) The retarded wave components may be ccmbined to produce

total extinction of the light when:

(1) the principal stress difference is zero or is

such as to produce a relative retardation

equal to an integral number of wave-lengths

of the light used. These lines are called

"isochromatics" or "fringes".

(2) the directions of the principal stresses at a

point are parallel to the optical axes of the

polariscope (polarizer and analyser). The

loci of su~h points are called tlisoclinics".

The mathematioal formulation of the "Stress-Optic Law" in two-

dimensions at normal incidence of light is given b,y:

R = r1 - r2 = C (P - Q) d

À N.f P - Q = N ë:d = d =N.F

•••••••••••••••••••••••••••••••

where:

••••••••••••••••••••••••••••••••

R = Relative retardation

r1= Retardation of wave parallel to the direction of

major principal stress, P

r2= Retardation of wave perpendicular to the direction

of minor principal stress, Q

(1)

(2)

P - Q = Principal stress difference (two-dimensional model)

N = Fringe order

À = Wave length of light waves (Angstrom)

C = Stress-optical coefficient of the model material

(Brewster = 10=13cm2/dyne)

10

d = Thickness of the material (inch)

f = Material fringe value ( lb x inch

inch2 Fringe order

lb

)

F = Model fringe value ( inch2 x Fringe order

2. Treatment of Photoelastic Data

Assumptions made in the stu4y19,20:

)

1) The "pillar"formed between the two rectangular openings

in the plate is analogous to a pillar between two stopes

in a mine. Furthermore, it is assumed that the stopes

are infinitely long in a direction normal to the plane

of the section, hence, a study of a cross-section provides

results with negligible error due to ignoring the stress

in the third dimension.

2) The study is therefore of pillard in plane stress con-

ditions. The assumption ignores any effects caused b,y

the extrusion of the Teflon load-equalizers (used to

achieve uniformly distributed loads on the models) in a

direction perpendicular to the plane of the plate. It is

believed that these effects, which would be confined to

the edges of the model, are very slight.

3) No buckling under load takes place since loading is

uniform along the third dimension perpendicular to the

plane of the plate.

11

Separation of the Principal Stresses:

To analyse the stress distribution in a plate under load, it is

necessar,y that the magnitude of the stresses be determined at ever.y

point. The stress at point "0" in a structure (Fig. 1) may be resolved

into major and minor principal stresses, "p" and "Q", (each pure com­

pression or tension) at right angles to each other. The value of shear

stress at any point in a plate, related directly to the principal stress

difference, CP - Q) is determined from the fringe photographs29•

In usual practice, a separation of stresses refers·ta individual

determlnations of "P" am "Q".

An iterative process of solving Laplace's Equation was used in

this work for determining the sum of the principal stresses. The

principal stresses "Pli and "Q" were resolved by adding am subtracting

the photoelastically-determined values of (P - Q) aM values calculated

for CP + Q)3,30•

Determination of (p + Q) from Laplace's Eguation9,14,16:

It was found that, for a plate subjected to a plane stress

system, the sum of the principal stresses at any point May be expressed

qy Laplace's differential equation, that is:

where S = P + Q

The function S(x,y) can be expanded into two Taylor's series:

- along the X-axis

12

Sy Sy

! ,il , , , \

Sx=O

~p P Q 0

~ DO -_~ Q=O

t Q-A--pbo

Sx=O

P

• ---Q t if. " r r S 1 ~

1 Sy

Fig. 1 Principal stresses "pH and "Q" in a model

1

--l------j 1 a 1 1 1

1 1 1 1 1 1 1 ---ï------1

y

l4--a~ 1 1 1 1 1 1

D[o-;:al- -r --

Fig. 2 Four surrourxiing equidistant points

13

14

1- S = So + (x - xo) s!§. + ..!. (x - xo)2 ~ + ..!. (x - Xo)3 ~ dx 21 dx~ 31 QXJ

1 4 d4S + 4i (x - Xo) ~ ••••••••••••••••••••••••••••• (3a)

- along the Y-axis

_ ( ) dS 1 ( )2 d2S 1 ( )3 d3s S - So + 1 - 10 dY + 2ï 1 - 10 d?' + 3i 1 - 10 dy)

1 ( )4 d4S ( ) + 4i y ~ Yo ~ .0........................... 3b

If we put x = ta an:i y = :a (Fig. 2), where "a" is sufficiently small,

we obtain;

along the X-axis (xo = 0)

•••••••••••••••••• (4)

Therefore;

•••••••••••••••••••••••••••• (4a)

Similarly, alo:l'lg the Y-axis (Yo = 0)

SB + S = 2S + a2 d2S D 0 dy2 ••••••••••••••••••••••••••••• (4b)

Together,

So = t (SA + SB + Sc + Sn) ••••••••••••••••••••••••• (5)

From this equation, it is seen that the value of the function,

S = P + Q, at every point in the field must be equal to the average of

the values at four surrounding equidistant points located as shawn in

Figure 3.

In general, i1' the spacings are not equal (Fig. 4), a similar

but more complex expression can be derived. This has the 1'orm

1 ~ ~ ~ ~ So = -1 + -1 (a(a+c) + b(b+d) + c(c+a) + d(d+b) ) •••••• (6)

ac bd

1 Il 1

A o c: II»

t-~

Fig. J A typical gridnet work

on a simple model (not

to scale).

1 B -r------ f---------1 b 1 1

A~I--------~~--~~~ t--- a---. 1 1 1 1

Fig. 4 Grid spacing near a curved

boundar.y. For convenience,

a is usually equal to b.

It is known that at a boundar.y there is only one principal stress

acting on a point (since the stress normal to the boundar.y must be zero),

so that, the sum 01' the principal stresses, (p + Q), at such a point is

numerically equal to their di1'1'erence, (P - Q). Values of (p - Q) are

obtained directly trom the isochromatic pattern. At ever.y grid point on

the boundaries (exterior and interior), the values 01' Sare known. From

these values, trial values 01' S at all grid points wi thin the boundaries

can be calculated successively b,y using Equation (5). The process

is iterated in a de1'inite sequence until the values of S become

15

stationar,y, i.e. the difference between the values of S of the last

iteration and those obtained in the previous iteration is negligible.

In this study, the acceptable difference was set at 0.001 of a fringe

unit.

The accurae,y of the results obtained from this method depends

upon30 :

1) The accuracy of the boundary values. AB noted, at points

on a free bouniary, (p + Q) is numerically equal to

(P - Q). In the case of a loaded boundary, this statement

is correct only when a uniform stress is applied nor.mally

to the boundary. In order to obtain uniform loads on the

models in this study, Teflon linings were inserted between

the mode1s and the platens of the loading frame.

2) An incorrect sign attached to (P + Q) at boundaries is

also a source of errors. For models of irregular shape

under non-uniform distributions of load (e.g. the study

of stress distribution in rock slopes, and in other con­

struction materials), the signs of (p + Q) are dii'ficu1t to

determine. In this study, since the openings are of square

and rectangular shapes, the signa of (P + Q) at their bound­

aries are clearly indicated. Negative signs are usually

given to (P + Q) acting in directions perpendicular to the

direction of uniaxial loads applied on the boundaries (de­

pending upon the locations of zero fringes), while other

values of (p + Q) are taken as being positive. (Note that,

in agreement with the convention generally used in photo­

elasticity, compressive stress is taken as being positive

16

and tensile stress as being negative.)

3) The accuracy of (p - Q) values within the model. Errors

are produced in estimating by interpolation the fractional

fringe orders at gr1d points. Additionally, the presence

of some residual stresses introduced by machining the

model also contributes to inaccuracies.

4) Overall accuracy also depends on the fineness of the

grid net work. In equations (3 and 4), the fifth terms

can be eliminated only when the spacing constant "a" 1s

sufficiently small. In the cases described herein, a

spacing of 1/8 in. was used; finer spacings could not

be used since the computer could not handle more than

a certain number of grid points (for reasons of memor.y

capacity).

The separation of the principal stresses by this method was

carried out on an IBM .360 Computer, using a progr~e developed by

Prof. J. E. Udd, MCGill University: some minor modifications were

made to the programme to adapt it to the author's requirements20,21.

The entire area of each model was evenly-di vided by 49 horizontal and

49 vertical lines at spacings of 1/8 in., or, by 65 horizontal and 65

vertical lines at spacings of .3/32 in. Since the capacity of the

computer was limited, input data for CP - Q) were taken only at grid

points within boundaries def1ned by the l3th and 36th vertical lines.

The computer programme is best illustrated by an example and 15 shown

in the Appel'ldix.

17

Model Material for Photoelastic Studies

Columbia Resin 39 was used as the model mate ri al in this

investigation because of its easy machinability, lack of inherent

stress and high transparency. Strain-creep is quite pronounced in

CR-39 and it is not possible to give ~ fixed stress-strain relation-

ship or modulus of elasticity without introducing the time factor.

Within the usual time limits of laboratory testing, though, it can be

said that the stress-strain curve for CR-39 1s linear to about 3000 psi

and th~~ the modulus of elasticity is about 250,000 psi30•

The properties of CR-39 are listed below:

Young's modulus, E; 3 x 105 psi

Poisson's Ratio,...u; 0.42

Strength (tensile); 6000 psi

Proportional limit; 3000 psi

Fringe Constant, f; 84 Lbs/in./fringe of principal stress difference

-6 / /0 Coefficient of Thermal Expansion; 72 x 10 ii:'ll., in. F

For a CR-39 sheet of tin. thickness, 333 psi is required to

produce one fringe. In order to obtain the magnitude of the stress

(in psi) at a~ point in the model, then, the fringe value for that

point must be multiplied b.Y 333.

Below the stress at the yield point, the material has a slight

optical creep over a period of time. This is true, however, only if

the load is maintained for times exceeding 15 minutes. Such not being

the case in these studies, however, no attention has been paid to the

affects of optical creep on the results presented. Since CR-39 also

develops time-edge stresses if allowed to age, the models were tested

18

and all measurements were taken the 5ame d~ they were prepared. It

must ba noted that soma models eut for the first series of tests were

used for the second series of tests since time-edge stresses were not

observed; That is perhaps due to the models having baen kept at fixed

humidity. The time-edge effect is normally associated with the hydro­

scopie properties of the model material.

Models

The size of the models was t in. x 6 in. x 6 in. In each case a

master template of the desired model was made from a thin sheet of alumin­

ium. Using the template as a guide, holes were first drilled in the plas­

tic plate using a standard banch-mounted electric drill with hardened bits,

and were then machinsd to final dimensions with a standard high-speed side­

milling machine. Great care was taken to avoid chipping and the production

of machining stresses in the model. Grid lines at 1/8 in. and 3/32 in.

spacings were then marked on the models. Models were thoroughly cleaned

bafore being inserted into the loading frame.

As noted, testing was completed within 15 minutes in order to

minimize optical creep and the time-edge effect20,30•

4. The Photoelastic Bench

A standard photoelastic bench, with rail-mounted components, was

used for this study2°,30• The polarizer and analyzer were graduated

in degrees and were capable of being potated about an axis conforming

with the optical path. Perpendicular tracks on which the quarter wave

plates could be moved in and out of the optical path were a design

feature of the instrument.

19

The biaxial loading frame used was connected to two ~aulic

pumps; one for applying loads in a horizontal direction and the other

for providing a vertioal component. The sides against which the model

rested were lined with Teflon.

5. Experimental Procedure

1) Models were placed in the loading frame &~ were adjusted

for proper orientation at right angles to the light beam.

2) Adjustments ta loading were made to obta1n symmetrical

stress patterns.

3) In order to minimize optical creep and deformation, the lowest

practical loads were used and suffi cie nt time was allowed to

elapse after loading to permit the rate of creep to diminish.

The use of relatively small loads also reduced deformations9•

4) When i t was found that a satisfactory stress pattern had

been produced, a first photograph of the isochromatics was

taken. The field of illumination was then changed to a

light background by rotating the analyzer through 900, and

a second photograph was taken.

