Stress Analysis Applied to Rock Failure Around Underground Openings

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STRESS ANALYSIS APPLIED TO ROCK FAILURE

AROUND UNDERGROUND OPENINGS

By Manhar J. Pandya

ProQuest Number: 10795915

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53903

A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Doctor of Science.

Signed:

Yvx«/>vk«v/v- . >ianhar J. Pandya

C/t w ià

Q W

Golden, Colorado .. .....

4-

1950

Approved:

uJ.

Clifton W. Livingston

Golden, Colorado Date

4%)^,

|

1950

FOREWORD AND ACKNOWLEDGEMENTS

In practice, it is found that much damage is done to the timbers and machinery by rock pressure alone, and various shapes of openings are progressively adopted to reduce this pressure.

Consequently, the

effect of the shapes of underground openings was first adopted for photoelastic research.

The old polariscope set-up at the Colorado

School of Mines was very inconvenient to work with, and it was very difficult to obtain reliable fringe pattern because of the difficulty in loading and an unsuitable light source; therefore, most isocnromatic pictures have been retouched.

It was also found out that the results

obtained by simple polariscope studies gaVe only the boundary stresses, which were by no means sufficient to provide an understanding of the failure of the opening.

Since the rock is twenty to twenty-five times

stronger in compression than in tension, the stress concentration factor obtained from polariscope work would not give the condition of failure found in metals. Further research in stress analysis by Stress-coat and failure of rock by model studies was carried out at the Colorado School of Mines and it was realized that stress analysis, physical properties of rocks, and failure observed in the field and at the laboratory in the School of Mines can be combined to give a reasonable picture of failure of rock. Since the stress analysis by Stress-coat was mostly for impact loads, stress analysis for static loads was necessary.

The writer also became

acquainted with the stress analysis work being carried out at the U. S. Bureau of Reclamation, Denver, in the photoelastic laboratory of the

Special Assignment section.

At the request of the Government of India

for the writer’s summer training at the Bureau under foreign trainees program, he gained admission to the Bureau of Reclamation, Denver.

Several

underground mining shapes have been completely analyzed for stresses at the Bureau, and a method has been derived for obtaining complete stress analysis for loads acting.ct any angle on the opening, once the stresses for horizontal, vertical, and shear loadings have been obtained after complete investigations l/.

Complete stress analysis of a horseshoe-

1/ Zanger, C. N., and Phillips, H. B., Stress distribution around a standard inspection gallery. Memorandum to Mr. J. J. Hammond, Bureau of Reclamation Special Assignment Section, Denver, September 28, 1948.

shaped opening by interferometer study was obtained.

With a view to

compiling the work of stress analysis for various shaped openings and reducing the duplications of work in future* the writer has liberally made use of the work done at the Bureau in this field.

Part of the work

done in civil engineering is also included, and the writer appreciates the cooperation of Professor T. A. Kelly, Colorado School of Mines, in connection with this work. It was realized from various discussion that no definite plan for the solution of rock failure problem had been set. continued his work of stress analysis.

Therefore the writer

Complète stress analysis for a

stope opening was carried out mathematically. Since rock failure is a function of several factors, the writer has compiled the results of stress analysis, and made one application of Mohr’s criteria of failure.

It is submitted here that this report may

be useful in further work in stress analysis, and may inspire useful

and constructive com; ents from those interested.

It is hoped that the

model failure studied at the Colorado School of Mines, stresses observed by Stress-coat under static and impact loads, and the present stress analysis may be combined with the proper field conditions, and the failures may contribute toward checking and testing the theories of failure of rock and may influence the future work in establishing a sound theory of rock failure. European investigators have contributed some of the recent and more acceptable theories of rock pressure around underground openings 2/.

2/ Schoemaker, R. P., A review of rock pressure problems: and Met. Eng. Tech. Pub. 2495, Nov. 1948.

All

Am. Inst. Min.

the rock pressure theories accept the arch formation above the opening but no satisfactory explanation has been obtained for the occurrance of the arch formation which evidently is connected with rock failure.

The

Mining Department of the Colorado School of Mines has tikcn a lead in the subject of rock failure, and there is no doubt that with the present set­ up and plans for the addition of new equipment such as pnotoelastic interferometer and SR-4 strain gages, much can be achieved towards the solution of rock failure. The writer takes tnis opportunity to express his sincere appreciation to Professor C. W. Livingston, head of the Mining Department, Colorado School of Mines, for his keen interest and continued help in offering the writer every opportunity to prosecute the present work. structive criticisms were indispensible to the writer.

His con­

The writer is

highly indebted to Mr. Carl N. Zangar, and Mr. H. 3. Phillips, Special Assignment section. Bureau of Reclamation, Denver Federal Center, who were always ready for certain clarifications of the operation of various instruments during the writer’s training there as well as on various other occasions.

Mr. I. E. Allen and Mr. W. J. Brown and Mr. H. J.

Kahm of the photoelastic laboratory. Bureau of Reclamation, very kindly helped the writer in recording, reading, and correcting interferometer results.

The co-operation of the Photoelastic Laboratory staff, the

Librarian, and other officials of the Bureau of Reclamation was very valuable in the preparation of this work.

The writer is very grateful

to Professor George Dobbins, University of Colorado, for extending all possible conveniences in pursuing the writer’s program. The writer wishes to acknowledge the assistance of the authorities of the Bureau of Reclamation, Denver, and its Technical Editorial Office for the sanction of the use of the material and data obtained by the writer during his training there.

Without this permission the present

compilation of the material would not have been possible.

The co-oper­

ation of the Bureau of Mines officials in rendering various reports available and allowing their free use in this work is very much apprec­ iated.

The writer is deeply indebted to the excellent literature on

photoelastic analysis by Dr. K. M. Frocht and Dr. Henry Favre, and the mathematical analysis on stresses by Martin Greenspan, which have served as the main source in submitting the resume of the theories involved in this work. The writer appreciates the kind service of Marchant Calculating Machine

Co. for lending the calculating machine. Professor E. 1. Fisher, English Department, Colorado Dchool of Mines, very kindly edited this manuscript and the writer is very grateful for his valuable suggestions.

Miss Mary Hoyt, Librarian, Colorado School of

Mines gave specification for typing, Chantirikn Pandya typed the manuscript work and proofread, and the final typing was done by Mrs. Johnson; to them the writer expresses his sincerest thanks.

TABLG OF CONTENTS

Formulation of Problem ........................................ Stress Analysis ............................................ Stress determination by Photoelastic Studies ........ Theory of Photoelasticity............................. Theory of Elasticity............................... Stresses at a point......... Strains at a point ............................. Stress-Strain Relations..........

1 4 9 10 12 12 21 21

Theory of Light.................................... Definitions..................................... Single Harmonic Rotions and their Compositions Monochromatic Light............................. Polarized Light ................................ Double Refraction................

22 24 26 2? 29 32

Theory of Polariscope ............... Standard Polariscope Set-up ..................... Stress-Optic L a w ....... Proof of Stress-Optic Law........................

35 35 36 40

Polariscope Study........ Fringes and Black Dots.................... Isoclinics.......... «......................... Stress Trajectories..............................

40 40 43 43

Stresses from Polariscope Study.................... ............. Shear Stresses ......................... Principal Stresses

48

Theory of Interferometer............. General..................................... Theory of Favre *s Method.......... Proof of the Theory ........ Description of Interferometer...... Adjustments of Interferometer............ ;.....

55 55 57 57 67 74

Model Materials.......... Specification for an Ideal Photoelastic material.e Common materials and Their Features...............

85 85 87

The Model......................................... Preparation of Photoelastic Model............... Effect of Thickness of the M o d e l ............... Transition from Model to Prototype ........ Stress Patterns................. Time-Edge Stresses......................... Stresses due to Creep........

43 49

92 96

97 97 99 99 100

Page Machining Stresses.......... Boundary Clearness............................... Setting of Model and Proper Loading. ....

100 100 101

Setting and Testing Polariscope...........

101

Loading Frame...................

102

Photography.................... Camera ...................... Film......................................... Numbering................................. Exposure....................... Isoclinics ............. Colored Stress Pattern........................... Developing .............................. Fixing....................... Reduction.............................. Printing ......................

104 104 104 107 107 107 107 108 108 108 109

Boundary Stresses From Polariscope............... General. .................... ...... . Calibration Test ............................... Pure Tension................................... Drift Opening Shape No.1 ........................ Drift Opening Shape No.2 ................. Drift Opening Shape No.3 ................... Drift Opening Shape No.4 ......................... Stope Opening Shape No.5 ......................... Inclined Load on Drift Opening Shape No. 3 ....... Tunnel Opening Shape No. 6 ..................... Deep Mine Opening Shape No. 7 ...................

HO

124 127 130 133 135

Reference to Polariscope Studies and Boundary Stresses Writer ’s Studies ....... Greenspan's Studies ....... Duvall1s Studies .... Bureau of Reclamation Studies ...................

143 143 143 147 152

115 115, 119 121

152 Stress Analysis Using interferometer.......... Loading F r a m e .............................. 154 Calibration Test ........ 154 Calibration Constants .............. 158 The Polariscope S t u d y ....... 159 The Interferometer Study .................... 162 Constants A, B, and K ........................... 162 Equilibrium Check ............................... ’ 168 The Stresses ................................... 169

Page Reference to Interferometer Studies ................ Stresses for Inclined Loading ................... Principal Stresses Around a Gallery ............

169 1?6 189

Stress Determination by Analytical Method ................ ........ Greenspan’s Solution for Holes in Plates Introduction ............. The Co-ordinate System ............................. .......................... The Stresses The Stresses Along Inner.Boundary ................. Curvilinear Co-ordinates ................ Values of Constants and Variables ............... Co-ordinates for the Values of a = 3.70, 3 = 0 degrees to 90 degrees .................. Co-ordinates for a = 1.14 ana 6 = 0 degrees to 90 degrees ................... Co-ordinates for a = 1.30 and -3 -0 degrees to 90 degrees .............................

197 197 197 198 199 205 214 214

220

Stresses at the Boundary..................... Equations for Stresses at Points on Boundary ..... Co-ordinates of Boundary Points ................. Boundary Stresses for Vertical Load ............ Boundary Stresses for Horizontal Load ....

221 221 223 224 225

Stresses in Neighborhood of the Opening ............ Stresses oa , op and Tap for a = 0.46 ?nd 3 - O to 90 degrees ...................... Stresses ca , op and T&3 for a = 0.70 and p - 0 to 90 degrees ................... Stresses oa , on and Tap for a = 1.14 and 3 = 0 to 90 degrees ........... Stresses oa , og and Tap for a =1.30 and |3 = 0 to 90 degrees .............................

227

Principal Stresses for Stope Opening ...........

218 219

227

235 239 244 248

Reference to Analytical Study .........................

254

Theories of Failure ..... General...... Resume of Rupture Theories............... Kaximum-Norml-Stress Theory ............ Maximum-i\ormal-Strain Theory........... Theory of Friction ................................ jXaximum-Shearing-Stress Theory ............ .'........ Rupture Criteria of 0. M o h r ........................ Maximum-Strain-Energy Theory....................... Hancky-Von Mises Theory .............

257 257 253 263 264 264 265 265 268 268

Page Comparisons ofTheories .............

Application of Mohr’s Criteria of Fnilure to Results of ......... Stress Analysis Conclusions

...................

269

269 276

Bibliography... »................................................

230

References............ ............................... ........

282

F O R M U LA TION

0 F

P R 0 B L E M

The failure of any mass can be considered as a function of two factors: 1. and 2.

The stress distribution in the mass Certain physical properties of the material of the mass.

Various theories of rupture take into account the above two factors, and it is only natural that the latter should be applied to the failure of rocks. The knowledge of metal failure, the laboratory tests of rock fail­ ure, and field observations make it possible to analyse the problem of rock failure further.

The stresses in the rock can be inherent and may

have been induced during its formation.

On the other hand the stresses

can be due to external forces such as the superincumbent load of rock formations, reaction due to lateral confinement and Poisson*s effect, unbalanced forces resulting from faults, folds, veins, excavations, disintegration due to temperature and moisture, and various other loads resulting from the phenomena not yet fully understood such as rocki

bursts.

These pressures may be observed as,

(a) Large sudden motions of the rock mass due to the release of unbalanced interior stresses. (b) Deformation of the rock mantle by the steady static pressures of inherent and external forces. and

(c) Rapid progressive deterioration of the rock by weathering

agents. The physical properties of the rocks vary widely since the char­ acter of the rock is dependent upon its geologic history.

The faults,

folds, veins, intrusions, cleavage and foliation make the rock inhomogeneous.

The varying conditions of pressure and temperature further

complicate the physical structure of a rock.

In general, however,

rocks can be arbitrarily classified, depending upon their density of compaction and generally homogeneous character, as 1.

Material without cohesion comprised

of loose rock such as

crushed stone, rubble, sand, crushed stone in the neighborhood of faults and sediments.

The effect of moisture makes this kind of rock especial­

ly dangerous. 2.

Compact rocks with fault surfaces not capable of resisting

tensile stresses, and include surface complexes which are weathered, eroded, and compacted. 3.

Compact material of plastic origin coming from the deeper

layers of mountain formation. 4«

This rock is essentially strong.

Naturally plastic rock at great depth.

All the classes of rock

come under this classification at great depth.

The above classification suggests that rock can be more brittle than cast iron or as loose as sand; certain sand stones can be quite elastic and can also be very plastic and viscous.

At the same time it is observ­

ed that the rock can stand a great deal of elastic strain in the general conditions observed underground.

It has been suggested that a program

for careful determination of the physical properties of rocks including triaxial tests and microscopic examination may yield valuable results. The stress distribution inside the rock masses due to the conditions of loading affect the physical properties of the rock, and the rock may yield due to crushing or compression failure, separation or tension failure, sliding or shear failure, or displacement failure, which is the mixture of the above normal and shear stress failures. The problem then reduces to the complete stress analysis of the

rock mass around excavations in elastic and plastic medium under static and impact loads as well as under the effect of temperature and moisture, and the knowledge of the complete physical properties of the rock.

The

effect of temperature and moisture becomes critical in porous and loose rocks encountered, in general, in the upper layers of the earth.

In

deep mines, spelling may occur probably because of the high geothermic gradient and moisture from the ventilation current.

Impact loads,

probably due to unbalanced stresses, generally occur in most of the deep mines and some shallow mines.

The plastic behavior of rock is a

very common phenomenon and is mostly encountered in sedimentary forma­ tions.

The rocks encountered in metal mining are elastic up to the

average depth mined today when under static load. Consequently, stress analysis of certain common shapes of mine openings in elastic rock media was adopted in the study.

The first

part of the study includes stress distribution at the boundary of the opening, and complete stress analysis around certain shapes.

The

literature on the stresses at the boundary of various shaped openings obtained by the Bureau of Mines investigations is also included.

Com­

plete stress analysis for certain shapes carried out at the Bureau of Reclamation has been included with a view to compiling in one report the investigations as applied to underground mining shapes.

Since

complete information on the physical properties of various types of rock is not available, application of the stress analysis was not possible.

Among the theories of failure of rock, Mohr1s criterion of

combined friction and strength failure is very frequently used.

