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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 fo rd Andrews B.A,Sc.. , University of Br i t i sh Columbia., 1961 A Thesis Submitted in Par t ia l F u l f i l l m e n t of the Requirements for the Degree of Master of Applied Science In the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1966 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n (Gordon C Andrews Department o f Mechanical Engineering The U n i v e r s i t y o f B r i t i s h C o l umbia Vancouver 8. Canada Date 13 May, 1966 ABSTRACT The general area of investigation was f i r s t 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 specific purpose of this project was to determine i f the shear stress along the edge of a plug i n a c i rcu lar 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 yielded a simple solut ion. The problem was then attacked experimentally by plane photo-e l a s t i c i ty and 32 configurations of six plane models were examined* The results showed that the shear stress was not uniform, but rose, to a high peak and then declined rapidly . Three-dimensional photoelastic techniques were also used and the results of f ive "stress-freezing" models confirmed this conclusion. Other s ignif icant conclusions concern the var iat ion of stresses with plug thickness; the use of f i l l e t s to strengthen the plug; and the diss ipation of shear stress with distance from the plug. Some discussion i s also made of the optimum design for a plug. PREFACE This thesis could not have been successfully completed without the generous assistance of many people. I.should l ike to record on paper my thanks to a l l of these people, and i n part icular to my supervisor, Professor William 0, Richmond, for his help and guidance; to the laboratory technicians,' John Hoar, Edward A b e l l , John Wiebe and Phi l ip Hurren for the ir many suggestions and the ir precise work i n preparing models and equipment; and to Miss Isobelle McCafferty for her valuable assistance- i n processing the photographs. I should l ike also to express my gratitude to the National Research Council , who provided the research grants which made this project possible. TABLE OF CONTENTS CHAPTER PAGE I. STATEMENT OF PROBLEM 1 Introduction . . . . . . . . . . . . . . . . 1 Purpose . . . . . . . . . . . . . . . . . 1 Problem Defined * 2 Assumptions * . 2 Variablels 3 Methods of Solution . 5 I I . THEORETICAL APPROACHES . . . . . . 7 Background from the Theory of E l a s t i c i t y . 7 Use of Complex Variables 9 Numerical Solution of Biharmonic Equation 11 I I I . DESCRIPTION OF PHOTOELASTIC POLARISCOPE . . 17 IV. EXPERIMENTS USING PLANE MODELS . . 20 Description of Models . 20 Design . . . . . . . . . . . . 20 F i l l e t Radii 20 Plug Thicknesses . 21 Material . . . . . . 23 Design of Loading System . . . . . . . . . 23 Requirements . . . . . . . 23 Model Support * . . . 24 iv CHAPTER PAGE Loading Head * 26 Pressure System . . . . . . . . . . . . 27 Experimental Prooedure « 29 Preparation of Models * . . . . . . . 29 Testing of Models- . . . . . . T . . . . 31 V. RESULTS OF PLANE EXPERIMENTS . . . . . . . 35 Observations 35 Specific Observations . . . . . . . . . 35 Patterns Observed , ' 37 Calculated Results . . . . . 38 Shear Stress along Plug Edge » . . . . 38 Maximum Shear Stress . 43 Isocl inics and Stress Trajectories . . 4-6 Estimate of Possible Error 47 Discussion of Results . . . . . . . . . 53 Uniformity of Shear Stress 53 Maximum Values of Shear Stress . . . . 53 Decrease i n Shear Stress .• 54-Optimum Plug Design . . . . . . . . . . 5^ VI- EXPERIMENTS USING THREE-DIMENSIONAL MODELS 58 Description of Models 58 Material 58 Model Design * 59 V CHAPTER PAGE Design of Loading System » 60 Requirements . . . . . . . 60 Model Support 60 Pressure System . . . . . 61 Experimental Procedure 64 Preparation of Models . . . . . . . . 64 Testing of Models . . . . . . . . . . 64 S l i c ing of Models 65 VII. RESULTS OP THREE-DIMENSIONAL EXPERIMENTS . 69 Observations 69 Patterns Observed . . . . . . . . . . . 69 Posit ion of Zero Fringe . . . . . . . 70 Calculated Results . . . . . . . . . . . 73 Uniformity of Shear Stress 73 Maximum Shear Streps . . . . . . . . . 73 Principa l Stresses i n Three Dimensions 74-I socl inics and Stress Trajectories . . 75 Estimate of Possible Error . . . . . . . 75 VIII. SUMMARY OF RESULTS 79 Comparison of Plane and Three-Dimensional Results 79 Uniformity of Shear Stress 79 CHAPTER v i PAGE Maximum Values of Shear Stress . . . . . . 79 Diffusion of Shear Stress 80 Conclusion . . . . . 81 BIBLIOGRAPHY 82 LIST OF FIGURES FIGURE PAGE 1. : Sectional View of Hole and Plug , . . . . . 4-2. Simplif ied Configuration for Theoretical Approaches 10 3. Transformation of Slot Configuration to Lower Half-Plane . . . . . . . . . 10 4* Network of Points for F in i te Difference Approximation of Biharmonic Equation . . . 13 5i Mathematical Model of Plane Configuration for Numerical Solution of Biharmonic Equation 13 6* Schematic Flow Diagram for Computer Program 15 7. Photbelastic Polariscope . . . . . . . . . . 17 8. Typical Model i n Support with Loading Head i n Place . . . . . . . . . . . . . . . . . . 22 9. Model Ready for Testing 25 10; Schematic Drawing of Loading System for Plane Models . . . . . . . . . . . . . . . 28 11. Photograph of Assembled Loading System for Plane Models 28 12. Model Aligned in Optical Path . 33 13. Polariscope, Model,Loading System,Ready for Testing ( » . . . " 33 14-. 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 v i i i FIGURE PAGE 15. Enlarged Fringe Patterns for Models with a Plug Thickness of 1/4- Hole Diameter and Various F i l l e t Radii . . . . . . . . . . . 40 16. Enlarged Fringe Patterns for Models' with a F i l l e t Radius of 1/8 Hole Diameter and Various Plug Thicknesses 4 . . . . . . . . 41 17. Shear Stress along Edge of Various Thick Plugs «. . . . . . . . . . . . . . . . . . 44 18. Shear Stress Along Edge of Plugs with F i l l e t Radius of One-Half the Hole Diameter and Various Thicknesses . . . . . . 45 19. Maximum Shear Stress as a Function of Plug Thickness 48 20. / Two Isoc l in ic Patterns for Plane Model No.4 49 21. Stress Trajectories for Two Configurations of Model No.4 < . . . .... . 50 22*' Non Dimensional Shear Stress Around Slot for Two Plane Models 57 23. Stress-Freezing Model Pr ior to Testing .'• . . 62 24. Loading System for Stress-Freezing Models . 62 25. Schematic of Loading System for Stress-Freezing Models . . . . . . . . . . . . . 63 26. Model and Loading System Prior to Testing . 66 i x FIGURE PAGE 2 7 . Stress-Freezing Apparatus During Test . . . 66 28* Method of S l i c ing Stress-Freezing Models . . 67 2 9 . Comparison of Fringe Patterns for Equivalent Plane and Three-Dimensional Models . .. . . 71 3 0 . Further Comparison of Fringe Patterns for Equivalent Plane and Three-Dimensional Models ; . , . . . . . . . . . . . . . . . 72 3 1 . Comparison of Shear Stress along Plug Edge for Plane and Three-Dimensional Models . . 