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Photoelastic investigation of the stresses at the edge of a uniformly-loaded plug in a cylindrical hole Andrews, Gordon Clifford 1966

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A- PHOTOELASTIC INVESTIGATION OF THE STRESSES AT THE EDGE OF A UNIFORMLY-LOADED PLUG IN A CYLINDRICAL HOLE by Gordon C l i f f o r d Andrews B.A,Sc.. , U n i v e r s i t y of B r i t i s h Columbia., 1961  A Thesis Submitted i n P a r t i a l F u l f i l l m e n t  o f the  Requirements f o r the Degree o f Master of Applied Science In the Department of Mechanical Engineering  We accept t h i s t h e s i s as conforming to the required standard  THE  UNIVERSITY OF BRITISH COLUMBIA April,  1966  In p r e s e n t i n g  this  thesis  inpartial  fulfilment of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y Columbia, for  the  r e f e r e n c e and study.  extensive granted It  I agree that  copying of this by t h e  gain  shall  make i t f r e e l y  I f u r t h e r agree that thesis  that  shall  permission  copying o r p u b l i c a t i o n  of this  Date  thesis  n o t be a l l o w e d w i t h o u t my w r i t t e n  Mechanical Engineering  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8. C a n a d a  Columbia  13 May, 1966  for  representatives.  (Gordon C  Department o f  available  f o r s c h o l a r l y p u r p o s e s may be  Head o f my D e p a r t m e n t o r by h i s  i s understood  financial  Library  of British  for  permission  Andrews  ABSTRACT The general area of i n v e s t i g a t i o n was  first  suggested by the problem of designing a plug to block off a tunnel that was to be f i l l e d with water.*  The s p e c i f i c  purpose of t h i s project was to determine i f the shear stress along the edge of a plug i n a c i r c u l a r hole could be considered uniform when the hole on one side of the plug was subjected to hydrostatic pressure*  Three mathe-r  matical solutions were attempted using the theory of e l a s t i c i t y , but none y i e l d e d a simple s o l u t i o n .  The  problem was then attacked experimentally by plane photoe l a s t i c i t y and 32 configurations of s i x plane models were examined*  The r e s u l t s  showed that the shear stress was not  uniform, but rose, to a high peak and then declined r a p i d l y . Three-dimensional photoelastic techniques were also used and the r e s u l t s t h i s conclusion.  of f i v e " s t r e s s - f r e e z i n g "  models confirmed  Other s i g n i f i c a n t conclusions concern  the v a r i a t i o n of stresses with plug thickness; fillets  to strengthen the p l u g ;  the use of  and the d i s s i p a t i o n of  shear stress with distance from the p l u g .  Some discussion  i s also made of the optimum design f o r a p l u g .  PREFACE This thesis could not have been successfully completed without the generous assistance of many people. I.should l i k e to record on paper my thanks to a l l of these people, and i n p a r t i c u l a r to my supervisor, Professor William 0, Richmond, f o r h i s help and guidance;  to the laboratory technicians,' John Hoar,  Edward A b e l l , John Wiebe and P h i l i p Hurren for t h e i r many suggestions and t h e i r precise work i n preparing models and equipment;  and to Miss Isobelle McCafferty f o r her  valuable assistance- i n processing the photographs. I should l i k e also to express my gratitude to the National Research C o u n c i l , who provided the research grants which made t h i s project  possible.  TABLE OF CONTENTS CHAPTER I.  PAGE  STATEMENT OF PROBLEM Introduction Purpose  1  . . . . . . . . . . . . . . . .  1  . . . . . . . . . . . . . . . . .  1  Problem Defined Assumptions  *  2  * .  2  Variablels  3  Methods of Solution II.  .  5  THEORETICAL APPROACHES . . . . . .  7  Background from the Theory of E l a s t i c i t y  .  Use of Complex Variables  9  Numerical Solution of Biharmonic Equation III. IV.  DESCRIPTION OF PHOTOELASTIC POLARISCOPE  11  . .  EXPERIMENTS USING PLANE MODELS . . Description of Models Design .  . . . . . . . . . . .  17 20  .  F i l l e t Radii  20 20 20  Plug Thicknesses Material  7  .  21  . . . . . .  23  Design of Loading System . . . . . . . . .  23  Requirements Model Support  . . . . . . .  23  * . . .  24  iv CHAPTER  PAGE Loading Head  *  Pressure System  26 . . . . . . . . . . . .  Experimental Prooedure  «  Preparation of Models  29 * . . . . . . .  Testing of Models- . . . . . . T . . . . V.  RESULTS OF PLANE EXPERIMENTS  . . . . . . .  Observations  29 31 35 35  S p e c i f i c Observations  . . . . . . . . .  Patterns Observed , ' Calculated Results  .  .  Maximum Shear Stress  35 37  .  .  .  Shear Stress along Plug Edge  38 » . . . .  .  38 43  I s o c l i n i c s and Stress T r a j e c t o r i e s  . .  4-6  Estimate of Possible E r r o r  47  Discussion of Results  53  . . . . . . . . .  Uniformity of Shear Stress  53  Maximum Values of Shear Stress Decrease i n Shear Stress Optimum Plug Design VI-  27  . . . .  .•  . . . ... . . . .  EXPERIMENTS USING THREE-DIMENSIONAL MODELS Description of Models Material Model Design  53 545^ 58 58 58  *  59  V  CHAPTER  PAGE Design of Loading System  »  Requirements  60 . . . . . . .  60  Model Support Pressure System  60 . . . . .  61  Experimental Procedure  64  Preparation of Models Testing of Models  . . . . . . . .  . . . . . . . . .  64 .  S l i c i n g of Models VII.  64 65  RESULTS OP THREE-DIMENSIONAL EXPERIMENTS  .  Observations  69 69  Patterns Observed . . . . . . . . . . .  69  P o s i t i o n of Zero Fringe  . . . . . . .  70  . . . . . . . . . . .  73  Calculated Results  Uniformity of Shear Stress  73  Maximum Shear Streps  . . . . . . . . .  73  P r i n c i p a l Stresses i n Three Dimensions  74-  Isoclinics  . .  75  . . . . . . .  75  and Stress T r a j e c t o r i e s  Estimate of Possible E r r o r V I I I . SUMMARY OF RESULTS  79  Comparison of Plane and Three-Dimensional Results  79  Uniformity of Shear Stress  79  vi PAGE  CHAPTER Maximum Values of Shear Stress Diffusion of Shear Stress Conclusion BIBLIOGRAPHY  .  .  .  .  .  . . . . . .  79 80 81 82  LIST OF FIGURES FIGURE  PAGE  1. :  Sectional View of Hole and Plug  , . . . . .  2.  S i m p l i f i e d Configuration f o r T h e o r e t i c a l Approaches  3.  10  Transformation of Slot Configuration to Lower Half-Plane . . . . . . . . .  4*  Network of Points for F i n i t e  10 Difference  Approximation of Biharmonic Equation . . . 5i  4-  13  Mathematical Model of Plane Configuration f o r Numerical Solution of Biharmonic Equation  13  6*  Schematic Flow Diagram f o r Computer Program  15  7.  Photbelastic  17  8.  T y p i c a l Model i n Support with Loading Head i n Place  9. 10;  Polariscope  . . .  . . . . . . . . . .  . . . . . . . . . . . . . . .  22  Model Ready for Testing  25  Schematic Drawing of Loading System for Plane Models . . . . . . . . . . . . . . .  11.  28  Photograph of Assembled Loading System for Plane Models  12.  Model Aligned i n O p t i c a l Path  13.  Polariscope, Testing  14-.  (  28 .  Model,Loading System,Ready  »  33  for . . . "  33  F u l l Size Fringe Patterns for Models with a Plug Thickness of about.Five. Hole Diameters and Various F i l l e t Radii  i . . *  39  viii FIGURE 15.  PAGE  Enlarged Fringe Patterns f o r Models with a Plug Thickness of 1/4- Hole Diameter and Various F i l l e t Radii . . . . . . . . . . .  16.  40  Enlarged Fringe Patterns f o r Models' with a F i l l e t Radius of 1/8 Hole Diameter and Various Plug Thicknesses  17.  41  Shear Stress along Edge of Various Thick Plugs  18.  4. . . . . . . .  «. . . . . . . . . . . . . . . . .  .  44  Shear Stress Along Edge of Plugs with F i l l e t Radius of One-Half the Hole Diameter and Various Thicknesses  19.  . . . . . .  45  Maximum Shear Stress as a Function of Plug Thickness  48  20. /  Two I s o c l i n i c Patterns f o r Plane Model No.4  21.  Stress T r a j e c t o r i e s f o r Two Configurations of Model No.4  22*'  < . . . ....  .  50  Non Dimensional Shear Stress Around Slot f o r Two Plane Models  57  23.  Stress-Freezing Model P r i o r to Testing  24.  Loading System f o r Stress-Freezing Models  25.  Schematic  .'• . . .  62 62  of Loading System f o r Stress-  Freezing Models 26.  49  . . . . . . . . . . . . .  Model and Loading System P r i o r to Testing  63 .  66  ix FIGURE  PAGE  27.  Stress-Freezing Apparatus During Test  . . .  66  28*  Method of S l i c i n g Stress-Freezing Models . .  67  29.  Comparison of Fringe Patterns f o r Equivalent Plane and Three-Dimensional Models . .. . .  30.  71  Further Comparison of Fringe Patterns f o r Equivalent Plane and Three-Dimensional Models ; . ,  31.  32.  . . . . . . . . . . . . . . .  72  Comparison of Shear Stress along Plug Edge for Plane and Three-Dimensional Models . .  76  I s o c l i n i c Patterns for Two Equivalent Models  77  CHAPTER I STATEMENT OF THE PROBLEM Ii  INTRODUCTION  During the construction of one of the Columbia River dams i n B r i t i s h Columbia, i t was necessary to block off a tunnel that had been bored through s o l i d rock and which was soon to be f i l l e d with a considerable head of water.  A concrete plug was designed by f i r s t  assuming that  a uniform shear stress would exist along the side of the plug where i t adhered to the tunnel wall and then c a l c u l a t ing the length of plug necessary to give a safe value of t h i s average shear s t r e s s .  The question of whether i t was  v a l i d to assume a uniform, average, shear stress along the plug-edge became the o r i g i n a l i n s p i r a t i o n f o r t h i s  research  project. I I . PURPOSE The purpose of t h i s i n v e s t i g a t i o n was to examine the configuration of a c i r c u l a r plug f i x e d i n a c y l i n d r i c a l h o l e , with hydrostatic pressure i n the' hole on one side of the plug and atmospheric pressure on the other, and to determine: 1.  whether the shear stress along the edge of the plug was uniform,  2.  what the maximum shear stress induced i n or around the plug for a given  hydrostatic  pressure, was, and 3.  how r a p i d l y the shear stress decreased as distance from the plug III.  increased.  