In this research, three series of tests were conducted on models

designed to simulate various ratios of opening width ta pillar width

with the same height of pillars (Fig. 5 through 38).

The types of openings considered here were squares and rectangles

and the ratios of opening width to pillar width used were 1.0, 3.0, 7.0,

11.0, 15.0 and 19.0. Opening width to pillar width was varied to

determined the nature of variations of stress concentrations.

In order ta determine possible influenoes of the nearness of the

20

\ l'

boundary of the models upon stress concentrations, an additional test

was designed to determine wh ether or not the edge of the plate would

affect the fringe pattern around openings. Before performing the

experiment, ratios of the distance from the edge of the plate to the

center of the outside hole, (A), to the haif-width of outside openings,

(R), were investigated.

The A/R ratio of aIl models in the first and second series of tests

(with the exception of a value of 2.7 in the 6 pillar case of the second

series) was greater than 4.0, the minimum ratio which Duvall recommended

as a result of his research8; It ___ if the plate width was such that the

distance from the edge of the pl"ate to the center of the outside opening

was equal to or greater than four times the radius of the outside open­

ing, the effect of the opening could be neglected".

In the third series of tests, the A/R ratio varied from 2.) to

1.5, both of which being less than the recommended ratio of 4. In order

to confirm that edge-effects were of minimal influence a further series

of tests was designed. Two models, inwhich A/R were 4.6 and 2.)(being

comparable in terms of number of openings and number of pi1lars wi th

those models for which A/R was previously 2.) and 1.5 respectively), were

cut and stress concentrations were compared with those from the previous

tests. The results showed very close agreement with the results from

the previous tests (third series) and thus, it was concluded that even

if openings in a plate were made so that the A/R ratio was less than 4.0,

the edge-effect of the plate upon the stress concentration around the

openings could be minimized under conditions of low load.

21

CHAPTER THREE

PRESENTATION OF RESULTS

As mentioned, three series of tests were corducted to study the

stress distributions in, aM around., mine pillars.

Results are presented in terms of extraction ratio and the ratio

between opening width and pillar width, (Wo/Wp), for each case (Fig. 60).

Some isochromatic fringe photographs are shown in photographs 1

through 16. The attention of the reader is drawn to the progressive

deve10pment of the fringe patterns.

Points of particular interest where the stress concentrations

were determined are 1abe1ed A, B, C, D, E aM F as shown on the "key"

given in Figures 39 through 55.

The stress distributions in and around the pillars am openings,

before and after cutting, are depicted in Figures 5 through 38. The

contours in these figures are expressed in terms of concentrations of

applied stress (which are dimensionless).

It should be noted that photoe1astic results are more convenient1y

expressed in terms of stress concentrations28•

Significant cross-sections for each case are shown in Figures 39

through 55.

Conventions used in the figures are;

1) loads are vertical and horizontal,

2) compressive stresses are positive, and tensi1e stressas

are negative,

3) the principal stress difference, (P - ~), is always

22

2.3

positive,

4) the heights of rectangular openings are maintained

constant throughout the series of tests.

t· .f"t Photograph 1. ,Isoobromatic Fringesaround 2opemngs(1 pillar),

Rati<t wo/Tllp=1, Vertical:. uni axial . Loading.

Photograph 2. Isochromatic Fringes around 2 openings (1 pUlar), . -

Ratio WO/Wp=l,Horizontal Uniaxial Loading.

Il

t

~ . 1

.. ~\ i ,

Î t t t t t t t t t Photograph 1. Isochromatic Fringes around 2 openings (1 pillar),

Ratio WO/Wp=l, Vertical Uniaxial Loading.

Photograph 2. Isochromatic Fringes around 2 openings (1 pillar),

Ratio WO/Wp=l, Horizontal Uniaxial Loading.

24

t

"- ,".:

Photograph 3. ~sochromatic F'riliges ar01U'Xi 4 o~~s (3p1llarâ),

Ra~o' :Wo/Wp=1, Vertical Ul:daxial Loading.

Photogl"'à:ph 4. Isochromatic Fringes around. 4 openings (3 pillars),

Ratio WO/Wp=1, Horizontal Uniaxial Loadirig.

li

(

t

.J; .'

t Î t t t t t t t t t Photograph 3. Isochromatic Fringes around 4 openings (3 pillars),

Ratio WojWp=l, Vertical Uniaxial Loading.

Photograph 4. Isochromatic Fringes around 4 openings (3 pillars),

Ratio \rJo lWp=l; Horizontal Uniaxial Loading.

25

t

1 ! :

i r 1 1 1

1

1

r r J 1 "

" , ,

~

t i ,

" "

~;B1Otogr~ph 5. Isochromatic Fringes aroundc 6' openiDglr(S pHlars}.

Ratio WolWp=1,Vertical. Uniaxtal Loading.

----,-.~-...,-- ----:--.,,---, ,-,'---"'--'~"',-"-"--' -,~-:,,:'---"

~ -.......

. __ ....... - . '--"'---

Photograph 6. Isochromatic Fringes around 6 openings (5 pUlars),

Ratio WO/Wp=1, Horizontal Uniaxial Loading.

'26

•

<tb

~

-

t

~ ~ ~

t t t t t t Photograph 5. Is ochromatic Fringes around. 6 openings (5 pillars),

Ratio WO/Wp=1, Vertical Uniaxial Loading.

Photograph 6. Isochromatic Fringes around 6 openings (5 pillars), '"

Ratio Wo/Wp=1~ Horizontal Uniaxial Loading.

26

•

t t .f t t. t .t t t Photograph 7. Isochromatic Fringes around 2 openings (1 pillar),

Ratio WO/Wp=3, Vertical Uni axial LÔading.

~_.

Photograph 8. Isochromatic Fringes around 2 openings (1 pillar),

Ratio WO/Wp=3, Horizontal Uniaxial Loading.

t

27.

\

t t t t t t t t t Photograph 7. Isochromatic Fringes around 2 openings (1 pillar),

Ratio WO/Wp=J, Vertical Uniaxial Loading.

Photograph 8. Isochromatic Fringes around 2 openings (1 pillar),

Ratio wojWp=J,·Horizontal Uniaxial Loading.

27

"~~ 1

.... "

1

1 1 1 1

1 1

1

1

1

~.

,

~I 1

1 1 ,

~! 1

Photograph 9. IsochromaticFril'1ges around 4 opeDings (3 piUars),

Ratio WoMp=J, Vertical ·UniaxiaJ. Loading.

Photograph 10. IsochI'anatic Fringes around 4 openings (J pillars),

Ratio WolWp=J, Horizontal Uniaxial Loading.

28 !

~.

l'

t' -~ j"" 't'. E.

: ~,. , *'

~

~

, , ~

\

~

.~

~

~

'i ,'\ y'

® i!()

~ r·

J ~

t t Photograph 9. Isochromatic Fringes around 4 openings (3 pillars),

Ratio WO/Wp=3, Vertical Uniaxial Loading.

Photograph 10. Isochromatic Fringes around 4 openings (3 pillars),

Ratio Wo/Wp=3, Horizontal Uniaxial Loading.

28

(.;:~) J. •

Photograph 11. Isochromatic Fringes around 6 openings (5 pillars),

Ratio Wofwp=3, Vertical Uniaxial Loading.

Photograph 12. Isochromatic Fringes around 6 openings (5 pillars),

Ratio WojWp=3, Horizontal Uniaxial Loading.

29

!

~ . :

, { ,i,'

j .. ' .... ~ "

. ,. ,- ... . ' .. .

_Photo~aph '13. Isochromatic, Fringes alter, the Extrs.ct10n of 1 p1llar,

Ratio.Wo/Wp='t\'for theCNtermost '.6Pening onthelett;

, Vèrticai Uniaxt&l Loading.

Photograph 14. Isochromat1c Fringes after the Extraction of 1 p1llar, . .-Ratio wà7Wp=7 for the outermost opening on the'left,

Horizontal Uni~al Loading.

ft_ I."t,,-,'

J ® ~~

J ~ ~ t t t t

t t t t t t t t Photograph 13. Isochromatic Fringes after the Extraction of 1 pi11ar,

Ratio WojWp=7 for the outermost opening on the 1eft,

Vertical Uniaxial Loading.

Photograph 14. Isochromatic Fringes after the Extraction of 1 pi11ar,

) Ratio WojWp=7 for the outermost opening on the 1eft,

Horizontal Uniaxial Loading.

30

~

t

i .

l~

' ...... ~ , ~ .

. < .. l" l

·1

1 1

r

. .lt. • ..' _ . . . _. ~>. . . _ ••........... _ . ...J PhCJt02!".liL1)h '15 •. Isochromatic:FriDges iller the:~actioD ot');p:1l.lU'~f

.,":

1 :

.Photograph 16.

bti,owo/Wp=15: f~r .;~out8mc)~:t' OP.ni1'1goD the 1eft,

o Vertic~ Un1~al Loading.· ,

Isochromatic Fringes after the Extraction of J pillars,

Ratio WO/Wp=15 for the outermost opening on the 1eft,

Vertical Uniaxia;I. Loading.

i 1

'E-I 1

·:.~l· , .

!

t f Photograph 15.

Photograph 16.

Î Î Î Î Isochromatic Fringes after the Extraction of 3 pi11ars,

;latio WO/Wp=15 for the outermost opening on the 1eft,

Vertical Uniaxial Loading.

Isochromatic Fringes after the Extraction of J pi11ars,

:fatio '.vojltip=15 for the outerrnost opening on the 1eft,

1ertical Uniaxia1 Loading.

31

! ! i 1·

i i,

i

1 1

i 1. i i

i: l \ 1 ~

1 1., 1:,1 l,:

" ,p ~ . ,) 1 î ~ , , Î: , ! i 1 , i 1 . 1 ~

f

- s- • t i * : * t ~ t t t t---~7 -\r----,'"'t< ,v

!-i t:'

,...0.50

0.7S ____

P stress Concentration

Q Stress Concentration

q~

"o~s

r- ': : 1 ~~ ~ ~ ~ ~ ~

Fig.S Contours of P and Q Stress Concentrations around 2 openings (1 pUlar)

Ratio Wo/Wp=l, Vertical Uniarlal Loading.

~

\.0.), N

b 1 b /;1 l· 1 f J. H ry ; e 1

~-~-- -------- ,"\.., ~---- --- ---- ----------- "V ----~----- ---I-E-r------r-----------------------------?I~· ~--------~~------------------------~

a "

r

P stress Concentration

__ 1.0__

~ Stress Concentration

\~

( r

--?1 IV ,1; , ~

Fig. 6 Contours or P and Q stress Concentrations around 2 openings (1 pUlar)

Ratio Wo/Wp=l, Horizontal Uniaxial Loading.

.~

~

~

e ( ) 6( l c:..pJ v e

t w t : t t .: t } t t-----....... 'vr--------t

P Stress Concentration

Q Stress Concentration

r K JK ': .JK . Ii : A\ JK . 1 Fig. 7 Contours ot P and Q stress Concentrations &round .3 openings (2 pillars)

Ratio Wo/Wp=l, Vertical Uniaxial Lo'ading.

~

~

o )Cir lLfloo e

-4 'V ~~ u~_~._~ UU}E-l ,"vi 'v 1

P stress Conoentration .- I.()O __ - ____ _

Q stress Concentration

~ >'V /V.. ~

Fig. 8 Contours ot P and Q Stress Concentration arourd 3 openings (2 pillars)

Ratio Wo/Wp=l, Horizontal Uniax:1al Loading.

'-.N

~

~

e 'i , oS 0y Cb.-V) e

t { {. : { t : t t t t-------.I'\.' >V..---------i

P stress Concentration

-o.SO

Q stress Concentration

.r JK ... ': ... Ji .: A X ·1 Fig. 9 Contours of P am Q Stress Concentration aroum 4 operdngs (3 p1llars)

Ratio· Wo/Wp=l, Vertical Uniaxial Loacling.

~ 0'"'

\,.,) .c-.

f - U'()v (r~ \' ! \' 1 \ e

'-1"'" ~

~-~ - .'V' ~. fE:--!---------------",..". .... 1"

P stress Concentration

- 1.0 .....