76 3 2 . I soc l inic 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 so l id 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 calculat-ing the length of plug necessary to give a safe value of this average shear stress . 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 or ig inal inspirat ion for this research project . I I . PURPOSE The purpose of th i s investigation was to examine the configuration of a c i rcu lar plug fixed in a c y l i n d r i c a l hole, 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 rapidly the shear stress decreased as distance from the plug increased. I I I . PROBLEM DEFINED Assumptions In order to reduce the number of variables to a manageable l e v e l f some assumptions were made. These are summarized as follows: 1J- Both the plug and the material through which the hole i s bored were assumed to be elast ic solids with no voids or faults 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 large* 3v The plug was assumed to be bonded Integrally with the c y l i n d r i c a l hole, or, in other words, the bond between the plug and the hole was assumed to have the f u l l strength of the material . 4* The plug was assumed to be a suff icient distance from the hole opening so that the stress d i s t r i -bution around the plug was unaffected by the hole opening* ' 5 Variables' These,assumptions permitted the scope of the re-search to be narrowed down to the two most important v a r i -ables that affect the stress distribution. These variables 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. The thickness of the plug i s an important factor i n i t s strength. As shown i n Figure 1, the thickness (h) can have any value greater than zero. It i s expressed throughout this thesis i n terms of the hole d i a T meter (d) or as a dimensionless ratio 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 significant effect on the concentra* tion of stress at t h i s corner. The radius can take any value from zero (for a square corner)to one-half the hole diameter. At this maximum diameter, the f i l l e t s on either side are tangent at the center of the plug,making the surface of the plug semi-spherical. Wherever reference i s made, i n this thesis^ to the f i l l e t radius, i t w i l l be expressed i n non^ dimensional form i n terms of the hole diameter. FIGURE 1 SECTIONAL VIEW OF HOLE AND PLUG 5 IV. METHODS OF SOLUTION It had been intended from the outset to use photo-e l a s t i c i ty i n determining the stress d i s t r ibut ion experi-mentally. Photoelasticity i s par t icular ly useful for complicated configurations and investigations 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 con-sulted and a wealth of information on techniques was discovered. However* no solutions of problems with a s imilar configuration were found, and the correlat ion of the results with other experiments could not be made. A summary of the methods of solution used i n this investigation i s as follows: Plane photoelast ic i ty. The reason for using a two-dimensional technique to examine a three-dimensional problem was, of course., that plane photoelasticity i s easier, faster and cheaper than three-dimensional photoelasticity*. Alsoy since the models are not destroyed (as they are in the "stress^freezing" method) during test ing, i t i s possible to re-check data or to a l ter the model and re-use i t . Most of this invest igat ion, therefore, concerned the plane approxi-mation. Three-dimensional photoelast ic i ty. The "frozen-stress" method of three-dimensional photoelasticity was used i n th i s investigation and was an important and essential 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 photoelasticity but no insurmountable problems were encountered, and the results were a very valuable comparison-for the plane results Theoretical approaches. Several attempts were made t f ind a theoretical solution that would predict the outcome of the experiments. These efforts only served to show that no simple theoretical solution existed. These theoretical approaches are discussed in deta i l i n the next chapter. CHAPTER II THEORETICAL APPROACHES The purpose of examining these theoretical approaches was to see i f simple solutions to the problem could be found for correlating the experimental results . None of the approaches yielded a simple solution, although i f suff icient time were devoted to advanced mathematical and numerical techniques, useful answers could be found with these methods. This chapter, therefore, describes the i n i t i a l work and the problems encountered. The f i r s t step was to simplify the configuration to two dimensions. Accordingly, a thin section passing along the center l ine of the hole,' and similar to Figure 1, was taken as a plane approximation of the three-dimensional case. By taking the case where the plug meets the wall in a square corner, and assuming that the plug was i n f i n i t e l y th ick , the configuration was further s implif ied to resemble a pressurized slot in the in f in i t e half-plane, 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 ca l theory of elasticity" 1" that the solution of two-dimensional problems 8 can be reduced to the integration of the biharmonic d i f ferent ia l equation: a** h**f >f • where 0 i s the Airy stress function, the stresses being defined by ' ^ (2), (3) and 7 ^ = — The general procedure i n solving for (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 satisfy the b i -harmonic equation, and the coefficients of these terms are determined by inserting the second derivatives of 0 into the equations for the stresses at the boundary of the body. In this part icular case, since a l l the boundary l ines were para l l e l to the x and y axes, the equations for the stresses at the boundary reduced to equations 2, 3 and 4. The only non-zero stresses around the slot were 6"x.= -P along the side and 6ij = -P along the bottom. A l l the stresses are zero along the x-axis, and approached zero as x and y got very large. This ruled out a stress function composed of polynomial terms since the derivatives would increase with x and y , and indicated either a grouping of more compli-9 cated terms or an i n f i n i t e trigonometric series . The search for a suitable stress function was r therefore, discontinued at this point . I I . USE OP COMPLEX VARIABLES An attempt was made to simplify the configuration using complex variables . The procedure proposed was: 1* to f ind a mapping function that would transform the slot configuration onto the lower half-plane, as shown i n Figure 3; 2. to solve the simpler z.