PROBLEM DEFINED  Assumptions In order to reduce the number of variables to a manageable l e v e l f some assumptions were made. summarized as  These  are  follows:  1J- Both the plug and the material through which the hole i s bored were assumed to be e l a s t i c s o l i d s with no voids or f a u l t s and the same modulus of e l a s t i c i t y  and y i e l d strength  throughout. 2.  The hole was assumed to pass completely through the body,, which was considered to be very l a r g e *  3v  The plug was assumed to be bonded  Integrally  with the c y l i n d r i c a l h o l e , or, i n other words, the bond between the plug and the hole was assumed to have the f u l l 4*  strength of the  The plug was assumed to be a s u f f i c i e n t from the hole opening so that the stress bution around the plug was unaffected hole opening*  material. distance distri-  by the  '5 Variables' These,assumptions permitted the scope of the r e search to be narrowed down to the two most important ables that a f f e c t the stress d i s t r i b u t i o n .  vari-  These v a r i a b l e s  are. the thickness of the plug and the f i l l e t  radius at the  point where the plug and the wall meet. Plug thickness. important  The thickness of the plug  f a c t o r i n i t s strength.  As shown i n Figure 1, the  thickness (h) can have any value greater than zero. expressed  i s an  It i s  throughout t h i s t h e s i s i n terms of the hole  meter (d) or as a dimensionless  dia  T  r a t i o of the plug thiokness  to hole diameter (h/d). F i l l e t " r a d i u s . The radius of the f i l l e t  at the corner  where the loaded surface of the plug meets the wall of the hole was found to have a s i g n i f i c a n t e f f e c t on the concentra* t i o n of stress at t h i s corner.  The radius can take any value  from zero ( f o r a square corner)to one-half the hole diameter. At t h i s maximum diameter, the f i l l e t s on e i t h e r side are tangent at the center of the plug,making the surface of the plug semi-spherical.  Wherever reference i s made, i n t h i s  t h e s i s ^ to the f i l l e t  radius, i t w i l l be expressed  dimensional  form i n terms of the hole diameter.  i n non^  FIGURE 1 SECTIONAL VIEW OF HOLE AND PLUG  5 IV. METHODS OF SOLUTION It had been intended from the outset to use photoelasticity  i n determining the stress d i s t r i b u t i o n experi-  mentally.  P h o t o e l a s t i c i t y i s p a r t i c u l a r l y useful  complicated configurations  and investigations  for  where the  stresses at many points of a model are to be found.  A l l the  major sources of information on stress analysis were consulted and a wealth of information on techniques was discovered.  However* no solutions  of problems with a  s i m i l a r configuration were found, and the c o r r e l a t i o n of the r e s u l t s with other experiments  could not be made.  A summary  of the methods of s o l u t i o n used i n t h i s i n v e s t i g a t i o n i s  as  follows: Plane p h o t o e l a s t i c i t y .  The reason f o r using a two-  dimensional technique to examine a three-dimensional problem was, of course., that plane p h o t o e l a s t i c i t y  is easier,  and cheaper than three-dimensional photoelasticity*. since the models are not destroyed (as  i t i s possible to  re-check data or to a l t e r the model and re-use i t . therefore,  Alsoy  they are i n the  "stress^freezing" method) during t e s t i n g ,  this investigation,  faster  concerned the plane  Most of approxi-  mation. Three-dimensional p h o t o e l a s t i c i t y .  The "frozen-stress"  method of three-dimensional p h o t o e l a s t i c i t y was used i n t h i s  i n v e s t i g a t i o n and was an important and e s s e n t i a l  approach  since i t was used to gauge the accuracy of the plane approximation.  The stress-freezing  procedure was time-  consuming and cumbersome compared to plane p h o t o e l a s t i c i t y but no insurmountable problems were encountered, and the r e s u l t s were a very valuable comparison-for the plane r e s u l t s T h e o r e t i c a l approaches.  Several attempts were made t  f i n d a t h e o r e t i c a l s o l u t i o n that would predict the outcome of the experiments.  These efforts  only served to show that no  simple t h e o r e t i c a l s o l u t i o n e x i s t e d . approaches  These t h e o r e t i c a l  are discussed i n d e t a i l i n the next chapter.  CHAPTER II THEORETICAL APPROACHES The purpose of examining these t h e o r e t i c a l approaches was to see i f simple solutions to the problem could be found f o r c o r r e l a t i n g the experimental  results.  None of the approaches y i e l d e d a simple s o l u t i o n , if  sufficient  although  time were devoted to advanced mathematical  and numerical techniques, useful with these methods.  This chapter, therefore,  i n i t i a l work and the problems The f i r s t two dimensions.  answers could be found describes the  encountered.  step was to simplify the configuration to Accordingly, a t h i n section passing along  the center l i n e of the hole,' and s i m i l a r to Figure 1, was taken as a plane approximation of the three-dimensional case.  By taking the case where the plug meets the wall i n  a square corner, and assuming that the plug was i n f i n i t e l y t h i c k , the configuration was further s i m p l i f i e d to a pressurized s l o t i n the i n f i n i t e h a l f - p l a n e ,  resemble  as shown i n  Figure 2. I.  BACKGROUND FROM THE THEORY OF ELASTICITY  It has been shown i n the c l a s s i c a l  theory of  elasticity" 1 " that the s o l u t i o n of two-dimensional problems  8  can be reduced to the i n t e g r a t i o n of the biharmonic differential  equation:  a**  h**f  >f •  where 0 i s the Airy stress f u n c t i o n , the stresses being defined by ' and  (2), (3)  ^  7^= — The general procedure i n solving f o r (p i s a t r i a l  and error process.  Usually a number of terms are chosen  which have the proper symmetry and which s a t i s f y the b i harmonic equation, and the c o e f f i c i e n t s  of these terms are  determined by i n s e r t i n g the second derivatives of 0 i n t o the equations f o r the stresses at the boundary of the body. In t h i s p a r t i c u l a r case, since a l l the boundary l i n e s were p a r a l l e l to the x and y axes, the equations f o r the stresses at the boundary reduced to equations 2, 3 and 4 .  The only  non-zero stresses around the s l o t were 6"x.= -P along the side and 6ij = -P along the bottom.  A l l the stresses are  zero along the x - a x i s , and approached zero as x and y got very l a r g e .  This ruled out a stress function composed of  polynomial terms since the derivatives would increase with x and y , and i n d i c a t e d e i t h e r a grouping of more compli-  9  cated terms or an i n f i n i t e trigonometric s e r i e s . f o r a s u i t a b l e stress function was r therefore,  The search  discontinued  at t h i s p o i n t . II.  USE OP COMPLEX VARIABLES  An attempt was made to simplify the configuration using complex v a r i a b l e s . 1*  The procedure proposed was:  to f i n d a mapping function that would transform the s l o t configuration onto the lower h a l f - p l a n e , as shown i n Figure 3;  2.  to solve the simpler z.-plane configuration using a stress function composed of two a n a l y t i c  ;  2  functions, 3*  as described by M u s k h e l i s h v i l i  ; and  to use t h i s s o l u t i o n to solve the given problem by employing the inverse- transformation.  The Schwarz-Christophel transformation was used since the r e c t i l i n e a r configuration was i d e a l l y suited to i t . However, the transformation resulted i n an e l l i p t i c of the second k i n d :  Z  integral  • .  FIGURE 2  SIMPLIFIED .CONFIGURATION.FOR THEORETICAL APPROACHES  :  FIGURE 3  TRANSFORMATION OF SLOT CONFIGURATION TO LOWER HALF-PLANE  /  11 This i n t e g r a l can be put into a form for which r e s u l t s are t a b u l a t e d , but i t was evident that the  approach  to the problem was beyond the scope of t h i s research  project,  and no further work was done. 3 Subsequently,  a s i m i l a r problem solved by Dohse  revealed that although a mathematical s o l u t i o n had been found using t h i s approach, extensive numerical work was necessary before the stresses could be c a l c u l a t e d . III.  NUMERICAL SOLUTION OF BIHARMONIC EQUATION  By d i v i d i n g the area of a two-dimensional  elasticity  problem into a network of points arranged i n a g r i d as shown i n Figure 4, the biharmonid equation (1) may be expressed i n the f i n i t e - d i f f e r e n c e  form:  + 05 + 07 + <&> + 0«  = O  ( 5 )  This equation may then be solved by using an e l e c t r o n i c computer for r a p i d c a l c u l a t i o n .  A program f o r the IBM 7040  computer was written using t h i s approach, and i s as  described  follows: Because of the symmetry of the plane configuration  i n Figure 2, only one-half was considered.  A large,  but  12 f i n i t e , mathematical model was u t i l i z e d , with a network of 20,000 p o i n t s , as,; .shown i n Figure 5, and the values of 0 , and The by  were c a l c u l a t e d at a l l points of the boundary.  values along the s l o t and upper boundary were obtained i n t e g r a t i n g equations  (1) and (2).  For the side and  lower boundaries, though, a stress function from Timoshenko"'' was used:  0 where r and 0  J^-r<9stn0 are shown i n Figure 5.  -i-Kre>tn<9  ( 6 )  It was assumed  that the pressure (P) at the bottom of the s l o t could be represented by a concentrated force at the o r i g i n since the distance from the point of a p p l i c a t i o n of the force to the boundary should be s u f f i c i e n t  f o r St* Venant 1 s p r i n c i p l e to  apply. A necessary row of points around the outside of the boundary were c a l c u l a t e d by extrapolating the nearest i n t e r i o r point using the slope at the boundary*  The i n t e r i o r  points were approximated, i n i t i a l l y , by a l i n e a r i n t e r p o l a tion.  