There­

fore a reverse procedure of combining the stresses with Mohr’s criterion of failure, which in general satisfies the combined shear and normal

4

stress failure, was adopted for one of the openings, and the shapes of failure arches were drawn for various static loads.

The general summary

of the methods of stress analysis and theories of failure of rock have also been included.

The justification of elastic rock under static

load as investigated here is that most metal mine opening failures are covered in this field and that a better application of the observed failure pattern is possible.

The problem has been divided into stress

analysis, theories of failure, and application of the Mohr's criterion of failure to the results of stress analysis.

STRESS ANALYSIS

Of several methods of stress determination, the foremost is the analytical method using mathematical theory of elasticity.

This theory

furnishes stress functions from which exact stresses and strains are calculated.

The mathematical difficulties multiply as* the boundaries

and the loads become irregular, and the partial differential equations become.unwieldy for practical purposes.

However, for geometrical

shapes and specially for the holes in the plates, easier solutions have been worked out.

The mine openings can be considered to repre­

sent holes in infinite media.

Stress analysis for one such opening

has been carried out. Experimental methods of stress analysis have been developed to obtain easier and quicker solutions for the engineers.

The brief

summary of these methods and their relative merits are considered J/.

3/ How to organize for experimental stress analysis: vol. 19, no. 4, pp. 113-136, April 1948.

Product Eng,

Brittle coating or stresscoat method requires the use of a spe­ cial paint that cracks under load.

It gives an over-all picture of

the surface stresses and their directions.

Its advantages can be

enumerated: (a) Tendency of strong sections to pass the load to weak regions is strikingly visible. (b) Idea of stress path is quickly obtained. (c) Stress distribution over the entire surface is obtained, making it possible to know where strain gages may be put to get exact strains. (d) Small critical regions near joints, fillets, and holes can be studied. (e) Impact loading can be used. Its limitations are: (a) No time scale in stresses and only maximum strains are recorded during cyclic loading. (b) Only one loading condition can be applied for one coating. (c) Temperature and humidity should be controlled in the laboratory. (d) Accuracy better than 20 to 25 percent requires great skill and experience. Mechanical strain gages or extensometers measure directly the deformations on a portion of the surface.

They are advantageous in

that : (a) They are useful' for short duration static measurements where remote measurements are not required, (b) Their cost is low, and they can be used over and over again. (c) They need not be calibrated. (d) Some extensometers can be used beyond the temperature range of other strain gages.

The following disadvantages make the application of extensometers limited: (a) More strain gages can not be used simultaneously as they become difficult to read. (b) Most of them are not useful for dynamic studies. (c) They can not be mounted on moving parts because their weight changes the natural frequency of the parts. (d) They are difficult to clamp securely to the test surfaces. The types of the mechanical strain gages are:

Huggenberger tensometer,

dial-type extensometer, Tuckerman optical strain gage, photoelastic extensometer and scratch gage. Electric strain gages are the most versatile, accurate, sensitive because they are small, continuously indicating and recording, compared to other strain gages.

They record strains by measuring electric

resistance or magnetic reluctance. 1.

They are of three types:

Magnetic or reluctance type have high output frequently not requir­

ing amplification, measure small displacements as well as strains, are durable, and can be permanently applied to parts under investigation. They are, however, bulky and necessitate screwing to the parts. 2.

Unbonded resistance wire gages (Stratham type) can record up to

+ 1500 microinches, are best for measuring displacements, measure small loads of 5 to 10 gms, and are advantageous over SR-4 gages for loads below 2000 psi.

They are not suitable for measuring surface

strains and require screwing or clamping. 3.

Bonded resistance-type gages (SR-4) are most frequently used.

(a) They are bonded close to the surface and hence are more accurate. (b) They measure one millionth of an inch per inch{strains up to one

7

percent o r more), and their calibration is steady under shock loads, explosive forces, or severe vibrations. (c) They are

cheap, andthe expense is not

repetitive.

(d) They are

very lightand can be applied

to, sealed, or mountedon

moving or stationary parts, as they stand vibrations up to 30,000 cps. (e) They can

be used inaerodynamics, vine

tunnels, and fluids.

(f) They are

free from hysteresis and have law creep value, andpaper

mounted or Bakelite mounted gages can stand temperature up to 500° F. (g) They are remotely indicating;i.e. their values can be recorded, photographed, radioed, or amplified for control purposes. Tneir limitation can be summed up as, (a) low output and amplification necessary specially for dynamic loads. (b) high cost of instruments, particularly for multiple recording anti dynamic loads. (c) required knowledge of electronic instrumentation for complicated stress study and dynamic loads. The X-Ray defraction method of stress analysis I*/ enables one to

4/ Norton, J. T., and Rosenthal, D., Recent contributions to x-ray method in the field of stress analysis: Proceedings of Society of Experimental Stress Analysis, vol. 5, no. 1, p. 71, 1941.

measure the distance between atoms and thus use interatomic distance as a gage length for the determination of elastic strains at the sur­ face.

A beam of monochromatic light falls at an angle and illuminates

the tiny grains of the specimen which neTract the light. The Refract­ ed cone is photographed, and from the diameter of the cone the spacing of the atoms is determined. is still limited.

The method is fairly new, and its field

The method was slow, and the maneuverability of the

instrument was limited.

These limitations have been overcome by high

tension transformer, control panel, and x-ray tube recording device. Photoelastic stress analysis utilizes the property of transparent isotropic materials to doubly refract light when loaded.

The advantages

of the method are as follows: (a) It gives stress distribution not only at the surface but throughout a section. (b) Stresses at the boundaries and at certain points are quickly obtain­ ed as are also the stress concentration factors.

Interferometer study

gives complete stress analysis for static loads. (c) It indicates stresses in otherwise inaccessible parts where- strain gages can not reach. (d) Tests can be completed and the stresses can be studied in small size models. (e) Value of stress at a point can be found which is equivalent to strain gage length of zero. The disadvantages of the method are:

.

(a) Operator must know higher mathematics for qualitative work.

However,

interferometer study eliminates this difficulty. (b) Two-dimensional models do not always reflect the conditions in threedimensional reality. (c) The three-dimensional photoelasticity may remove the above objection, but for the present it is beyond the range of average engineering group. (d) Dynamic and impact values are hard to get,though attempts are made in this field. Among the experimental methods of stress analysis, photoelastic analysis stand foremost as it gives point by point stress analysis.

The

strain measurements by SR-4 gages have been extensively used by the Bureau of Mines, but they give average value of the strain over the gage length.

The stresscoat gives crack lines which are difficult to read.

For laboratory studies the photoelastic method gives the results which come closest to the results obtained by the mathematical theory of elas­ ticity.

Photoelastic analysis of several shapes of mine openings was

conducted by the writer and is included.

STRESS DETERMINATION BY PHOTOELASTIC STUDIES

Like other methods of stress-analysis, photoelasticity* depends upon the effect of stresses on other properties of the material. Photoelasticity can be considered a science which deals with the ef­ fects of stress upon light passing through a transparent material or being reflected by an opaque material.

Its application to engineer­

ing consists of the converse process in which stresses are inferred from ihe.optical effects. Photoelastic analysis is a method of determining stresses usual­ ly in a transparent model of the prototype loaded statically or dynam­ ically, by using polarized light.

Obviously the study of photoelas­

ticity requires the knowledge of light in the branch of physics and and elasticity in the branch of mechanics J>/. —

':*»

.

-

-

j/ Alexander, N., Photoelasticity, p. 1, Rhode Island State College, Kingston, R.I., Jan. 1936.

From Phot or Phos, a Greek word meaning "light”, and elasticity ”equilibrium of perfectly elastic bodies under the action of external forces."

10

THEORY OF PHOTOELASTICITY The discovery of temporary double refraction in transparent iso­ tropic materials laid the foundation of the science of photoelasticity, ’ The phenomenon of the breaking up of a light ray into two rays in certain crystals as the permanent or intrinsic property of these crystals was known to the early physicists.

Seebeck 6/ seems to have been the first

6/ Seebeck, Schweiggers Journal, t. 7, p. 288.

to observe double refraction in stressed isotropic material.

Double

refraction existed as long as the material was stressed, and was the temporary or extrinsic property of the material.

Sir David Brewster %/,

%/ Brewster, David, Philosophical Transactions of the Royal Society, London, 1814-1816.

independently of Seebeck, carried out the first important experiments in this field.

A. Fresnel 8/ carried out research on temporary double re­

ft/ Fresnel, A., Note sur la double refraction du verre comprime*, Annales de Chimie et du Physique, t. 20, pp. 376-383, 1822.

fraction of glass and confirmed most of Brewster *s findings.

Wertheim 2/>

9/ Wertheim, Sur la double refraction temporairement produite dans les corps isotropes, et sur la relation entre 1*élasticité rfiéchanique et élasticité optique, Annals de Chimie et de Physique, 3eserie, t. 40, p. 156, 1854.

made the first quantitative experiments and found the law tying relative

11

retardation of doubly refracted rays with the difference in principal stresses, thickness of the material, and the optical constant of the material.

Neumann 10/ in 1841 made some remarkable studies in the field t

10/ Neumann, Die Gesetze der Doppelbrechung des Lichtes in comprimierten oder ungleichformig erwarmten unkrystallinischen Korpern, Gesammelte werke, Bd III, Teubner, Leipzig, 1912.

of temporary double refraction, and established the general theory of temporary double refraction of transparent isotropic bodies under threedimensional stresses.

Neumann's general theory was confirmed by numerous

physicists, notably E. Mach, W. Voigt, Kerr, and Pockles. In studying the temporary double refraction, the nineteenth century physicists had sin mind principally the research in optical phenomena as a function of the mechanical phenomena or the interior stresses.

However,

in the last half of the past century, attempts were made to solve the problem of interior stresses as a function of optical experiments by Leger, J. Kerr, and C. Wilson.

It was only in 1900 that the complete

solution of the problem of two-dimensional stresses was obtained by Augustin Mesnager 11/.

He established optical-mechanical method for the

11/ Mesnager, A., La deformation de solides, Congrès international de Méthodes d ’essai de matériaux de construction, t. 1, p. 149, Paris,

1900.

determination of two-dimensional elasticity.

Persistent and life-long

labors of E. G. Cocker and L. N. G. Filon 12/ contributed mainly to the

12/ Cocker, E. G. and Filon, L.. N. G., Treatise on Photoelasticity, Cambridge University Press, London, 1930.

development of the science of photoelasticity. New development in larger and better polarizers and in better model materials has made further researches possible.

Since the theory of

photoelasticity requires the study of the theory of light and theory of elasticity they will be considered now.

The excellent text book on

photoelasticity by M, M. Frocht 13/ has made the following abstract easy

13/ Frocht, M. M., Photoelasticity, vol. 1, John Wiley and Sons, Inc., New York, 1941.

for the clear understanding of photoelastic analysis, and unless other­ wise stated, the following work has been taken directly from this book.

THEORY OF ELASTICITY The mathematical theory of elasticity concerns itself with the stresses produced by temporary displacements of points so long as the external force exists, and as such the theory holds within the elastic limits of the Hooke's law.

There are two kinds of,displacements:

the rigid-body

displacements which consist of pure translation, pure rotation, and tAeir combination and are called general plane motion; and the displace­ ments which give rise to change in shape and size of the body.

The

latter type of displacements gives rise to elastic deformations, and strains beyond the elastic range called plastic deformation in which the size and shape of the body are permanently deformed.

The following

resume of the theory holds for two-dimensional stresses only unless expressed otherwise.

Stresses at a Point The stress o at a point P across a definite area M

is defined

13

by the expression a = Limit AA4 0

AF , where AF AÂ

denotes the increment of the force on the area AA surrounding the point P.

The stress o is for convenience resolved into normal stress, which

may be tensile Or compressive, and shear stress, normal and parallel to the area AA, respectively.

Since photoelastic analysis for the most part is a study in plane or two-dimensional stresses, a rectangular element of area AA in a model may be considered. Fig. 1, p. 17.

The general two-dimensional

stress system on this element can be defined by two normal tensile and/or compressive stresses and two tangential or shear stresses.

A

simple deduction proves that cross shears are equal on a rectangular I element. If the element is oriented parallel to x-axis and y-axis and the compressive and tensile stresses are considered negative and positive, respectively, normal stresses parallel to these axes can be designated as ♦ ox and £ Oy.

A line passing through the comers

where tangential stresses on two adjacent faces meet is called the shear diagonal.

The shear diagonal passing through the first and the

third quadrant of the element represents positive shear +‘*Xy, and the other diagonal represents negative shear -t ^ .

An element acted upon

by ox , Oy, and T^y stresses is completely defined; however, if this element is rotated to a plane parallel to AA, Fig. 1, p. 17, inclined at an angle +0 to the x-axis, the stress distribution on this element is changed to: o9 = P-f r A -gy 2 and

tq = -~

ay

+ '

Px-USE 2

Sin 29 -

Cos 29 *►

Cos 29

Sin 29

14

where oq is the normal stress and Tq is the tangential stress in the direction 8.

For getting the stresses on the other plane of the element,

9 is replaced by 0 + ~

degrees.

If any of the stresses or the angle

are negative, their signs are changed wherever they occur.

The above

equations are easily derived by drawing a polygon of forces and taking their components in the direction of 0, and by using trigonometric relations.

Now if a new set of axes X 1 and Y* are taken parallel to

direction 9, and 9 + ^ « 6 1, Og becomes o^t and, eg» becomes Oy,, and Tg or t @ i becomes Tx iyt.

Mohr[s Circle For the given values of o , a and t , there always exists one x y xy value of 9 when Xq vanishes; then the oQ and Og, are the only stresses acting on the element.

These stresses acting on the planes where shear

stresses have vanished are called principal stresses, p and q.

When

Td = 0,

° -X

sin 29 » T%y Cos 29

Tan 20 -

(1) x ” °y

therefore,

2Txy o -* Ov i■ ""-r— Cos 29 * ;■...4A—"1'J g '""■1 -"-r" , uos Rr" V ^°x ” + ^Txy / (°x “ °y) * ^*Txy

sin 29 «*

*

hence

2

°x + °y . , /o (ox ~ «y) + ^cy Oq ■ p » -*-5— ^ + 1 /2 y/i°x - o J ‘ + 4t

y'2 +

p - °x * °y

♦ 1 /2 / ( o x - Oy)2 (2)

also

Tgt » q » 0x g

- 1 /2 1/ (°x “ °y)^ +

It will be seen from Equation 1 and 2 that given a , a and t , the x y ,xy

15

principal .stresses p, q and the angle 0 at which they act can be derived~ mathematically.

A graphical solution is obtained by the construction of

Mohr’s Circle, Fig. 2, p. 17» and DD’ =

Along the o and r axes, 00’ = ox> 0E1 = Oy

are laid down in the directions to agree with the signs of

the stresses.

A simple convention is followed for convenience, in that

T^y is drawn above or below the point of ox according as

is acting

upward or downward on the right hand face of the element; i.e., r^y line is drawn below ox if x^y is negative and x%y is drawn above ox if x ^ is positive. With the center C of E ’D* and radius CD, a circle is drawn cutting o-axis at A and B.