-plane configuration using a stress function composed of two analytic ; 2 functions, as described by Muskhelishvili ; and 3* to use this solution to solve the given problem by employing the inverse- transformation. The Schwarz-Christophel transformation was used since the rec t i l inear configuration was idea l ly suited to i t . However, the transformation resulted in an e l l i p t i c integral of the second kind: Z • . FIGURE 2 SIMPLIFIED .CONFIGURATION.FOR THEORETICAL APPROACHES : FIGURE 3 TRANSFORMATION OF SLOT CONFIGURATION TO LOWER HALF-PLANE / 11 This integral can be put into a form for which results are tabulated, but i t was evident that the approach to the problem was beyond the scope of this research project, and no further work was done. 3 Subsequently, a s imilar problem solved by Dohse revealed that although a mathematical solution had been found using this approach, extensive numerical work was necessary before the stresses could be calculated. I I I . NUMERICAL SOLUTION OF BIHARMONIC EQUATION By dividing the area of a two-dimensional e l a s t i c i t y problem into a network of points arranged in a gr id as shown i n Figure 4, the biharmonid equation (1) may be expressed i n the f inite-difference form: + 05 + 07 + <&> + 0« = O ( 5 ) This equation may then be solved by using an electronic computer for rapid ca lculat ion. A program for the IBM 7040 computer was written using this approach, and i s described as 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 points, as,; .shown i n Figure 5, and the values of 0 , and were calculated at a l l points of the boundary. The values along the slot and upper boundary were obtained by integrating equations (1) and (2). For the side and lower boundaries, though, a stress function from Timoshenko"'' was used: 0 J^-r<9stn0 -i-Kre>tn<9 ( 6 ) where r and 0 are shown i n Figure 5. It was assumed that the pressure (P) at the bottom of the slot could be represented by a concentrated force at the or ig in since the distance from the point of application of the force to the boundary should be suff icient for St* Venant1s pr inciple to apply. A necessary row of points around the outside of the boundary were calculated by extrapolating the nearest in te r io r point using the slope at the boundary* The in ter ior points were approximated, i n i t i a l l y , by a l inear interpola-t i o n . The biharmonic f inite-dif ference equation, i n the form: <& = o.4(<ft+<&+03+<k) - O - l (0^08 + 0o+ <A*) - 0 . 0 5 ( > 5 + 07 4-0, +0„) (?) was then applied to each point and a more accurate value of 7 8 2 6 9 3 0 1 5 10 4 12 l l FIGURE 4 NETWORK OF POINTS FOR FINITE-DIFFERENCE 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 applied, successivelyj to every in ter ior point of the grid, ' with the process being repeated u n t i l nearly stationary values were obtained. The program, which i s shown schematically i n Figure 6, ran successfully, after the usual minor errors were eliminated. • It carried'out 100 iterations i n 28 minutes, then calculated and plotted a graph of the stresses along the edge of the plug. The accuracy of these results were, however, unacceptable. The change i n (p during the last i t e ra t ion was 50 times as great as the l imit of 1% of the difference between points which had been set. The possible error on the f i r s t difference was therefore i 2 5 $ , and, on taking the second derivative, Was much larger. This unsatisfactory result could probably be corrected by using advanced techniques to make the values converge more rapidly . However, this program represented only one configuration in the range to be investigated,' and the work necessary to improve the program and extend i t to other configurations was considered excessive. This method was, therefore, discontinued, and no theoretical results were obtained for correlat ion of the photoelastic experiments. 15 DIMENSION MATRIX FOR 0 AND MATRICES FOR M iM. AROUND BOUNDARY 3x v 6\j  READ PRESSURE- (P) 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 CALCULATE: EACH EXTERIOR POINT BY EXTRAPOLATION T RE-CALCULATE EACH INTERIOR VALUE OF 0 USING EQ* (7) CALCULATE <5^%V ALONG PLUG EDGE USING E Q ^ ( 2 ) ^ . ( 4 ) IN NUMERICAL FORM T PRINT S i ,Tj^ PLOT GRAPH OF c S ^ T ^ V S DISTANCE ALONG PLUG ETJQE S T O P FIGURE 6 SCHEMATIC FLOW DIAGRAM FOR COMPUTER PROGRAM CHAPTER III DESCRIPTION OF PHOTOELASTIG POLARISCOPE Although the experimental techniques were different for preparing and stressing the two types of photoelastic models, the same polariscope was used to observe the fringe patterns on both the plane models and the " s l i ces " from the three-dimensional models. This chapter, therefore,, i s devoted to describing the apparatus which was common to these experiments. The polariscope was model No.401 manufactured by the Polarizing Instrument Company. It i s shown schematic-a l l y in Figure 7 and had the following character is t ics : Light Source The polariscope was designed to operate with maxi-mum efficiency using monochromatic green l ight with a wave-length of 5^ -61 angstroms. A 100-watt, mercury vapour pro-ject ion bulb was used; i t emitted a high-intensity mercury o spectrum which has 54-61 A as i t s brightest v i s ib l e l i n e . F i l t e r s The l ight emitted from the projection bulb passed through a blue glass heat f i l t e r ; a ground glass diffusing plate ; and Wratten f i l t e r s 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 ine was v i s i b l e on the developed plates , showing that the f i l t e r s were, indeed, effective. Since the f i l t e r s were situated between the lamp and the polar izer , minor imperfections i n the glass plates had no effect on the photoelastic fringe patterns. Polarizing and quarter-wave plates The polar izer , quarter-wave plates and analyzer were 4# inches i n diameter. The polarizer and analyzer were "Polaroid" laminated i n strain-free glass* The quarter-wave plates were polystyrene, also laminated i n strain-free glass. The assembled optical path gave c lear , sharp, fringe patterns and the only problem encountered i n the use of the polariscope was the lack of a linkage to permit simultaneous rotation of the analyzer and polarizer 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 ight was formed into a 4}( inch dia-meter para l l e l beam by a condensing lens and a coll imating lens mounted in the lamp-holder. Upon emerging from the analyzer, this pa ra l l e l beam entered a collecting-condensing lens and passed d i rect ly to a 35n™ Exacta camera equipped 19 with a telephoto lens (focal length = 135 mm). Positioning of models The models and " s l ices " were placed on a central frame midway between the polarizer and analyzer. The bed of the frame could be moved horizontal ly and v e r t i c a l l y for accurate positioning of the models i n the optical 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 thin section or " s l i c e " taken along the centre l ine of the hole and passing through the plug and the body containing the hole. 