Table Of ContentThe Pennsylvania State College
The Graduate School
Department of Mineral Engineering
Division of Mining
Application of Dimensional Analysis to the Control
of Strata Surrounding a Mine Opening
A thesis
by
Arthur William Brune
Submitted in partial fulfillment
of the requirements
for the degree of
Doctor of Philosophy
August 1952
Approved:
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Date * Chief, Division of Mining Engineering
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Auo us /9'f?' L
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Date ' Head, Department of Mineral Engineering
i
ACKNOWLEDGEMENTS
The author wishes to express his gratitude to all those who have
aided materially in the preparation of this thesis.
The work was performed under the direct guidance of Professor
A. W. Asman, Chief, Division of Mining, to whom acknowledgement is herewith
made for the excellent criticism and helpful suggestions that he offered,
as well as granting the use of the reproducing machine for the making of the
graphs and drawings. Professor D. R. Mitchell, Head, Department of
Mineral Engineering, suggested the thought which became the basis of the
thesis. The picture of the model was taken by Dr. F. L. duBreuil,
Assistant Professor, Division of Mining. Thanks sire due to Mr. A. W.
Bitner, Instructor, Division of Mining for tlje remarks and criticism that
he tendered. To Mr. Yoh Han Pao, Department of Engineering Mechanics,
acknowledgement is made for his helpful advice, which he was ever ready to
offer, concerning this study.
Through the courtesy of Dr. J. Marin and Mr. S. S. Eckley,
Department of Engineering Mechanics, use of the Baldwin switching unit and
of the Baldwin strain indicator was possible. Dr. J. W. Fredrickson of
Dow Chemical Company, formerly Chief, Division of Metallurgy of this
institution, kindly permitted the use of the hydraulic testing machine.
And finally, but far from least, sincere thanks are rendered to
Mrs. Elda M. Brune for her patience and deep understanding which she showed
toward the author throughout the entire period during which the study was
performed.
I
ii.
TABLE OF CONTENTS
V.mL
PREFACE..................................................... 1
ROCK PRESSURE................................................ 4
Failures of Rock Underground ................. • . . . . 13
Subsidence............................................ 14
NEED FOR THIS STUDY........................................ 16
MODEL STUDIES OF MINES..................................... 18
Fayol's Work.......................................... 18
Bucky's Work ................................ 19
DIMENSIONAL ANALYSIS....................................... 22
Advantages........................................... 22
Principle.............................................. 22
Application to This Study................................ 26
Min6 Opening and Strata............................ 26
Model Material .............. 26
Loading of Models ......................... . . . . . 32
STRAIN GAGE............................................... 38
Principle............................... 38
Application ........................ . . . . . 39
MATHEMATICAL CRITERION FOR ROCK FAILURE......................... 43
PROCEDURE................................................. 48
Elastic Properties of Model Materials .. .............. • 48
Modulus of Elasticity ............................ 48
Poisson's Ratic......................... 48
Method of Loading Models ................................. 50
iii
TABLE OF CONTENTS (cont.)
Z£S®
Measurement of Strain............................... 52
Experimental Arrangements . . . . . . . . . . . . ....... 52
RESULTS................................................ 53
DISCUSSION.............................................. 76
Modulus of Elasticity ......................... 76
Poisson’s Ratio .......... . 77
Strains ................ . . . . . . . . 78
Stresses...................................... . . 79
Mohr Envelopes of Stress............................ 81
Stresses at the Sides of Ribs............ 85
Roof Stresses................. 86
Depth of Overburden..................... 90
CONCLUSIONS............................................ 91
APPENDIX................................................ 94.
BIBLIOGRAPHY............................................ 127
iv.
LIST OF ILLUSTRATIONS
Plate Page
1 Photograph of model assembly................. vii.
Figure
1 Floor pressure normal to longwall face . . . . ............ 9
2 Surface movements, underground convergence, and pack load
for longvall mining . . . . . . . . . . . . . . . . 9
3 Coal mine entry . . . . . . . .......................... 27
A Model 1 ............................................. 29
5 Model 2 ............................................. 30
6 Model 3 ............................................. 31
7 Schematic diagram of strain-gage circuit ............. ... 41
8 Position of strain gages on model ....................... 42
9 Rectangular prism loaded on four sides.......... 44
10 Mohr circle of stress......................... 4.6
11 Mohr envelope of stress ............... 46
12 Method of loading beams for determining moduli of elasticity • 49
13 Plan of base plate, supports, and model................. 51
14 Mohr envelopes of stress in prototype, left coal rib, model 1. 57
15 Mohr envelopes of stress in prototype, right coal rib, model 1 58
16 Mohr envelopes of stress in prototype, left coal rib, model 2. 59
17 Mohr envelopes of stress in prototype, right coal rib, model 2 60
18 Mohr envelopes of stress in prototype, left coal rib, model 3* 62
19 Mohr envelopes of stress in prototype, right coal rib, model 3 62
20 Vertical stresses in prototype, left coal rib, model 1 . . . . 64
21 Vertical stresses in prototype, right coal rib, model 1 . . . 65
V.
LIST OF ILLUSTRATIONS (cont.)