Then 0A * p, OB * q and angle ACD * 20.

The proof of the circle is easily obtained by considering!

0A » OC + CA = OC + CD = 0C + ✓ CD'2 + DD'2

-

+ yf°x_^y)2 +

+ 1/2 / (0X - Oy)2 ♦ Ur^r

“ P Similarly

OB = q, and

DD1 . A 2xxv ■ggj ■ tan 20 “0 “ . 0

•

When p, q, and 0 are given, a reverse procedure for Mohr's circle can be followed to obtain a# and x^, Fig. 3, p. 17.

After obtaining

points A and B, a circle with center C of AB is drawn.

Next a line CD

making an angle 20 with o-axis is laid and a perpendicular from D on o^axis marks the point D ’; then 0D' * Oq and DD' * Xq.

Since the theory

of the circle has been proved, a formula for Gq and Xq in terms of p, q, and 9 can be derived from Mohr's circle. o0 * 0D* • OC + CD' = 0C + CD Cos 20 * 0C + CA Cos 29

- £ -^ a 2

+ £-=_a 2

Coe 29

Results from Mohr’s Circle A number of conclusions follow directly from Mohr’s circle: 1.

If an element is subjected to normal stresses only without shear, then the only planes free from shear are the outside or given planes or planes parallel to them; these shearless pjlanes are called ”principal planes” and the stresses acting upon them are called ”principal stresses” denoted here by p and q.

2. The principal stresses p and q are respectively the greatest and the smallest of the normal stresses in an element. 3.

The shear stresses on mutually perpendicular planes are numer­ ically equal.

4.

00 + 001

5.

Of the two stresses oy and Cg*, the one which makes the smaller

=

p + q

acute angle with the direction of p is algebraically greater of the two. 6.

Maximum shear stress circle

max equals * f the radius of the Mohr's

T mn„

and acts on planes inclined 45 degrees to the

principal stresses. 7.

The normal stresses o^ on planes of maximum shear are equal, given by

8 . Stresses on planes inclined equally on both sides of p or q

are equal.

17

— "y

(a) ri i

ES II

a$

i si

il ¥ s

Fig* Sketches Showing the Determination of the Magnitudes and Directions of Principal Stresses for Given General Systems of Two-Dimensional Stresses, by Means of M ohr’s Circle.

I nr

D'

5 i l -i

ni i s

(0,0) « 0, since p « q « 0

, jfl(0,0) 3fr (0,0) 3, * — =~---- p + -q 1 dp ^ dq lfI (f.p) dp

the equation becomes,

, a, ana £ £ l _ ^ £ > , b dq

•

3^ * ap ♦ bq

Similarly, due to the symmetry of the equation,

3g = bp +

aq

Introducing the thickness t ofthe plate,3^ and 3g become,

3, « atp + btq 1 3^ = atq + btp

II.

) ) )

(1)

Further demonstration of the theory can be obtained by basing

the arguments on Neumann's theory: Since unstressed material is isotropic, indices of refraction are

the same in all the three axes x* y , and z.

The ellipsoid of indices

is then reduced to a sphere represented by the equation,

(x^ ♦

♦ z^) * 1

(2)

v being the velocity of light in all the three directions.

After load­

ing of the model, the indices in the three direction change, changing the velocity of light in different directions.

The ellipsoid of indices »

is then given by,

v ^ x 2 + v22y2 + v^2z2 = 1

(3)

v^, v^ and v^ being the velocities of light in three principal directions

In addition to the optical change in indices, the physical dimen­ sions of the material also change when the material is loaded.

As

considered earlier in the theory of elasticity, the strains of the deformations are given by normal and shear strains in different direc­ tions.

In the principal directions, the shear strains vanish and the

principal strains ex , e ,- and

are left along the principal axes of

the ellipsoid of indices. It has been stated earlier that if 0 produces a strain of e , x x* corresponding strains in y and z directions are given by By * ez • -uex . If a body is acted upon by o^, o^, and the strains ex ,e^,

are given

in x, y, and z directions,

by:'

(4 )

ex - e-L - u(e2 ♦ e^) and similar equations for

and ezJ

in x direction;and -us2 and -ue^ are Oj respectively

in x

is the strain produced by 0 ^ thestrainsproduced

direction. Now,when thebody

by o2 and

is not deformed*

i.e* ex “ ey *

= 0, velocity in all directions is v, and the axes of

the sphere of indices are proportional to v^. axes are

o o proportional to , v2 , and

After deformation, the

2 ,and their relation with

deformations sx , e^., and ez can be given by:

\ z - V2 - C ^ x

-

and similar equations for Vg

C2u (62 ♦e3)

CLe1 2

- v

2

2

and

(5)

2 - v .

2 2 Since and C2 are constants for adjusting the values of v^ - v , •p 2 2 2 v2 - v , Vo v , and u is a constant for the material, the above equation can be written as,

Tl2 " V l

*

a2(el * V

♦ v2

(6)

and two more symmetrical equations. Take a^ > 2kv, a2 » 21v and since a^ and a2 are small In magnitude, the square root can be taken by

approximation. The new equations then be­

come, - V + ke1 ♦ l(e2 + Eg)

)

v2 - v ♦ ke2 ♦ l(e3 ♦ e^)

j )

) (7)

) v

- r * ke3 ♦ l(ei ♦ e2)

)

where k and 1 are new constants of proportionality between the change in velocities due to load and the change in thickness due to load.

This

is Neumann’s hypothesis. If V is the velocity of light in vacuum, n is the index of refrac­ tion when the material is unstressed, and n^, n2, ry, are the indices of refraction in three principal directions of the stressed material

V - ™

"

%

" %

62

n - ni Then,

therefore, n-n^ •

nn^

ke1 > i(e0 + Bo)

n

%

Because k and 1 are compared to velocities of light.

v1

and.

n-n-i « n

~

V

I 6! *

and

v (E2

vx

*=

V

(8)

+ e3 )

In a two-dimensional stress system of 0 ^ and 3

i.e, putting

■ 0 in the equations of the type (1) considered in elasticity under

strains at a point.

1

.pKo, E' 1 - UCo) (9)

E2 ‘ I any one of the principal directions, in terms of the stresses. Fig. 38, p. 64.

A

similar instrument is used in the Bureau of Reclamation,Denver Federal Center, Denver, Fig. 39, p. 69. The writer worked for the Bureau in the summer of 1949, and the following material has been derived from the experience obtained there, as well as from various technical memoranda available in the Bureau of Reclamation Library. The interferometer consists of a cast aluminum U-shaped base on which various optical pieces are suitably mounted, and an optical rail on which a light source and a condensing lens are mounted.

As the

interferometer measures the variation of the wave length of light, minute variations in the distances of the optical pieces make large movements of the interference bands.

Hence, the east base is made thick

and heav^ and mounted on springs to absorb vibrations,* Fig. 40, p., 69. The movement of the interference bands is clearly visible when the instrument is slightly pressed on the sides*

Furthermore, the apparatus

is so delicate that changes in temperature cause the movement of the bands j the room temperature is controlled and a maximum change of only 5* F. is allowed in the room. The optical parts in the apparatus include the following, Fig. 41, P. 69,: 1.

A point source of high intensity plane polarized monochromatic

light, with a marker in the center of the point source:

as only point

by point analysis of stresses is made, and the plane polarized vibra­ tion is made to pass through the principal directions, plane polarized point source of light S is used.

In the telescope the point source

69

Fig. 41 O l o M - u » of oortlj* aed opt leal flat parallela.

is magnified and hence a marker iÿ used to show the center of the source for reference.

Because much of the light is also absorbed and reflected,

a high intensity mercury lamp is desirable to make interference dark bands clearly visible. 2.

A condenser mounted on the optical rail:

the condenser is

moved forward and backward so that there is a pin-point focus of light in the model. 3.

The optical parallel

and

with thin aluminum plating on

one face such that light is both reflected and refracted:

the depth

of the coating is gauged so that 30 percent of the light is reflected and 30 percent refracted, the rest being dispersed and absorbed when the L

1

and L, are at 45° angle to the path of the incident beam. o

light beam S is split up by

The

into an outside path going around the J.

model and an inside path going through the model.

The two plane polar­

ized rays. Fig. 42 (A), (B), p. 71, are again brought together at and the interference phenomenon is developed at L^, Fig. 42 (C), p. 71. 4.

Clear glass optical parallels

and

is mounted on a

spindle and equipped with a tangent arm and a tangent screw.

As the

tangent screw is turned in, the effective length of the glass through which the outside light path must travel increases.

The change in

the length of the outside path can be made equal to the change in the length of the inside path caused by the model under load.

A micrometer

dial, previously calibrated to read directly the changes in the optical paths in terms of pounds per square inch, bears against an agate insert on the tangent arm.

The purpose of

is to give the same optical dis­

tance to the inside path as the outside path where 5.

The two optical parallels

and

has been inserted.

which are given a full

*

71

, - EY E

PIECE

MICA HALF W A V E PLATE OUTSIDE POINT

SOURCE

PLANE

PATH-

OF -MODEL

POLARIZED

MONOCHROMATIC LI 6 H T —

. "-MICA

HALF WAVE

PLATE

INSIDE

OPTICAL

PATH-'

PATH

OF I N T E R F E R O M E T E R

;

OUTSIDE

PATH

INSIDE P A T H

i

i

i

REPRESENTATION OF LIGHT WAVES

•RAY I R A Y 2-^

'INTENSITY VARIATION

APPEARANCE OF INTERFERENCE BANDS IN EYEPIECE

M..42

PHOTOELASTIC OPTICAL

J.t.S.

5-2-40

INTERFEROMETER DIAGRAMS

• .. >•

288-D-M28

first surface aluminum coating such that they act as full reflecting mirrorsi

the reflected ray from optical parallel

is totally reflect­

ed by L,, and the refracted ray from L after passing through L0 is 4 i d totally reflected by L^. 6.

Two half-wave plates, cut from one mica sheet, arranged on

either side of the model and so connected by a piano wire around the inside U of the aluminum base that when one of the quarter-wave plates is moved, the other plate also movesî The two plates are set in reverse position with respect to each other.

The half wave plate has

the property of turning the plane of vibration of the incident ray. The plane of vibration of the incident ray and the plane of vibration of the emerging ray are symmetrical about the plane determined by thé ray and the optic axis of the plate.

Suppose the plane polarized ray

is vertical and the principal direction at the point under analysis in the model is inclined say +25® to the vertical.

Then the optic

axis of the first half wave plate is turned till it is inclined +12*30* and the vertical vibration of the incident ray is turned +25°, so that the emerging vibration from half-wave plate is coincident with the principal direction.

The second half-wave plate reestablishes the

vertical vibration which interferes with the vertical vibration of the outside-path ray. 7.

An eye-piece or telescope with crosshairs to view the inter­

ference bands;

interference bands appear because of the interference

of the two rays even when there is no model interposed between half­ wave plates. Fig. 42 (C), p. 71, in the U-shaped space provided. The crosshairs of the telescope and the interference bands are visible in the eye-piece and are shown in Fig. 42 (D).

8.

A cylindrical holder with rectangular window in which a rectan­

gular piece of the same material and the same thickness as that of the model:

this holder is placed between the optical mirror L

glass parallel

and clear 4 and the rectangular piece of material equalizes the

outside light path with the inside light path in which the unstressed model has been introduced. 9.

An arrangement for introducing a polaroid in front of the

eye-piece, and an arrangement for cutting off the outside path of the light in front of the optical parallel L,: the polarizing axis of this ' o Polaroid is horizontal so that the vertically polarized inside path ray is completely extinguished, when there is unstressed model in the U-shaped space.

Since the outside path ray is cut off, no light will be visible

in the telescope, but when the model is stressed, light will come in the telescope and by rotating the half-wave plates, complete extinction is obtained.

The angle of rotation of the half-wave plates gives half the

value of the inclination of the principal direction to the vertical. Another extinction is obtained at 45 degrees to the first angle and marks the position of the second principal direction. All the optical pieces, L^, L^, L^, L^, L^, L^, are mounted in cylindrical brass boxes with rectangular openings and four screws by which any optical piece can be rotated around the vertical axis as well as horizontal axis.

The holder which holds the rectangular piece of

model material has a screw feed arrangement so that the piece can take larger motions around vertical axis as well as horizontal axis.

The

half-wave plates are mounted on circles graduated in degrees and minutes with verniers at one end to read the angles. Fig. 41, p. 69. The loading frame rides on two rails on the table in the U-shaped space.

74

The loading frame has vertical and horizontal screw feeds so that the model in the frame can be moved 0.05 inches vertically or horizontally for analyzing various points in the model, while the inside path ray is fixed in position.

Adjustments of Interferometer

initial Adjustment

This adjustment is the most complete and thorough of all the adjustments and is necessary only when the instrument is completely out of adjustment.

The adjustment makes use of nearly parallel rays of

light to the interferometer* table and carefully levelled.

The interferometer is placed on a high The six optical flats are removed from

their supports by unscrewing the three capstan nuts which hold each flat.

Care should be taken to protect the flats from finger marks,

scratches, etc.

A transit or level is adjusted to the same height as

the optical flats. Indoor Adjustment 20/:

It is not always easy to obtain a distant

20/ Staff of photoelastic laboratory, Adjustment of the interferometer, indoor target method, Bureau of Reclamation, Denver, Sept. 1, 1948.

object for obtaining parallel rays.

It may be necessary to take out

the apparatus for outdoor adjustment, and the delicate set-up is dis­ turbed when bringing the instrument back into the laboratory.

The

indoor method developed at the Denver branch of the Bureau of Reclam­ ation makes use of several targets and can be called the indoor target method. Equipment: The equipment necessary is

75

1. A room with a clear floor area of about 30 sq. ft. 2. A surveyor’s level or transit 3. A precision level 4. A 50-foot tape measure 5. An extension cord and reflector lamp for illuminating the targets. 6. Screwdrivers and crescent wrenches for adjusting mounts on the interferometer base. 7» A sturdy table to hold the interferometer. 8. Two targets:

one with two centers at a distance equal to the distance

equal to L^L^, on the same horizontal center line, Fig. 43, p. 76, and one target with three centers a distance apart equal to or

and

and L^L^, all located on the same horizontal plane with

the center of the flats. Fig. 44, p. 76. Preliminary Steps: outlined here.

The preliminary steps (before adjustments) are

Fig. 44, p. 76, shows schematically the steps outlined

below to be taken in adjusting the interferometer. 1.

Set up the interferometer on the table near one corner of the room with L nearest the corner. 4 level, and free from vibrations.

2.

Remove all optical flat mounts from the interferometer base except

3.

The set-up should be rigid,

and L^.

Set up the transit about 8 feet in front of

in' line with

L^L^, and at the same elevation as the horizontal centerline through the mounts for the optical flats. 4.

How swing the transit in a horizontal plane about its vertical axis and establish horizontal lines on the walls opposite those adjacent to the interferometer and parallel to L L

and to

76

.Target A

Target B

El

0 ISv

F ig . A3

.Target E

..Target 0

..-Target C

ÿ

El

0

ED

Flg. 44

ALIGNMENT

TARGETS

INTERFEROMETER ADJUSTMENT

Aueusr ieeê

o

- f c l - is

77

Y v 5,

Having established these horizontal lines, the flats can all be set such that the planes of the flats are perpendicular to the horizontal plane through their centers.