~1.5

.E-

a.U

-o,so

Q Stress Concentration

~ /v I~ k--

Fig. 10 Contours of P and Q stress Concentration around4 openings (3 pi:LlaI·s)

Ratio Wo/Wp=l, Horizontal Uniaxial Loading.

\.t.) "'-l

.' . -J" 'r v e •

t J ~ : ! ~ : ! ! t 1 • ~ ~ 1

p stress Concentration

(,0

____ o~

Q Stress Concentration

f TT· f t t ft·· f, ;. t Fig. 11 Contours or P am Q Stress Concentration aroum 5 openings (4 pillârs)

Ratio Wo/Wp=l, Vertical Uniaxial Loading.

\JÇ 0<;

~

S' (u . - 1 e

~ N .~ ~ . . ..

P Stress Concentration

---·/,0

_------ '·0 __

·\'o~

:;?'$" ç"S" ~.o i·o ?o()

l.'iP . 2.'10 O'S ~~

o

-0.5 -0:5"'.

Q Stress Concentration

~ 1f 11 jE-

Fig. 12 Contours of· P and Q Stress Concentration around 5 openings (4'pillars)

Ratio Wo/Wp=l, Horizontal Uniaxial Loading.

.'-/

'1 _

.v

\..) \0

• V '--JI v

+ J, 4 : .J, .J, t ~ ~ ··t tu "v -v ,

P stress Concentration

,0'5 ~o·S_

Q Stress Concentration

f l' t f f l' t t .1' .. 1 Fig. 13 Contours of P am Q Stress Concentration aroum 6 openings (5 pillars)

Ratio Wo/Wp=l, Vertical Uniaxial Loading.

e /? è)

g

u '-./ J

" LI e

~ ~ ~~~ _~ _________ n~_n __ ~ _ ~_n __ ~

P stress Concentration

----~--I.o ------/ --~------------------

"'-- _ _ -D.,~'- . ,0·$

, 0''> ~'

o. Q stress Concentration

~ .... v lE--

Fig. 14 Contours ot P am. Q. stress Concentration aroun.:l 6 openings (5 pil1ars)

Ratio WoMp=l. Horizontal. Uniax1al Loading.

((P~

:r

., " 1 . \.. / .-t ' ~ .~ k ~ ~ k, L t

. . . -

P Stress Concentration

-D:5"""--__

Q Stress Concentration.

f t 11 1 t t ft + Fig. 15 Contours of P and Q Stress Concentration around 7 openings (6 pillars)

~atio Wo/Wp=l, Vertical Gniaxial Loading.

/' ç'

( (b) ( r-I -

P stress Concentration

~------______ --------o~s ----~

(.0 "'()~"" ______ :-____ _

G.,

0'0 ~_ •. ,s

#0.2' __ -0·9 - ___ -0'$

Q Stress Concentration

-;.J ~J -r F--

Fig. 16 Contours of P and. Q Stress Concentration arouM 7 openings (6 pUlars)

Ratio Wo/Wp=l.. Horizontal Uniaxial Loading.

I~

~

• ~( l) v e

r t '" : t t ~ i .j, 1 J--------..... .'l.-' ------1

P stress Concentration

------- ---r- .J----r---!.!.~

r x • ; 1 X : x 1 . ·1 Fig. 17 . Contours of P and Q stress Concentrations arouM 2 openings (1 pillar)

Ratio Wo/Wp=3, Vertical Uniarlal Loading.

./~

f:

~"- \ / \'\ e .{ v~.

4 'V' /v --- _u -:--~

1 .V IV 1

P stress Concentration

~ o' ,

QStress Concentration .0,<.0

~ ~ ~ ~

Fig. 18 Contours of P am Q stress Concentrations aroum 2 openings (1 pillar)

Ratio WoMp=3, Horiz·ontal Uniaxial Loading.

&

,"

" \

'.:

·~ ( 1.-) '..-J _' /

'"

v ;'ie ."

t . . t t ~. t·· * .'. !. i i ~ r IV , IV 7 1

U z,o

~-=-~ ",~--,·':':"'-"-:'~n-J '. . -ur

P Stress Concentration

_._-~ .. -.

-o,U ".'.

.... -

Q Stress Concentration

"~ · . t • . t t t t·· f' tA l "Fig. 19 Contours of P am Q Stress Concentrations arourd 3 openii2gs (2 pillars) "

: ~ ,Ratio Wo/Wp=3, Vertical Uniaxlal'Load1ng.

lï~:

-'

t

"

.. ,

1

".)LV 1 1, \ ' .• :~- .. .,. ,v>~

.\

"~I 'le .~< ~ 1 .... • .... ' l '

P Stress Concentration

1.0

1.5------ J

':, t

,~I

: --:;. -Oost» ~.JS

·0 •• ,

Q Stress Concentration .

:,,~ .... ,... k:' . . .- : ~ .

. "

Fig. 20 Contours or P and Q St-ress Conce~trations around)' openings (2 pUlars) .

Ratio WoMp='J.i Horizontal Uniax:l.al Loading.·

.'~

- ( 1 1") \

-l' l '~I ) v e

t { { ~ { }: { .j, t ç ".,' ." ... ' j il

P Stress Concentration

---1.'5

Q Stress Concentration

r x x : 1 x : ~ x xl

Fig. 21 Contours of P and Q Stress Concentrations around. 4 openings (3 plllars)

Ratio Wo/Wp=3, Vertical Uniax1al Loading.

~~

~

• 'e

~ ~.' .~,----- ~ 1 .1/ Iv 1

P stress Concentration ~o hO

-O.2S' ___________ _

.-~,s~."\ Q Stress Concéntration

~

Fig. 22 Contours of P and Q Stress Concentrations around 4 openings· (3 pillars)

Ratio Wo/Wp=3, Horizontal Uniaxial Loading.

.k-

... -......

$

'-? '-- 1 v -e:

t ~ ~ . ~. { , . ~. , ~ t

A { ... :)

P Stress Concentration

Q Stress Concentration

~ '~~J

+ l' t {. t t .;: '1' .... '1' t Fig. 23 Contours of P and Q stress Concentrations aroum 5 openings (4 pillars)

Ratio Wo/Wp=3, Vertical Uniaxial Loading.

'- ..Je::

\.n o

Ct 7\ 1/ ._, v: \.\ e

~ 11 '" 1<-P stress Concentration

/·0 -------....

- -------. \,s ----___ ----- "5'

-----200 ......

Q Stress Concentration

Fig. 24 Contours of P and Q Stress Concentrations aroun:i .5 openings (4pillars)

Ratio WoMp=3. Horizontal Uniaxial Loading.

Q..-,...::..I

\.n ......

. -

b C-S- J 'V

e

t { { . ~. { !. ~. !.., t 1 ( ~< ,v , 1

PStress Concentration

QStress-Concentration

+ ft 1 t f t f t + Fig. 25 Contours or P an::l Q' Stress Concentrations around 6 openings (5 pillars)

Ratio Wo/Wp=3, Vertical Uniaxial Loading •

z..5~ .~

'iG

o \.....; / fil e U'-.JJ

4 ',," ______ n_ ------ _n ,''''' -JEE--1 IV IV 1

1O P Stress Concentration \.0

o HS

0') {'

,~

Q stress Concentration

~ ~ ~ ~

Fig. 26 Contours of: P am Q stress Concentrations around 60penings (5 pillara)

Ratio Wo/Wp~3, Horizontal Uniaxial. Loading.

\.n '-'>

rzLb / v e

t i { ~ i i ~, t t J li 'v '!o' J 1

P Stress Conoentration

Q stress Concentration

r ~ ~ ; x '" : '" ( '" =l 'Fig. 27 Contours of P and Q Stress Conoentrations around 7 openings (6 pillars)

Ratio WoMp=3t Vertical Uniaxial Loading.

2*

':S-

e - j. \ j,

'2~1 1) , ; J ',! . 1,_ tJ ; .. ~ fe

~-_ .. -

1 =====~;;;;;::tc ~~~........ ==---. '" _-========~IE-1

P Stress Concentration

----_l'O~ --...... ~I'O

,.!J

, 1 ~ __ r·/ ) ( 0'0-----/ ___ 0. 0 -

-/l'ilS

-D·~S _________ _

------- -D'S

,o·s Q Stress Concentration

--?1 .'V' . /V ~

Fig. 28 Contours 01' P am Q Stress Concentrations arouni 7 openings (6 pillars)

Ratio Wo/Wp=3, Horizontal Uniaxial Loadingo

~

',' ~

" " v' ·:e '.: ..

f { .~ :, . { { :, ,j, ,J, t 1 l 'V .'l.' 1

P Stress Concentratiçn

Q Stress Concentration

l ~ : . l x 1: 1 .... li: .AC :

Fig. 29 Contours of P and Q Stress Concentrations after the Extraotionof 1 pillar

Ratio WoÎWp=7 for the outermost· opening on the left, Vertioal Uniaxial Loa~ing.

·i7~ j

V\

'"

((: ) e

~. --- ---.--- -------~ "V ~

L ~ ~ \ 1

P Stress Concentration LO

____ i·O 1,0_

'.0--

-l -0.7S,

.0.1$

Q stress Concentration

-?1 l'V' . /V k--

Fig. 30 Contours of P and Q Stress Concentrations after the Extraction of 1 pillar

Ratio Wo/Wp=7 for the outermost opening on the left, Horizontal Uniaxial Loading.

.. $0":.

Vl -.;)

..

... ) !< '. /1 :···e

t ~ { ~ { { ~ . . ~ ~ t 1 1 \ \ aI"" 1 >v _ 7 1

1·11 "I.-SO

r.-o ;.r':::::7! ;;:- i:j's .o-,s

... ,s ;:;-

Q

P Stress Concentration

1.0 .

_.L-

Q stress Concentration

l iK À ': ïK x·: JK . Jf\ 1 Fig! 31 Contours of P and Q Stress Concentrations after the Extraction of 2.pillars

Ratio Wo/Wp=ll for the outermost opening on the left, Vertical Uniaxial Loading.

~)~

\.J\ ex>

fi ,

/( '_ " l e.·

~--- - ----------- ------ ---,'V< ------------- ~, ~

< ~ ~ < 1

P stress Concentration 1.0.

_'.0---- ./I.D~ -_~. ~,.o lcE-

- ~ ".~ .. r./ ~~

__ :.1$- -

Q Stress Concentration

--?1 ;,v.'V ~

~ig. 32 Contours of P and Q Stress Concentrations after th~ Extraction of 2 pillars

Ratio Wo/Wp=ll for the outermost opening on the left, Horizontal Uniaxial Loading.

>~

\J\ \0

{ ' l' . " '. J ··e

t * ,), ~ -], ~ . [' , , ,: , 7~ ,), ,), ·t

P Stress. Concentration

r;c' ,.0

Q Stress Concentration

+ t t ~. t t ;:.. l' t·~ Fig. 33 Contours of P and Q Strèss Concentrations after the Extraction of 3 pi11ars

. Ratio Wo~p=15 for the outermost opening on the 1eft, Vertical.Uniaxial Loading.

> ( 0-) c:::

'" o

;\' ··e )'~,

---===i v "'V __ u _. _. - ---. ---~

1 ....... ·~v "'v' " . 1

P stress·Concentration 100

~--------------------1,0--

~~/ Q Stress Concentration

~ .~ ~ ~

Fig. 34 Contours of P and Q Stress Concentrations after the Extraction of 3 pillars

Ratio WoMp=15 for the outermost opening on the 1eft, Horizontal. Uniaxial Loading.

~ ....

" < • r '. ') 1 /. e

t } { ~. ~ }.!. { . ~ t

P stress Concentration

o 0.0

c:=;:;::::>

~:.:~-~=~======~-~ :1,0__ -,/ -_ I:1S -- .-::---~ __ -ro ~--------- ~~\ " /.'