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 high. This size was chosen to take advan-tage of a recently-bui l t loading frame which proved to be very convenient during the tests . The upper slot was one inch wide and four inches long on a l l models. Measuring from the bottom of the s lo t , there was a distance of 3# to 5 hole diameters to the boundary of the model. It was presum-ed that this was adequate clearance to avoid interference from loca l stresses at the model edge, and t h i s , indeed, proved to be true. F i l l e t Radii In order to observe the effect on the stress d i s t r ibut ion as the f i l l e t radius was varied, four models 21 were designed, each with a different f i l l e t radius. These models were numbered as follows: No. l — zero f i l l e t radius (square corner); 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 imilar to models No. l and No .2. These four f i l l e t r a d i i provided a var iat ion over the complete range possible,, from the smallest to the largest possible radius. The f i l l e t s were the same on both sides of the plug. Plug Thicknesses Each model, when newly-made, was rectangular, with a single slot on i t s upper edge to which pressure was applied. This represented the configuration for an extreme-ly thick plug. The plug thickness was then reduced by cutting a slot i n the lower edge of the model to simulate the extension of the hole. The plug thickness could then be further reduced by lengthening this s lo t , as shown i n Figure 8. HYDRAULIC CO FIGURE. 8 TYPICAL MODEL IN SUPPORT WITH LOADING HEAD IN PLACE 23 Material The plane models were made from ]i inch thick Columbia Resin 39 (also cal led CR-39 and Homalite 911) which was purchased in strain-free sheets with polished surfaces. The y i e l d strength of CR--39 i s about 6,000 ps i and the modulus of e l a s t i c i ty i s about 250,000 p s i . The stress-optical constant (or fringe constant) was taken as 89 p s i / f r inge/ inch, which was the average of five tests taken using the material on hand. However, an important factor that was not real ized at the beginning of the tests was that CR-39 i s subject to serious strain-creep at high loads. 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 period, but this omission does not seem to have seriously affected the resul 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. I I . DESIGN OP LOADING SYSTEM Requirements The loading system for the plane models was designed to do two things: 24 1. apply a uniform pressure to the upper notch which simulated hydrostatic pressure i n a ho le v and, 2. support the model so that i t reacted to th i s pressure l ike an in f in i t e s l i ce rather than a f i n i t e 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 restraining effects on a s l i ce of a three-dimensional configuration, and yet not so r i g i d that loca l stresses developed at the perimeter of the model. The support consisted of a three-sided, 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 tr ips of 1/16 inch thick rubber were used to give a more uniform pressure. Plast ic spacers were also inserted, where necessary, for more uniform clamping. This support prevented the large deformations and bending stresses i n the plug that would have resulted i f the model had been unrestrained when pressure was applied. Also , examination of the model under load revealed no FIGURE 9 MODEL READY FOR TESTING 26 s ignif icant loca l stresses around the perimeter at any time. Loading Head A uniform pressure was applied to the upper slot using a loading device s imilar to those discussed by 5 D u r e l l i , Lake and Tsao^,. i n which a gum rubber tube was inf la ted 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 Dure l l i et alt were made of s teel 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 bursting. An adaptation was therefore made to avoid this requirement. Two 3 by 5-inch plates of #-inch thick plexiglas were bolted to the loading head on either side of the tube, with a s l id ing clearance over the model, and the need for precise positioning was eliminated. The plexiglas did not affect the stress pattern, mainly because i t i s re la t ive ly insensitive 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 lot* Any further increase in a i r pressure was transmitted to the 27 slot through the tubing, and no other part of the loading head applied any force to the model. The pressure read from the gauge did not, of course^ agree with the actual pressure on the s lo t . Dure l l i et a l . carried out several tests on the ir device and showed that the relat ionship between the gauge pressure and the effect-ive (or actual) pressure was l inear , except i n a small range around 25 to 30 ps i when the tube firstexpanded. The loading head was tested in a s imilar manner and the relationship also proved to be l inear , with a slope of uni ty , for 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 inear relationship to zero actual pressure* However, a simpler method (described in the test procedure) gave the same accuracy and was preferred* Pressure System The system for Controll ing the pressure supply to the loading head is shown i n Figures 10 and 11. The pressure source was a 2,000 ps i tank of compressed a i r which was regulated by a standard high-pressure regulating valve, and pressure readings were made from a 0-2,000 ps i FIGURE 11 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 f lexible "Strato-flex" hose from the tank to the regulator was subject to 2,000 ps i and the pressure on the model side occasionally reached 350 ps i during testing,, I II . •EXPERIMENTAL PROCEDURE Preparation of Models The CR-r39 for the models came from the manufacturer with glassr-like surfaces that required no further pol i shing. It was necessary, then, merely to cut the material to the outline shown i n Figure 8. This was more d i f f i c u l t than i t f i r s t seemed, since CR^39, unlike "Lucite" or "Plexiglas" , i s a very b r i t t l e plast ic to machine, and small edge chips show c lear ly on the patterns and obscure the fringe order on the boundary* There i s also a problem with residual machin-ing stresses i f the heat generated by the cutter i s excess-ive . The f i r s t f ive models were cut on a mi l l ing machine using a sharp end-mill rotating at about 1600 rpm. The s ixth model was shaped using the method suggested by Dolan and Murray^, and described i n deta i l by Lee, Meadows and 7 TaylorV. This method was also used to cut out the lower s lots on a l l models. A comparison of the two methods i s 30 given i n this section. •The mi l l ing operation was a long., precise operation which gave dimensions accurate to a few thousandths and sharp edges with no chips. However., many precautions were required to avoid residual machining stresses. These pre-cautions^ recommended by Dolan and Murray^, included using an a i r jet directed on the tool to cool i t ; using sharp mi l l ing cutters reserved solely for 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 th i s care, the result ing models showed unacceptably^-large machining stresses of 1)£ fringe orders. The alternate method, although i t gave less precise dimensions, was much faster and gave negligible machining stresses. The procedure required an accurate metal template which was f ixed to the 01^39 using special masking tape, adhesive .on both sides* The unwanted plas t ic was cut away by a small jig-saw to within about 1/16-inch of the tem-plate , 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 , cutting teeth.. The cutter was mounted i n an ordinary dr i l l -pres s and the template was guided by a c i rcu la r pin threaded into the dr i l l -press table d i rect ly 31 below the cutter* Actually., two pins were used:- a "roughing" pin about 1/64-inch