This may be done

by sighting the*telescope on each of the flats separately as they are installed in the adjustment sequence.

They should then

be adjusted so as they are rotated about their vertical axes, the image of the horizontal lines on the walls does not change in elevation with respect to the horizontal cross-hair in the transit. mounts.

Adjust the flats by use of the capstan nuts on the Check the adjustment after final locking of the

capstan nuts. 6.

Fig. 4 6 , p. 78, shows the position of the flats in their mounts with respect to the center of the mount.

-

First Adjustment;

After establishing the line L L, swing the transit 1 4 through 90 degrees and establish a point on the wall on the horizontal plane through the center of the flats.

Measure the horizontal distance

from the center of the transit tothe center of L^.

Measure off this

same distance from the point justestablished on the wall and place the center of target B over this point. of L ly.

1

This will place target B in front

and target A in front of L on the lines L„L and L L , respective4 1 3 4 0 Now the plane of the flat

will be at 45 degrees with the direction

L L when the inage of target B is seen in the transit. Next rotate the 1 3 flat until the centerlines of target A coincide with the vertical and horizontal cross-hairs of the transit and with the horizontal and vertical centerlines of target B. and

This completes the adjustment of

and they may be locked in place.

78

From li gh t source

Silvered —

I

S il v e r e d

To observer

fl& . 46

POSITION

OF

FLATS

INTERFEROMETER

*ueueT

ie«e

IN

MOUNTS

ADJUSTMENT

O-FtL-le

79

With

and

locked in place, establish a point by use of the

transit and L^, about 8 feet in front of Second Adjustment: Place mount imately parallel to the flats in

on the line L^L^.

in position with the flat approx­

and L^.

position previously established in front of

The transit is moved to the on the line

and is

adjusted to the same horizontal plane established on the walls of the room.

Mount the triple target. Fig. 44, p. 76, in line with L^L^,

LgL^, and L^L^.

Now, with the transit, set

so that targets C and E

on the triple target give coincident images on the vertical and horizon­ tal cross-hairs in the transit.

This completes the adjustment and

should be locked in position. Third Adjustments

Place

in position with the flat approximately

parallel to the other flats already adjusted.

Complete the adjustment

by obtaining coincident images in the transit from target C of Fig. 44 by the two separate paths L.L.L, and L L L,. 1 4 o 1 3 o

The two images of the

target must again fall on thevertical and horizontal cross-hairs of the transit priorto locking the flat in position. Fourth Adjustment;

Place L .in position and obtain coincident 2

imâges in the transit of targets C and D shown in Fig. 44 through the two separate paths L-L Lz and L L L.. l 4 ° 23 o Fifth Adjustment *

Place L^ in position and make adjustments

similar to the fourth adjustment except coincident images must be ob­ tained from targets C and D on the triple target through paths L L L, 1 4 o an approaches closest to the ideal photoelastic material and is very popular in this country.

This material is

water clear, easily machinable, isotropic, constant in properties at room temperatures> and optically sensitive.

It has a linear stress-strain

relation up to about 6000 psi., linear stress-fringe relation up to about 7000 psi., a modulus of elasticity of about 615,000 psi., tensile strength of 17,000 psi. for 5-minute loading, a tensile strength of 12,500 psi., for long-time loading, and a Poissonfs ratio of 0.365. Gelatin: Gelatin is the most sensitive photoelastic material, being 600 times more optically sensitive than bakelite, but it varies consid­ erably with gelatin concentration, glycerin content, and temperature. Because of its high optical sensitivity, it finds special use for the study of the stresses produced by body forces, as in soil mechanics, earth dams, and tunnel problems.

89

Columbia Resin 23/:

CR-39:

is chemically termed allyl diglycol carbonet*

23/ Coolidge, D. J« Jr., An investigation of the mechanical and stressoptical properties of Columbia Resin, CR-39, Proc, of the Soc. for •Experimental Stress Analysis, vol. 6, no. I., pp. 74-82,* 1948.

Commercially obtainable sheets of CR-39 are cast-polymerized between glass plates and 14 percent shrinkage in volume occurs.. Stock material l/4 -in. thick was found to vary between 0.225 in. to 0,2 6 2 in. 1 Stress-strain relationship: Within the usual time limit of labora­ tory testing, the stress-strain curve for CR-39 is linear to about 3000 psi. with a modulus of elasticity of 250,000 psi.

Introduction of

the time factor makes pronounced the effect of strain-creep, and the plastic does not have any fixed stress-strain relationship or modulus of elasticity, Fig. 47, p. 90. Ultimate tensile strength:

The material will withstand over

7000 psi., in short-time application of load, but only 55 percent to 60 percent of this ultimate strength should be expected over a long period of time. Fig. 4 8 , p. 90. Temperature effect:

At high temperatures of 110 C, the ultimate

strength of CR-39 is roughly 1000 psi., or one-seventh the value at room temperature of 25 C.

Again, a drop of 55 percent to 60 percent of

the strength occurs after 1 to 2 hours under stress. Fig. 49, p. 90. Strain at rupture : There is no fixed value of strain at rupture, which seems to increase with time of application of load.

The elonga­

tions at rupture range from 3.9 percent to 5.6 percent in tensile

specimens. Moisture loss and absorption:

The time-edge effect has been shown

90

SOQO-

$000-

Ssooo

S P tC M tN

*17

MOO

000$

oozo

0090

IN /IN

47 Stress-strain curve for CH-39.

T I M E - M IN U T E S

Fig. 48 Variation of strength in CR-39 with time of application of static load.

S «00

400

4 03 t im e

SO

SO

- m in u t e s

Fig. 49 Variation of strength in CR-39 with time of application of static load at elevated temperature (110°C).

91

to be due to loss or absorption of moisture at the edge surfaces. CR-39 absorbs up to 1.33 percent by weight of moisture after long periods of time in water-saturated air.

In dry condition, the resin loses 0.44

percent by weight of its absorbed moisture or volatile organic matter. In room conditions, CR-39 gives up or loses moisture from time to time, depending upon the humidity of the surrounding atmosphere. Fig. 50, p. 93. Time edge effect:

A model cut in dry atmosphere, when brought in a

humid atmosphere, will swell at the edge, and compression effect will develop at? the edges up to a depth of about 1/32 inch to l/l6 inch. Shrinking and therefore tension will develop in a period of time if the model cut in humid air is brought in a.dry atmosphere. * Stress-fringe relationship: The stress-fringe relationship of CR-39 can be considered linear up to a stress slightly below 3000 psi., keeping the time factor constant. Fig. 51, p. 93. Optical creep:

The stress coefficient is not constant in CR-39> as

optical creep of 11 percent, 14 percent, and 26 percent has been observed in the 1000- to 3000-psi. range of pure bending with 15-minute, 30-minute, and 3-hour constant loadings respectively.

However, the stress concen­

tration factor is found to be within 6 percent during 3-hour loading, Table 1, p. 93. Other properties:

CR-39, also called Allite, is available in highly

transparent sheets of various thicknessesé

It is quite hard and strong,

and its optical clarity equals that of glass.

It has a high optical

sensitivity and is available at moderate costs; thus it is a very popular photoelastic material.

It comes with a polished, highly-glazed surface

which does not show crazing under stress (i.e. developing small cracks

92

under tension) even after 3 weeks of loading.

It is more brittle in

machining than Lucite or bakelite and requires careful cutting. A table of characteristics of different materials is shown. Table 2, p. 94. THE MODEL

Preparation of Photoelastic Model

The model material used in all the experiments was Allite because of its high transparency, polished surfaces, and high optical sensitivity. As CR-39 is very brittle, careful machining is necessary.

First a

suitably sized plate is cut from the Allite sheet by machine saw, leaving 0.1-in. margin while cutting.

Allite sheets of suitable size are avail­

able, and for polariscope work 6 x 6 x l/4-in. pieces of clear pyroline or lumerine are cemented on either side of Allite by acetone.

Three

thin lines perpendicular to each other and within 0.1 in. of the edges of the plate are scratched on one piece of pyroline by a scriber.

The

!distance between two parallel lines is measured by a 100-divisions-toan-inch scale, and an exact distance obtained is laid off from the third line.

The fourth line at the above distance is laid parallel to the

third line.

The cemented plate is then placed in a milling machine

and levelled by means of a surface-gage, using the top edge line as the mark on either eide of the block that holds the plate.

The block is

tightened, and a suitable mill is introduced in the machine.

When the

motion of the cutting mill is counterclockwise and the work advances to /the right, it is referred to as inward side milling, whereas the combination of the left feed and counterclockwise mill motion is called outward side milling.

It.has been observed that with inward side milling

there is considerably less chipping of the brittle Allite and the machining

s

. ...

-

iz e .e e

2 * * D*YS

i

s

I

40

I etciMm V (iNo*v 140 T IM* - D A Y »

Fig. 50 Kffcrt of diffvn'nt atmoephrric conditions on the weight of CR-39.

3500

2500

O 4» CO

%" designates manufacturer *s specifications* All figures are approximate and are given in an attempt to classify the materials*

^

K*B*

i l l

8

Norton Resin

I I

6

Methyl Methacrylate n

a

Lucite

1

°

Glass

6

copolymer

8. o «

S

h

Vinyl

®

h

Vinylite VS 1310

1 3 1 £

°.

w

! 1

5,500

1

2,750

Strain Creep

N.

Fyralln

1,850

Poisson*s Ratio

§ 's

Monsanto C.N. 205D

»

Catalin

Modulus B Psi 8. o

PhenolFormaldehyde

Elastic Limit Psi

i* l l

Bakelite BT 48004

Tensile Strength Psi

E "s

Rubber

Chemical Classification

Ip

Gelatin 1 (Water 65# glycerin 1

Material

RSPCRT CH M irm iA L S KSBEARCH

94

i ° „

A

95

stresses are smaller than when outward side milling is done.

Small

cuts of 0,01 in. are taken each time till the marked line is reached. The last three or four cuts should not exceed 0.005 in. produce heat, chipping, and machining stresses. of the plate are similarly milled.

Large cuts

The other three edges

The plate may be clamped at the

open ends so that pyroline plates do not come off while milling& In the interferometer work, however, the model prepared in the above fashion created indistinct interference fringes.

It was found

that acetone affected the polished surface of Allite, and the practice of cementing the Allite has been discontinued.

The faces of the Allite

are taped so that the polished surface does not get scratched in machin­ ing.

For obtaining clear isoclinics in the polariseope plexiglass models

may be used. The milled plate has sharp and smooth edges.

The plate is taken

out and the center is carefully marked by drawing diagonal lines with the scriber.

The desired shapesof opening or openings

marked symmetrically about the center.

a re

carefully

Openings should be equidistant

from the two parallel edges of the plate.

In the case of mine openings

in an infinite medium it has been found that a plate four times the maximum width or height of the opening represents an infinite medium. In a 6 x 6 x 6-in. plate the height or the width, whichever is greater, should not exceed 1.5 in.

The shape of the opening is very carefully

laid out, correct to l/lQO in. as a line called the A-line of the opening.

A smaller shape, about l/l6 in. smaller all around, is then # laid inside the opening. In the case of a model in which the outside edges are to be cut to a shape like that of a dam, a l/l6-in. line is laid outside the actual shape.

This extra line is called the B-line,

gnd while the model is being cut with a saw, it acts as a guide beyond which a saw cut should not be made* After the A- and B-lines for the correct shape of the opening have been laid out, a l/4-in. hole is drilled in the center of the opening* Next a saw is passed through this hole, and a rough cut with the saw is made up to the B-line.

The material between the A- and B-lines is very

carefully filed by very fine machine files.

The model is taken out and

the inside edges are further smoothed by fine sandpaper... The precautions necessary in machining the models are to have sharp tools, to keep moderate speeds, to take small cuts, and to reserve the tools for photoelastic materials only.

Precise machines are available

for working irregular shapes, but suitable techniques are developed through experience only.

For better results, the model may be observed

through a polariseope to check the machining stresses.

Machining

stresses penetrate up to about l/l6 to 1/32 in. at the edges and are usually deeper and smaller than time edge effect. »

Effect of Thickness of the Model

The true plane stress system can exist only in very thin plates. In thicker plates, both the magnitude and the direction of the stresses may vary from layer to layer.

Except at free boundaries where all the

stresses are either normal or tangent to the boundary, the directions of the principal stresses and their magnitude obtained photoelastically are the average over the thickness.

In practice, plane stress problems

can be produced in plates in which the linear dimensions axe about three to four times as great as the thickness. The optical effect of variations in thickness of a model is to

97

produce indistinct and discontinuous fringes, in a stress pattern, and to introduce vagueness in the isoclinics.

The fringes may merge with one

another, and it is difficult to determine fringe order at points of stress concentration. follow.

The isoclinics become less black and more difficult to

Hence model thickness should not vary greatly from the average

thickness.

Transition from Model to Prototype

The question arises, in transferring the results from a transparent model to a prototype, as to the influence of the material on the stress distribution.

There is an abundance of theoretical and experimental

evidence proving that, in a two-dimensional stress system, the stress distribution is generally independent of physical constants so long as the materials are homogeneous and isotropic, the bodies are either free or subjected to constant body forces, and the stresses are within the elastic limit.

In the cases of an infinite plate with a concentrated

load of plates with holes where the applied multiple or concentrated load does not reduce to a couple or zero, and of multiple connected bodies with unequal loads, a correction for Poisson*s ratio of the materials is necessary.

Even in these cases the correction is small

enough to be neglected, as the correction (in the larger stresses) is about 10 percent which is well within the limit of experimental error. The theoretical and experimental proof that stress patterns are independent of Poisson*s ratio can be obtained by comparing numerous results obtained theoretically in various investigations 24/*

For

24/ Op. cit. (A Treatise on photoelasticity), pp. 128-130 and 501-524*

98

example, in Fig. 52, heavy lines show the stresses obtained in a bakelite model, and the dotted lines show stresses from a steel model, whose Poisson’s ratios are 0.39 and 0.25 respectively.

Similar results were

obtained by comj>aring strain-gage results in aluminum plate with photoelastic results in bakelite 25/.

25/ Frocht, M. M., Photoelasticity, vol. II, pp. 179-201, John Wiley and Sons, New York, 1948.

Furthermore it can be shown that in the case of concentrated loads or multiple connected loads, Poisson’s ratio does not make appreciable difference in stresses.

The stresses in an infinite plate under the

action of a concentrated load are given by:

or

(3 + u) P* cos 9 4%tr (1 - u) P f cos 0

°9

TrQ

t+ntr

(1 - u) P* sin 6 4%tr

where P* is the load, r and 6 are the coordinates, and t is the thicki ness. From these equations it follows that the ratio of the radial stresses or in bakelite and rock respectively is (3 + u)/(3 + u ’), and the ratio of the other stresses od and

is (1 - u)/(l - u 1).