The biharmonic f i n i t e - d i f f e r e n c e equation, i n the form:  <& = o.4(<ft+<&+03+<k) - O - l (0^0 + 0o <A*) +  8  -0.05  (>  5  +  0 4-0, +0„) 7  (?)  was then applied to each point and a more accurate value of  7  9  8  2  6  3  0  1  10  4  12  5  ll  FIGURE 4 NETWORK OF POINTS FOR F I N I T E - D I F F E R E N C E APPROXIMATION OF BIHARMONIC EQUATION  FIGURE 5 MATHEMATICAL MODEL OF PLANE CONFIGURATION FOR NUMERICAL SOLUTION OF BIHARMONIC EQUATION  14 0 was obtained by averaging the twelve closest  points.  This procedure was a p p l i e d , s u c c e s s i v e l y j to every i n t e r i o r point of the g r i d , ' with the process being repeated nearly stationary values were obtained.  until  The program, which  i s shown schematically i n Figure 6, ran s u c c e s s f u l l y ,  after  the usual minor errors were eliminated. • It c a r r i e d ' o u t 100 i t e r a t i o n s  i n 28 minutes, then  c a l c u l a t e d and p l o t t e d a graph of the stresses along the edge of the p l u g .  The accuracy of these r e s u l t s were,  however, unacceptable.  The change i n (p during the l a s t  i t e r a t i o n was 50 times as great as the l i m i t of 1% of the difference between points which had been s e t . e r r o r on the f i r s t  The possible  difference was therefore i 2 5 $ , and, on  taking the second d e r i v a t i v e , Was much l a r g e r . This unsatisfactory  r e s u l t could probably be  corrected by using advanced techniques to make the values converge more r a p i d l y .  However, t h i s program represented  only one configuration i n the range to be i n v e s t i g a t e d , ' and the work necessary to improve the program and extend i t other configurations was considered excessive.  to  This method  was, t h e r e f o r e , discontinued, and no t h e o r e t i c a l r e s u l t s were obtained for c o r r e l a t i o n of the photoelastic  experiments.  15  CALCULATE: EACH EXTERIOR POINT BY EXTRAPOLATION DIMENSION MATRIX FOR 0 AND MATRICES FOR M iM. AROUND BOUNDARY 3x v 6\j  R E A D PRESSURE- (P)  T RE-CALCULATE EACH INTERIOR VALUE OF 0 USING EQ* (7)  CALCULATE <5^% ALONG PLUG EDGE USING E Q ^ ( 2 ) ^ . ( 4 ) IN NUMERICAL FORM V  T  CALCULATE BOUNDAR Y VALUES OF USING EQUATIONS (2) ,(3) t~ (G)  C A L C U L A T E INTERIOR V A L U E S  OF (p ROUGHLY BY LINEAR INTERPOLATION  PRINT S i , T j ^  PLOT GRAPH OF c S ^ T ^ V S DISTANCE ALONG PLUG ETJQE  STOP  FIGURE 6 SCHEMATIC FLOW DIAGRAM FOR COMPUTER PROGRAM  CHAPTER III DESCRIPTION OF PHOTOELASTIG POLARISCOPE Although the experimental techniques were f o r preparing and s t r e s s i n g the two types of  different  photoelastic  models, the same polariscope was used to observe the  fringe  patterns on both the plane models and the " s l i c e s " from the three-dimensional models.  This chapter, therefore,,  devoted to describing the  apparatus which was common to  these  is  experiments. The polariscope was model No.401 manufactured by  the P o l a r i z i n g Instrument Company.  It i s shown schematic-  a l l y i n Figure 7 and had the following  characteristics:  Light Source The polariscope  was designed to operate with maxi-  mum e f f i c i e n c y using monochromatic green l i g h t with a wavelength of 5^-61 angstroms. j e c t i o n bulb was used;  A 100-watt, mercury vapour pro-  i t emitted a h i g h - i n t e n s i t y mercury o  spectrum which has 54-61 A as i t s brightest  visible line.  Filters The l i g h t emitted from the p r o j e c t i o n bulb passed through a blue glass heat f i l t e r ; plate;  and Wratten f i l t e r s  a ground glass  diffusing  58 and 77 between two glass  17  FIGURE 7 PHOTOELASTIC POLARISCOPE  18 plates.  Tests were run on the lamp and f i l t e r  assembly  using a spectroscope and only the green (54-61 A*) l i n e was v i s i b l e on the developed p l a t e s , showing that the were, indeed, e f f e c t i v e .  Since the f i l t e r s  filters  were  situated  between the lamp and the p o l a r i z e r , minor imperfections i n the glass plates had no effect  on the photoelastic  fringe  patterns. P o l a r i z i n g and quarter-wave  plates  The p o l a r i z e r , quarter-wave plates and analyzer were 4# inches i n diameter.  The p o l a r i z e r and analyzer  were " P o l a r o i d " laminated i n s t r a i n - f r e e  glass*  wave plates were polystyrene, also laminated i n glass.  The assembled o p t i c a l path gave c l e a r ,  The quarterstrain-free sharp, fringe  patterns and the only problem encountered i n the use of the polariscope was the lack of a linkage to permit  simultaneous  r o t a t i o n of the analyzer and p o l a r i z e r for easier viewing of the i s o c l i n i c s . Lens system and camera The f i l t e r e d l i g h t was formed into a 4}( inch d i a meter p a r a l l e l beam by a condensing lens and a c o l l i m a t i n g lens mounted i n the lamp-holder.  Upon emerging from the  analyzer, t h i s p a r a l l e l beam entered a collecting-condensing lens and passed d i r e c t l y to a 35n™ Exacta camera equipped  19 with a telephoto lens ( f o c a l length = 1 3 5 mm). P o s i t i o n i n g of models The models and " s l i c e s " were placed on a c e n t r a l frame midway between the p o l a r i z e r and analyzer.  The bed  of the frame could be moved h o r i z o n t a l l y and v e r t i c a l l y f o r accurate p o s i t i o n i n g of the models i n the o p t i c a l path. The frame i s not shown i n Figure 7» but can be seen i n Figure 9•  CHAPTER IV EXPERIMENTS USING PLANE MODELS I.  DESCRIPTION OP MODELS  Design The plane models simulated a t h i n section or " s l i c e " taken along the centre l i n e of the hole and passing through the plug and the body containing the h o l e .  The  models, therefore, resembled Figure 1 , as can be seen from Figures 8 and 9 • The outer dimensions of the models were 7 7 / 8 inches wide by 9 inches h i g h .  This s i z e was chosen to take advan-  tage of a r e c e n t l y - b u i l t loading frame which proved to be very convenient during the t e s t s .  The upper s l o t was one  inch wide and four inches long on a l l models. from the bottom of the s l o t ,  Measuring  there was a distance of 3# to 5  hole diameters to the boundary of the model. It was presumed that t h i s was adequate clearance to avoid interference from l o c a l stresses at the model edge, and t h i s , indeed, proved to be t r u e . Fillet  Radii In order to observe the effect  on the stress  d i s t r i b u t i o n as the f i l l e t radius was v a r i e d , four models  21  were designed, each with a different f i l l e t r a d i u s .  These  models were numbered as f o l l o w s : N o . l — zero f i l l e t radius (square c o r n e r ) ; No.2 — f i l l e t radius equalled one-eighth the hole diameter; No*3 — f i l l e t radius equalled one-quarter the hole diameter; No.A- - - f i l l e t radius equalled one-half the hole diameter* Two replacement models were also made, s i m i l a r to models N o . l and N o . 2 . These four f i l l e t r a d i i provided a v a r i a t i o n over the complete range possible,, from the smallest largest possible r a d i u s .  to the  The f i l l e t s were the same on both  sides of the p l u g . Plug Thicknesses Each model, when newly-made, was rectangular, with a single s l o t on i t s upper edge to which pressure was applied.  This represented the configuration f o r an extreme-  ly thick plug.  The plug thickness was then reduced by  c u t t i n g a s l o t i n the lower edge of the model to the extension of the h o l e .  The plug thickness could then  be further reduced by lengthening t h i s s l o t , Figure 8.  simulate  as shown i n  HYDRAULIC  CO  FIGURE. 8 TYPICAL MODEL IN SUPPORT WITH LOADING HEAD IN PLACE  23  Material The plane models were made from ]i inch t h i c k Columbia Resin 39 (also c a l l e d CR-39 and Homalite 911) which was purchased i n s t r a i n - f r e e surfaces.  sheets with polished  The y i e l d strength of CR--39 i s about 6,000 p s i  and the modulus of e l a s t i c i t y i s about 250,000 p s i . stress-optical  constant  The  (or fringe constant) was taken as  89 p s i / f r i n g e / i n c h , which was the average of f i v e t e s t s taken using the material on hand.  However, an important  f a c t o r that was not r e a l i z e d at the beginning of the t e s t s was that CR-39 i s subject to serious loads.  strain-creep at high  Because of "fais, the fringe constant depends on the  length of time between applying the load and observing the fringe pattern.  No attempt was made to standardize  this  delay p e r i o d , but t h i s omission does not seem to have s e r i o u s l y affected the r e s u l t s ,  and the measured value of  the fringe constant agrees with the value obtained by 4 Clark  with a few seconds'  delay between loading and observ-  ing the model. II.  DESIGN OP LOADING SYSTEM  Requirements The loading system for the plane models was designed to do two t h i n g s :  24 1.  apply a uniform pressure to the upper notch which simulated hydrostatic pressure i n a h o l e v and,  2.  support the model so that i t reacted to t h i s pressure l i k e an i n f i n i t e s l i c e rather than a finite  plate.  Model Support The model support, as shown i n Figures 8 and 9, held the model r i g i d enough to simulate the r e s t r a i n i n g effects  on a s l i c e of a three-dimensional c o n f i g u r a t i o n ,  and yet not so r i g i d that l o c a l stresses developed at the perimeter of the model.  The support consisted of a three-  s i d e d , pin-connected frame.  The model rested on the h o r i -  zontal beam and was. then clamped along the sides and bottom by the frame.  Wherever the metal touched the model,  s t r i p s of 1/16 inch t h i c k rubber were used to give a more uniform pressure.  P l a s t i c spacers were also i n s e r t e d ,  where necessary, for more uniform clamping. This support prevented the large deformations and bending stresses i n the plug that would have r e s u l t e d i f the model had been unrestrained when pressure was a p p l i e d . A l s o , examination of the model under load revealed no  FIGURE 9 MODEL READY FOR TESTING  26 significant  l o c a l stresses around the perimeter at any time.  