larger than the cutter, and a "finishing." pin the same diameter as the cutter* The depth of the last cut could, therefore, be controlled some-what* The templates were made from 1/16-inch steel for accuracy and durabi l i ty but i t i s believed that 1/8-inch aluminum-would have been as suitable and easier to make* This method gave residual stresses of less than half a fringe order at the maximum* The cause of this difference was most l i k e l y the rapid, shallow cuts taken by the Pratt & Whitney cutter, although the need to clamp material t i ght ly during mi l l ing may also be a factor. .The excessive stresses on the or ig inal f ive models were a l l at the f i l l e t s at the bottom of the s lots . •Consequently, they were easi ly eliminated by lengthening the slots about X-inch using the second method of cutt ing, Testing of Models A to ta l of 32 configurations of the six plane models were stressed and examined in the course of this project . Models No. l and i t s replacement, No. l-X, both with square corners, broke during their f i r s t tes t . Models No.2, 3, 4- and a duplicate of No.2, ca l led 2-X, were tested 32 using the following general procedure: The loading head was s l i d Onto the upper slot of the model', and then threaded onto the end of a hydraulic piston which was part of the main loading frame* (This piston permitted fine adjustment of the loading head i n the v e r t i c a l d i rec t ion) . The model support was then clamped to the model, with special 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 optical path, as shown i n Figures 12 and 13*' The regulating valve was then opened and the expansion of the tube was watched closely* The pressure at which the tube expanded and f i r s t touched the bottom of the slot could be easi ly observed, and was recorded, as well as the pressure.at which the tube completely f i l l e d the s lo t . The average of these two pressures gave the "expansion" pressure described earlier^ and the actual pressure on the slot was obtained by subtracting this 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 ight f i e l d s , taking care not to change the loading or focus between exposures. Photographs of the i soe l in ic s every 5 ° through 9 0 ° were also taken, and FIGURE 12 MODEL ALIGNED IN OPTICAL PATH •34-when this was completed, the pressure was rel ieved* The apparatus was then dismantled* and the lower slot lengthened using the template method o-f cutting described e a r l i e r . Bits of masking tape usually stuck to the p l a s t i c , so the model was washed quickly i n soap and water and dried thoroughly. The test procedure could then be repeated. Overnight, and during long delays between te s t s , the models were stored in a vat f i l l e d with aviation o i l to prevent edge effects from- forming. CHAPTER:V RESULTS OP PLANE EXPERIMENTS I. OBSERVATIONS This section describes some of the more important observations made during the tests on the 32 plane config-urations. Specif ic Observations Model N p w l . The signif icant characterist ic of Model No'.l was the intense stress concentration at the sharp corners of the s lot * On the very f i r s t test of the ser ies , th i s model cracked under a pressure of 150 ps i with only 8 fringes c lear ly v i s i b l e . A replacement model, No*1-X, was made and tested1, and the photograph of the result ing pattern (Figure 14(a)) was enlarged and examined under magnification* Again., only 8 fringes were v i s ib l e c lear ly , indicating a safe load. However, many minute fr inges , with spacing too fine to be resolved, existed i n the l/64-inch or so around the corner. Within minutes after the photograph i h Figure 14(a) was taken, model No.lrrX also cracked diagonally from the left-hand corner at an angle of 51° from the hor i -zontal , s imilar to model N o . l . The unfortunate loss of these two models emphasized the extremely high stress con-36 eventration at the sharp corners as well as the fact that almost a l l of the physical properties of CR-39 depend on the length of time under load'. Coolidge showed that the. ultimate tensi le strength of CR-39 may be as high as 89.00 ps i for short applications of load* but for long-duration loading the strength decreases and may possibly be as low as 4,000 p s i . .The calculations for models No. l and No.l-X i n the next section were based on an estimated stress difference of 4*000 psi at the moment of fracture, and this i s , indeed, considered-conservative. Model No.2. The results of model No.2 with plug thicknesses of 2d or less are not considered re l i ab le since the model was inadvertently permitted to dry In a i r at high temperatures, and bad edge effects formed as a resul t . A duplicate model, Np.2-X, was then made, and a typ ica l pattern is shown i n Figure 14(b). Models No.3 and No>4. The strength ofthese models was their most s ignif icant character i s t ic . It required about 300 ps i to give a fringe order of 9 on most configura-tions of model No.4* This; i s quite a contrast to model.; No . l and No. l-X, both of Which cracked below 233 p s i . The f i r s t tests on both model No.3 and No*4 with a plug thickness of 5d were unreliable since the pressure was 37 applied to the model before the model support was firmly-tightened. This resulted i n a pre-stressing of the model and the calculated results are consequently somewhat high. Patterns Observed A selection of photographs of the various i so-chromatic fringe patterns observed i s shown in Figures 14, 15 and 16. Effect of varying f i l l e t radius. Figure 14 shows the typica l f u l l 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 slot (the top of the simulated plug)* .These patterns remained essential ly the same as the plug tiickness was reduced to about l ^ d . Below this value, however, the patterns gradually changed, and Figure.15 shows enlarged photographs of the patterns when the plug thickness was #d (the smallest thickness examined in d e t a i l ) . 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* 3 8 Effect of varying; plug thickness. Figure 16 shows the effect on the fringe pattern (enlarged in this Figure) as the thickness of the plug i s reduced from 4d to #d»" The patterns are almost ident ica l for thicknesses of 4-d, 2d and l)£d*. but change, gradually when the thickness i s reduced below this valuer The stresses are so high for a thickness of $d, that roughly half the pressure gives the same maximum fringe order as the thickest plug. The trans-i t i o n below thicknesses of l}£d i s discussed further i n the next section.. I I . CALCULATED RESULTS Shear Stress Along Plug-Edge The shear stress (*7xij) along the edge of the plug was calculated using two fundamental equations:: from photoelastic theory: < - 4 = n c (8) where: are the pr inc ipa l stresses i s the isochromatic fringe order i s the model thickness i s the fringe constant (89 psi / fr inge/ inch) n t c and from basic strength of materials: J v a J where Q i s the angle between the x-axis and the direct ion of cS[ as obtained from the i s o c l i n i c patterns (counterclockwise pos i t ive) . I 