----r.i~ ~~ ...

~~~s- ~==-----. --.' -D,iS

~--;:i'S- \ ... ~~ ~v Q.

"

conutration . ()

..:~o

+ t .1'. 1 t if f l' if t Fig. 35 Contours of P and Q Stress Concentrations alter the Extraction of 4 pillars

Ratio Wo/Wp=19 for the outermost opening on the left, Vertical Un;iaxial Loadingo

') ';~

0'\ 1\)

i 1 <::. /...... / e

-~r-- u ----- -- • • 'V ---------- "'\.~-. ---- ~

P stress Concentration

_,,0 _ '.0--____ _ ----- ..-'

~~----------------­----------------------.;ZJ) ______________ __

Q Stress Concentration

--?1 .'V' ,1. ~

Fig. 36 Contours of P and Q Stre·ss Concentrations after the Extraction of 4 pillars

Ratio Wo/Wp=19 for the outermost opening on the left, Horizontal Uniaxial Loading.

L &O:Z 7. '-..J

~ \..0)

- r· ...-.[ .. .., •. -;"'> {~ .. , .. - ~. "·e

~ _~ ~ ... ___ ~ _. __ J } ,~ , } t n n ~ --------_ .. _~._--~ - -------

~---------------____ ------~l ~~(------------------------~

Q Stress Concentratio

P stress Concentration '.

r 0.0 ____

+ l' l' .; .. tif· ~ l' t . ~. Fig. 37 Contours of P and Q Stress Concentrations after the Extraction of 5 pillars

Vertical Uniaxial Loading.

' ..

~ 7~

~ ~

, 'I:---:---? -.. 'f'-é"~.-'-·· F

,e >~~

--7(1-~==================~'\..' ,"V' ---~ 1-, ''''v' IV \

P Stress Concentration \.0 1.0

_ ------1.0 ______ 1.0 _______ -"---------

/ I.~

~~~~ ~.o- ~ -ce:

• .. '5 0_0

-0.15_ ____ -o.1S_

Q Stress Concentratj.on

-7i "V /V ~

Figo 38 Contours of P and Q stress Concentrations after the Extraction of 5 pillars

Horizontal Uniaxial Loading.

~

" \ .:

? CI' ':'e

s::: o .r-! 01.) ct!

~ ID o s::: o

U

III III ID ~ +l

6

5 4

3

2, l

CI) -2

~ ~

Section !!::!i'

P(Sx = 0)

""'"'O--~ 0 0--- A_.8-~~(Sy = 0)

Q(Sx = 0)

v

!~Yi_!li ii t ~ ~ ~ Sy ~ ~ ~ - ----=-1 . . 1 I<E-

H b D A_,_ 1\ D C 1 H'

-<>1 ED DE _. _________________ 1-

Sx ~

î Î Î SyÎ Î Î Î Î, Î Î Sy. Î ' Î Î

Sx oE--

Section y=y.'

Fig. 39 Distribution of Principal Stress Concentra.tion along the Horizontal and Vertical Sections.

1 pillar, Wo/Wp~l, Uniaxial Loading.

5

0\ 0\

6) . ~!... 2f e

Section !:!::!l'

1 5

!::: 4 0 on 3 ....., C1l

13 2 ~

~l ~ ,- 1 ....

!::: ------' 1

.... -----~ PC Sy=O ) o 0

~ -1 /' ..... - ------"'L-Q(Sy=O) Q(-Sx=O) Section Y::Y..' ~I 2 .....,-

CI)

-3- . 1 1

i 1 1

~ .~ ~ t 1 ~Sy ~ ~. ~

1 ~ ! ~sy ~ ~ ~

1

i i

H~I 1 ! I~H' 1

(t_ ,--n B œ

~I El DE I~ -4 -1 \1)._ 1 2 3\ 4 5

S41 1 I~ 1. l,

: •

t t t t tsy t t l", l' l' tsy t t t Q(Sy=O)~

..

Fig. 40 Distribution of Principal Stress Concentration along the Horizontal and Vertical Sections.

2 pillars, Wo/Wp=l, Uniaxial Loading.

Q-..,. .~

~ -~

l.f ~~ )J 1 e ~

Section l!:!i' 6

s:: 5 0

:0 4 III

13 3 s::: ~ 2 s::: 8 1

---1 i : 1

tIl 0 tIl ,C)

13 -1 ri)

-2

li: i

! i

V·~ ~ i d 1---.1--~ ! 1

, , 1 1 • 1

DIB ' (

Section Y..::!'

Q(Sx=O)

---? -0-0 ~

~I } I~ --~--------- ------.---~

t t t t tSy t t t t tsy t t t

Fig. 41 Distribution of Principal Stress Concentration ~ong the Horizontal and Verti~al Sections.

3 pU1ars, Wo/\o/p=l, Uniaxia1 Loading

~

~

6

s:: 5 0

:f34 CIl

~ 3 al tl 2 s::: 8 1 tIl

~ 0 r.. ~-l

-2

-3

.. , . Lf· r

Section !!::li'

i\. Ji ........ V' IVI iVI ( .)- 1 vi l' 1

Q SX~O . : 1 :: i 1 1

: ; 1 1 1 l ,1 1 . IV l ,

~

;

i 1

Section y.::y..'

_~_LL!irLLU_L_LJ-_LL~I~SYr~J ~--~ : 1 ! 1 . 1 r . I~

--l-_C ·B_I 1 AD· H~H' ED 0 o-D LJ~~ ____ ~

-.;.r H .... ~

Sx ~

. . Sx ~

L ._---------------t t t Sy t t t t ,t tsy 1 'f l'

1 t

Fig. 42 Distribution of Principal Stress Concentration along the Horizontal and Vertical Sections.

4 pillars, Wo/Wp=l, Uniaxial Loading

.:6 ':~ .

6

O· -.!

$

6

§ .5 .r-!

~ 4 ~

~ 3 (l)

g 2 o U l

.. 4- ( ~ . ,/

Section !!::!!'

ry 1 1

~ 0 b';a\~W~~:i-J..#.ji·~·MI-~4 il l~~l ~~ b~~ ~-l

-2

-3

~ ~ J Sy ~ ~ S~ J ! ~ l J ~ J J J

H

.-- VI 1 l,. 1 \ , 1· i

~ I! 1 1 : i 1 1 1. ~ l 'c B_! ,_, D ~ Po. D , i --ri 1

ED . D D D U~----~ H'

Sx ;x ~'

t t t . t Sy t t t t V' t t t Sy t . t' t t

e .

~ection Y=Y..'

Q(Sx=O)

e.-P(SY=O)

Fig. 43 Distribution of Principal Stress Concentration along the Horizontal and Vertical Sections •.

5pillars, Wo/Wp;l, Uniaxial Loading

-... c

~

7 ~ 6 o

::J 5 rd H

04.l 4 ~ ID

g 3 8 2

·- ..... L/ \·f· b f' 1

Î

Section H-H'

* l ·~t J~~.P~;;:~tL.j:iG4L.l~~~'~~ Q(Sy=O) bo~LH~~bIDL~aJ' ~ f CI) ]

-1

-2

··e

Section V-V'

~ ~ i J . : ~ Sy ~ : ~ ! ~ : ~ f'V ~ : J 1 ~ s~ t 1 ~ i ~ 1 ~ r - - - -~. - - - - - - - ..;.., - - - - ...;. - -:... - -:... - -. - -:. -r Q(Sx=O ! : . : :. 1 i ! 1 ! 1

-?I i.!, : i ! 1 . 1 i : I~ H Ile B : • . i AD! liB C l ,H'

~!ED'D ,0 D 0 D Dl~ S-.2L... 1 t 1 Sx ~ . . I~

~------~----~-~ ---------~--~ ·1 1 t t Sy 1 t t 1 v' t t t Sy t t . t· 1 . Fig. 44 Distribution of Principal Stress Concentration along the Horizontal and Vertical Sections.

6 pillars, .Wo/Wp=l, Uni~al Loading

--. -

'='l .....

e

7 6

~ 5 o

.,-i 4 ~ b J ~ ~ 2 ~

8 1

p(Sx=O)

I! ( 1 3 ,J

r

Section !!::!i' .

~-""~P(Sy=O) 1/) _. -.7,/,-.LIII .... ·~··~, •• A.·........ ... I/)O~ . ~ -1 ~~-4-~:t_A-~.*"')K 1\1\ r-·}~-#'.fs.-~.*."".*· .. f .. "t' ~.-j&ï#

CI) -2

-3 1

~

-;. H

'0 e

Section V-V' -

~

e D ADe 1 H' -----ED-D~ "+- -J2-~--:'1 { 1 l , 'cf' 1

Sx -:;.

t ~_~-i-~-~-~-~-i_~-i-;~ Fig. 45 Distribution of Principal Stress Concentration along the Horizontal and Vertical Seotions. . .

1 pillar, Wo/Wp=3, Uniaxial Loading

\

~

- LI6 .. 2 ).' i

Section l:!:!!'

.... ~

~~ . ~ 5 \ o ,

::J '~ ""'-_ _ _ _ A P(Sr-O

) . .~/ ~ ...... _-------•.•. _-Q(Sy=O)

al o s:: o

ü

[J)

0 [J)

al H ..., -1 CI)

-2 Q(Sx=O)

-3

~ t

~

H_

~

~ ~ Sy ~ ! 4 ~ ; J [~ : ~ ; J, $y J, --J, - - - - i - - - -: - ;,... - -j- - -' - - - -

i : 1 i I~ , l,

C BA D 'B c ED-D-O lE-

s~ I~ -------------~-------t 1 ~t t t t f t f~ 1 1

Section Y::!'

Fig. 46 Distribution of Principal stress Concent~ation along the Horizontal and Vertical. Sections.

2 pillars, Wo/\'Ip=3, Uniaxial Loading

·"e -~ ',rJ

-..J \.,)

-- W j

t " i '1

") )"' .~ e -R

8

7 Section H-H' s:: 6 0

·rI 5 ii1 1-< 4 ~ CD

:3 ()

s:: 0

(.) 2 CI)

1 CI) CD 1-<

0 ~ CI)

-1

-2

~ t

~

H ... ~

Sx ~

~ t

~ [\ '1 j~ . . ........ i \ "

1 ......

1 .. · ................ ~P(Sy=O)

Q(SX=O)~ \."....J' I~ ~ il: 1 1 1 1

; 1 1 1. 1

- i - ii-Sy i _,!'_i -1 tl!- i -lil- i ~Ii. _1._ ~ . 1 l' ! 1 : lE-

. 1 . IC BI 1 D 1. D! C' ,~ .

E D D D DE . .1_: 22 1 î 1. 1 ~ 1 4 1 te. 1 â

J Sx . ~

-T-t Syt-T-t-t.:t-t~tSY T-t- t

Section Y.::!'

Fig. 47 Distribution of Principal Stress. 'Conc'entration along the Horizontal and Vertical Sections o

:3 pillars, Wo/Wp=:3, U~axial Loading

"t

9 8

~ 7 .~ 6 +l

~ 5 ..., s:: 4 ID () s:: o 3 u

L' i/ 10 .' tt l' -,

..5

~:ection !!::li'

1 : i Section' V-V' i ! i _

V"

* ~ ~ s~ ~ , ~ ~ ~ ; ~ ~ , ~ : ~ Sy ~ : ~ t ~,~~-v, ~

~I ~J

S41 '1- 'I~ ,f t t syt -t- f-f-~- t ÎSY Î Î t

V' '

Fig. 48 Distribution of Principal stress Concentration along the Horizontal and Vertical Sections.

4 pillars, Wo/Wp=J, Uniaxial Loading

e ,~

~\

~

8

7 86

.r-!

~5 1-<

oIJ S:::4 (!) ()

§ :3 t.)

11)2 II)

fI -f.J

e '-'; ) r -;>

Section l!:!i 1

CI) Q' ~~ Q ~""',~0-S lM 1'*',~ a Il fd-éj''''~ IP ~""',tJ a s 8 è 1""',,, ~ Il fi "~ r' ~a , ' J

-1

-2 • ' 1

1 i 1 , VI, 1

-3

,.e

Section Y.::!'