Loading Head A uniform pressure was applied to the upper s l o t using a loading device s i m i l a r to those discussed by 5  D u r e l l i , Lake and Tsao^,. i n which a gum rubber tube was i n f l a t e d by compressed a i r and forced evenly against the model*  Figure 8 shows a broken-out section of the loading  head, with the gum rubber tube v i s i b l e .  • .  The loading devices suggested by D u r e l l i et  alt  were made of s t e e l and r e l i e d on precise alignment of the model edge to prevent the rubber tube from expanding around the model and b u r s t i n g .  An adaptation was therefore made  to avoid t h i s requirement.  Two 3 by 5-inch plates of #-inch  t h i c k p l e x i g l a s were bolted to the loading head on e i t h e r side of the tube, with a s l i d i n g clearance  over the model,  and the need for precise p o s i t i o n i n g was eliminated. The p l e x i g l a s d i d not affect the stress p a t t e r n , mainly because i t i s r e l a t i v e l y i n s e n s i t i v e compared to GR-39• To apply pressure to the s l o t , compressed a i r was applied to the gum-rubber tube from the system shown i n Figure 10*  At about 25 to 30 p s i , the tube reached i t s -  y i e l d point and expanded, touching the perimeter of the s l o t * Any further increase i n a i r pressure was transmitted to the  27 s l o t through the tubing, and no other part of the loading head applied any force to the model. The pressure read from the gauge d i d not, of course^ agree with the actual pressure on the s l o t .  D u r e l l i et a l .  c a r r i e d out several t e s t s on t h e i r device and showed that the r e l a t i o n s h i p between the gauge pressure and the  effect-  ive (or actual) pressure was l i n e a r , except i n a small range around 25 to 30 p s i when the tube  firstexpanded.  The loading head was tested i n a s i m i l a r manner and the r e l a t i o n s h i p also proved to be l i n e a r , with a slope of u n i t y , f o r gauge pressures over 4-0 p s i .  This meant that  the actual pressure could be found from the gauge reading simply by subtracting the pressure required to expand the tube*  This :"expansion" pressure was found at f i r s t by  extrapolating tbe l i n e a r r e l a t i o n s h i p to zero actual pressure*  However, a simpler method (described i n the test  procedure) gave the same accuracy and was preferred* Pressure System The system f o r C o n t r o l l i n g the pressure supply to the loading head i s shown i n Figures 10 and 11.  The  pressure source was a 2,000 p s i tank of compressed a i r which was regulated by a standard high-pressure  regulating  v a l v e , and pressure readings were made from a 0-2,000 p s i  FIGURE 1 1 PHOTOGRAPH OF ASSEMBLED LOADING SYSTEM FOR PLANE MODELS  29 gauge with scale markings every 10 psiw  The system was  designed to withstand high pressures^ since the "Strato-flex"  flexible  hose from the tank to the regulator was  subject to 2,000 p s i and the pressure on the model side occasionally reached 350 p s i during testing,, III.  •EXPERIMENTAL PROCEDURE  Preparation of Models The CR-r39 for the models came from the manufacturer with glassr-like  surfaces that required no further p o l i s h i n g .  It was necessary, then, merely to cut the material to the o u t l i n e shown i n Figure 8. first  This was more d i f f i c u l t than i t  seemed, since CR^39, unlike " L u c i t e " or  i s a very b r i t t l e p l a s t i c  "Plexiglas",  to machine, and small edge chips  show c l e a r l y on the patterns and obscure the fringe order on the boundary* ing  There i s also a problem with r e s i d u a l machin-  stresses i f the heat generated by the c u t t e r i s  excess-  ive . The f i r s t  f i v e models were cut on a m i l l i n g machine  using a sharp end-mill r o t a t i n g at about 1600 rpm.  The  s i x t h model was shaped using the method suggested by Dolan and Murray^, and described i n d e t a i l by Lee, Meadows and 7 TaylorV.  This method was also used to cut out the lower  s l o t s on a l l models.  A comparison of the two methods  is  30  given i n t h i s  section.  •The m i l l i n g operation was a long., precise  operation  which gave dimensions accurate to a few thousandths and sharp edges with no c h i p s .  However., many precautions were  required to avoid r e s i d u a l machining s t r e s s e s .  These pre-  cautions^ recommended by Dolan and Murray^, included using an a i r jet  d i r e c t e d on the t o o l to cool i t ;  using sharp  m i l l i n g cutters reserved s o l e l y f o r use with p l a s t i c ;  and  taking small cuts of ,003-inches or less on the f i n a l passes*  Nevertheless, i n spite of t h i s care, the r e s u l t i n g  models showed unacceptably^-large machining stresses of 1)£ fringe  orders. The alternate method, although i t gave l e s s  precise  dimensions, was much f a s t e r and gave n e g l i g i b l e machining stresses.  The procedure required an accurate metal template  which was f i x e d to the 01^39 using s p e c i a l masking tape, adhesive .on both sides*  The unwanted p l a s t i c was cut away  by a small jig-saw to within about 1/16-inch of the template,  and the model was then trimmed to size using a }4~inch  diameter Pratt & Whitney tungsten carbide cutter with 48 shallow, h e l i c a l , c u t t i n g teeth..  The cutter was mounted i n  an ordinary d r i l l - p r e s s and the template was guided by a c i r c u l a r p i n threaded i n t o the d r i l l - p r e s s table d i r e c t l y  31 below the c u t t e r *  Actually., two pins were u s e d : -  a  "roughing" p i n about 1/64-inch larger than the c u t t e r , and a "finishing." p i n the same diameter as the cutter*  The  depth of the l a s t cut c o u l d , therefore, be c o n t r o l l e d somewhat*  The templates were made from 1/16-inch s t e e l f o r  accuracy and d u r a b i l i t y but i t i s believed that 1/8-inch aluminum-would have been as suitable and easier to make* This method gave r e s i d u a l stresses of l e s s than h a l f a fringe order at the maximum*  The cause of t h i s  difference was most l i k e l y the r a p i d , shallow cuts taken by the Pratt & Whitney c u t t e r , although the need t o clamp material t i g h t l y during m i l l i n g may also be a f a c t o r . .The excessive stresses on the o r i g i n a l f i v e models were a l l at the f i l l e t s  at the bottom of the s l o t s .  •Consequently, they were e a s i l y eliminated by lengthening the s l o t s about X - i n c h using the second method of c u t t i n g , Testing of Models A t o t a l of 32 configurations of the six plane models were stressed and examined i n the course of t h i s project.  Models N o . l and i t s replacement, N o . l - X , both  with square corners, broke during t h e i r f i r s t  test.  Models  No.2, 3, 4- and a duplicate of No.2, c a l l e d 2-X, were t e s t e d  32 using the following general procedure: The loading head was s l i d Onto the upper s l o t of the model', and then threaded onto the end of a hydraulic p i s t o n which was part of the main loading frame* p i s t o n permitted fine adjustment vertical direction).  (This  of the loading head i n the  The model support was then clamped to  the model, with s p e c i a l care to ensure that i t was centered and uniformly supported*  The pressure system was connected  to the loading head, and the main loading frame was moved so that the model was centered i n the o p t i c a l path, as shown i n Figures 12 and 13*'  The regulating valve was then  opened and the expansion of the tube was watched c l o s e l y * The pressure at which the tube expanded and f i r s t touched the bottom of the s l o t could be e a s i l y observed, and was recorded, as well as the pressure.at which the tube completely f i l l e d the s l o t .  The average of these two pressures gave the  "expansion" pressure described e a r l i e r ^ and the actual pressure on the s l o t was obtained by subtracting t h i s amount from the gauge reading. The pressure was then increased u n t i l 8 or 9 fringes were observed on the model*  The fringe pattern was photo^  graphed with both dark and l i g h t f i e l d s , taking care not to change the loading or focus between exposures.  Photographs  of the i s o e l i n i c s every 5 ° through 9 0 ° were also taken, and  FIGURE 12 MODEL ALIGNED IN OPTICAL PATH  •34-  when t h i s was completed, the pressure was r e l i e v e d * The apparatus was then dismantled* and the lower s l o t lengthened using the template method o-f c u t t i n g described e a r l i e r . the p l a s t i c ,  B i t s of masking tape usually stuck to  so the model was washed quickly i n soap and  water and dried thoroughly. be repeated.  The test procedure could then  Overnight, and during long delays between  t e s t s , the models were stored i n a vat f i l l e d with a v i a t i o n o i l to prevent edge effects from- forming.  CHAPTER:V RESULTS OP PLANE EXPERIMENTS I. OBSERVATIONS This s e c t i o n describes  some of the more important  observations made during the t e s t s on the 32 plane c o n f i g urations. Specific  Observations Model N p w l . The s i g n i f i c a n t c h a r a c t e r i s t i c  of  Model No'.l was the intense stress concentration at the sharp corners of the s l o t *  On the very f i r s t  test of the  s e r i e s , t h i s model cracked under a pressure of 150 p s i with only 8 fringes  clearly v i s i b l e .  A replacement model, No*1-X,  was made and tested 1 , and the photograph of the r e s u l t i n g pattern (Figure 14(a)) was enlarged and examined under magnification*  Again., only 8 fringes were v i s i b l e c l e a r l y ,  i n d i c a t i n g a safe load.  However, many minute f r i n g e s ,  with  spacing too fine to be r e s o l v e d , existed i n the l / 6 4 - i n c h or so around the corner. Figure 14(a)  Within minutes after  the photograph i h  was taken, model No.lrrX also cracked diagonally  from the left-hand corner at an angle of 51° from the h o r i z o n t a l , s i m i l a r to model N o . l .  