Page
22 Vertical stresses in prototype, left coal rib, model 2 ..... 66
23 Vertical stresses in prototype, right coal rib, model 2 . . , • 67
24- Vertical stresses in prototype, left coal rib, model 3 ..... 68
25 Vertical stresses in prototype, right coal rib, model 3 . .. . 69
26 Z rtress in roof of prototype, model 1 . . . . . 70
27 X stress in roof of prototype, model 1 ....................... 71
28 Z stress in roof of prototype, model 2 . . . . . . . ......... 72
29 X stress in roof of prototype, model 2 ....................... 73
30 Z stress in roof of prototype, model 3 .............. 74
31 X stress in roof of prototype, model 3 . . . . . . 75
32 Load-deflection curve for Tenite I-001-S2.................. 118
33 Load-deflection curve for Tenite II-218-S . 119
34 Load-deflection curve for Lucite HC-201 . . . . . .......... 120
35 Overburden-ey curves for Tenite II-218-S, model 1 ............121
36 Overburden-ez curves for Tenite II-218-S, model 1 . . . . . . . 122
37 Overburden-ey curves for Tenite II-218-S, model 2 . . . . . . . 123
38 Overburden-ez curves for Tenite TI-218-S, model 2 .......... 124
39 Overburden-ey curves for Lucite HC-201, model 3 . . . . . . . . 125
40 Overburden-ez curves for Lucite HC-201, model 3 . . . . . . . . 126
vi.
LIST OF TABLES
Table p«pr»
1 Arrangement of Models ............ . . . . . . . . . . . . . 26
2 Specific Gravities and Densities of Plastics ............... 32
3 Loads Applied to Models .............................. 36
4 Elastic Properties of Model Materials ................... ♦ 50
5 Stresses in Prototype, Mine Model 1, Condensed Table . . ... 54
6 Stresses in Prototype, Mine Model 2, Condensed Table . . ... 55
7 Stresses in Prototype, Mine Model 3, Condensed Table ....... 56
8 Equations of Mohr Envelopes of Stress ........ 63
9 Moduli of Elasticity of Plastics....................... 77
10 Polsson's Ratios of Plastics • • • • • • • • • • • • • • • • • 77
11 Test of Mine Model 1 ................................... 94
12 Test of Mine Model 2 .................................... 97
13 Test of Mine Model 3 .....................................100
14 Stresses in Prototype, Mine Model 1 ....................103
15 Stresses in Prototype, Mine Model 2 ....................107
16 Stresses in Prototype, Mine Model 3 .................... 133
17 Flexural Test of Tenite I-001-S2...........................115
18 Flexural Test of Tenite II-218-S...........................116
19 Flexural Test of Lucite HC-201.............................117
1.
PREFACE
In underground mines the rock surrounding the openings made by
man fall eventually* To state that they can be maintained indefinitely is
untrue because , as certainly as nature abhors a vacuum, likewise nature
abhors an opening made beneath the surface of the earth* Of course, it
must be admitted that the rock around some natural underground openings,
l*e., caves, has not collapsed after centuries* Nevertheless, even such
openings will eventually fail after erosion has continued long enough.
Mast manmade underground openings are not in strata that are
strong enough to support themselves as is true of caves* Furthermore, in
caves the openings have a natural arched shape with the weaker rock having
been' removed by solution, leaving the stronger behind* However, the open
ings in mines usually are larger than those in caves, and the former are
made in rock regardless of the weakness or strength of the material, the
single criterion for the mine opening being that it be in a material that
can be extracted at a profit, with the opening frequently being wider than
those found in oaves. The concomitant result of such an arrangement is
that manmade openings underground collapse more readily* Thus man has
found it necessary to support the openings that he has made, although
frequently the support is eventually inadequate, and it too fails*
As a result, studies and observations of various kinds have been
made of rock contiguous to mine openings, but it is admitted readily that
man really does not know much as to what happens after an opening has been
made underground* ^he purpose of this study is to gain some insight into
what does occur underground by investigating the action of strata that
surround a mine opening* It is an established fact that the roof, ribs or
2.
walls, and floor of many mine openings fall. The roof might fall into the
opening quickly or slowly and in masses or small layers. The ribs or walls
might enter the original opening by spelling, and the floor might come
into the opening suddenly or, as is true of mines where a clay constitutes
the floor, by heave. Each such action is a failure of rock, and the pri
mary purpose of this study is an attempt to determine how the rocks act
when loaded by overburden. The customary failure of rock is a relatively
slow, gradual process eventually resulting in bursts under some conditions.
This study is restricted to this slow action of rock, but the model was
not loaded to failure.
In order to investigate a mine structure it is rather difficult
to work with the prototype, i.e., the mine structure itself. Fortunately,
there is a tool whereby it is possible to work with a model of the prototype;
that tool is known as dimensional analysis. With it a model of convenient
size can be devised, and the model can be tested. From the resulting
action in the model the action of the prototype can be determined, although
that is not exactly true in this case because the model does not precisely
simulate the prototype; consequently the results are not completely
applicable to the prototype.
Plastics have been used in this study as model materials. Other
materials can be used as well; however the variety of plastics available,
most of which have relatively low moduliof elasticity and other precise
physical properties, make them quite suitable for use as model materials.
The model represented a simple, rectangular mine opening in a
series of beds, under-clay, coal, shale, coal streak, through sandstone,
arranged as horizontal strata. A particular plastic material was used for
m