~ I~,_ !._!l_~~~1,_ ~-!~l_!._:l_l~ii-i~ll~ Q(s=O;; . li, l 'e à : 1 D AD: Be 1 1 H·-l----~--- .. --=- lE EI~ _lr~ --I-J 1',11

Sx J Sx ; tI ~ ~ , ce- Q ; L _______________________ ~ (~ 1 t t t f syt t t t t tsy t Î f é l

. " .. P(Sy=Q) V'

Fig. 49 Distribution of Principal Stress Concentration along the Horizontal and Vertical Sections

5 pillars, Wo/Wp=3, Uniaxial Loading

c ........

""l ~

9 8

7 § 6

.r-! +l

~5 +l

~4 t)

83 u tIl2 tIl

~1

8 J v (, f

,.-,

Seotion li::1i'

+> ÇIl 0 ~"~9M0 ~ I~\~ e J..,f.~ ~ QI ~:l""'~ ~ ~J 1"""~~o0'0~"~f''''', 6 .. e·~ ~ P:-:;/ -1

Q(Sx=O) i • . 1 V : ; , · Seotion V-V' , S . , si' -~r:.-i_i·._i_i._Yi_!_i_[~_i_:1Y l_!l; __ i_~ Q(SX=1°)' \

, , :' , ; l ,~

--::'1 ' " ;, ;. ' I~ • 1 l ' • -. , t- le B' .' A D" ! B Cil H' .~ \

,DDDDDD[ ~~-1-1-T-T:T-l-l-~-1-1-T-l-l-~~

Y' V' Sy Q(Sy=O)'

2 ~-''!..,

~ ~

; ~

tp(sy=O)

Fig. 50 Distribution of Prinoipal stress Conoentration along the Horizontal and Vertioal Seotions.

\. 6 pillars, Wo/Wp=3, Uniaxial Loading

e

4 6

--..J

~

-..J "'-l

8. ) P . r /" \,

l ' . ·:,0 7 j "" -, ~':., ',( 1 It /'"

Fig. 51 Distribution of Prinoipal Stress 10

9 8

Section 1:!::!i' Concentration Along the Horizontal

s:: 7 o ::36

111

b5 s:: ~4 s:: o °3

-1

-2

P(Sx=O)

i 1

mld Vertical Sections. Wojwp=7,

Extraction of 1 pillar,

Uniaxial Loading.

l' ~ j, -- ri

, ':!:.'! 1 Section Y.::Y.' i Sy , ; 1 .! Sy: 1

r • ! . ,I-E- iS r ~j- ~ - ±- - ~ _[i:_ i -'~[_ i -li _l_,i I_! -il~ Q(Sx=O)--.~~ ~I Iii ," •

H F, D A A . i i 1 r----,c 1

~l El °1 D D D LJ--l~ ._ Sx 1 l 1 Sx ~ ,--? ~"I ; L _________________________ ~ , ~ ~

t t t Sy t t t t t t tSY t 'f t Q(Si=O/ \P(SY=O) V'

8

---..j r: .......

10

~

fIt

11 10 Section H-H~

9

8

c 7 0

:D 6 Cl! ~, .5 -g ~ 4 c 8 3 (/) (/)

.. , ............. .. ... - ....... _ ............ ,. ...... -. .-ID $..! +' Ul

Q(Sx=O) -2

~ t.,.-- r) -- /) r-c:. '1 .' e

. Fig..52 Distribution of Principal Stress

Concentration along the Horizontal

and Vertic~ Sections. Wo/Wp=ll,

Extraction of2 pillars,

Uniax:i.~ Loading.

Section V-V' -V ., .

rl- ~ _L Sy L_ ~ -1-1~ _! _'l_ .r~:!_ ± _.! -,~ Q(Sx=O)

~J . H, 1 F· DA: i . B C l ,H~ ____ .

--?/, El . 1 D D D u;«-u_~ ~ ... __ ." 12

Sx l·· ., Sx 1

~~t-t-t Syt-t-t-5-t-t-ts;T-t-fJ~ Q(~O;) (:y=O)

-....> --...;::.

~

'> ) ,'" r " ," ..: -. , 'C 1 . e

Fig~ 53 Distribution of Principal Stress

13 1- ~ Concentration along the Horizontal 12 11 Section li=!!'

10

and Vertical Sections. Wo/Wp=15,

Extraction of 3 pillars, '9

§ 8 Uniaxial Lo~ng.

.r-!

~ 7 $..,

~ 6 Q)

g 5 o u 4 !Il

P(sx=O)

~3 ~ +' 2 l} ~ ~t-~\L7 '~"XM«"~ '" '-. .• 7 . . ~ I~Hll 0 Q+-G ; Il & : l~~ \l -...e .... o .. ;r., • .... J., ~'œ~

........... ................... _ ....... ..-......... _ ................. ...

.. 1

-2 V ,

~ ,~ ~ Sy J ~ ~ ~ t j ~ : 1 Sy ~ ~ ~. Q(Sx=O)

r-------------------------ï ' E-

H 1 ~ : F D· AB' C l ,H'

_1 El . 1 D D ;E-1 1

s~~ __ ' _____ ' ________ -l-- _______ ...!~x t T fsyT T fT f' T fsyT T f

-"2,

V'

\ '\;

Section y::y..'

...... , , .. -

4 8 ........ , ... , ,. ~ , -

CP(Sy=O)

~ ,

(» o

~

\ "! :';. i1 .' . 1:- , 1 1 ···e

14

13

12

11

10

9 1::8 o

Section H-H'

Fig. 54 Distribution of Principal Stress

Concentration along the Horizontal

and Vertical. Sections.· Wo/Wp=19,

Extraction of 4 pillars,

·rI

~7 ~ -1.)6 1:: ID

85 o °4

P(Sx=O)

~r t'j' .., ID 3 " .......................... _ ~ __ -.. __ ........ _ --' 1-. ___ .... _ ....... - ......... ...---_ +> l' rn 2 , u. .... P(Sy=O)

1 _~.' ~ . .

.if \ , , ,

Uniaxial Loading.

.......... ....... o k% .\ ~ $ ~ 19 (il li ~ S (il e e QI Il e e a e ® li e 8 8 ~ Il li e 9 (1 ~ III elle Il ID 8 1jr .·X·. ~ ~ • • • ~ t'i: "-y,

-1 ... ~)(. .. ~-(;. !; i ;;:-«o..~

-2 ~..?Q(Sx=O)

V Section y::y..'

~ r 1'_1_1 ~y~ ~ ! _l_l_l~Yl_l_Jl- ~ _ J 1_ Q(Sx=O)

H. : F . DAC 1 • .H~ __

SX~: E l ' 1 D __ !~~_: . ~I ... 1. .. I~ L ____ -.:.. __ · _________________ ..J .. &1[4 ..

t t t Syt Î Î Î Î Sy Î t ,1 t t . P(sy=o)

8

,:'>,---...

co ....

14

13 12 Il

10

9 S::8 o ,,,' ~7 b 6 s:: Il)

25 o t.>4 III

~3 $..,

~~ 1

" "7 ., ..... 1.' j.': ) l' ',,-. ..,-

Fig. 55 Distribution of Principal

Section !!.::li'

1 ! J ! Il

. p(Sy=O) ,'J~, ., s ...... '11 '

~ ............... -............. _ ... ~ .• -....r.. ... ---~-_ ...... - ...... ..;.r."",."",-" 11:'\ 1 i f\ ..........

1

Stress Concentration along

th~ Horizontal and Vertical

S~ctions. Uniaxial Loading,

Extraction of 5 pillar~.

0lr.::: '~'~~ 131 ~ 9 e 1iI~~ >èJ ~0~ ~Q' ~H06'~ (lge0~ ~'f.. 'cy> : ',~,1( .. .)("rrl(,,',('

B (S:x.=;O.) • ~i Q (Sy--O) , ___ " >:i .. -" b" '. ~c;-.. 12" .... ~" 0.... .. a <7"" "h~ ... _/ 1

V

-1

-2 Section y::y.'

~ : ~ ·1 Sy ~ . ~ 'V ~ ~ i ~ Sy ~ ~ r~ Q(S~ ~i-~- -- ~ -- -- ~ - - -~ ~- - ~ ~ --- - - - -l~,. ~~\~

~ ~: E [- - - - ~_H_I ____ ?·· >_~.;--'It.. ï?:.- 1

~~ 1 . 1· 1 *"" sx ,

Lt - t - t: t - t - t - t-T- t -L~T-T - ~!.J Q(Sy=o~~f Cp(Sy=O)

<:><)

~

<XI l'V

5

4

3

2 ' .

, ... ,J

P:/SitionA (Sx=O) ()z 0 o 0

<J-_ ct (1 J Position B - --_ (Sx=O) ----_ .. _6 __ .-'~,

JJ- - ,. • .,.... -A • ct ./ '~, /,

()

// Position C -;1 ~" .L:-A .,

, (s -0) ~. 'IJ.--'- ....... /~ :y= , /- ~.""A......5Position C ,t .. '.. (Sx=O)

_ ....... a+... )( +"''-I., ....... ~ ... ,......, -."'_ ......... f. ~ ••••• ···~Position E

''''*_'_'_'_*::-' /' * ~ (Sx=O) ",' .' ...... " / .,./ '. Position A ...... ! ___ ., ',. ~ (Sy=O)

'2 .. .. J --~ ..... :-

J

~ '., ~Position B Position D * (Sy=0)

(Sy=O)

j 1 1 !~ 5 6

Number of Pillars

Fig. 56 Relation between Stres~ Concentration and Number of

Pillars in First Series of 'rests (Wo/Wp=l).

8,3

')~

12

11

10

9

8

7

6

5

4

s:: 3 0 -ri

~ 1-1

.l-l s:: 2 Q) C)

s:: 0

C,) 1 'if) 14 ol) ~ ~ 0 CI'

... 1.

,.2

... 3

Sx=O; Vertical Uniaxial Loading

Sy=O; Horizontal Uniaxial Loading

Position A (Sx=O) o

<1- _-CJ, .-",.- .". ". , ,

'CJ--. ~Position B -- --(1 (Sx=O ) ....a_----â ,

~.,. , 1:::.-_ - -â-- , , t/ Position C

" (Sx=O) ~+.. n. - --

-- ".. A .+-P ·t· E ... _. __ ._.c.;._A __ "~'~+""'-•••• • OSlo l.on ,_.-.-- -~ - .. ·-'0 (S 0) + ••••• _ +_-----+' "X=

" A_position C

\'Positio:t1 A (Sy=O)

Position E--.. (Sy=O) "'\ ...... ----+ ----+_., .• ,..+-,.. + ...­~ .....

~ ...... + * __ o_~ ., .. ", ...qc_._.~-.... .Lo_'-'-,.,...""'" *-~-.- :Ic:_ o_o Position D

(.5x=O )

Number of Pillar~

(Sy=O)

Fig. 57 Relation between Stress Concentration and Number of

Pillars in Second Series of Tests (Wo/Wp=3).

84

~ 0 .,.. ~

~ a> ~ 0

(,)

(Il (Il

a> J.t +' CI)

10

9

8

7

6

5

4

3

2-

1

0

, ... 1

,·2

-;

P-Vertical 10ading (Wo/Wp=J)

A

(,p-VertiCal Loading (W6/Wp=1)

X

/ (Wo/Wp=l) (Wo/Wp=l )

A

~-Vartical Loading CQ-HOrizontal Loading

! >1.. __ -* , - .. ~_ .• - ... .,. .,..-*-- ---\ -_ ... *- - ---..®- - - .....()-\..,~-:; ... ;::(p ....... ~---=~--~ -::::ç ;:>-_- . ..f:!.... li

I.t\- - I:J ~ \~ ... Bori~ont~ Loadi.ng ·~Q ... VE'rtiCal Loading

(Wo/Wp=J) (Wo/Wp~J) ~

1 2 J 5 6

Number of PilJ.ars

Fig. 58 Relation between Stress f;oncentrations 400 Numbet'

of Pillars for both Series of Tests.