39 ( a ) Square Corners, 235 p s i . NN-7 (b) F i l l e t Radius = d RR-4 g, 210 psi , (c) F i l l e t Radius a 4-' Y-12 200 ps i (d) F i l l e t Radius = d 2 ' T-175 •11 p s i . FIGURE 14 FULL SIZE FRINGE PATTERNS FOR MODELS WITH A PLUG THICKNESS OF ABOUT FIVE HOLE DIAMETERS AND VARIOUS FILLET RADII * EE - 3 MM-2 S i -z ^  j / J \ - m _ - - i n _ j - T i _ j - CL ( c ) F i l l e t R a d i u s = ^,130 p s i . ( d ) F i l l e t R a d i u s = -|, 100 ps FIGURE 15 ENLARGED FRINGE PATTERNS FOR MODELS WITH PLUG THICKNESS OF ONE-QUARTER HOLE DIAMETER AND VARIOUS FILLET RADII 41 s s - 9 ss-17 (a) Plug Thickness = 4d, 200 p s i . (b) Plug Thickness = 2d, 200 ps TT-3 UU-4 (c) Plug Thickness = 1,5&, 200 p s i . ( d ) P l u g Thickness = d, 200 p s i . UU-12 . UU-14 (e) P l u g Thickness = ^, 200 p s i . ( f ) Plug Thickness = ^, 110 p s i FIGURE 16 ENLARGED FRINGE PATTERNS FOR MODELS WITH A FILLET RADIUS OF ONE-EIGHTH THE HOLE DIAMETER AND VARIOUS PLUG THICKNESSES 42 The var iat ion i n shear stress along the edges of thick plugs with various f i l l e t r a d i i i s shown, i n Figure 17* The maximum point for the square configuration i s based on the estimate of fa i lure stress described ea r l i e r . Several important conclusions can be drawn from this graph: 1 . None of the four distributions 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 is small since the fringe order i s about 0 . 5 i h this region. 2. The maximum value of the shear stress and the gradient of the curve are both less for larger f i l l e t r a d i i . 3. The point of maximum shear stress occurs at a distance along the plug edge equal to about one-half the f i l l e t radius. 43 The variat ion of the shear stress (Pxy) along the edge of the plug as a function of the plug thickness i s shown i n Figure 18 for model No.4 with f i l l e t radius equal to This model had the least uneven dis tr ibut ion of shear stress, but the graph shows that the reduction of plug thickness does not make i t more uniform. This graph dould not be plotted for model N o . l , but from extrapolation of the results of the other models, this conclusion evident-ly applies to model No. l as wel l . Maximum Shear Stress The maximum fringe order, i n every configuration, was on the boundary of the s l o t : - i n the corner, for model N o . l , and near the center of the f i l l e t for the others .r For an accurate estimate of the maximum fringe value, the orders (from both l ight and dark f i e l d photographs) were plotted 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 8 and 9 with 0 equal to 4 5 ° . The var iat ion of this 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 this graph: 1. There i s evidently a t rans i t ion of some sort FIGURE 17 SHEAR STRESS (Txy) ALONG THE EDGE OF FOUR PLUGS WITH § = 5 AND SPECIFIED FILLET RADII a •' o .2. .4 . 6 • & i - O 1.2 |<4 DISTANCE. ALONG EDGE OF PLUG (IN HOLE. DIAMETERS) •FIGURE 18 SHEAR STRESS (?xy) ALONG EDGE OF PLUGS WITH FILLET RADIUS OF ONE-EIGHTH HOLE DIAMETER AND VARIOUS THICKNESSES 46 around 1 or Vfc. h o l e d i a m e t e r s . For t h i c k n e s s e s g r e a t e r than t h i s t h e maximum shear s t r e s s i s c o n s t a n t , and f o r v a l u e s l e s s than t h i s , the s t r e s s i n c r e a s e s r a p i d l y . 2. The maximum shear s t r e s s depends on the f i l l e t r a d i u s . As the r a d i u s i s i n c r e a s e d , the s t r e s s f o r equal p l u g t h i c k n e s s e s decreases i n every case* 3* The p o i n t s become more s c a t t e r e d as the f i l l e t r a d i u s i s de c r e a s e d . S i n c e the maximum s t r e s s e s were a p p r o x i m a t e l y e q u a l f o r each t e s t , t h i s s c a t t e r i s most l i k e l y due t o the d i f f i c u l t y of making s m a l l r a d i u s f i l l e t s (and square c o r n e r s ) a c c u r a t e l y . T h i s emphasizes the dependence of the maximum shear s t r e s s on the- f i l l e t r a d i u s . Another 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 f r i n g e o r d e r s a l o n g the r a d i a l l i n e was t h a t the f r i n g e o r d e r , i n every c a s e , d e c r e a s e d t o l e s s t h a n one w i t h i n a d i s t a n c e o f 0.5 t o 1.0 h o l e d i a m e t e r s , depending on the model ( w i t h i n s i g n i f i c a n t e x c e p t i o n s ) . I s o c l i n i c s and S t r e s s T r a j e c t o r i e s The i s o c l i n i c p a t t e r n s were r o u g h l y s i m i l a r f o r a l l c o n f i g u r a t i o n s 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 soc l in i c s for two configurations of model No.4. show this change most c l ear ly , and they are included as Figure 20. The i s o c l i n i c pattern for plug thicknesses equal or greater than l#d resembled sketch (a), and for thicknesses of d or less, resembled sketch (b). The important difference between them is the' elimination of the isotropic point. This shows up again c lear ly in Figure 21 which gives the stress tra jectories determined from the i soc l in i c s in Figure 20. This change i s discussed in more deta i l further on i n the thesis . I I I . ESTIMATE OF POSSIBLE ERROR The requirements for accuracy i n photoelastie q ' experiments are: (1) a well-made model with no machine stresses or edge chips; (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 effects . These requirements were met f a i r l y well i n this invest igat ion, with a few exceptions. The models were f a i r l y well made, as described ea r l i e r , and the only LEGEND S Y M B O L M O D E L Q - 1 SR — l-X -A - 2 A — 2 - X ® - 3 o - 4-© - S T R E S S - T RADIUS SQUARE-SQUARE-- i : i I 2 3 4-PLUG THICKNESS (lN HOLL DIAMETERS) FIGURE 19 MAXIMUM SHEAR STRESS ON VARIOUS PLUGS AS A FUNCTION ' OF PLUG THICKNESS 4 9 (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 = Ifi (b) 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 ~ = 1 Angles Counterclockwise from V e r t i c a l FIGURE 20 TWO ISOCLINIC PATTERNS TENSION COMPRESSION: (b) S t r e s s T r a j e c t o r i e s - Model 4, ^ = 1 FIGURE 21 STRESS TRAJECTORIES FOR TWO CONFIGURATIONS OF MODEL No.4-51 s ignif icant problem was a residual machine stress of possibly 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 application of uniform pressure to the s lo t . The edge effects were avoided f a i r l y well by immersion of the models in aviation o i l when not i n use, but as they were repeatedly tested^ they gradually developed some edge effects . The combination of the edge effects and machining stress together was as much as one-half of a fringe order on some tests . This would have l i t t l e effect on relat ive results , since a l l models were affected i n about the same way, but on calculated values (where usually 8 or 9 fringes were present) the possible error i s - 1/4 fringe order or - 3 $ . An important source of possible error was the neglect to avoid the effects of creep by having a standard delay period between loading and photographing the modal. Most photographs were taken within a few seconds after stressing the model, but a delay of a minute or so i s 4 possible. Clark showed