The unfortunate l o s s of  these two models emphasized the extremely high s t r e s s con-  36 eventration at the sharp corners as well as the fact  that  almost a l l of the p h y s i c a l properties of CR-39 depend on the length of time under load'.  Coolidge  showed that the.  ultimate t e n s i l e strength of CR-39 may be as high as 89.00 p s i f o r short applications of load* but f o r long-duration loading the strength decreases and may possibly be as low as 4,000 p s i .  .The c a l c u l a t i o n s for models N o . l and No.l-X  i n the next section were based on an estimated stress difference of 4*000 p s i at the moment of f r a c t u r e , is,  and t h i s  indeed, considered-conservative. Model No.2.  thicknesses  The r e s u l t s  of model No.2 with plug  of 2d or less are not considered r e l i a b l e  since  the model was inadvertently permitted to dry In a i r at high temperatures,  and bad edge effects formed as a r e s u l t .  A duplicate model, Np.2-X, was then made, and a t y p i c a l pattern i s shown i n Figure 14(b). Models No.3 and No>4.  The strength ofthese  was t h e i r most s i g n i f i c a n t c h a r a c t e r i s t i c .  models  It required  about 300 p s i to give a fringe order of 9 on most configurations of model No.4*  This; i s quite a contrast to model.;  N o . l and N o . l - X , both of Which cracked below 233 p s i . The f i r s t  t e s t s on both model No.3 and No*4 with a plug  thickness of 5d were u n r e l i a b l e since the pressure was  37 applied to the model before the model support was firmlytightened.  This r e s u l t e d i n a pre-stressing  and the c a l c u l a t e d r e s u l t s  of the model  are consequently somewhat high.  Patterns Observed A s e l e c t i o n of photographs of the various chromatic fringe patterns  iso-  observed i s shown i n Figures 14,  15 and 16. Effect  of varying f i l l e t r a d i u s .  the t y p i c a l f u l l  Figure 14 shows  size isochromatic patterns for the four  different f i l l e t s when the plug thickness was about 5d. The zero fringe can be observed on each photograph at about one hole diameter from the bottom of the s l o t the simulated plug)*  (the top of  .These patterns remained e s s e n t i a l l y  the same as the plug tiickness was reduced to about l ^ d . Below t h i s v a l u e , however, the patterns gradually changed, and Figure.15 shows enlarged photographs of the patterns when the plug thickness was #d (the smallest examined i n d e t a i l ) .  thickness  The zero fringe was no longer v i s i b l e  on the models, and the f i r s t  order fringe i s outermost i n  each of the photographs of Figure 15.  It i s evident from  these two Figures that the stress concentration i s greatest for the square-cornered model, and gradually decreases as the f i l l e t radius i s  increased*  38  Effect the effect  of varying; plug thickness.  Figure 16 shows  on the fringe pattern (enlarged i n t h i s  Figure)  as the thickness of the plug i s reduced from 4d to #d»" The patterns are almost i d e n t i c a l f o r thicknesses 2d and l)£d*. but change,  of 4-d,  gradually when the thickness  reduced below t h i s valuer  is  The stresses are so high for a  thickness of $d, that roughly h a l f the pressure gives the same maximum fringe order as the thickest p l u g . i t i o n below thicknesses next  The trans-  of l}£d i s discussed further i n the  section.. II.  CALCULATED RESULTS  Shear Stress Along Plug-Edge The shear stress (*7xij) along the edge of the plug was c a l c u l a t e d using two fundamental from photoelastic  equations::  theory:  <-4  =  n  c  (8)  where: n t c  are the p r i n c i p a l stresses i s the isochromatic fringe order i s the model thickness i s the fringe constant (89 p s i / f r i n g e / i n c h )  and from basic strength of  materials:  J v a J where Q i s the angle between the x-axis and the d i r e c t i o n of cS[ as obtained from the i s o c l i n i c patterns (counterclockwise p o s i t i v e ) .  I  39  ( a ) Square Corners, 235  (c)  F i l l e t Radius  psi.  a 200 4-'  psi  NN-7  Y-12  d  (b)  F i l l e t Radius = g,  (d)  F i l l e t Radius = d  RR-4  210 p s i ,  T-•11 psi. 2 ' 175  FIGURE 14 FULL SIZE FRINGE PATTERNS FOR MODELS WITH A PLUG THICKNESS OF ABOUT FIVE HOLE DIAMETERS AND VARIOUS FILLET RADII  (c)  Fillet  * EE-3 MM-2 R a d i u s = ^,130 100 p s S i -z ^ p s i j . (/ dJ )\ F- mi_ -l- li ne_ jt- R T ai d _ ij u- s = -|, CL  FIGURE 15 ENLARGED F R I N G E PATTERNS FOR MODELS WITH PLUG THICKNESS OF ONE-QUARTER HOLE DIAMETER AND VARIOUS F I L L E T R A D I I  41  ss-9  (b) Plug Thickness  ss-17  (a) P l u g T h i c k n e s s  = 4 d , 200 p s i .  = 2 d , 200 ps  (c) Plug Thickness  TT-3 = 1,5&, 200 p s i . ( d ) P l u g T h i c k n e s s  UU-4 = d, 200 p s i  (e) P l u g T h i c k n e s s  . UU-12 = ^, 200 p s i . ( f ) Plug Thickness FIGURE 16  . UU-14 = ^, 110 p s i  ENLARGED FRINGE PATTERNS FOR MODELS WITH A FILLET RADIUS OF ONEEIGHTH THE HOLE DIAMETER AND VARIOUS PLUG THICKNESSES  42 The v a r i a t i o n i n shear stress along the edges of t h i c k plugs with various f i l l e t r a d i i i s shown, i n Figure 1 7 * The maximum point f o r the square configuration i s based on the estimate of f a i l u r e stress described e a r l i e r . Several important conclusions can be drawn from t h i s graph: 1.  None of the four d i s t r i b u t i o n s could be con-  sidered constant or uniform along the edge of the plug.  Each has a maximum value which decreases to  zero within one diameter, i n spite of the fact  that  the plugs are a l l about 5 diameters i n thickness. The stress most l i k e l y becomes negative beyond one diameter, but the magnitude i s small since the f r i n g e order i s about 0 . 5 i h t h i s region. 2.  The maximum value of the shear stress and the  gradient of the curve are both less f o r l a r g e r fillet 3.  radii.  The point of maximum shear stress occurs at a  distance along the plug edge equal to about oneh a l f the f i l l e t  radius.  43 The v a r i a t i o n of the shear stress (Pxy) along the edge of the plug as a function of the plug thickness  is  shown i n Figure 18 for model No.4 with f i l l e t  equal  to  radius  This model had the least uneven d i s t r i b u t i o n of  shear s t r e s s , but the graph shows that the reduction of plug thickness does not make i t more uniform.  This graph  dould not be p l o t t e d f o r model N o . l , but from extrapolation of the r e s u l t s of the other models, t h i s conclusion evidentl y applies to model N o . l as w e l l . Maximum Shear Stress The maximum fringe order, i n every c o n f i g u r a t i o n , was on the boundary of the s l o t : -  i n the corner, f o r model  N o . l , and near the center of the f i l l e t f o r the others .r For an accurate estimate of the maximum fringe value, orders  (from both l i g h t and dark f i e l d photographs)  the  were  p l o t t e d along al.line drawn perpendicular to the boundary at the point of maximum order*  The maximum value was then  converted to maximum shear stress using equations with 0 equal to 4 5 ° .  8 and 9  The v a r i a t i o n of t h i s maximum shear  stress as a function of the plug thickness i s shown for each model i n Figure 19. Several conclusions can be drawn from t h i s 1.  graph:  There i s evidently a t r a n s i t i o n of some sort  FIGURE 17 SHEAR STRESS (Txy) ALONG THE EDGE OF FOUR PLUGS WITH § = 5 a AND SPECIFIED FILLET RADII •'  o  .2.  .4  .6  •&  i-O  1.2  |<4  DISTANCE. A L O N G E D G E OF P L U G (IN HOLE. DIAMETERS) •FIGURE 18 SHEAR STRESS ( ? x y ) ALONG EDGE OF PLUGS WITH FILLET RADIUS OF ONE-EIGHTH HOLE DIAMETER AND VARIOUS THICKNESSES  46 a r o u n d 1 o r Vfc. h o l e  diameters.  g r e a t e r than t h i s t h e  maximum s h e a r  constant,  and f o r v a l u e s  increases  rapidly.  2.  The maximum s h e a r  radius.  For thicknesses  less  stress  than  the stress  As t h e r a d i u s i s i n c r e a s e d , t h e s t r e s s f o r i n every  case*  The p o i n t s become more s c a t t e r e d a s t h e f i l l e t  radius  i s decreased.  were a p p r o x i m a t e l y is  this,  d e p e n d s on t h e f i l l e t  equal plug thicknesses decreases  3*  stress i s  most l i k e l y  radius This  S i n c e t h e maximum s t r e s s e s  equal f o r each t e s t ,  due t o t h e d i f f i c u l t y  fillets  (and square  o n the- f i l l e t  o f making  o f t h e maximum  every 0.5  along the r a d i a l case,  t o 1.0  decreased hole  insignificant Isoclinics The  to less  diameters,  shear  was t h a t t h e f r i n g e than  fringe  order, i n  one w i t h i n a d i s t a n c e o f  depending  on t h e m o d e l  (with  exceptions).  and S t r e s s isoclinic  configurations  line  small  radius.  A n o t h e r o b s e r v a t i o n made w h i l e p l o t t i n g t h e orders  scatter  corners) accurately.  emphasizes t h e dependence  stress  this  Trajectories p a t t e r n s were r o u g h l y  similar  for a l l  of a l l models e x c e p t i n g minor changes f o r  different f i l l e t r a d i i and excepting one important change when each model was reduced from a thickness of l}£d to d . The i s o c l i n i c s f o r two configurations of model No.4. show t h i s change most c l e a r l y , and they are included as Figure 20.  The i s o c l i n i c pattern for plug thicknesses  greater than l#d resembled sketch (a), of d or less, resembled sketch  and f o r  equal or thicknesses  (b).  The important difference between them i s the' e l i m i n a t i o n of the i s o t r o p i c p o i n t .  This shows up again  c l e a r l y i n Figure 21 which gives the stress  trajectories  determined from the i s o c l i n i c s i n Figure 20. i s discussed i n more d e t a i l further on i n the III.  This change thesis.  ESTIMATE OF POSSIBLE ERROR  The requirements f o r accuracy i n photoelastie q ' experiments  are:  (1) a well-made model with no machine  stresses or edge c h i p s ;  (2) good boundary v i s i b i l i t y ;  (3) loading apparatus that does not change as the model deforms;  ( 4 ) avoidance of creep;  and (5) prevention of  edge e f f e c t s . These requirements were met f a i r l y well i n t h i s i n v e s t i g a t i o n , with a few exceptions.  