85

6

5

J .0.5

,...., CI ......,

§ or! III s:: Q)

E-t

-1.0

-2.0

---1 pillar(P)

~6 pillars(Q)

-~:::-: ... ~ ..... -::"::~-5 pillars (Q) ..... -, ... --0--- 4 pillars (Q ) •..••••. __ 1 pillar(Q)

. 3 pillars(Q) 2 pillars(Q)

50 60 70 80 Percent Recover,y

Vertical Loading (Sx=O)

3 pillars(P)

~~5 p~llars(P)

~/' x-....... 2 pJ.llars(P)

"'-1 pillar(P)

o~ 11.: .,' ." 4 pillars(P) .' " /

/

" ~A.~6 pillars(P)

~ .. 0,.......... ./3 pillars(Q)

~. ~5 pillars(Q) ...... r.:;~ __ ,6 pillars(Q) ')C. .... ~ ~ ~ 4 pillars(Q) ~ ~2 pillars(Q)

)(----1 pillar(Q)

50 70 Percent Recover,y

Horizontal Loading (Sy=O)

Fig. 59 Maximum P and Q stress Concentrations as a Function of

Percent Recover,y for various Number of Pillars.

86

g7

;::: c::-

ori

~ ~

~ Ci) 0 ::: 0 u III III Q)

b tI)

14

1:3

12

11

10

9

8

7

6 .

.5

4

:3 .

2

i

0

~j.

-2

-:3

Extraction Ratio

= WOW~ Wp x 100

1

-~~Hol~zontal Loading

(Q-lïO,-.J.zontal Loading

'A?, //

(

Q-Vertical Lomng

70 80

o

90

Fig. 60 Maximum P and Q stress Concantrations a~ a Function of

Percent Recovery with Systematic Extraction starting

wi th 5 pillars.

87

14

13

12

11

10

9

8

'? -.

r,: ~

~ 0

:0 5 r.li ~ "~ i ~ .::J) ,tj. ; :0 ~ 0 0

'(1) :3 Dl ID ,fj :"n

!f" !

-1'

-2

Position A (S.x=O) ~ ./Posi tion F *" (Sx=O)

/ "-Il " ...

/ GPosition B (Sx=O)

*/ <» X ~/ ; ~

/ ~ ,.., Position E '~- ,- - ,- ,-()-'" (Sx=O)

,"~' ,

/ ~b / ~

~/ ,~ .' ~ 6. ... '

, " x~. ~~ , ~Position C

88

.' /- . .",'" (Sx=O)

~.ll IY~~~' ~à----- - 6 LPosition F , ~~ (S,y--O) ,,~J ><:...... ~ ./Posi tion C

kL ...... , - -- .,.-- .... 1" ( -0) I~ ' .... *-- - ..... _~--------- ... Â __ .-·~ sr-ja '... .......... -........* .... ~ ... -

~ ------$ O_p ·t· A _._... ~. • os~ ~on t' ............ ~ ~ • .,.,..... (0) r ~ ........ > - • .",. • ....- sr-\~ . i::::$_._._._._,.-._._.\:-

Position B Position D (~) (~O)