Poisson's ratios u = 0.365 for bakelite and u ’ * 0.20 for rock.

and

Taking

99

In other words > in average rock the maximum or radial stresses are about 5.2 percent smaller, and the other small components are about 21 percent greater than bakelite in the case of concentrated loading. Similar results were obtained by Bickley 26/, who made an extensive

26/ Bickley, W. G., The distribution of stress round a circular hole in a plates Royal Society of London, Philosophical Transactions, A, vol. 227, pp. 3S3-415, 1928.

mathematical investigation for a circular hole in a plate; he statesi It is seen that the effects of Poisson*s ratio are, from a practical standpoint, negligible. Consequently, experimental results, obtained from stresses in nitrocellulose or glass models, can be accepted as giving valid results for similar members or structures of steel, since the effect of the difference of Poisson*s ratio is probably less than the order Qf the possible experimental error. The general effect of increasing u is seen to be in the direction of easing the stresses. However, the models used by the writer for the experiments were only holes in infinite plates which were subjected to uniform static loads at the plate boundaries only.

When concentrated loads were applied, no

loads were applied at the inside boundaries.

In all these cases, there­

fore, physical constants have no effect in the plane stresses obtained.

STRESS PATTERNS

Time-Edge Stresses

The stresses due to time-edge effect are quite small, and are produced after several hours.

This effect shows that for accurate

boundary results the model should be photographed immediately after machining, and no permanent set of models with stress-free edges can be

100

made except for demonstration purposes#

Stresses Due to Creep

The phenomenon of continuously increasing deformation under constant load is known as strain-creep.

Similar effects are observed during

unloading, and a reverse effect is noticeable after the load is removed. An analogous phenomenon in the fringe order under constant load is known as "optical creep.

Significance of creep lies in the fact that the

fringe order becomes a function of time, and in a complicated stress system it may alter the relation of stresses at various points. It can be said that within a two-hour test, the effect of creep on stress distribution in Allite is negligible for the stresses below 3000 psi*

It has been observed that the creep takes place within the

first 15 minutes after loading, and the photographs Should be taken after 15 to 20 minutes.

Of course, lower stresses on the model decrease

creep; therefore, the maximum fringe order for Allite should not exceed 7 fringes.

Machining Stresses

These stresses can be reduced by Careful machining.

The machining

stresses on the tension side produce a characteristic looping or curling of the ends.

On the compression side the fringe order is increased, and

the fringes tend to flatten out.

Boundary Clearness

The boundary vagueness has three principal causes, and the ideal conditions for removing this vagueness can be stated here.

1.

The cross sections of thg models 'must be rectangular.

Non-

re ctangular sections are the results of poor machining in which the cutting edge of the tool is not normal to the surface of the model, 2.

The beam of polarized light must be parallel; otherwise shadows

will be cast on all edges. 3»

The rays must be parallel to the edges of the model.

When the

rays graze the edges, shadows are cast which are referred to as space effect.

Setting of Model and Proper Loading

The model should be perpendicular to the beam of light, and the light rays should be parallel to the edges of the model. can be achieved by suitably turning the model.

These results

Where space effect can

not be eliminated altogether, it is desirable to move the model about its vertical and horizontal axes till the space effect is removed from the important portions in the model. In order for the fringes to be sharp and continuous, stresses or strains should be two-dimensional; i.e. the load should be uniformly distributed.

In order to avoid improper distribution of the loads,

equalizers of cardboard, cork, rubber, or leather should be used between the load and the model. LIBRARY

SETTING AND TESTING POLARIS COPE

COLORADO SCHOOL OF MINES GOLDEN, COLORADO

Anyone working with photoelasticity must acquire the proper tech­ nique for rapid setting of polariseope.

These general suggestions can

be 'made: 1.

Alignment: First make the optical bench quite horizontal, and

.PHOTOEIASTIC ANALYSIS OF STRESS AROUND A GALLERY LOADING MECHANISM

then align the polarizer, analyzer, quarter-wave plates, lenses, light source, and camera so that their centers lie on one line parallel to the long axis of the optical bench. 2.

Clear screen field:

Hake sure that the fields on the screen

is of uniform intensity and well defined, and for this purpose the mixed set-up of the polariseope should be used.

Adjust the screen and other

elements so that screen field is circular with a sharp outline, and free from colored or black blotches.

Generally, delicate adjustments of the

lamp clears the screen field of blotches*

»

*

3*

Spherical aberration:

No polariseope is acceptable if it

distorts the shape of the model or stress pattern*

A simple rectan­

gular network in the position of a model should show no geometric distortion on the screen, 4*

Check zero reading:

A polariseope may produce extinction for

both plane and circular setting, and yet the zero reading on the scales may be incorrect, and faulty isoclinics may result.

This can be

verified by checking a known zero isoclinic such as a cross in a sym­ metrically loaded disk or rectangular block. 5*

Stress pattern test:

A tension model as large as the diameter

of the field should be loaded and viewed at the center as well as at the edges; variation in the space effect can then be readily observed for each setting. 6.

Provision for enlargement and reduction:

The movement of the

camera towards the convergence of rays should produce reductions and the movement of the camera away.from the convergence of rays should produce enlargements. 7*

Cleanliness:

Optical pieces should be kept clean, and a cloth

102

cover for each piece should be provided, Ô.

Quarter plates:

They should be readily accessible and remov­

able for recording isoclinics.

LOADING FRAME

It is obvious that a suitable straining frame is essential for photoelastic studies, as the photoelastic problems are different from material testing problems.

The loading frame should be so devised that

it can be adopted to various problems to be studied in photoelastic work.

The writer’s experience with the frames suggests that this

point can be best brought about by the photographs obtained in the first several polariseope experiments. The first loading frame was too large to be handled by one person and the lever ratio was only 1:4 which made it inconvenient to handle larger models which require larger loads.

The Allite model l/4-in.

thick has a model fringe value of 356 pounds per inch, and to produce a good fringe pattern required at least a 200-lb. to 300-lb. load at the end of the loading device.

The frame had only vertical movement

but no horizontal movement. The new frame was devised after the fashion of the one used in the photoelastic laboratory in the Bureau of Reclamation, Denver.

It consists

of two vertical bars fixed in a heavy base with >a rectangular frame be­ tween the bars.

The base is movable in the horizontal direction and is

graduated to give a moment of 0.01 in.

A horizontal piece connecting the

two bars at the top holds a gear for vertical movement, and is also graduated to give a movement of 0.01 in.

The rectangular frame is held

between the bars by two horizontal metal pieces, and is fixed with the

202.6 LBS.

y/ /STRESSES ON\ INNER SURFACE 2000

/STRESSES ON OUTER SURFACE

4000 1-2000

52 * '

Comparative Curves of lioumlary Stresses from Bakelite anti Steel Models Obtained from Photoelastic and Isopaehic Patterns.

104

upper vertical gear.

The loading machine is flexible in the sense that

any rectangular frame can be fixed in the machine.

Any kind of loading

mechanism such as vertical point load or uniform load, horizontal load, or triangular load, can be fixed on the rectangular frame for particular problems. This rectangle can be discarded and a new one can be used for another kind of problem without much expense. Fig. 53, p* 105. The loading device on the new machine was suggested by an idea used at the University of Colorado photoelastic laboratory.

A circular move­

ment of a wheel near the camera operates the load on the loading bar, and minute variations in loads can be made.

In the beginning a spring

balance may be used, but the wheel can be calibrated to give direct reading of loads. Fig. 54, p. 106.

In this way it is possible for one

person to use and operate polariseope.

PHOTOGRAPHY

Camera

An 8 x 10-in. camera is well suited for photoelastic work.

The

camera should slide along on the optical bench and should have adjust­ ments for rising and moving longitudinally and transversely.

Film

For the green monochromatic light from a mercury-vapor lamp, Eastman Kodak Super Ortho Press (safety) films are well suited in photo­ graphing stress patterns.

Films are loaded in total darkness, but a

15-watt incandescent light in a Wratten safelight lamp No. 2, with a Wratten safelight glass, series 2, may be used.

The film holders,

sheaths, kits, and holder slides should be absolutely clean and dust

105

aacj

53#

New loading frame

106

Fig. 54.

Loading device

107

free.

The film should be lightly brushed on the emulsion side with a

high quality camel1s-hair brush to remove any dust left in manufacturing process.

Numbering

Before loading, the silver strip on the cover slides of the holders are serially numbered.

An envelope may be made out carrying the correct

number of the negative, the type of the model and the test, the magnitude and the type of loads, the dimensions, the material, the fringe value, and the necessary fringe orders.

The negative and prints can then be

placed in an envelope and suitably filed.

Exposure

The time of exposure can be determined by experience or by the use of low-scale-reading exposure-meter.

For exposures of less than one

second, automatic shutters are necessary.

The diaphragm opening varies

with the time of exposure and the intensity of light source.

For clear

and sharp fringes, smaller diaphragm opening is helpful.

Isoclinics

*

Isoclinics can be sketched directly on à piece of paper placed on the ground glass of the camera.

For photographing isoclinics, a white

light source may be used and isochromatics are eliminated by deliberate overexposure of the Ortho Press film, or high loads which cause washingout of colors. Colored Stress pattern Colored pictures can be obtained by using white light source with

108

Eastman Kodachrome Professional (Safety) film. Type B, with a 60-watt incandescent light, and for images one-half the model size, an exposure of l/3 second was found necessary.

Developing

It is advisable to have the developing room adjacent to the polariscope room, so that development can be carried out immediately.

The

model and the loads should not be changed or disturbed before prompt development shows that a good negative is obtained.

Fine-grain developers

such as Glycin and Agfa plenachrome are very satisfactory, and the tem­ perature of developing and fixing solutions should be maintained within 60 to 70 F.

With the use of slow developer and super ortho press film,

the films can be inspected at will in the same safety red light and the time of development varied to compensate for small errors in exposure.

Fixing

After the film has been developed, it is rinsed in coal water for about 5 seconds, after which it is immersed in a chrome-alum hardening bath for about 3 minutes and then placed in a fixing bath for about 1/4 to 3/4 hour.

It is then washed in circulating water for an hour

in a deep tray.

If the temperature of developer and hypo is kept at

70 F, no alum hardening bath is necessary.

Reduction

Though it is difficult to intensify underexposed or underdeveloped negatives, better contrast is obtained by reducing solution such as a ferricyanide-hypo reducing solution, which is known as a subtractive

or surface cutting reducer.

The negative can be placed in a reducing

tank directly after fixing.

To avoid streaks, always rinse the negative

in cold water before holding it up for inspection.

Printing

Eastman Kodak developing formula D-72, and contact printing Azo single weight, glossy printing paper in contrasts F-3, F-4, and F-5 seem to give veiy satisfactory results in printing.

The printing-box

light should be well diffused and of low enough intensity to insure an exposure of 10 seconds or more.

After exposure, the print is developed

in a tray of D-72 solution for 1 to 2 minutes, placed in a short-stop solution of water and acetic acid for 1/2 minute, and then fixed in a hypo for l/2 to 3/4 hour.

Contact printing can be carried out under the

yellow light of 10-watt incandescent bulb in an Eastman safelight lamp. Model B, with a Wratten safety glass, series 0A, over the developing tray.

After 1 hour of washing in cold circulating water the prints can

be mounted on ferrotype plates previously waxed or polished.

If fine

talcum or Simoniz wax is used and the plates are covered with large sheets of blotting paper and are allowed to dry overnight, the prints can be easily removed from*ferrotype plates without subsequent curling of prints.

BOUNDARY STRESSES FROM POLARISCOPE

General

... Several différât sections of opening shapes used frequently in underground mining were studied, including tall or wide, rectangular shapes with circular corners, rectangular openings with semicircular roof, a stope opening, a horseshoe-shaped tunnel opening, arid an elliptical shape.

/

The openings Were made in 6 x 6 x 0»250-in. Allite

plates, such that the Width of the plates were five times the maximum width or maximum height of the opening*

Thus, the holes in the plates

represented underground openings in an infinite medium of rock.

The

plates were loaded with static vertical load, or horizontal load, or both, and in one case the plate was loaded at an angle. Since the load is applied only to the outside boundaiy, there is an equal and opposite reaction on the parallel edge of the model in each case, so that the total load vanishes.

The hole in the plate

can then be considered to be an opening in an infinite medium which is not laterally confined.

It is not necessary to consider the effect of

Poisson's ratio of the model when transferring the stresses from the model to the prototype.

The boundary of the hole is free from external

loads, so that the only force active at the inside boundary is the tangential stress* either p or q.

The fringes at the inside boundary then give

The boundary stresses can be drawn by determining the

stresses in psi. at the points where the fringes touch the inner boundary of the hole* * The self-supporting arch occurs between the two singular points obtained at the boundary of the hole.

It has been found from the theory

of elasticity that such an arch extends up to av^T - a from the center of the roof, if the average radius of the opening is a, 27/« These

27/ Kafadar, A* D., An investigation of the stress distribution around underground openings by photoelastic method: Colorado School of Mines, Master’s Thesis, p. 133>1943.

arches are shown by dotted lines for each shape in the graphs of stress distribution.

However, this shape of arch exists only for openings in

rocks which are hot laterally confined.

Calibration Test

The model under load viewed in the circular polariscope will show isochromatiCs.

The value of those fringes in terms of psi. is determined

separately by a calibration.

If most of the models are loaded in com­

pression, it is desirable to take the value obtained from the compression test.

However, the fringe value of the material should not vary too

much in compression, tension, or bending tests on the calibration member.

Pure Compression V

If a,model with its width and thickness small compared to its length, is loaded in a direction of its longer axis, the only stress at the center of the model is the applied stress parallel to its longer axis.

No shear stresses should be applied, and there should be no

bending of the model.

The value of maximum shear at any point in a

model is given by , vmax ** nP **

# where n is the fringe order at ^ the point*and F is the model fringe value. In the model under pure .

compression, q ■ 0.

Therefore

.

F * JBS& «■ JL. ** £ x i- * n

2n

A

2n

psi., 2An —

where A is the area of the model and P is the applied load.

The material

fringe value is defined as the model fringe value of a model of one-inch thickness, so that the material fringe value becomes f * Ft, where t is the thickness of the model. The compression test of a piece of Allite 0.250 in. thick, 1.125 in. wide and 4 in. long was carried out, Fig. 55 (a), p. 113, and Fig. 55 (b), p. 114.

Observed data Fringe Order n

Scale Reading, lb.

Increment of Load, lb./fringe

The lever ratio was 4s1.

50

Model Fringe Value * F -

0 1

50

2

100.5

50.5

?p. * ft_ _ _ _ _

2 x 1.1 2 5 x 0 .2 5 0

50

3

150.5

2

101

1

50

51

0

0

90

* 356 psi. 50.5 Material Fringe Value » Ft = 356 x 0.250 1

50.5

50.5

- 89 pel. 2

100

49.5

3

151

50.5

2

100.5

50.5

1

50

49.5

0

0

Average

50 50

113

&, 4>

r-

11 IS Io

o

. oa u

H IA ^ . -H

O

-I < z
c >o

>o

JAM.

«■

m

Unit Load for Interferometer

In the interferometer study, the load is applied each time a read­ ing is taken.