The models were  f a i r l y well made, as described e a r l i e r , and the only  LEGEND MODEL  RADIUS  -  1  SQUARE-  SR  —  l-X  A  -  A  —  2-X  ®  -  3  o  -  ©  -STRESS-T  S Y M B O L Q  PLUG  I  2  THICKNESS  -  2  SQUARE-  -  i  : i  4-  3 4(lN HOLL D I A M E T E R S )  F I G U R E 19 MAXIMUM SHEAR STRESS ON VARIOUS PLUGS AS A FUNCTION ' OF PLUG THICKNESS  49  (a) I s o c l i n i c  P a t t e r n f o r Model No«4 w i t h  (b) I s o c l i n i c  P a t t e r n f o r Model N o , 4 w i t h ~ = 1  Angles Counterclockwise  from V e r t i c a l  FIGURE 20 TWO  ISOCLINIC  PATTERNS  = Ifi  TENSION  COMPRESSION:  (b)  Stress Trajectories  - M o d e l 4, ^ = 1 F I G U R E 21  STRESS T R A J E C T O R I E S FOR TWO  CONFIGURATIONS OF MODEL No.4-  51  s i g n i f i c a n t problem was a r e s i d u a l machine stress of p o s s i b l y one-rquarter fringe  order.  The design of the loading apparatus ensured good boundary v i s i b i l i t y , and the deformation of the model had no effect  on the a p p l i c a t i o n of uniform pressure to the  slot. The edge effects were avoided f a i r l y well by immersion of the models i n a v i a t i o n o i l when not i n use, but as they were repeatedly tested^ they gradually developed some edge e f f e c t s .  The combination of the edge effects and  machining stress together was as much as one-half of a fringe order on some t e s t s .  This would have l i t t l e  on r e l a t i v e r e s u l t s , since a l l models were affected about the same way, but on c a l c u l a t e d values 8 or 9 fringes  were present) the possible  effect in  (where u s u a l l y  error i s -  1/4  fringe order or - 3 $ . An important source of possible neglect to avoid the effects  error was the  of creep by having a standard  delay period between loading and photographing the modal. Most photographs were taken within a few seconds  after  s t r e s s i n g the model, but a delay of a minute or so i s 4 possible.  Clark  showed that the fringe constant  by 4 to <6°/o i n t h i s i n t e r v a l , so a possible  decreases  error of i 2 Or  52  3# would apply to the c a l c u l a t e d r e s u l t s ,  and the r e l a t i v e  values as w e l l . The estimates of the possible error on making measurements  i s as f o l l o w s :  Fringe Order.  Both l i g h t and d a r k - f i e l d photographs  were enlarged 2)£ times and examined under magnification. The values on e i t h e r s i d e , where a p p l i c a b l e , were averaged, and the p o s s i b i l i t y of error i s considered to be i 0.2 of a fringe order, or - 2.5$ on most measurements. Gauge pressure.  The gauge was accurate to £ 5 p s i ,  and since most pressures were about 200 p s i or higher, there i s a possible  error of - 2,5^.  Plug thickness.  This was measured, where p o s s i b l e ,  on the photographs using a scale with 50 d i v i s i o n s to the inch.  The e r r o r , with 2# times enlargement of the f i l m ,  i s obviously n e g l i g i b l e . F i l l e t radius.  The f i l l e t r a d i i on the s l o t s were  cut using a template as described.  There i s a p o s s i b i l i t y  of error during the machining process,  but i t i s  difficult  to estimate i t as a percentage. Angle Q .  The angles measured from the i s o c l i n i c  53 patterns are subject to error since the i s o c l i n i c pattern was p l o t t e d every 5°•  It i s believed that i n t e r p o l a t i o n of  the angles was accurate w i t h i n - 2° although a small error may have been introduced i n t r a c i n g the i s o c l i n i c s . percent error i n s i n ©  The  would depend ..on the .angle.  IV. DISCUSSION OF RESULTS Uniformity of Shear Stress The r e s u l t s of the plane experiments  demonstrate  c o n c l u s i v e l y that the shear stress i s not uniform along the edge of a p l u g .  Figure 17 shows t h i s to be so,  of the f i l l e t r a d i u s .  independent  However, i t may also be concluded  that a square-cornered plug has the highest value and the sharpest decline i n shear stress along i t s  edge,  and that  a large f i l l e t reduces the maximum value of 7x\j a n d makes the d i s t r i b u t i o n less uneven. Maximum Values of Shear Stress The important stresses i n the strength of the plug are the stresses on the boundary; s i n c e , as Heywood"^ points out, f a i l u r e  almost i n v a r i a b l y begins on the surface and  seldom from w i t h i n .  In t h i s case, f a i l u r e could begin along  the edge of the plug ( e s p e c i a l l y  i f the bond to the tunnel  wall were not as strong as the material i t s e l f )  or at the  point of maximum stress (which coincides with the plug-edge  54  only for a square-cornered  plug).  The shear stress CTJOJ) along the edge of the plug i s shown i n Figure 17 for various t h i c k plugs,  and the  maxima vary from about 2$. to 9 times the applied hydrostatic pressure, depending on the f i l l e t  radius.  The maximum shear stress on the plug boundary i s shown i n Figure 19 as a function of the plug t h i c k n e s s , and i t may be concluded that t h i s stress changes l i t t l e thicknesses  for  over l)£d, but increases r a p i d l y when the t h i c k -  ness i s decreased below one hole diameter. Decrease-in Shear Stress It may be concluded, with regard to Figure 1 7 , that the shear stress along the edge decreases to zero within one hole diameter, and remains small beyond that p o i n t ,  especi-  a l l y for t h i c k plugs. From observations  made when p l o t t i n g the  fringe  orders r a d i a l l y from the points of maximum s t r e s s ^ i t may be concluded that the' fringe order (and thus, the maximum shear s t r e s s ) decreases to about one-ninth within a distance of 0 . 5 to 1 . 0 hole diameter (with i n s i g n i f i c a n t  exceptions).  Optimum Plug Design Although i t was not part ofthis i n v e s t i g a t i o n ^ the  55 optimum plug design can be concluded d i r e c t l y from these results. Figures 17 and 19 show conclusively that lower stresses exist (on plugs of equal thickness) when a large f i l l e t radius i s used.  The optimum plug would, therefore,  have a f i l l e t radius equal to one-half of the- hole diameter. Figure 19 also indicates  a very small decrease i n  stress as the plug thickness i s increased from Vfi to 5 hole diameters.  The explanation for t h i s t r a n s i t i o n at Yfi hole  diameters depends on the presence of an i s o t r o p i c point at the zero fringe (on a l l four models) as shown i n Figure 14. This point does not p e r c e p t i b l y move as the plug thickness i s reduced from 5 to l)£d.  However, a further reduction from  l^dto d removes t h i s p o i n t , and the stress t r a j e c t o r i e s change, as shown i n Figure 21. a zero stress difference  The i s o t r o p i c point represents  and therefore  zero shear  stress.  Consequently, i t may be concluded t h a t , f o r a given hydros t a t i c pressure, the s t r e s s - t r a j e c t o r y pattern w i l l as the plug thickness i s increased to l}£d.  The i s o t r o p i c  point w i l l then appear, and the t r a j e c t o r i e s w i l l The gradients  change  stabilize.  along the t r a j e c t o r i e s also s t a b i l i z e to a  degree, since the pressure i s constant,  and the p o s i t i o n of  the i s o t r o p i c point does not evidently change as the t h i c k -  ness i s further increased.  The benefit  of a thickness  greater than l)£d i s therefore marginal, and the optimum plug would be l)£d t h i c k . Figure 22 has been included f o r comparison of the maximum shear stress around the optimum plug with what might be c a l l e d the " o r d i n a r y " , t h i c k , plug.  In spite of the thickness  of the  square-cornered square-cornered  p l u g , the highest maximum shear stress i s 160$ of that • on the optimum p l u g .  (a) Plug Thickness = 5d  (b) Plug Thickness = 1.5a  FIGURE 22 NON-DIMENSIONAL SHEAR STRESS AROUND SLOT FOR TWO PLANE MODELS  -CHAPTER V.I EXPERIMENTS USING THREE-DIMENSIONAL MODELS The  three-dimensional photoelastic  approach used  i n t h i s i n v e s t i g a t i o n was the "frozen-stress"  technique i n  which a model i s stressed at a high temperature and the r e s u l t i n g fringe pattern becomes f i x e d i n the p l a s t i c ,  even  when the load i s removed after c o o l i n g . I.  DESCRIPTION OF MODELS  Material "Hysol 6 , 0 0 0 - O P " ,  a commercially-prepared material  with s t r e s s - f r e e z i n g  p r o p e r t i e s , was used for the models.  It i s a transparent,  amber p l a s t i c ,  and the material on  hand was cast i n a 4--inch diameter c y l i n d e r a few feet The mechanical properties of Hysol 6 , 0 0 0 - O P ,  long.  as described  by the manufacturer, include a t e n s i l e strength of 3,54-0 psi  and a modulus of e l a s t i c i t y of 2 , 1 9 0 p s i at  which i s the s t r e s s - f r e e z i n g The  (critical)  270°F,  temperature.  fringe constant at t h i s temperature was l i s t e d as 1 . 3 5  p s i / f r i n g e / i n c h , b u t . a t e s t showed the material on hand to have a value of 3 * 7 3 p s i / f r i n g e / i n c h . may  The large  have been the r e s u l t of a s l i g h t l y - l o w  difference  stress-freezing  temperature, or the r e s u l t of ageing, since the material  59 had been i n stock f o r several  years.  Model Design The models simulated a c y l i n d r i c a l s e c t i o n , removed from a large e l a s t i c a hole.  body, containing the plugge.d p o r t i o n of  The shape can be seen i n the broken-out section of  the model i n Figure 2 5 . Five models were made i n a l l .  They were a l l c i r c u -  l a r cylinders, 4 inches i n diameter and 6 inches i n l e n g t h , cut from the c y l i n d e r of Hysol.  Holes were bored i n one or  both ends of the c y l i n d e r s , with various f i l l e t r a d i i at the corner where the wall of the hole met the simulated plug. The f i r s t  model had a 5/8-inch diameter h o l e , as  shown i n Figure 2 3 , but the following models had one-inch diameter h o l e s .  