that the fringe constant decreases by 4 to <6°/o i n this in terva l , so a possible error of i 2 Or 52 3# would apply to the calculated resul t s , and the re lat ive values as wel l . The estimates of the possible error on making measurements i s as follows: Fringe Order. Both l ight and dark-f ie ld photographs were enlarged 2)£ times and examined under magnification. The values on either side, where applicable, were averaged, and the pos s ib i l i ty 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 ps i or higher, there i s a possible error of - 2,5^. Plug thickness. This was measured, where possible, on the photographs using a scale with 50 divisions to the inch. The error , with 2# times enlargement of the f i l m , i s obviously negl ig ible . F i l l e t radius. The f i l l e t r a d i i on the slots were cut using a template as described. There i s a po s s ib i l i ty of error during the machining process, but i t i s d i f f i c u l t 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 plotted every 5°• It i s believed that interpolation of the angles was accurate within - 2° although a small error may have been introduced i n tracing the i s o c l i n i c s . The percent error i n s i n © would depend ..on the .angle. IV. DISCUSSION OF RESULTS Uniformity of Shear Stress The results of the plane experiments demonstrate conclusively that the shear stress i s not uniform along the edge of a plug. Figure 17 shows this to be so, independent of the f i l l e t radius. However, i t may also be concluded that a square-cornered plug has the highest value and the sharpest decline in 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 nd makes the di s tr ibut ion less uneven. Maximum Values of Shear Stress The important stresses i n the strength of the plug are the stresses on the boundary; since, as Heywood"^ points out, fa i lure almost invariably begins on the surface and seldom from within. In this case, fa i lure could begin along the edge of the plug (especially 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 thick 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 thickness, and i t may be concluded that this stress changes l i t t l e for thicknesses over l)£d, but increases rapidly when the thick-ness i s decreased below one hole diameter. Decrease-in Shear Stress It may be concluded, with regard to Figure 17, that the shear stress along the edge decreases to zero within one hole diameter, and remains small beyond that point, especi-a l l y for thick plugs. From observations made when plott ing the fringe orders rad ia l ly from the points of maximum stress^ i t may be concluded that the' fringe order (and thus, the maximum shear stress) decreases to about one-ninth within a distance of 0 . 5 to 1 .0 hole diameter (with ins ignif icant exceptions). Optimum Plug Design Although i t was not part ofthis investigation^ the 55 optimum plug design can be concluded d irect ly from these resul t s . 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 this t rans i t ion at Yfi hole diameters depends on the presence of an isotropic point at the zero fringe (on a l l four models) as shown i n Figure 14. This point does not perceptibly move as the plug thickness i s reduced from 5 to l)£d. However, a further reduction from l^dto d removes this point, and the stress trajectories change, as shown i n Figure 21. The isotropic point represents a zero stress difference and therefore zero shear stress. Consequently, i t may be concluded that, for a given hydro-stat ic pressure, the stress-trajectory pattern w i l l change as the plug thickness i s increased to l}£d. The isotropic point w i l l then appear, and the trajectories w i l l s t ab i l i ze . The gradients along the trajectories also s tabi l ize to a degree, since the pressure i s constant, and the posit ion of the isotropic point does not evidently change as the thick-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 thick . Figure 22 has been included for comparison of the maximum shear stress around the optimum plug with what might be cal led the "ordinary", th ick , square-cornered plug. In spite of the thickness of the square-cornered plug, the highest maximum shear stress i s 160$ of that • on the optimum plug. (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 this investigation was the "frozen-stress" technique i n which a model i s stressed at a high temperature and the result ing fringe pattern becomes fixed i n the p l a s t i c , even when the load i s removed after cooling. I. DESCRIPTION OF MODELS Material "Hysol 6 ,000 - O P " , a commercially-prepared material with stress-freezing properties, 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 cylinder a few feet long. The mechanical properties of Hysol 6 ,000 -OP, as described by the manufacturer, include a tensi le strength of 3,54-0 psi and a modulus of e l a s t i c i ty of 2 , 1 9 0 psi at 2 7 0 ° F , which i s the stress-freezing ( c r i t i c a l ) temperature. The fringe constant at this temperature was l i s t e d as 1 .35 ps i / f r inge / inch , but.a test showed the material on hand to have a value of 3*73 p s i / f r inge / inch . The large difference may have been the result of a s l ightly-low stress-freezing temperature, or the result of ageing, since the material 59 had been i n stock for several years. Model Design The models simulated a c y l i n d r i c a l section, removed from a large elast ic body, containing the plugge.d portion of a hole. 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 length, cut from the cylinder of Hysol. Holes were bored in one or both ends of the cyl inders , 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 hole, as shown i n Figure 2 3 , but the following models had one-inch diameter holes. The f ive models were numbered as follows: 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 . 2 5 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 did not cover the f u l l range of var ia-60 t ion 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 invest igat ion. II . DESIGN OF LOADING SYSTEM Requirement The loading system was required to provide a uniform pressure, that would not vary with time or tempera-ture , to the hole in 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 controlled 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 thin rubber membrane to apply pressure to the hole (s imilar to the gum-rubber tube method for the plane experiments). The model and support could, then, have been separated s l i g h t l y , avoiding extraneous stresses due to clamping or d i f ferent ia l expand s ion . 