The air-piston lifts up the load acting at the pan, and

therefore only this changing load acts on the model and moves the interference fringes. The variable load on the model » 528.23 lb. Load on the model in psi. * -— --- * 429.455 0.250 x 4.92 Factor for unit load, K » -■■ = 0.0023285 429.455 Therefore

KA - -8.32 x 0.0023285 - -19.37 x 103 KB « 11.28

X

0.0023285 = ♦26.27 x 103

Equilibrium check

An equilibrium check is applied to find out whether the total load at the cross section perpendicular to the applied load is of the same amount as the total load.

Several points on both sides of the opening

on an axis perpendicular to the applied load are analyzed in the interferometer, and the stresses given at the top of the data sheets.

are obtained by using the formulae Prototype principal stresses for

l-psi. load are first obtained by using factors KA and KB in the formulae for

and o^.

0^ or the stresses perpendicular to the axis

or the equilibrium line are calculated and plotted against the axis for each point.

The graph for each loading is drawn, and the area under the

graph is found by plenimeter.

By taking the proper scale for load and

the distance, the area under the curves were found to be 1.05 psi. for » Vertical load and 1.013 psi. for vertical load, which were within the

169

experimental error. Fig* 61, 82, pp. 170, 171 respectively.

The Stresses

The boundary stresses were obtained from the isochromatics. stresses —

These

i.e. the fringe order multiplied by the model fringe value —

were divided by the load on the model of 445.12 psi.

The boundary stresses

are drawn for vertical and horizontal loading of 1 psi., and the effect of Foisson's ratio and hydrostatic condition are also shown in Fig. 83, * p. 172. Principal stresses for the model can be obtained by using the formulae for Og and

given at the top of the data sheets.

Prototype

principal stresses for a load of 1 psi. were obtained by using constants KA and KB instead of A and B in the same formula.

The normal and the

shear stresses along the reference axes were obtained by using the formula given for ox , Oy, t ^ .

The value of the prototype principal

stresses and their angle, and the stresses ox , Oy, and

are calculated

and shown for vertical and horizontal load of 1 psi. in Figs. 77-80, pp. 164-167. The principal stresses and their angles are drawn in Figs. 84, 85, pp. 173, 174.

REFERENCES TO INTERFEROMETER STUDIES

Stress Analysis for a Gallery

The photoelastic staff of the Bureau of Reclamation carried out an interferometer study for a 5 x 7-ft. gallery 29/.

Fig. 86, p. 175,

29/ Zangar, C. N., and Phillips, H» B., Stress distribution around a standard inspection gallery: Memorandum to Mr. J. J. Hammond, Bureau of Reclamation, Denver, September 28, 1948.

Equilibrium curves for the section along the horizontal axis of the model under vertical load of 1 psi.

W « t) © k •P p CO CO 1 1 I I 1 1 t 09 09

Xf c #) b( w

«

CO ©

k p CO . .

.

O P P -AM 220-1.74-127 -107-OH -•12 208-217 -182 -128 -103 ty -A72-SJO 248-181 -MS -086 F.V -yn -217 -187-III -078■0*7 $ F -2.4»-1.40•1.00-044-014 -002 l.tf ♦104♦048♦080♦082 ♦048♦034 J/ ♦700♦128♦128 ♦071 ♦042►028 ♦700 «2M •220♦187 ♦138♦118 ♦700♦180♦104♦088♦028 ♦007 o' 0 0 0 0 0 N? -TOO-180 -104 -088 -028-007 0|*' -7.00 288•220-187 -IS8-I.M V -*00 -MS -128-071 -042-028 -034 o,r -104 -048-080-082 -023 Vo.46-010 -028-OSS *14» ♦1.40♦108♦6441♦0J4*002 I'1S2L ►047 T.r; ii!L ♦187♦II ♦>72 11» 248♦181 ♦US ♦088 ♦288 ♦817'♦I.82ÏJE ♦103 M . ♦1.74♦127♦107♦081 AM W ♦086 ♦078 JUL 1Ü 1 U5L

4,4* 0 0 0 0 j0 0 0.33-024-0.03+0.13+0.21 B.X' 0 C,W 0,-0.34-0.24-0 .06j+0.l7+035 o,v' 0 -0371-0.31-0.031+0.15+030 E,U‘ 0 -0.311-0.34-020-0.02+0.2C F,T' 0 -0.38j-0.49-0.3a-0.26,-019 G,S' 0 -0.32|-0.4i-0.571-0.50-0.33 H,r 0 -0.63j-0.63-0.64-0.57-0.43 1,0" o•♦0.l8j+0.2«♦0.25*0.15+0.07 0 ♦I.SStl.46♦1.12♦0.77t0.47 K,0* 0 ♦021*018-0.01-018-0.30 L.N' 0 ♦l.2l!fl.42+l.35|+0.»4+0.56 M.y 0 0 0 0 ;0 0 N.L' 0 -1.21-1.42-135-0.94-056 o,*' 0 0.21-0.19♦0.0lj+0.l9j+0.3C P,J' 0 -1.39-1.41-1.12-0.77{-0.47 0,1' 0 -0.M-0.21-0.2H-0.I5-0.07 0 •►0.63|♦0.69+0641+0571+0.43 8,6' 0 +0.32♦0.49♦o.sH+o.scj+o.ss 7,F 0 ♦0.3»♦0.49+0.3»+0 .261+0.19 u.r 0 +0.31+034+0.20+002-0.20 v.of 0 ♦0.37+0.31*0.0$-0.l5j-0.30 w.C 0 ♦0.34+0.26♦0.06-0.I71 -0.35 X|8‘ 0 ♦0.33+0.24+0.031-01510.21

NOTES (♦) Indicotes compression ( - ) Indicotes tension

Table 11

shear stress alons reference lines

STRESS COEFFICIENTS FOR LOAD OF ILS. PER SO. IN. Ai A cy 0.V E.U' F.T' 6.6 m,»; 1.0 jy K.O'

L.W « y 6,1-

O.K' F.y 0.1' *y 88

T.F' U.É' V,0'

w.c K.8

OMTANCC OUT FROM CALLCRY 1.0 2.0 30' 40 05 0 0 -020 -0 38 -0.74 -106 -1.38 0 - u s -1.20 -1.24 -1.21 -1.14 0 -0 74 -077 -077 -070 -0.59 0 ♦006 -0.06 -0.14 -OH -0.08 0 +0 25 +0 37 ♦0 49 +0 51 +051 0 +079 +090 ♦100 ♦102 ♦ 100 0 +0.53 +0 78 ♦1.04 ♦1.26 +1 34 0 +0.30 +0.54 +0.92 +121 +1 37 0 +045 +0.64 +0.88 +105 ♦ 117 0 +0 84 ♦091 +0.92 ♦0.92 +0.91 0 +0.14 +0 15 ♦0.11 ♦0 06 «0.06 0 -0 91 -1.08 -1.15 -1.15 -1.12 0 -0 72 -0 82 -0 88 -092 -0.96 0 -091 -1.08 - 1 15 -1.15 -I 12 0 4014 ♦015 ♦OH +0.06 +006 0 ♦0 84 +0 91 ♦092 +0.92 ♦0.91 ♦ 1 06 +1.17 0 ♦045 +064 0 +0 30 +0.54 +0.92 ♦ 1.21 ♦ 1.37 0 +0 53 ♦0 78 ♦106 +1 26 +1 34 0 +0 79 ♦080 ♦IOC +1.02 ♦ 1.00 0 +025 +037 ♦049 ♦0 51 +0 51 0 +006 -006 -0.14 -0 II -0 08 -0 0 -0 *4 - 0 -0 70 -0 59 0 -1.15 -1 20 -1.21 -1 14

+0.88

77 77 -1.24

notes (♦ I

-------

184

on (horizontal load of 75 psi.), Oy (vertical load of 200 psi.), and TXy (shear load of 40 psi.).

For example, the normal stress on can be

found for the point on line A at a distance 0 from the boundary by adding normal stresses due to on » 75, Oy » 200, and a « o x nx x

* 40 as follows,

the stress concentration factor for the static load in the direction of ox ,which is

horizontal, at a point

on the line A and at a distance 0, Therefore

o^

»75 x 4.00 (from Table 6, p. 182) “ 300

Similarly,

o^

• 200 x -1.00 (from Table 3> p. 181)

- -200

and

o_« 40 x 0.00 (from Table 9, p. 183) * 0 nxy so that the total on “ 300 - 200 =» 100 psi.

In the same way the stresses on for various points on lines A, D, and H wereobtained,Table 12.

Tables 13 and 14 give

and on£ the shear stresses respectively.

the tangential stresses,

Fig. 91, p. 1&5, shows the kind

of loading on the opening for this particular example.

Table 12 - The normal stresses o„ Stresses due to d„

Distance out

o

0

+300

-200

0

+foo

•5h

+212

—68

0

♦144

l.Oh

+165

-20

o

+145

2. Oh

+114

0

0

+114

3.Oh

+99

♦4

0

+103

4. Oh

+92

+10

0

+102

X

y

o_. xy

n

185 Fig. 91.

The stresses ox , o and at the center of the area before the opening is made due to the inclined vertical load and the effect of Poisson1s ratio.

63-3

t>Si

ay

as'Sio'rv

C.0'>r)(3ir' cSS ion

186

Table 12 (contd.) Distance out

o

Stresses due to o„

0

>91

,5h

0_

xy

Eo n

; >174

-245

+20

+82

+158

-106

+134

Ih

>74

+146

-87

+133

2h

>64

>134

-65

>133

3h

>58

>132

-52

>138

4h

>54

+126

-41

> >139

0

•81

>436

•18

>337

. -32

+378

+4

>350

Ih

-15

>336

+10

>337

2h

>7

>286

+13

>306

3h

+20

>258

+11

>289

4h

+26

>234

>9

>269

Txy

Zdt

.5h

X

y

'

Table 13 - The Tangential stresses 1

Distance out

.

'

Stresses due to 0x

°y

0

0

0

0

0

.5h

29

4

0

33

Ih

38

26

0

64

2h

36

76

0

112

3h

24

118

0

142

4h

17

142

0

159

187

Table 13

(contd.)

Line

Distance out

H

Stresses due to °y

°x

Txy

Iot

0

0

0

0

.5h

17

28

-15

30

lh

23

46

-12

57

2h

28

64

-1

+91

3h

31

76

+6

113

4h

32

82

+12

126

0

0

0

o

o

•5h

-6

18

-2$

-13

lh

0

+22

-26

-4

28

"26

♦14

25

-23

+36

35

"17

+56

12 3h

0

Table 14 - The shear stresses 0^

Line

Distance out in feet

Stresses due to °x

°y

°xy

*nt

Oh

0

0

0

0

0.5h

0

0

-8

-8

l.Oh

0

0

-15

-15

2. Oh

0

0

-3 0

-30

3.Oh .

0

0

"42

-42

4. Oh

0

0

-55

-55

A

188

Table 14 (contd.)

Line

Distance out in feet

Stresses due to BOnt

°x

°y

0

0

0

0.5h

-45

+72

+2

+29

l.Oh

-53

>120

-2

+65

2. Oh

-57

+132

-6

+69

3.Oh

-57

♦124

-4

+63

4. Oh

-57

+116

-3

+56

0

D

°xy 0

0

0

0

0

0

0.5h

-4

0

+12

+8

l.Oh

-5

0

+22

+17

2.Oh

-5

0

+37

+32

3.Oh

-5

0

+48

+43

4. Oh

-5

0

+55

+50

H

Stress Analysis for Wide Opening

The stress analysis for a wide opening of height to width ratio of 1:1.31 under vertical pressure was also carried out at the Bureau of Reclamation 30/.

The interferometer study was compared to the analytical

30/ Zangar, C. N., and Phillips, H. B., Photoelastic determination of stresses around the elevator lobbies in Hungry Horse Dam: Memorandum to L« G. Puls, Bureau of Reclamation, Denver, September 9> 1948.

189

solution (using Martin Greenspan's method)> and the results were found to be in agreement except at sharp comers where Martin Greenspan's solution does not apply.

Analytical results are omitted here, but the interferometer

results are given in Figs. 93> 94, 95, pp; 191-193.

Fig. 92, p. 190, shows

the boundary stresses by pol&risoope as compared to the boundary stresses by analytical method.

Principal Stresses Around a Gallery

The magnitude and directions of the principal stresses for various combinations of the vertical and horizontal load are shown in Figs. 96-98, pp. 194-196, as obtained from the Bureau of Reclamation Studies 31/.

31/ Russel, Frank M., Design data for gallery stresses as determined by means of Plaster-Celite models: Technical Memorandum No. 555, Denver, June 16, 1937.

These figures can be construed as representing the effect of Poisson's ratio and hydrostatic pressure.

The stresses ox , Oy and

are also

given in the tables accompanying the figures. Fig. 96 , p. 194, shows the change in the boundary stresses as the side pressure increases.

In Figs. 97 (a), 97 (b), 98 (a), and 98 (b),

the point of hydrostatic pressure 32/ moves up as the side pressure

22/ McCutchen, W. R., The behavior of rocks and rock masses in relation to military geology: Masters Thesis, Colorado School of Mines, September 14, 1948.

increases.

Since lateral pressure varies from one half of the vertical

pressure to that equal to vertical pressure, these figures give a true picture of stress distribution for most rocks.

190

NOTES ( - ) Indicate* compre'tion. (♦ ) Indicate* tention. Strct* *hewh i* for a uniform load of I lb. per *q. inch actinf vertically. Stre*« in left half of lobby tome a* in right half. Analytical stret* will be *ome in all four quadrant*.

.Analytical curve

---.E *perim en tol curve.

! LOBBY 1 ‘ -Symmetrical about t .

» SIS1

(+)

ITAESS SCALE - « DIMENSIONAL STUDY IN LBS. PER SO. I NCH

DIMENSION SCALE IN FEET

IN LBS. FEN SO. INCH

Fig. 92 HUNGRY ANALYSIS

SECTION

HORSE

DAM

OF STRESS AROUND ELEVATOR

PENPCNOICULAR

TO

AXIS

LOBBY

OF DAM

PRINCIPAL STRESS AT SOWNDRY COMPARISON OF PHOTOELASTIC STRESS AND ANALYTICAL A U aU IT

104 0

STR

191 LOCATION OF REFERENCE

NOTES

LINES

A B C

S trts s u shown are for a uniform load of lib per sq inch acting vertically Stress in left half of lobby some as in right h a lf.

1

_i

STRESS SCALE - 2 DIMENSIONAL IN LBS. PER SO. INCH

»#; oi

0 ! « » STRESS SCALE - 3 DIMENSIONAL IN LBS. PER SO.INCH

* - Symmetrical about t

10 DIMENSION SCALE IN

D IS T A N C E

OUT

FROM

LOBBY

IN

6— I

STUDY

STUDY

I» FEET

FEET

r~

t

p\

r

F ig . 93 HUNGRY

HORSE

DAM

'SECTION PERPENDICULAR TO AXIS OF OAM AUGUST, 1*4#

STRESS NORMAL TO REFERENCE LINES

447-P E L -10

192

A

B

NOTES Stresses shown ore for o uniform lood of i pound per square inch acting vertical 1/ Stress m left half of lobby same as in right half.