The f i v e models were numbered as f o l l o w s :  No.SF-1  Square cornered plug with Ij-"" = 5«6  No.SF-2  F i l l e t radius = f ,  §  = 3.25  No.SF-3  F i l l e t radius = § , §  =3.25  No.SF-4  F i l l e t radius =- § , §  =1-5  No.SF-5  F i l l e t radius = | , | ~ =' 1.0  The models d i d not cover the f u l l  range of v a r i a -  60 t i o n of plug thickness and f i l l e t r a d i i ,  since they were  intended as a.check on the accuracy of the more thorough plane i n v e s t i g a t i o n . II.  DESIGN OF LOADING SYSTEM  Requirement The loading system was required to provide a uniform pressure,  that would not vary with time or tempera-  t u r e , to the hole i n the model.  Moreover, since the fringe  constant of Hysol i s so low at 2 7 0 ° F , the pressure had to be very accurately c o n t r o l l e d In the range of 15 to 25 p s i . Model Support The model was supported -by a simple rectangular frame, as shown i n Figures 23 and 24-, and the uniform loading was applied to the hole by a i r  pressure.  It had been intended to use a t h i n  rubber membrane  to apply pressure to the hole ( s i m i l a r to the gum-rubber tube method f o r the plane experiments).  The model and  support c o u l d , then, have been separated s l i g h t l y , avoiding extraneous sion.  stresses due to clamping or d i f f e r e n t i a l expand  However, the rubber membranes simply could not take  the high temperature and burst around 2 0 0 ° F . The cross-beam of the model support was consequently  61  cemented to a large gasket-rubber  washer, which, i n t u r n ,  was cemented to the model to prevent a i r leakage. rubber helped to minimize the effects  The  of thermal expansion,  but the cement was rather unyielding and stresses developed i n the v i c i n i t y of the hole opening. Pressure  System In order to meet the requirement f o r  pressure, a s e l f - r e g u l a t i n g from some p l a s t i c  accurate  pressure source was constructed  tubing and a standard,  low pressure,  weight gauge t e s t e r , as shown i n Figures 24 and 2 5 .  dead-  This  device had the advantages of utmost accuracy and the a b i l i t y to l e t the air. expand and contract while maintaining constant pressure.  Since the gauge t e s t e r i s f i l l e d with o i l ,  a chamber had to be i n s e r t e d i n the tubing to prevent the o i l from entering the model.  This reduced the r i s k s  of  b o i l i n g the o i l or possibly a f i r e i n the oven from o i l leaks. The gauge t e s t e r could not cope, however, with large a i r leaks i n the l i n e s , so an a i r compressor was also required to r e p l e n i s h the a i r side of the system*  The  compressor was only s e r i o u s l y required for model No.SF-5 since leaks were n e g l i g i b l e on the other models. The pressure system was rather makeshift,  but i t  FIGURE 23 STRESS-FREEZING MODEL PRIOR TO TESTING  FIGURE 24 LOADING SYSTEM FOR STRESS-FREEZING MODELS  63  A.IR COMPRESSOR  DEAD-WEIGHT GAUGE- TE.&TEJR.  FIGURE 25 SCHEMATIC OF LOADING SYSTEM FOR STRESS-FREEZING MODELS  64 worked very w e l l , and gave constant,  accurate,  pressure  to the models. III.  EXPERIMENTAL PROCEDURE  Preparation of Models The only model preparation was boring a hole i n one or both ends of the c y l i n d e r after i t had been cut from the stock.  The hole i n model No.SF-1 was d r i l l e d ,  then machined, using an end-mill f o r the square corners. The  other models were a l l turned on a lathe using ordinary  t o o l s and c u t t i n g procedures.  The s t r e s s - f r e e z i n g proced-  ure was preceded, i n each t e s t , by a 16'to 18-hour annealing  period which permitted the cement to harden and i t  Is  presumed that the machining stress was removed either by the annealing or the s t r e s s - f r e e z i n g  itself.  Testing of Models The models were assembled, as shown i n Figure 26, with a copper tube passing through a hole i n the oven to the pressure system. to 1 4 0 ° F and l e f t  The oven temperature  f o r 16 to 18 hours.  was  increased  The a i r pressure  was then applied to the model by turning the loading screw of  the gauge t e s t e r with the appropriate  piston.  The temperature  weight on the  was then increased to 270°F and  maintained there u n t i l the heat had thoroughly soaked i n t o  65  the model. Model No.SF-1 was heated f o r 2 2 hours, but the pattern was not very sharp, so the heating period was increased to 3 5 or 40 hours f o r the other models. The c o o l i n g period was c r i t i c a l , since thermal stresses had to be avoided. was reduced by 4 ° F / h o u r .  I n i t i a l l y , the  temperature  After a few hours, the rate was  increased to 8 or 1 0 ° F / h o u r , and when the temperature was below 2 0 0 ° F , the rate was increased to 3 0 or 4 0 ° F / h o u r . The arrangement  of the apparatus during the t e s t  i s shown i n Figure 2.7* S l i c i n g the Models After the s t r e s s - f r e e z i n g  heating c y c l e , each model  was s l i c e d along the axis of the h o l e , as shown i n Figure 28, y i e l d i n g a section s i m i l a r to the plane models.  The  s l i c e was cut about 3 / 8 - i n c h t h i c k and reduced tb 1 / 4 - i n c h on the m i l l i n g machine, using a f l y - c u t t e r . was taken at 9 0 ° to the hole at Its  Another section  mid-epoint.  The s l i c i n g  was done very precise/ly, with s p e c i a l care to avoid heat" and machining s t r e s s e s .  The surface of the s l i c e was then  p o l i s h e d using fine emery immersed i n "Brasso" metal p o l i s h , and the f i n a l buffing was done with a soft c l o t h and "Brasso".-  66  FIGURE STRESS-FREEZING  27  APPARATUS  DURING  TEST  FIGURE 28 METHOD OF SLICING STRESS FREEZING MODELS  68 The resultant, piece was. not quite transparent,  but  i t became so when.wette-d completely with o i l , and the f r i n g g pattern was then photographed.  In every case, the  model was s l i c e d , machined,- polished and photographed within 8 to 12 hours, i n order to avoid edge e f f e c t s . The models were then stored i n o i l f o r possible use1, and t h i s proved .very effective effects.  future  i n preventing edge  CHAPTER VII RESULTS OF THREE-DIMENSIONAL EXPERIMENTS I. OBSERVATIONS Patterns Observed Isochromatic fringe patterns f o r four of the f i v e s t r e s s - f r e e z i n g models are shown i n Figures 29 and 30 together with patterns from the equivalent plane models f o r comparison.  The s i m i l a r i t y of the plane and t h r e e -  dimensional patterns* i n general, i s evident, although important differences  exist*  of the f i v e models i s as Model No.SF-1.  A.more d e t a i l e d d e s c r i p t i o n  follows: The intense stress concentration  at the square corners of the model i n Figure 29(a) i s very s i m i l a r to that found on the plane model.  The fine  fringes  In the area around the Corner could not be r e s o l v e d , and t h i s model was, therefore,  of no value f o r numerical  calculations. Model No.SF-2. The second model, with a f i l l e t cl. radius of ^ , had a c l e a r but unsymmetrical isochromatic fringe p a t t e r n .  The lack of symmetry was due to a drop of  cement which was a c c i d e n t a l l y permitted to f a l l and was not found u n t i l the model was s l i c e d .  \  i n the h o l e , It  adhered  70. to  t h e bottom o f t h e hole,- on t h e l e f t  in  t h e s l i g h t l y h i g h e r maximum f r i n g e v a l u e  Figure  side,  and  resulted  as s e e n i n  29(c). M o d e l NOfcSF-3. The t h i r d m o d e l h a d a v e r y  fringe pattern.  faint  The c a u s e f o r t h i s i s n o t k n o w n , a n d  although the fringe  orders  t r a s t was t o o l o w t o w a r r a n t  were  publishing  M o d e l s No.SF-4- a n d S F - 5 . s h o w n i n F i g u r e 30.  counted  e a s i l y , the con-  here.  The l a s t t w o m o d e l s a r e  No p r o b l e m s w e r e e n c o u n t e r e d  i n their  preparation or testing. P o s i t i o n of Zero  Fringe  I n each of the photographs of the m o d e l s i n F i g u r e s 29 approximately plane  models.  a n d 30,  stress-freezing  t h e outermost f r i n g e  t h e same p o s i t i o n a s t h e z e r o f r i n g e I t was, a t f i r s t ,  s h o w e d i t t o be t h e f i r s t - o r d e r f r i n g e .  over  outer-  investigation  T h i s was a s u r p r i s -  r e v e l a t i o n , i n d i c a t i n g that the f r i n g e p a t t e r n extended a greater area i n the three-dimensional  t h e plane c o n f i g u r a t i o n , The  on t h e  believed that this  m o s t f r i n g e was t h e z e r o o r d e r , b u t f u r t h e r  ing  i sin  aero  case  c o n t r a r y t o what h a d b e e n  than i n expected.  . f r i n g e i n t h e t h r e e - d i m e n s i o n a l mode-Is was  along the outer circumference m o t t l e d b y a b a d edge e f f e c t .  found  o f t h e c y l i n d e r w h e r e i t was  71  N-15  h  R-8  (a) h Ihree-Dim.Model No.SF-1, (b) Plane Model No.l-X,# = 5, ^ = 5*6, Square Corners, 15 p s i . Square Corners, 100 p s i .  PP-9  (c^Three-Dim.Model No.SF-2, fr = 3.25, Radius = d, 25 p s i .  h ~ (d) Plane Model No.3, # = 4> F i l l e t Radius = d, 200 ps 4 Y  1 2  FIGURE 29 COMPARISON OF FRINGE PATTERNS FOR EQUIVALENT PLANE AND THREEDIMENSIONAL MODELS  72  (c) h Three-Dim.Model No,SF-5, - r = 1 , Radius = d, 20 p s i A  8  XX-1  w-9  (d) Plane Model No*2, fh » 1, F i l l e t Radius = d, 120 p a i 8  FIGURE 30 FURTHER COMPARISON OF FRINGE PATTERNS FOR EQUIVALENT PLANE AND THREE-DIMENSIONAL MODELS  73 II,  CALCULATED RESULTS  Uniformity of Shear Stress The shear stress CT^)  a  -^nS  the edge of the plug  was c a l c u l a t e d for models No.SF-3 and No.SF-4- and p l o t t e d i n Figure 31 along with the equivalent plane  configurations*  •The c a l c u l a t i o n was done' i n the manner described f o r the plane models. The r e s u l t i n g graphs reach a peak of shear s t r e s s then drop off sharply, i n a s i m i l a r manner to the plane models, but the peak value i s only 1/3 to 1/2 the peak value of the plane shear s t r e s s »  The stress-freezing  also show a l e v e l l i n g - b f f of the shear stress at  models  . 