iPOSit:E (Sy=O)

~~~------ ~----..... ,~~------I::._."", •. ~ ........ ,- ~ )(

'" '-." A. (POsition D ->t.l..--'-,-,--.--A-,_. (Sx=O)

,?:tUai:' Removing 1 ---. -. -. -. ---A. ;started ....

-3~~~--~~---~~~~~~~~~~~~~~~~~~------1 5 10 15 20

-Ratio of Opening Width to Pillar Width (Wo/Wp)

Fig. 61 Stress Concentrations in Pillars formed b,y Rectangular

Openings as a Function of the Ratio of Wo/Wp with

Systematic Extraction starting with 5 Pillars.

... -J.~ " )

INTÉRPRETATION OF EXPERIMENTAL FINDINGS, SUMMA.~Y AND RECOMMENDATION

In this study since ditficulties were encountered in counting

the fringes.at corners because ot the small radii of curvature the

data are subject to some error. In determining the relationships

between stress concentration and the ratio ot opening width to pillar

width ,and percent recover,y, however, averages of values at similar

points were employed. This procedure should minimize the consequences

of errors in individual determinations.

In order to obtain results ot greater accuracy, studies ot all

the cases were made in finer detail by enlarging the areas around

the holes.

An inspection of the values obtained trom the tests shows that

the trends of stresses are not smooth. This can he explained through

ditficulties in counting the tringe orders. In the cases involving

5 and 6 pillars, values are generally lower compared with those in

wl1ich l to 4 pillars were used. This i5 due not only to the smaller

size of model openings O/S" x 3/S" x 9/1611 rather than -f" x t" x 5/S"),

but aIso to the fact that Many fringes at high stress concentrations

cause difticulties in interpreting experimental results.

1. First Series of Tests (Wo/Wp=l)

The ratio of opening width to pillar width was 1 to 1 and trom

1 te 6 pillars were used. Six models were made. "Extraction Ratio"

was 50 percent.

Leads were applied vertically and horizontally, and stress

89

concentrations were deteX"mined !,J.sing photo'i>llastic techniques described

previously at positions A, B, C" D and li: as shmm in Figu..."'"8 56. Con­

tour drawings of stl~SS ooncentrations in the models ~re shawn in

Figures j through 16. The data partaining to this series o~ tests

is presented in tabular form in Table 1.

Vertical Loading (Sx=O, MO)

Stresses at positions A, B and Care tangential to the boundar,y

of the openings, and are of the same sign as the applied stress (that

is to s~, compressive). It appears that stress concentrations at

positions B (outermost edge of the outer pillar) and C (outermost corner

of the outer opening) do not vary appreciably a.s the number of pillars

is increased (Fig. 56 ). At position D, stress 'concentrations remained

constant throughout the series of tests but were of opposite sign

(Tension).

Generally, in each case, the stxsass -concentration at position A

(edge of the central pill;ar) 1s the maximum that is produced. It can

be noted that the stresses at A araà1wqs compressive, am that their

concentrations tend to increase as the number otpillars 15 inoreased

to a total of 5. This IIlight be due to ms.xinnull bending stress ocourring

near the IIlid-span of the panel as a result of the vertical deflection2•

Such a mechanism oould be used to account for variations in pillar

loading similar to those observed. In t.his series of experiments the

rate of increase of stress concentrations at A is rather gent.J.e;

varying from 4.46 to 4.96.

90

Loading No. of Condition Pillars

. Vertical­uniaxial (Sx=O)

1

2

3

4

5

6

Hori z ont al- 1 'uniaxial (sy=O) 2

3

4

5

6

Table 1

Stress Concentration at Position ABC D E

4.46 3.72 -1.19 2.73

4.71 4.26 3.85 -1.10 2.74

4.70 4.18 3.71 -1.14 2.81

4.79 3.73 3.22 -1.21 2.64

4.96 4.23 3.31 -1.24 2.64

4.96 3.72 3.01 -1.13 2.71

2.23 2.98 2.16 -1.12

2.26 2.66 3.66 2.41 -0.87

2.66 2.66 3.83 2.46 -0.93

2.56 2.20 2.93 2.05 -0.59

2.58 2.78 3.57 2.42 -0.63

2.17 2.01 2.80 1.86 -0.50

Max. Stress Conc. "p" "9" 4.46

4.80

4.94

5.08

5.37

5.04

2.98

3.66

4.00

-1.19

-1.10

-1.14

-1.21

-1.24

-1.13

-1.12

-0.50

Stress concentrations in a plate containing from l to 6 pillars

between square openings. Ratio of opening w:i.dth to pillar width;

Wo/Wp=l. Loads applied vertically and horizontally to the line

defined by the centers of the openings. "Extraction Ratio" is 50

percent (for the locations of positions A, B, C, D and E, see Fig.

39 through 44).

~:

91

1. Negative values represent tensile

stress concentrations.

centrations at siulilar positions

2. Stress concentrations at positions

are the averages of stress con-

~ C BAD ABc E-

~ ~q D J~JA D 8D~~ -+t t t t t f t t t t t~

Sy Sy General Positions in a Plate in the model.

Horizontal. Loading (SEO, Sx#JJ

The sarna models were used for tests with uniaxial. horizontal.

loading conditions. In Table l, it will be seen that stress con­

~entrations at positions A and B show slight~ irregular variations as

the number of pillars is increased. The lI1agnitudes of stress con­

centrations show ver,y~ variations. The maximum st~ess concentration

'was round at position C (theAuter corner of the outerrnost opening) in

each case. It can be seen that the concentration of major principal.

stress at C ranges between 2.98 and 3.83.

When compared with the vertical. loading condition, the general.

trend of the concentration of ~jor principal. stress shows a lower and

more gentle rate of increase. At the sarna time, the concentration of

minor principal stress, which is tensile, at position E decreases while

the number o~ pill~s i5 increased.

2. Second Series of Tests (Wo twp=3)

In these tests, one quarter of each pillar (in the foregoing

models) wes l'smoved from each side to produce an opening width to

pillar wldth ratio of 3 't.o 1. Extraction ratio was therefore 75

percent~ As berore, loads were applied vertically and horizont~.

Drawings of the m.odel al"e shown in Figures 17 through 28. The data

resulting from this series of tests is shawn in Table 2. Sorne photo­

graphs of fringe patterns are shown fo~ cases involving l, 3 and 5

pillars (photographs 7 through 12). Stress concentrations were

determined at positions A, B, C, D and E being similar locations as

described first series. Thus, the results have a comparative value.

92

The relationships between principal stress concentration and

~rcent recover,y, and ~un stress con~entration and number of

piLlars are :3hown in Figures 57, 58 Uld 59. Figure 56 may be compared

'with Figure 57 :tOr the fi~st series of tests. The atress concentrations

&:.~ similéU' although higher concentrations were form.ed and gradients

wel~ more intense than those in the ti~st series.

Vertical cLoading . (sx=O,. srlo)

Umer this loading condition, a concentrat,ion of major principal­

tit.)!"eSS ls formed at position A (on either side of the central pillar).

It "'Tfls observed thatstresses at A were aJ.ways positive (compression)

and that the concentration incl"eased as the number of pillars was

increased from 1 to ,t~,. The ,rate of increase is found to be higher than

that nQted in 1"9sultsobtained trom the tirst series ot tests at the

same position, A. Concentrations at A varied from 5.90 to 8.30 as

compa:red with 4.46 to 4.96 in the tirst series.

'rhe magnitudes of stl'8SS concentrations at positions B, CandE

show oruy ôlight i:i:'regulcn- variations. Compared with the data obtained

lrolll t,he first i3xp9:t"imellts, it can he seen that higher stresses at

eom.'PRr#lbl~ :po3it..ions aiso occl:cr-red in the second series of tests (Fig. 59).

The stress at position D, a concentration of minor principal

;;tress, is tangential to the bounda:ry of the opening 8Jld the magnitude

decreases from -1.97 to -1.32 (in tension) as more pillars ,are formed.

Horizontal Loading (Sy=O, SxiO)

The maximum concentration of major principal stress developed at

the outer corner of the outer opening (position C) in each case. The

93

Û) \~ , ,

Loading No. of Condition P1l1ars

Vertical­Uniaxial (Sx=O)

1

2

'3

4

5

6

Horizontal- 1 uni axial (Sy=O) 2

3

4

5

6

Table 2

Stress Concentratio.'- Position ABC LI 'E

5.90 5.23 -1.97 3.59

6.72 6.72 5.17 -2.07 3.62

7.56 7.06 5.55 -1.79 3.53

8.30 7.25 5.71 -1.56 4.20

7.90 6.05 4.21 -1.34 3.89

7.91 5.81 4.39 -1.32 4.04

2.37 3.69 3.16 -1.58

2.39 2.39 3.82 2.15 -1.34

2.35 2.35 3.92 2.46 -0.78

2.40 2.40 3.84 2.02 -0.67

2.17 2.17 4.06 2.72 -0.8?

1.55 1.55 3.10 1.86 -0.41

Max. stress Conc. "P" '''9"

6.40

8.82

8.42

7.91

4.22

4.30

4.47

3.84

4.34

3.10

-1.66

-1.47

-1.49

-1.58

-1.34

-0.89

-1.15

-0.98

-1.03

Stress concentrations in a plate containing from 1 to 6 pillars

between rectangular openings. Ratio of opening width to pillar

width; Wo/Wp=3. "Extraction Ratio" is 75 percent. Loads applied

vertically and horizontally to the lins defined by the centers of

the openings (for the locations of positions, see Fig. 45 through

50).

~:

94

Sy Sy 1. Negative values represent tensile

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ stress concentrations. ~Cd ADA B c~ ~ E 0 D 0 DEoE-.. 2. stress concentl'ations at positions ~ C SAD A 6 c~X

are the averages of stress con-

centrations at similar positions in

General Position in a Plate the model.

Table 3

Wo/Wp Loading No. of pillars Extraction Stress Concentration at Condition Extracted Ratio Ratio ~~2 A B C D~4~

Vertical- 0 1 50.0(1) 4.96 4.23 3.31 -0.99 uni axial (Sx=O, 0 3 75.0(2) 8.05 6.05 4.21 -1.26 . s'riO)

1 7 87.5 9.14 7.03 4.92 -1.69

2 11 91.7 10.67 7.11 4.98 -1.71

3 15 93.7 12.28 7.91 5.83 -2.17

4 19 95.0 13.89 6.55 -2.16

.2 100.0~2t 12.12 -2.22

Horizontal- 0 1 50.0(1) 2.58 2.78 3.57 2.78 uniaxial (Sy=O, SxjO)

0 3 75.0(2) 2.17 2.17 4.06 2.71

1 7 87.5 . 2;.46 2.46 3.07 2.33

2 11 91.7 2.79 2.79 3.49 2.42

3 15 9:h7 3.00 3.00 3.37 2.40

4 19 95.0 2.90 3.62 2.75

2 100.0(]) 3.50 2.80

Stress concentrations at different extraction ratios as pillars are removel

plate initially containing 5 pillars. Opening heights were constant (for

positions, see Figures 51 through 55).

General Positions in a Plate

!12: (1) and (2) results

first and second

(3) 100% recovery il

from a mining panel

and 5 pillars).

(4) Negative signs

(5) represents the

similar positions

:traction .tio (%)

0.0{1 )

5.0(2)

7.5

:>.0(1)

5.0(2)

7.5 .

).0

).00)

A

4.96

8.05

9.14

10.67

12.28

13.89

2.58

2.17

2.79

3.00

2.90

3.50

Table 3

Stress Concentration at B C D(4)

4.23

6.05

7.03

7.11

7.91

2.17

2.46

2.79

3.00

4.21

4.92

4.98

5.83

6.55

3.57

4.06

3.07

3.49

3.37

3.62

-0.99

-1.26 .

-1.69

-1.71

-2.17

-2.16

-2.25

2.78

2.71

2.33

2.42

2.40

2.75

2.80

: Positions (5) i E F

2.64

3.89

~ 5.34

6.12

7.66

7.72

; 9.74

-0.63

-0.87

-0.74

-0.98

-0.90

.-1.01

-1.26

7.08

7.82

11.24

9.64

13.12

3.38

3~49

3.75

3.26

3.50

Max. stress Concentration "P" "9"

8.42

9.14 '",

10.67

13.32

13.89

13.49

3.57

4.34

3.07

3.49

3.75

3.26

3.50

-1.24

-1.47

-1.97

-1.99

-2.33

-2.16

-2.25

-0.79

-0~98

-0.86

-0.98

-1.05

.1.01

-1.26

lt extraction ratios as pillars are removed in sequence from a

.ars. Opening heights were constant {for the locations of

l 55).

:::l c

.ate

~: (1) and (2) results (Wo/Wp=l and 3) obtained from the

first and second series of tests.

(3) 100% recovery in the table indicates total extraction

from a mining panel (initia1ly comprised of 6 openings

and 5 pillars).

(4) Negative signs indicate tension.

(5) represents the average of stress concentration at

similar positions in the model.

96

by joining the centers of the openings. Table 3 presents the data at

the position shown in Figures 51 through 55.

Vertical Loading (Sx=O, SyfO)

In Table 3 it will he noted that stress concentrations at a11

positions increase more rapidly as additional pillars are recovered~

The variations are greater than the rate of increase noted when

extraction ratio was increased from 50 to 75 percent (first and second

series of tests). The minor principal stress "QtI was found to bel

tensile at both the top and bottom of the openings with the maxinnu~

value occurring at position D (mid-span). High concentrations of

nmjor principal stress were observed near the corners of the openings

of position A as shawn in Figures 51 through 55.

'l'able 3 demonstrates that the maximum concentration of major

principal stress at position A varies in magnitude trom 9.14 to 13,,89

!1S t.he :i~ecovery is increased from 75 to' 95 percent (or, Wo/Wp ratio 1$

:tncrea~ed trom 7.0 to 19.0). At the same time, the maximum concen­

trat.ion of minor principal stress, at position D, increased fï'om ... 1 .. 69

1~t:.I .. 2~25 (Fig. 61). In a1l cases, the maximum stresses were developed

.'i.:r\')iU)(l. a. pillar adjacent to the opening expanded by the removal of ta

p~.lla.J:' 6'

.[ol'izontal Loading (SeO, Sx;l:O)

An increase of stress concentrations at a1l positions was

.,bserved for both major and minor principal stresses. Refe:rring t..o

T.able 3; the maximum concentration of major principal stress, at.

position A, varies in magnitude from 2.46 to 3.50 and the maximum

97

concentration of min or principal stress, at position E, range from

-0.74 to -1.26. Tensile stresses occurred strongly in the walls

closest to the edge.

Table 3, and Figures 60 and 61 show how stress concentrations var,y

with an increase in percent recover,r together with an increase in the

Wo/Wp ratio in the case of 5 pillars (or 6 openings).

4. Calculation of the Maximum stress Concentration

It has been found that the maximum stress concentration increases

ver,r rapidly as percent recovery is increased (Fig. 60). Duvall derived

the following empirical equation from bis eXperlmental data (circles and.

ovaloids wi th vertical-uni axial loading conditions):

100 2 . K = S + 0.09 {( 100 _ R) - 1 )

~:

K = Maximum stress concentration in pillars

s = Maximum stress concentration around a single opening

R = Percent recover,r = Wo/{Wo +Wp) x 100