?

Z

±

I0 M 4 '

STRESS SCALE - 2 DIMENSIONAL STUDY IN LBS PER SO INCH

,-Symmetricol about

t

STRESS SCALE - 3 DIMENSIONAL IN LBS PER SO INCH

-------------- H

STUDY

9M 3

DIMENSION SCALE

IN FEET

O P 0 N• M LOCATION OF REFERENCES LINES

D IS T A N C E

OUT FROM

LOBRY

IN

FEET 10

M

_i .J

.zrrrr:

"_L

:î_" Ü_ L .

L.

Fig. 94 HUNGRY ANALYSIS

SECTION

HORSE

OAM

OF STRESS AROUND ELEVATOR

PERPENDICULAR

TO

AXIS

LOBBY

OF DAM

STRESS TANGENTIAL TO REFERENCE LINES AUOUST, I MO

It

14

N O TES

LO CATION OF REFERENCE U N E S

A B C

Stresses shown ore for o uniform lood of I pound per squore inch octing verficolly Stresses in le ft half of lobby some os in right h a lf

STMCSS

S C A L E - t DI MENSI ON AL STUDY IN LOS PEN SO IN CH

?

!

?

?

10.584.

STRESS SC A LE - 3 DI MEN SI ON AL STUDY IN LSS PER S O INCH O S 10 IS

_____I I_____I

»

DIMENSION SCALE IN FEET

-1

For reference lines on le ft side o f center line, sign o f shear is reversed.

•9583' J

O

P

O

N

M DISTANCE.OUT PROM LOBBY IN FEET r

1

6

i

i

•

10

,

k

14

ri 1 !__

F

i

i

1 ri L_ 1 1

■

1

t

1 -------------P 0 ____

1

L ------------- 1------------- 1-------------- 1-------------L ---------- J --------------- 1

P |~ L - j—

J

1 “T ■ 1— -

— r>

— u_

1 Fig. 95 HUNGRY a n a l y s is

SECTION

of

HORSE around

P E R P E N D IC U L A R

SHEAR AUOUST. 1*4#

stress

DAM

elevato r

TO A X IS OF

lo bby

DAM

S T R E S S A LONG R E FE R E N C E L IN E S 4 4 7 - PEL-12

\9k

0 I « â ■ 10 111 1 1 > » > 1 » I ‘ I

J F ig . 96

SCALE OF STRESSES IN LB. FEN SO. IN.

LOADS APPLIED SYMMETRICAL WITH GALLERY

EXI

ION LESENO

•Tweeses f l o t t e o aw ay f a o m o a l l c a y o f c n i n o o c n o t c cow» IN D IC A T E #

OmeCTION OF F N IN C IFA L •TRE#»C# IN •TNUCTUW t WITHOUT O ALLIN V .

195

196

A*» E2=d. H-8

h

x=

+ ■ Ilèl:?!

/•

‘x - X'

%:

:d f:

v +*

M

skl-sf $5 ib sê* -Bili « 'Ml 65 SS |S 188 i8tq SI$8 1888 sBst :*• itm f??p f ‘ Ç1 ïï■f 18;;s::::!8ïi issu s; S8S8

8

/■ / = x=

*

S

A*! $

Xb

v-t

/»

*fs

i l =:| i t N •5! i$t! iU •et ;Sse 6"! t; ?e|! « i # !!ÎS ;r ifi 18ï ;fi 188* «f i •?«‘fs??? =: | f ?!ff ‘if?? e !•!•tee Is T i-

*

S £

88

Ü ss

II

"f.

If!

^ L

11} II!

\ H 2’

?f lilîf IIIIJ • •'»■>-> i + t>o ®

i*

MkuWnwa siiZMüa^i

$ Vî

m

? ir

■■■■■■

SfeSi.ljSiîSSSIiÉi S

i M

^

S

ü

s:;;;;:;;::::::::::

STRESS.DETERMINATION BY ANALYTICAL METHOD

The theory of elasticity provides an analytical method for obtaining complete analysis of stresses in an elastic medium.

Of the mathematical

methods of stress analysis for symmetrical holes in a plate, the stress analysis for a circular hole by Professor Kirsch 33/ seems to be the ---- — —

r-—

' —

-----------

—

: ---

: ------

'

33/ Og* cit.

easiest (see p.

)*

An exact solution for the stresses in the neighbor­

hood of several symmetrical-shaped holes was obtained by Martin Greenspan 34/.

34/ Greenspan, M., Effect of a small hole on the stresses in a uniformly loaded plate: Quarterly of Applied Mathematics, vol. 2, no. 1, pp. 6071, Brown University, 1944. This method forms the basis of the writer's solution for an oval-shaped stope opening, and hence Greenspanfs paper is reproduced here in part.

GREENSPAN'S SOLUTION FOR HOLES IN PLATES

INTRODUCTION

The stresses can be obtained for a hole having any boundary of which the equation can be expressed in the parametric form,

x * p cos 0 + r cos 33,

y ■ q sin £ - r sin 33

(1)

The plate is supposed in a state of generalized plane stress, the stress at points remote from the hole having the constant normal components ox = sx , 0^ « Sy, and the constant shearing component

= t

.

198

Equation 1 represents a closed curve having symmetry about the x-axis and about the y-axis.

For certain values of p, q, and r the

curve is simple, i.e., it does not cross itself.

By adjustment of the

values of p, q, and r a variety of simple closed curves is obtained, including a good approximation to an ovaloid and a good approximation to a square with rounded comers, as well as exact ellipses (r * 0) of any eccentricity.

The approximate ovaloid is obtained by taking

p « 2.063,

q « 1.108,

r = 0.079,

(2)

The approximate square is obtained by taking p - q * 1,

r * - 0.14,

(3)

THE COORDINATE SYSTEM

The solution of the problem is simplified by the use of a system of curvilinear coordinates (a, 3) such that Equation 1 of the boundary of the hole reduces to the form a * ao. Such a system is obtained by writing x * (ea + abe"0 ) cos 3 > ac^e

cos 33>

)

y ■ (ea - abe"01) sin 3 - ac^e

a sin 33»

)

S

W

For constant a, say a0. Equation 4 reduces to Equation 1 for the boun­ dary of the hole, where

p «* ea° ♦ abe"^0 ,

q .■ ea° - abe"^^,

r * ac^e ^a°.

(5)

From Equations 2 and 5 it is easily calculated that for the approx­ imate ovaloid of Equations 1 and 2,

ea ° = I*585/

ab ■ 0.758,

ac^ * - 0.314.

199

By keeping ab and ac^ fixed and varying a and 3 the appropriate coordin­ ate system for the ovaloid is obtained. shown in Fig. 99, p. 200.

This coordinate system is

The appropriate systems for the approximate

square with rounded corners are similarly obtained. Figs.100,101, p. 201.

THE STRESSES

In the absence of the body forces, the equations for the stresses, oa . Op (normal components) and

'(shearing component) in the plate

subjected to a system of plane stresses Sx , S^, and

o_ fç

cos 23 ^ A-y^ cos 4B)

= 2C-^(A^q ^ +

aCgCAgQ

can be derived:

+

^22

c° s

20

♦

A2^

cos

4P + Ag^ cos 63 )

* SC^A^q ♦

A^2 cos 23 + A ^ cos

4B > ^36

cos-^3)

♦ 2 C^(A^q +

A^2 cos 23 +

43 +

cos 63 )

cos

1

+ 0jg(A«pQ * A ^ cos 23 ♦ A ^ cos 43)

-”4 " ^

- 2 O^(A^2

sin 23 ♦

sin 43

+ A55 sin 63 )

- 2 Cy(Ay2

sin 23 +

A ^ sin 43

+ A ^ sin 63 )

- 2 Cd(Ad2

sin 23 +

A ^ sin 43

♦ A ^ sin 63 )

B^2cos 23 + By^ cos 43 + B ^ cos 63 ) > 202(820 + B22 oc s 23 ♦ cos 43 + Bg^ cos 63 ) 2Ci(Bio +

- 2 C^(B^q ♦

B^2 cos 23 +

- 2 C^(B^q +

B^2

o cs

cos

23 + B ^ cos

43 + B^^

cos 63 )

43 + B ^

cos 63 )

- C^(B^q + B ^2 cos 23 ♦ B ^ cos 43)

(6 )

200

201

OfOS O 7

77

f l * . 100 . Coordinate system for problem of square hole with roumled corners, sides of s ^ square parallel to Cartesian axes.

Fig. 101

Coordinate system for problem of square hole with rounded corners, diagonals parallel to ("artesian axes.

202

-20,(8 sin 23+ o o2

Bz,sin 43 ♦ 04

B,. sin 63 ) 00

+ 20 y(By2 sin 23 ♦

By^ sin 43 +

By^ sin 63 )

♦ 203(832 sin 23 +

83 ^ sin 43 ♦

B33 sin 63 )

M ^ » 120^(0^2 Gin 23 +

sin 40 ♦

(7)

sin 60)

h - 2 C2 (D22 sin 20 ♦ D2^

sin 43 + 826 8in

20^(832 Gin 20 +

sin 43 ♦ 83 ^ sin 63 )

- 2C^( 0^2 sin 20 ♦ D, y 44

sin 40 ♦ D//; sin 63 ) 46

♦

♦ 205(852 sin 20+ 85^ sin 43) - 205(830 +862

cos 20 ♦

83 ^ cos 43 ♦ 833 cos 63 )

2C y(D yQ

♦ B y2

COS

20 ♦

B y^ COS

-2 0 3 (8 3 0

♦832

COB

23 ♦

8 3 ^ COS 4 3 ♦ 8 3 3 COS 6 3 )

♦

43

♦ B y£ COS

60)

(8)

[ 2 = e261 ♦ A

h

2^ 211 ♦ 9a2c6e"6a - 2ab cos 0 . \ , ♦ ôa^bc^e”^ 1 cos 20 - 6ac^e~^ cos 43»

in which Aio *

+ 4a^b^ - (24a^c^ > IBa^b^c^ - a^b^Je*^1 ♦ 24a^b^c^e"e^a

- 81a^,c ^ c “'^ a > '12

- 4 labe^0

- (3a^bc^ - a?b?)e'm^a + (9a^bc^ - 3a^b^c^)e”^ct + 9a4bc9e-10*| ,

* 2(3ac? + a2b2 - ba^b^c^e*"^1 + 9a^c^e”^1), A20 « 2ab - (ba^bcr^ - a^tP )e*”^ 1 ♦ 30a^bc^e’*^1, A22 “ - 2 e20 - (4 ac3 - a ^ e ^ t

6 (a2 c6 - a^b2 c^)e~^- 18a3c ^ ”10^ ,

A 24 ■l ab - Sa^bc^e”^0 ♦ 9a^bc^e"‘^a>

^26 * " 6a^c^e“^a ,

(9)

203

A^o = 3abe2a - ISa^bc^e*"^,

A^2 * -(e^1 - 13ac^ + 3a^b^ ♦45a2c^e*e^a ),

^34 * ab®201 " 9a^bc^e“^a , '

A ^ * 3ac^,

A^q ** 3 abe"e^a - 3 a^bc^e*"^a ,

A^9 * - ^3 - (9 ac^ -a^b^)e"^

A ^ * abe"^0, ♦ 3a^bc^e“^a ,

A ^ * - 3ac^e"^a ,

A^o " e^a “ a^b^e"^ - 27a^c^e~^a ,

- 9 a^c^e"^j,

A^^ * - 12a2bc^e"^a ,

A ^ * 6ac^e"^, A^2 *

+(a^b^ + 4ac3)e~2ci + ôa^c^e*^® + 18a3c9e“*^^®j, 9a3bc^e”^a ),^66 * * Üa^c^e""^,

A ^ * - (ab -2a2bc^e‘“^a -

A72 “ e^a * 3(a2b^ + 5ac^) + 45a2c^e*"^a > ^76 “ *"3ac^,

Ay^ * - (abe^® - 9a^bc^e"e^a), A32 " 3 + (a^b2

♦ 9ac3)e~4a - 9a2c^e"d® >_

A84 * " (abe"2a

♦ 3a2bc3e"6a),

A ^ - 3ac3e”‘Z*a ,

®10 * a ^ ♦ 4a2b2♦ (24a2c^ + 6a^b2c^ + a^b^)e""^® - 24a^b2c^e“"®a - ÔlaAc^e^-2®,

8^2 ” • 4jabe2®

- (3a2bc^ - a^b^Je"2® - 9a^bc^e“^® +18a^bc^e“^^®J,

« - 2(9a

e“,^a> B22 * 2^2(acP - a2b2)e“2® - 3(4a2c^ ~ a^b2c3)e“*^® + 9a^c^e*e^^a”j, B2^ - ab - I6a2bc3e“ ^® ♦. 9a3bc^e"^a>

B^q - 3abe2a -

l^a^c^e"*2®,B^2 » - (e^ -l$ac^ +3a2b2 + 45a2c^e”^a),

B^^ - abe2*1 -9a2bc3o“2®, B40 * 3abe"2a - 3a 2bc^e™,^a > B

44

B ^ « 6(acPe"^ - 2a2c^e” ^a) ,

« abe”2® ♦ 3a2bc3e”^ia,

B^^ = 3ac3, ~ |3”(9ao^ - a2b2 )e™^°’ - 9 a2 c^e"B®J, B,,* - 3ac3e"e^a , 40

204

®50 m e^a * a^b^e"’^a - 27a^e”^a >

*.-* 12a^bG^e"^,

- ôac^e”^01,

b62 - 2|2ac3e*2a ♦ 3(a3b2c3 ♦ 4a2c6)e*^a ♦ 9a3c9e“10aJ,

B ^ * - (ab + 4a2bc3©*,4a - 9a3bc^e“^a), B ^ * - ôCacPe”^1 ♦ 2a^c^e"^a), B72 B76 B84

D12 D16 D22 D24 °32 d34

Q4a ^ 3(5a(P ♦ a^b^) + 45a^c^e"*^, - 3ac3,

- - (abe^0, - 9a^bc3e“^a),

Bg2 * 3 ♦ (9ac3 + a2b2 )e"*^a - 9a2c^e“^a ,

- (abc” 261 ♦ 3a2b(Pe"^a ) ,

Bg^ « 3ac^e“^a ,

(lOa^bc^ ♦ a^b3c3)e”^a - 3a^bc^e“^^a,

- 2(ac3 + 3a^c^e*"Ba),

- a2bc3e"2a,

e2* ♦ (2ac3 ♦ a^b2)e"^1 - 3(2a2c^ ♦ a3b2c3)e*"^a ♦ 9a3c9e™^^ 4a2bc^e"^a ,

- 3(ac^e“2a ♦ a2^ © - ^ ) ,

> 15ac3 ♦ 3a2b2 + 45a2c6e“i*a , - (abe20, - 9a2bc3e“2a) ,

■* - 3ac3,

D42

3 ♦ (9ac3 + a ^ 2)©"^01 - 9a2c^e“^a, ^44 “ ” (abe*201 + 3a2bç3e“^a),

D46

3ac3e"e^a,

D^2 * ab - 3a2bc^e™^a ,

ôac^e”201,

DgQ « 12a^bc^e~®a,

D54

D62 D64 D70 D74 DS0 D84

- je2^ — (2ac3 + a2b2)e™,2a' — 3 (2a2c^ ♦ a3b2c3 )e*™^01 — 9a3c9e“^ c*”|, 4a2bc3e“^a ,

Dgg * - 3(ac3e“2a - a2©^©**^1),

3 (a b e ^ - Sa^cPe-2 ™),

D?2 - - J e ^ - 3(5ac3 - a2b2) ♦ 45a2c6e~1*°j,

ab©20, - 9a2bc^e*,'2a,

" 3a Ci - 1/4(3% ♦ Sy),

Dgg ■ - 3ac^e“^a* C3 - - 1/4(3% - Sy),

C7 - - l/2T%y,

(10)

205

- 2(1 - ac^e”^ao)C2 * ab(Sx ♦ ^y) - 6 ^ 0 (5% - Sy), 4(1 -ac3©’^ © ) ^ - 4a2bc3e"2ao(Sx + Sy ) - (e^o > 3ac3)(Sx - Sy),

- 2(1 - ao3©""^o)C^ * j©2^© - (a©3 - a^b2)©"201© ♦ (3a2©^

+

(10)

a3b2©3)e"^o

- 3a3c9e"10aoj(Sx ♦ Sy) - 2ab(Sx - Sy), (1 ♦ *c?«~J*o)C6 - e ^ o T

,

- 2(1 ♦

ac3e ^ o ) C g - (e^o - 3ac3)T

,

The case ac3©”^61© ■ ♦ 1, for which some of the 0*8 in Equation 10 are infinite, does not correspond to a simple curve for a * a0 and hence is excluded.