3 d , and  the stress does not pass through z$ro as i t does i n the equivalent plane case, Maximum Shear. Stress The maximum fringe orders were found i n i d e n t i c a l locations on the three-dimensional and plane models.  Using  techniques of enlargement and magnification again, the maximum fringe order was determined f o r each model except No.SF-1.  The fringe orders were converted to shear stress  i n the manner stated previously for plane models, and p l o t t e d on Figure 19 for comparison.  74 The f i r s t  observation from Figure 19 i s that the  r e s u l t s are, again, 1/3 to 1/2 of the plane values'. However, among themselves,  models N o « S F - 3 , SF-4 and SF-5  ( a l l of which had a f i l l e t radius equal to ^) show a r e l a t i v e r e l a t i o n s h i p s i m i l a r to the equivalent plane models. Model No.SF-3 had a maximum shear stress higher than that of SF-2, (which had a f i l l e t radius of ^ ) ,  This  r e l a t i v e r e l a t i o n s h i p i s also i n agreement with the r e s u l t s obtained by the plane experiments. P r i n c i p a l Stresses i n Three Dimensions A s l i c e was taken out of the stress-freezing  models  at 9 0 ° to the axis of each pressurized hole and half-way along i t .  The r a d i a l and tangential stresses were assumed  to be maximum at t h i s p o i n t , since the hole opening and the v i c i n i t y of the plug both had r e s t r a i n i n g effects which reduced these s t r e s s e s . The difference  i n p r i n c i p a l stresses was c a l c u l a t e d  f o r each l a t e r a l s l i c e and compared to the difference  in  p r i n c i p a l stresses at the point of maximum fringe order on the s l i c e along the hole a x i s .  In every case (except for  Np *SF-1, which could not be determined) the stress difference at the point of maximum fringe order was greater by 6 to 20$.  Two conclusions can be drawn from t h i s :  75 1*  Since the hydrostatic pressure equals one of  the p r i n c i p a l stresses on the wall of the hole:, and i t was the same at both points examined, then the p r i n c i p a l t e n s i l e stress at the f i l l e t must be greater than the-tangential stress-around the h o l e , 2.  I f the maximum shear stress i s considered the  c r i t e r i o n f o r f a i l u r e , then the plug would f a i l before the e l a s t i c body containing the hole would burst * I s o c l i n i c s and Stress  Trajectories  There was an important difference between the s t r e s s t r a j e c t o r i e s of the plane and three-dimensional models:there was np i s o t r o p i c point observed on any of the threedimensional models*  A comparison of the i s o c l i n i c patterns  f o r two equivalent models i s shown i n Figure 52* The reason f o r t h i s difference l i e s i n the methods of supporting the models,  The plane models had support along the sides and  bottom, whereas the three-dimensional models were supported only along the bottom, III.  ESTIMATE OF POSSIBLE ERROR  The accuracy of the three-dimensional experiments was- not quite as good as the plane i n v e s t i g a t i o n .  .The Hysol  DISTANCE: A L O N G E.DQE OF P L U G (IN HOLE- D I A M E T E R S )  (a)  h = 1.5 d  (b)  h = d '  1.0  FIGURE 31 COMPARISON OP SHEAR STRESS (Txy) ALONG PLUG EDGE FOR PLANE AND THREE-DIMENSIONAL MODELS  ISOCLINIC PATTERNS FOR TWO EQUIVALENT MODELS MEASURED FROM VERTICAL, COUNTERCLOCKWISE POSITIVE "  78 material had been on hand for several years, : and had i n c l u s i o n s and i n c o n s i s t e n c i e s , fringes  i n the photographs.  as shown by the i r r e g u l a r  The i s o c l i n i c l i n e s were  p a r t i c u l a r l y d i f f i c u l t to t r a c e .  They were n o t - l i n e s , but  r e a l l y vague, shadowy areas. In the heating phase of the test', the c o n t r o l was very accurate,  temperature  since an e l e c t r o n i c c o n t r o l l e r  with a thermocouple sensor was used, which could detect + o fluctuations aneously.  of — 3 F i n the a i r temperature almost  The possible  instant-  error i n temperature c o n t r o l was  consequently n e g l i g i b l e , although the design of the oven was such that temperature differences  of as much as 5 or  10 degrees may have existed between the top and bottom of the model. The error i n pressure a p p l i c a t i o n i s  considered  n e g l i g i b l e because of the accurate c o n t r o l provided by the dead-weight gauge t e s t e r * The greatest possible  source of e r r o r , however,  was the value of the fringe constant.  On the actual c a l c u -  l a t i o n of i t from r e s u l t s of a t e s t beam* there i s an estimated possible  error of - 7$«  compared to the manufacturer's  However, i t s high value  s p e c i f i c a t i o n i s unexplained.  .CHAPTER VIII SUMMARY OP RESULTS I.  COMPARISON OF PLANE AND THREE-DIMENSIONAL RESULTS The s t r e s s - f r e e z i n g experiments confirmed some of  the plane r e s u l t s and disagreed with others.  The most  Obvious difference was the missing i s o t r o p i c point i n the i s o c l i n i c pattern.  This was a r e s u l t  of the stress t r a -  j e c t o r i e s i n the model, which, i n t u r n , are subject to the method of supporting the model. approximations  Since both approaches  were  of the actual case, the actual i s o c l i n i c  pattern may be a compromise between the two* This discrepancy d i d not, however, affect agreement  the  on more important points., l i s t e d below:  Uniformity of Shear Stress In both cases 1 the shear stress- (7xtj ) along the :  edge of the plug rose to a peak value and then dropped off q u i c k l y , and i t i s , Uniform^.  therefore,  not v a l i d to consider i t  • •  Maximum Values of Shear Stress The maximum value of the shear stress (7$aj ) :  w  a  s  shown to be between 2}£ and 9 times the hydrostatic pressure  80  (depending on the f i l l e t radius) f o r a t h i c k plug i n a plane modelw  The three-dimensional case i n d i c a t e d that the  s t r e s s was a c t u a l l y much l e s s , possibly 1/2 or 1/3 of the plane v a l u e s .  This reduction was expected, since the three-  dimensional configuration i s much stronger than the plane arrangement * The maximum shear stress at any point of the plug was found to occur at the corner f o r the square-cornered plug and at about the mid-point of the f i l l e t f o r other plugs.  The value of t h i s maximum shear stress was shown  (1)  to decrease as the f i l l e t radius was increased;  (2)  to be f a i r l y constant at plug thicknesses greater than  l}£d;  and ( 3 ) to increase r a p i d l y as the plug thickness was  decreased below l)£d..  These conclusions were made i n the  plane experiments, and the r e s u l t s  of the f i v e  stress-  freezing experiments agreed, although the actual values were 1/3 to 1/2 of the plane values. •Diffusion of Shear Stress The shear s t r e s s ,  (TXLJ)  w  a  s  found to decrease tb  zero within one hole diameter i n the plane experiments. The three-dimensional tests showed a s i m i l a r r a p i d dropoff, it  but the value s t a b i l i z e d above zero.  may be concluded that the shear stress  In e i t h e r case,( 7 x u ) drops off  81 to a r e l a t i v e l y small amount within one hole diameter, regardless of the plug thickness or f i l l e t  radius.  This can also be concluded by simple observation of the isochromatic fringe patterns*  since almost every pattern  shows that 90$ of the fringe orders are within one hole diameter of the point of maximum order.  The fringe  order,  and thus the shear s t r e s s , decreases to a small percentage i n the distance  of one hole diameter. II.. CONCLUSION  The preceding section has summarized the experimental answers to the problems stated i n Chapter I . The.se conclusions apply to a plug bonded i n t e g r a l l y i n a hole, through a s e m i - i n f i n i t e e l a s t i c body, subject, of course, to the s i m p l i f y i n g assumptions made i n that  chapter-.  82  LITERATURE CITED 1.  S. Timoshenko, Theory of .Elasticity•,• McGraw-Hill Book C o . , New York, 1934.  2.  N i l . M u s k h e l i s h v i l i , Some Basic Problems of the Mathematical Theory of E l a s t i c i t y , t r a n s l a t e d by J . Radok, Groningen, Holland, P.Noordhoff ,• L t d . , 1953.  3<.  F . E * Dohse, "A Problem of the Notched Half Plane i n the Mathematical Theory of E l a s t i c i t y " , Doctoral Thesis, University of I l l i n o i s , 1962.  4.  A . B . J . C l a r k , " S t a t i c and Dynamic C a l i b r a t i o n of a Photoelastic Model Material-, CR-39.", Proceedings of the Society f o r Experimental Stress A n a l y s i s , V o l , X I V , N o * ! , 1956, p.195-  5*  A . J . D u r e l l i , R.L* Lake and C . H . Tsao, "Device f o r Applying Uniform Loading to Boundaries of Complicated Shape", Proceedings of the Society f o r .Experimental Stress A n a l y s i s , V o l . X I , N o . l , 1953* p.55.  6.  T . J . Dolan and W„M« Murray, " P h o t o e l a s t i c i t y Fundamentals and Two-Dimensional A p p l i c a t i o n s " , Handbook of Experime-ntal Stress A n a l y s i s e d i t e d by M. Hetenyi, John Wiley & Sons, New York, 1950,  7.  B.R. Lee, R. Meadows, J r . , and W.F. T a y l o r , "The Photoelastic Laboratory at the Newport News Shipb u i l d i n g and Dry Dock Company", Proceedings of the Society f o r Experimental Stress A n a l y s i s , V o l , V I , N o . l , 1948, p.83.  8.  D i ' J . Coo] i.dge, J r . , "An Investigation of the Mechanical and Stress-Optical Properties of Columbia Resin, CR-39"i Proceedings of the Society for Experimental Stress A n a l y s i s , V o l . V I , N o . l , 1948, p*74.  9.  M.M. Frocht, R. Guernsey and D. Landsberg, "Photoe l a s t i c i t y .- A P r e c i s i o n Instrument of Stress A n a l y s i s " , Proceedings of the Society for Experimental Stress A n a l y s i s , 1953, V o l . X I , N o . l , p.,105*  10.  R.B. Heywood, Designing by P h o t o e l a s t i c i t y , H a l l , L t d . , .London, 1952.  Chapman and  

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