He mantioned in bis report8 that the ab ove equation ncan be made to .

fit the data for openings of aqy Shape by adjusting the value of S to

suit the shape of opening being studied" •

. A comparison of the author's experimental data and the results

obtained from Duvall' s empirical equation is given in Table 4.

It is concluded that the equation can provide a rough approxi-

mation of the maximum stress concentration until recover,y reaches 87.5

percent. For recoveries greater than about 92 percent the t'Wo 'Values:

are nClt comparable. It is suggested that addi tional studies!lrlil ':'equ.1.~"&à

98

~

to obtain a more complete knowledge of methods b.1 which stress concen-

trations around multiple rectangular openings might be predicted.

Table 4

Comparison of values calculated from Duvall's equation

and author's experimental results.

Percent MarlllIUlll stress MaxillIUlll stress concentration Recovery concentration around in the pillars

(R) a square opening(S) K values Author's results

50.0 3.2 (4.69) 3.47 (4.96) 4.96

7.5.0 3.2 (4.69) 4.55 (6.04) 8.42

87 • .5 3.2 (4.69) 8.87 (10.36) 9.14

91.7 3.2 (4.69) 16.00 U7~~) 10.67

93.7 3.2 (4.69) 23.70 (2.5.19)' 13.32

.22.0 J.2 ~4.62l J2·1O ,40·,2°l 1î·82

~: 1. S = 3.2 1s taken from Duvall's report7•

2. The values of S in parentheses were adjusted to 4.69 to

produce the same stress concentration as the author's results

at 50 percent recovery.

,. Sl1IllI!1&Y' of the Results

A two-dimensional photoelastic study of stress patterns for the

t.hree series of tests designed to simulate selected "Room-and-Pillar

Mining Systems" reveals that:

1) The two key parameters influencing the stress distribution quite

obviously are the ratio of Wo jwp and the number of pillars, liN" •

It seerns, though, that "N" loses its significance when it is

greater than four.

99

. 2) For either loading situation studied, at Most positions in all

models the major principal stress, "p .. , was found to be compressive

and the rr.dnor principal stress, "Q", to be tensile.

3) Umer vertica).-uniaxial loading, the maximum concentration of

major prineipal stress occurs in the centermost pillar on, or near

the rib df the pillar. If pi1lars are removed in sequence from one

side of a panel to the other, the maximum stresses are developed

around. the pillar adjacent to the expanded opening. Under the sarne

loading conditions, the maximum stress concentration can be approxi­

mated (up to 87.5 percent recover,y) by using the equationdeveloped

by Duvall8•

4) For horizontal loading, when the number of pillars exceeded one,

the Most critical stress was developed at the outer corners of

the outermost openings

Thus, when a new opening is created in deep mines, the

stresses in the interior mining zones will be relieved to soma

extent.

5) Under horizontal loading the maximum concentration of major princi~

pal stress remained riearly equal (about 3.5) as the number of open­

ing was increased in the cases of Wo/Wp = 1 and 3.

stress concentrations around multiple openings, therefore,

can be reduced by developing a row of openings in the diraction of

application of the major principal stress.

6) As the ratio of Wo/Wp ~ncreases, the average stress ooncElntl"at.ioll

in the pillars increases at a higher rate than for that &t lower

Wo/Wp ratios.

100

Similar results to those presented he rein are reported b.Y pre­

vious investigators (see the page 4 for the summar,y of the previous

work).

It must be noted that some of above results (l, 3, 4 and 6)

agree with those which Duvall found for the cases of circular and

ovaloidal openings.

6. Recommendations for Further Study

1) The above study has been limited to two-dimensional problems; i.e.

the pillars considered are rib pillars. For cases involving square

pillars and other complicated Room-and-Pillar î.fining Systems, the

proximity of the additional two pillar boundaries necessitate a

three-dimensional photoelastic approach.

2) This study may be extended to include other paramGters such as ne",

defined as the angle between the line joining the centers of the

openings and the horizontal, since this condition (9 f 0 for dipping

deposits) is often encountered in the field.

3) Since stress concentrations at the corners of openings ara greatly

affected by the fillet radius, additional studies concerned with

various ratios of r/Wo (the radius of curvature of a opening to

the openingwidth) would be desirable.

4) The effects of biaxial loading on the sarne model can be studied

by using a superposition technique developed b.Y O. B. Nair and

Prof. J. E. Udd of McGill UniversitylO,23.

5) An attempt to study the problem of multiple op'anings using mathe­

matical techniques would represent a valuable contribution.

101

102

6) A verification of aU experimental results by comparison with

actual field coMitions will be required in order to permit an

application of this data to mining operations involving multiple

openings.

1. Caud1e, R. D. and

Clark, G. B.

2. Coates, D. F.

3. Dally, J. W. and

Riley, W. F.

4. Davies, J. J. L.

5. Denkhaus, H. G.

6. Dure1li, A. J. and.

Riley, W. F.

BIBLIOGRAPHY

"Stresses around Mine openings in soma

simple Geologie Structures"

University of Illinois, Engineering

Experiment Station Bulletin No. 430,

Vol. 52, 1955.

"Rock Mechanics Princip1es"

Dept. of Mines and Technical Surveys,

Mines Branch MOnograph 874, Queen's

P.rlnter, ottawa, Canada, 1965.

"Experimental stress Analysis"

McGraw-Hill Book Co., New York, 1965.

"Fillars Applications &nd Limitations

in Underground Mining"

M. Eng. Thesis, McGill University,

1959, Montreal, Canada.

"The Application of the Mathematical

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103

7. Duvall, W. I.

8. Duvall, W. I.

9. Frocht, M. M.

10. Green, A. E.

Il. Hete~, M., Ed.

12. Hoek, E.

"stress Analysis applied to Underground

Mining Problems"

Part l - Stress Analysis applied to

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U. S. Bureau of Mines Rept. of Investi-

gations 4192, 1948.

"Stress Analysis· appll.ed to Underground

Mining Prob1ems"

Part II - Stress Analysis applied to

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U.S. Bureau of Mines Rept. of Investi-

gations 4387, 1948.

nPhotoe1asticity"

Vols. l and II

John Wiley and Sons, Inc., New York,

1941(I) and 1948(II).

"General Bi-harmonie Analysis for a plate

containing circu1ar holes"

Roy. Soc., Proc. 176A, 1940, pp. 121-139.

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John Wiley and Sons, Inc., New York, 1950.

"Experimental study of Rock-Stress Problems

in Deep-level Mining"

Experimental Mechanics, Pergamon Press,

New York, 1963.

104

1 \'"

\',':' ,

"'" 1 " ,

13. Hoek, E.

14. Ho1ister, G. S.

15. How1and, R. C. U.

16. Jessop, H. T. and

Harris, F. C.

17. Ling, C. B.

18. Merri11, R. H.

"The Application of Experimental

Mechanics to the Study of Rock stress

Prob1ems encountered in Deep-level

Mining in South Africa"

Int~ Cong. Exp, Mach., Nov., 1961.

"Experimental s'tres\ Analysis Princip1es

and Methods"

Cambridge at the University Press, 1967,

pp. 184-196.

"Stress in a Plate containing an Infinite

Row of Ho1es"

Roy $ Soc., Proc. 148A, Feb., 1935,

:pp~ 471-491.

IiPhotoe1asticity: Princip1es and

Methods"

C1ea:Cfer ... Hume Press Ltd., Lomon, 1949.

~On the Stresses in a Plate containing

two Circular Ho1es"

Journ. App. Physics, Jan., 1948, pp. 77-82.

'~Design of Underground Mine Openings,

Oi1 Shale Mines, Kif1e, Colorado"

U. S. Bureau of Mines Rept. of Investi-

gations 5089, 1954.

105

106

19. Mind.lin, R. D. nA Review of the Photoelastic Method

of stress Analysis"

Journ. App. Physics, Vol. 10,

l. No. 4, April, 1939, pp. 222-241.

II. No. 5, ~ 1939, pp. 273-294.

20. Nair, O. B. nphotoelastic Analysis of stress in and

around Mine Pillars"

M .. Eng. 11cGill Thesis, 1964, Montreal,

Canada ..

21.. Nair, O. B. and "Determination of Principal Stresses

Udd, J .. E .. in Single Pillars~sing an Iterative

Proc. of the 6th Symposium on Rock

Mechanics, University of Missouri at

Rolla, 1964, pp. 391-406.

22. Nair, O. B. and "Stressas around Opanings in a plate

Udd, J. E. due to Biaxial Loads' through a Super-

positioning Technique"

Proc. of the Rock Mechanics Symposium,

University of Toronto, Toronto, Jan.,

1965, pp. 121-135.

23. Obert, L. "Measurements of Pressures on Rock

Pillars in Underground Mines"

Part l, U.S. Bureau of Mines Rept. of

Investigations, 3444, 1939.

107

24. Obert, L. and "Rock MSchanics and the Design of the

Duvall, W. I. structures in Rock"

John Wiley and Sons, Ltd., New York,

1967.

25. Obert, L., Duvall, W. l., "Design of Underground Openings in

and MSrrill, R. H. competent Rock"

u.s. Bureau of Mines Bulletin 587, 1960.

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Part 2. Accident Review, Ventilation,

Ground Control, and Related Subjects.

1961, pp. 40-66, 89~1l7.

27. Panek, L~ A. "Stresses about Mine Openings in a

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Edwards Brothers~ Inc~, New York, 1951.

28. Savin, G. N. »stress Concentration around Holes"

Di'llision 1: Solid ruld Structural Mechanics,

V'·,l .. l, pp .. 392 .. 421, Pergamon Press,

1961 ..

29. Sinclair, D. and IIPhotoelasticity and its application to

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Trans. AIME, Coal Division, Vol. 139,

1940, pp. 224-252.

30. Talapatra, U.

)1. Trumbachev, V. and

Melinikov, ,E.

32. Gyenga, M. and,

Coatas. D. F.

"Photoelastic Stress Analysis of the

End of a Borehole"

M. Eng~ 1'hesis, McGill University,

1968, Montreal,. Canada.

J1Dist:r.ïbution of Stress in the inter­

vaning Pillars .Q.t Medium and Steep

Dips"

4th Int. Cor~erenee on Strata Control

and Rook Mechanics, New York, 1964,

pp .. :t .. 7.

"Development of' Stress .Analysis by

Photoalasticity for Slopes"

Unedit6d Div:1.sional R.eport, FMP 67/B-MRL,

Minas Branch, ottawa, 1967.

lOB

APPENDIX

Computer Programme

FORTRAN IV G LEVEL l, MOD 1 i"lA IN DATE = 68273 10/20

0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012

0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023

0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048

@049

C DETERMINATION OF FIELD STRESSES AT DIFFERENT POINTS OF A PLAT~. C FIELD STRESSES AROUND 4-PILLAR UNDER UNIAXIAL HORIZONTAL LOADING

DIMENS ION At 49,49) ,8 (49,49) ,e (49,49) ,0 (49,49) ,E (49,49), 1 F ( 49 ,49) , G ( 49,49) , H (49,49) , X ( 49 ,49 )

DO l 1 = 1,49 DO l J=1,49

A(I,J)=OoO S(I,J)=üoO

C(I,J)=OoO D(I,J)=OoO

E(I,J)=OoO F(I,J)=OoO

G(I,Jl=OoO X(I,J)=OoO

l H(I,J)=OoO C READ IN BOUNDARY VALUES

kEAD(5,lO)(All,J) ,J=1,49) 10 FORMAT(16F502)

RE AD ( 5 , 11 ) (A ( l , 1 ) , 1 = 1 ,49 ) 11 FLiRMAl(FS o2)

READ(S,12)(A(49,J),J=1,49) 12 FJRMAT(16F502)

REAO(S,13)(A(I,49),I=1,49) 13 FOR1~Al( F502)

DO 8 1=6,44 8 REAO(S,14)(A(I,J),J=23,27)

14 FORMAT (5F502) C REAO IN PHOTOELASTIC VALUES OF (P-Q)

DO 22 1=1,49 22 REAO(S,23)(C(I,J),J=13,37) 23 FURMAT(16F502)

WRITE(6,45) 45 FOKMAT(lHl)

WRITE(6,49) 49 FUKMAT(lH ,34HTHE BOU~OARY VALUES ARE AS FOLLOWS) .

DO -Tl 1=1,49 41 WkITE(6,42)(-Â(I,J),J=13,36) 42 FORMAT{lH ,24F502)

wRITE<ô,52) 5 2 FOR lU T( 1 Hl) 50 DU 30 1=2,48

DO 50 J=2,48 ,\jA=J-l 1~8=J+ 1 I~C= 1-1

NO=l+l IF{J-23) 101,70,70

70 IF{J-27) 71,7:\',10J. 71 IF(I-61 101,72,7.2 72 IF(I-12) 80,80,73 73 IF(I-14) 101,74,74 74 IF(I-201 dO,30,75 75 IF(l-22) 101,76,76 76 IF(I-Zèl SO,SO,77

FORTRAN IV G LEVEL 1, MOD 1 MAIN DATE = 68273 10/2

0050

tt0 051

52 053

0054 0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081 0082 0083 0084 0085 0086 0087 0088 0089 0090 0091 0092 0093 0094 0095 0096 0097 0098 0099 0100 0101 0102

8 103

77 IF(I-30) 101,78,78 78 IF(I-36) 80,80,79 79 IF(I-38) 101,82,82 82 IF(I-44) 80,80,101

101 A(I,J)=Ow2S*(A(I,NA)+A(1 ,NB)+A(NC,J)+A(NO,J» 80 CONTINUE

DO 102 1 = 2 , 48 DO 102 J=2,48 X(I,J)=ABS(A(I,J)-B(I,J»

XA=.X(l,J) B ( 1 , J ) =A ( 1 , J )

IF(XA-00001) 103,50,50 102 CONTINUE 103 WRITE(6,107) '" 107 FORMAT(lH ,45HTHE CALCUL~TED VALUES OF (P&~) ARE AS FOLLOWS)

DO 109 1=1,49 109 l'JRITE(6,110) (A( 1 ,J) ,J=13 ,36) 110 FORMAT(lH ,24F502)

wRITE(6,203) 203 FORr~AT(1H1)

WRITE(6,204) 204 FORMAT(lH ,48H THE PHOTOELASTIC VALUES OF (P-Q) ARE AS FULLOWS)

DO 205 l=l,L:-9 205 WRITE (6,206)(C(I,J),J=13,36) 206 FORMAT(lH ,24F502)

00 208 1=1,49 Da 208 J=1,49

208 D(I,J)=(A(I,J)+C(I,J))/2000 WRITE(6,53)

53 FORi~A T (lHl) WRITE(6,209)

209 FURMAT(lH ,42H THE CALCULATED VALUES OF P ARE AS FOLLOWS) DD 210 1=1,49

210 y.J RIT E ( 6 , 211 ) (D ( 1 , J ) , J = 13 , 36) 211 FURMAT(lH ,24F502)

.00213 J=1,49 DO 213 1=1,49

213 E(I,J)=(AlI,J)-C(I,J))/2000 \~ RIT E ( 6, 300 )

300 FOKMAT (lH1) WRITE (6,30U

301 FORMAT(lH ,42H THE CALCULATED VALUES OF Q ARE AS FOLLOWS) DO 302 1=1,49

302 WRITE(6,303)(E(I,J),J=13,36) 303 FOk~AT(lH ,24F502)

DO 305 1=1,49 DO 305 J = 1,49

305 F(I,J)=C(I,J>l2000 WRITE(6,54)

54 FURf"lAT(lHl) ~RITE(6,306)

306 FORMAT(lH ,36H MAXIMUM SHEAR VALUES ARE AS FOLLOWS) DO 307 1=1,49

307 WRITE(6,30S)(F(I,JJ,J=13,36)

FORTRAN IV G lEVEl l, 1.,00 l. MAIN DATE = 68273 10/20

0104 308 FURMAT(lH ,24F502)

.105 DL=OoO 106 DO 111 J=1,49

0107 111 OL=DL+A(1,J)+A(49,J) 0108 AVLD=DL/98 0 0 0109 WRITE(6,40i) 0110 401 FORI'IAT (1Hl) 0111 WRITE(6,309) 01.12 309 FORr'1AT ( tH ,39HAVLD IS AVERAGE lOAD ON LOADED BOUNDARY) 0113 WRITE(6,311)AVLD 0114 311 FORMAT( lrl ,6H AVLD=,F502) 0115 DG 402 1=1,49 0116 DO 402 J=1,49 0117 402 G( 1, J )=D (l,J )JAVLD 0118 WRlTE(6,403) 0119 403 FORMAT(lH ,4lH P STRESSES IN CONCENTRA TI ONS OF lOAD ARE) 0120 DO 404 1=1,49 0121 404 WRITE(6,405)(G(I,J),J=13,3é) 0122 405 FORMAT( lH ,24F50 2) 0123 WRITE(6,406) 0124 406 FORMAT<lHl) 01.25 DO 407 1=1,49 0126 DO 407 J·=l, 49 0127 407 H( I,J)=E(I ,J)/AVLD 0128 WRITE(6,40S) 0129 408 FORMAT(lH ,41H Q STRESSES IN COI\:CENTRATIONS OF lOAD ARE) 0130 DO 409 1=1,49 0131 409 W RITE ( 6 , 410) (H ( 1 , J ) , J= 13 ,3 6) 0132 410 FORr"AT ( l H ,24F5 0 2) 0133 CALL EXIT 0134 END

TOTAL MEMORY REQUIREM~NTS 016464 BYTES