THE STRESSES ALONG INNER BOUNDARY

The tangential stress in the boundary a ■ aQ is

°t ” The equation for

- a0*

can be obtained as

-4 ■ 4 G1 le^o 4- a^b2©^2^© - 9a2c^e~^a° - 2ab cos 2gJ h0

L r

*

-1

♦ 4C2|abe-®ao - (1 - 3ac3e-^ao)cos 2gl - 4C6Jl ♦ Sac3»"*0® in which

8inj2P,

(11)

hQ denotes the value of h fora * a0.

Substitution into Equation 11 of hQ from Equation 9, of C^,

C2, and

from Equation 10, and replacement of the constants a, b, c, and a0 by their

J(p2 +

values obtained from Equation 5 gives, finally

6rq) sin2 3 ♦ (q2 t 6rp) cos3 - 6r(p ♦ q) cos2 23 ♦ ^r^j°t

206

(Sx ♦ Sy)(p2 sin2 p + q2 cos2 p - 9r2) - T ^ C p + q f P ♦ q t 2r 8ln 2P

. (P2 - q2)(Sy + Sy) - (p * q) (Sy - Sy) (p - 3r) sin p - (q - 3r)co8 pll. p + q - 2r

The boundary stresses for different shapes are given in Figs.102, 103 ,104, PP. 207, 208.

THE APPLICATION OF GREENSPAN1S METHOD Completestresses for 1:4 and withrounded

a stope opening withheight

to width ratio

corners were obtained bypurelyanalytical

method.

BOUNDARY OF THE STOPE

Required Boundary

The equation for a hole given below or its nearest approach to the boundary should be first obtained using equation,

x ■ p Cos p + r Cos 3P y * q Sin p - r Sin 3P This can be done by trial-and-error method

—

—

—

Fig. 105# The required boundary of the hole Trial-and-error Solution

1.

From Curve 1, when p ■ 0,

x * 2, and y - 0.0

p - ~ , x - 0, and y » 0.5

(12)

207

%

102(a) Qyaiojd hole, tension parallel to long axis. Distribution of stress along ovaloid Ixnmdary. dashed curve shows the distribution of stress for the rase of an elliptical Ixiundary having the M K ratio of major to minor axis and the same rectified length as the ovaloid lioundary. The

Pig. 1 0 2 (b ) Ovaloid hole, tension parallel to short axis. D is trib u tio n o f stress along th e ovaloid Ixn m il.ir t he dashed curve shows the distribution of stress for the rase of an ellip tic a l Im un dary having tinWee ratio of major to minor axis and the same rectified length as the ovaloid !>oundarv.

208 'f

«Square" hole, tension parallel to side. The dashed c u r v é shows the distribution o f stress f o r rectiSed length as the "square” boundary.

t

Fig. 104

th e

case o f a c ir c u la r

b o u n d a r y h a v in g th e :

103 “S q u a r e " h o le , te n s io n p a r a lle l t o diagonal. hole, The d a s h e d curve show s th e d is t r ib u t io n o f "très hr the c s t o f a c ir c u la r b o u n d a r y h a v in g th e sam e Wtiied length as the "square" b o u n d a r y .

F ig .

209

Therefore

2 * p ♦ r 0.5 * q + r

1.5 * p - r q * 0.5 - r Also, when 3 » 7 degrees, x - 2,

y * 0.25

Sin 0 - 0.122

Cos 0 - 0.993

Sin 30 - 0.356

Cos 30 - 0.934

Therefore:

0.25 “ 0.122q - 0«358r

but

q - 0.5 - r

so

0.25 - 0.122 (0.5 - r) - 0.356r = 0.061 - O.46r 0.169 » - 0.48r r - - 0.394 p - 2 * 0.394 - 2.394 q - 0.5 + 0.394 - 0.694

Solving for co-ordinates of the hole from the Equation, x - 2.394 Cos 0 - 0.394 Cos 30

)

y - 0.894 Sin p * 0.394 Sin 3P

)

) p - 5 degrees

x - 2.394 (0 .996 ) - (- 0.394) 0.966 - 2.00 y » 0.894 (0.087) - (- 0.394) 0.259 - 0.18

p - 7 degrees

x - 2.394 (0.992) ♦ (- 0.394) 0.934 - 2.00 y - 0.894 (0.122 - (- 0.394) 0.358 - 0.25

P - 1 0 degrees

x - 2.394 (0.985) - 0.394 (0.777) - 2.03 y - 0.894 (0.225) - 0.394 (0.629) -0.45

p - 16 degrees

x -

2.04,

y - 0.54

P - 20 degrees

x -

2.05,

y « 0.65

(2)

210

Curve 1.

Required boundary in first quadrant.

W

Curve 2.

Curve obtained from Trial 1.

Curve 3.

Curve obtained from Trial 2.

211

Plotting the co-ordinates obtained by solving Equation 2, Curve 2, p. 210 is obtained. 2. To remove the irregularity obtained in Curve 2, a point P is taken in Curve 2 where there is maximum irregularity, p. 210. The point P shown is used for finding the angle and x and y coordinates to get a suitable equation of a curve approaching more nearly the required curve. The point

P taken on therequired

curve

imposes a condition on the new equation that the point P must fall where it is shown. At the point P, Curve 2, x - 1.5

y * 0.5

tan 3 -

w 0.333

3 • 18

degrees 26 minutes

Sin 3 = 0.30620

Cos 3 *0.94869

33 * 5 5

degrees 18 minutes

Sin 33 ■ 0.82214

Cos 33 * 0.56928

Therefore:

1.5 « p x 0.94869 0.5 -

q x 0.30620

+ -

r x 0.56928 r x 0.82214

« 0.30620 x (.5 - r)

- r x 0.82214

« 0.15310 - 0.30620r

- 0.82214r

0.3469 - - 1.128r r « - 0.307 p - 2.307 q * 0.807 Solving for co-ordinates: 3 - 5 degrees

x - 2.307 (0.996) - 0.307 (0.966) - 2.00 y - 0.807 (0.087) + 0.307 (0.259) - 0.15

3 - 7 degrees

x - 2.00 y - 0.21

Solving for other values of 3, x and y co-ordinates did not fall near

212

the required boundary, and Curve 3, p. 210, was obtained, 3. At the point P, Curve 3, x * 1,00

y * 0.5

3 * 26 degrees34 minutes

Sin 3 * 0,44724

Cos 3 * 0.89441

Cos 33 ” 0.17880

Sin 38 * 0.98389

33 ■ 79 degrees42 minutes

'

tan 3 - ^ q q * °»500

0.5 - q x 0.44724 - r x 0.98389 »

(0.5 - r) x 0.44724 - r x 0.98389

» 0.22362 - 0.44724r - 0.98389r 0.27638 - - 1.43H3r r - - 0.193 p - 2.193 q * 0.693 Solving for co-ordinates with several values of 8, the curve obtained did not confirm to the required boundary. Curve 4, p. 213. 4. At the point P, Curve 4> x ■ 0.5

y ■ 0.5

8 * 45degrees 38 - 135 degrees

Sin 8 ■ 0.70711

y » 0.5 - 0.70711q - 0.70711r v 0.70711 (0.5 - r) - 0.70711r • 0.353555 - 0.70711r - 0.70711r 0.146445 - 1.4l422r

P ” 2.104 q * 0.604

v . P

Cos 8 ■ 0.70711

Cos 38 = -0.70711 Sin 38 - 0.70711

x - 0.5 - O.TOTTlp + 0.70711r

r * - 0.104

tan 8 * §^4 * 1.00

233

/

Curve 4.

Curve obtained from Trial 3.

Curve 5.

Curve obtained from Trial 4.

The equation for the boundary becomes, x « 2.104 Cos ; - 0.104 Cos 3P

)

) y » 0.604 Sin p + 0.104 Sin 33

)

Co-ordinates from Equation 3, 3

Cosp

Cos 33

5

0.996

7

X

Sing

Sin 33

0.966

0.087

0.259

2.00

0.08

0.992

0.934

0.122

0.358

1.99

0.11

10

0.935

0.866

0.174

0.500

1.98

0.16

13

0.974

0.777

0.225

0.629

1.97

0.20

16

0.961

0.669

0.276

0.743

1.95

0.24

0.500

0.342

0.866

1.93

0.30

20

;\O.-9W:0

ÿ

y

30

0.366

000

0.500

1.000

1.82

0.41

40

0.766

-0.500

0.643

0.866

1.66

0.49

50

0.643

} —0.866

0.766

0.500

1.44

0.51

70

0.342

—0.866

0.940

-0.500

0.81

0.52

SO

0.174

-0.500

0.985

-0.866

0.42

0.50

90

000

-0.000

1.000

-1.000

0.00

0.50

"

Plotting the above co-ordinates. Curve 5, p. 213* was obtained, which very nearly approached the required stope opening.

CURVILINEAR COORDINATES

Values of Constants and Variables

2.104 - p ■ e* + a b e ~ * .......... .(4) p ♦ q » 2e 0.604 ■ q * e

Adding,

- abe"®

2.708 - 2ea

.... (5)

or

ea - 1.354

215

so

a « 0.3031,

e*"® » 0.7385,

and e"30, * 0.4028

putting these values in Equation 4, 2.104 - 1.354 + 0.7385 ab and

or

ab - 1.016

r - -0.104 * ac^e“3a

therefore,

-0.104 ■ 0.4028 ac3 ac3 * -0.258

The curvilinear co-ordinates are obtained by keeping ab and ac constant as obtained above, and varying a and p.

The general equation

for the ap-co-ordinates becomes, x - (ea +

l.Olèe”®) cos p - 0.258e"3a cos 3P

)

y * (ea -

l.OlSe"0 ) sin p ♦ 0.258e 3a sin 3P

)

The co-ordinate system obtained by Equation 6 is shown in Fig. 106, p. 216, With different values of ea and by changing the value of p from 0 degrees to 90 degrees for each value of ea , points beyond the boundary of the hole are obtained at which stresses can be calculated.

Table 15.

a

ea

Tabulated Values of a-variables used in Computations

e2*

V*

e"°

e-2a

e-4a

e-6™

e"8® e"10a

e'12®

0.3031 1.354 1.833

3.362

0.739 0.545

0.297

0.162

0.088 0.048

0 .0 2 6

0.46

1.584 2.509

6.297

0.631 0.399

0.159

0.063

0.025 0.010

0.004

0.70

2.014 4.055

16.445 0.497 0.247

0.061

0.015

0.004 0.0009 0.0002

1.14

3.127 9.777

95.583 0.320 0.102

1.30

3.669 13.464

0.0105 0.0011 0.0001

181.271 0.273 0.0743 0.0055 0.0004

0

0

0

0

0

216

O Fig. 106

CO-OBDIMATK STSTEM TOR THE STOPE OPBilRG

217

Table 16. Values of 3 -variables Cos 23 Cos 48 Cos 63 Sin 23 Sin 43 Sin 63 Cos 3

3

(Ss 33 Sin 9 Sin 39 •

3$ 0

degrees

1.000

1.000

1.000

5 degrees

0.985

0.940

0 .8 6 6 0.174

0.342

10 degrees

0.940

0 .7 6 6

0 .5 0 0 0.342

20 degrees

0 .7 6 6

0.174 -0 .5 0 0 0.643

35 degrees

0.342 -0*766 -0 .8 6 6 0.940

0

0

1 .0 0

0

0

0.#%087

0.259

0*643 p . 866 0.985

0 .8 6 6 0.174

0.500

0 .8 6 6 0 .9 4 0

0.500 0.342

0 .8 6 6

0.643 - O . W Ô 0.819 -0.259 0.574

0*966

50 degrees -0.174 -0.940

0 .5 0 0 0.985 -0 .3 4 2 - o . W 0.643 -0 .8 6 6 0 .7 6 6

0*500

60 degrees -O.500 -0.500

1 .0 0

70 degrees 43.766

0.174

0.500 0.643 -0 .9 8 5

75 degrees -0 .8 6 6

0 .5 0 0

80 degrees -0 .9 4 0

0 ,7 6 6 -0.500 0.342

0

0.985

0 .8 6 6 -0 .8 6 6 '

1.00

-1 .0 0

O

0 .5 0 0 -1 .0 0

0 .8 6 6

>0

Q.j$6é 0 .3 4 2 -0.866 0.940 -0 .5 0 0

0 .5 0 0 -0 .8 6 6 «îlyoo .0 .2 5 9 -0.707 0 .9 6 6 -0.707

’ V- ^ 4 4 i

c

J

1 tH

fH

t-1 tt

°tB

" 0 degrees

4.480 x 1 - 0 .6^2 - 2 .4 9 4 - 2.499 x 1 * 0.845 x 1

°tp

- 5 degrees

-

ta

10 degrees -

0 .8 4 0

- 4.569

: a i l ' K 0 . X * 0.8 4 5 x ~ 4~ "

~ 4-?6°

f .4 9 4 - 2 !499'x o \ w > * 0.8 4 5 x ÔTf R " o t m ' 4 ,5 ° 6

Jta ■ 20 degrees * §^7^

*

?t3

35 degrees - q ]||| - 0.897

225

2 !135 * ■°'675»

°tp “ 50 degr*9S *

°tg “ 60 degrees - -2.332 . -0.363 3.321

„ °tp -

70 degrees -^ 2 Z | . -0.894,

- 75

°t0 *

80 degrees ■

■ -0e884,

*35. degrees

°t3 "

90 desree8 -

- -0.877

Boundary

to

S3

g

Ioo

(0 OO •rl

(2

03

"3 E

O ti d V ts

£ H

b

î