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, in turn, was cemented to the model to prevent a i r leakage. The rubber helped to minimize the effects 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 for accurate pressure, a self-regulating pressure source was constructed from some plast ic tubing and a standard, low pressure, dead-weight gauge tester , as shown i n Figures 24 and 25. This device had the advantages of utmost accuracy and the a b i l i t y to let the air. expand and contract while maintaining con-stant pressure. Since the gauge tester i s f i l l e d with o i l , a chamber had to be inserted i n the tubing to prevent the o i l from entering the model. This reduced the r isks of bo i l ing the o i l or possibly a f i r e i n the oven from o i l leaks. The gauge tester could not cope, however, with large a i r leaks i n the l ine s , so an a i r compressor was also required to replenish the a i r side of the system* The compressor was only seriously required for model No.SF -5 since leaks were negl igible 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 we l l , and gave constant, accurate, pressure to the models. I I I . EXPERIMENTAL PROCEDURE Preparation of Models The only model preparation was boring a hole i n one or both ends of the cylinder 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 for the square corners. The other models were a l l turned on a lathe using ordinary tools and cutting procedures. The stress-freezing proced-ure was preceded, i n each tes t , by a 16'to 18-hour anneal-ing period which permitted the cement to harden and i t Is presumed that the machining stress was removed either by the annealing or the stress-freezing i t s e l f . 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. The oven temperature was increased to 1 4 0 ° F and le f t for 16 to 18 hours. The a i r pressure was then applied to the model by turning the loading screw of the gauge tester with the appropriate weight on the pis ton. The temperature was then increased to 270°F and maintained there u n t i l the heat had thoroughly soaked into 6 5 the model. Model No.SF-1 was heated for 2 2 hours, but the pattern was not very sharp, so the heating period was increased to 3 5 or 40 hours for the other models. The cooling period was c r i t i c a l , since thermal stresses had to be avoided. I n i t i a l l y , the temperature was reduced by 4 ° F / h o u r . 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 test i s shown i n Figure 2.7* S l i c ing the Models After the stress-freezing heating cycle , each model was s l i ced along the axis of the hole, as shown i n Figure 28, y ie lding a section s imilar to the plane models. The s l i ce was cut about 3 /8- inch thick and reduced tb 1 /4-inch on the mi l l ing machine, using a f ly -cut ter . Another section was taken at 9 0 ° to the hole at Its mid-epoint. The s l i c i n g was done very precise/ly, with special care to avoid heat" and machining stresses. The surface of the s l i ce was then polished using fine emery immersed i n "Brasso" metal po l i sh , and the f i n a l buffing was done with a soft c loth and "Brasso".-66 F I G U R E 27 S T R E S S - F R E E Z I N G A P P A R A T U S 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 fringg 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 effects. The models were then stored i n o i l for possible future use1, and this proved .very effective i n preventing edge ef fects . CHAPTER VII RESULTS OF THREE-DIMENSIONAL EXPERIMENTS I. OBSERVATIONS Patterns Observed Isochromatic fringe patterns for four of the five stress-freezing models are shown i n Figures 29 and 30 together with patterns from the equivalent plane models for comparison. The s imi lar i ty of the plane and three-dimensional patterns* i n general, i s evident, although important differences exist* A.more detailed description of the f ive models i s as follows: Model No.SF-1. The intense stress concentration at the square corners of the model i n Figure 29(a) i s very s imilar to that found on the plane model. The fine fringes In the area around the Corner could not be resolved, and this model was, therefore, of no value for numerical ca lculat ions . Model No.SF-2. The second model, with a f i l l e t cl. radius of ^ , had a clear but unsymmetrical isochromatic fringe pattern. The lack of symmetry was due to a drop of cement which was accidentally permitted to f a l l i n the hole, and was not found u n t i l the model was s l i c e d . It adhered \ 70. t o t h e bottom o f t h e hole,- on t h e l e f t s i d e , and r e s u l t e d i n 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 as seen i n F i g u r e 2 9 ( c ) . Model NOfcSF -3 . The t h i r d model had a v e r y f a i n t f r i n g e p a t t e r n . The cause f o r t h i s i s not known, and a l t h o u g h t h e f r i n g e o r d e r s were c o u n t e d e a s i l y , t h e c o n -t r a s t was t o o low t o w a r r a n t p u b l i s h i n g h e r e . Models No.SF-4- and SF-5. The l a s t two models a r e shown i n F i g u r e 30. No p r o b l e m s were e n c o u n t e r e d i n t h e i r p r e p a r a t i o n o r t e s t i n g . P o s i t i o n o f Zero F r i n g e I n each o f t h e p h o t o g r a p h s o f t h e s t r e s s - f r e e z i n g models i n F i g u r e s 29 and 30, t h e outer m o s t f r i n g e i s i n a p p r o x i m a t e l y t h e same p o s i t i o n as t h e z e r o f r i n g e on t h e p l a n e models. I t was, a t f i r s t , b e l i e v e d t h a t t h i s o u t e r -most 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 i n v e s t i g a t i o n showed i t t o be t h e f i r s t - o r d e r f r i n g e . T h i s was a s u r p r i s -i n g r e v e l a t i o n , i n d i c a t i n g t h a t t h e f r i n g e p a t t e r n e x t e n d e d o v e r a g r e a t e r a r e a i n t h e t h r e e - d i m e n s i o n a l c a s e t h a n i n t h e plane c o n f i g u r a t i o n , c o n t r a r y t o what had been e x p e c t e d . The aero . 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 f o u n d a l o n g t h e o u t e r c i r c u m f e r e n c e o f t h e c y l i n d e r where i t was m o t t l e d by a bad edge e f f e c t . 71 N-15 h R-8 (a)hIhree-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 Y ~ 1 2 (d) Plane Model No.3, # = 4> F i l l e t Radius = d, 200 ps 4 FIGURE 29 COMPARISON OF FRINGE PATTERNS FOR EQUIVALENT PLANE AND THREE-DIMENSIONAL MODELS 72 (c)hThree-Dim.Model No,SF-5, - r = 1 , Radius = d, 20 psi A 8 XX-1 h w - 9 (d) Plane Model No*2, f » 1, F i l l e t Radius = d, 120 pai 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 -^ n S the edge of the plug was calculated for models No.SF-3 and No.SF-4- and plotted i n Figure 31 along with the equivalent plane configurations* •The calculat ion was done' in the manner described for the plane models. The result ing graphs reach a peak of shear stress then drop off sharply, i n a s imilar 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 models also show a leve l l ing-bf f of the shear stress at .3d, and the stress does not pass through z$ro as i t does i n the equivalent plane case, Maximum Shear. Stress The maximum fringe orders were found i n ident ica l locations on the three-dimensional and plane models. Using techniques of enlargement and magnification again, the maximum fringe order was determined for each model except No.SF-1. The fringe orders were converted to shear stress i n the manner stated previously for plane models, and plotted on Figure 19 for comparison. 74 The f i r s t observation from Figure 19 i s that the results are, again, 1/3 to 1/2 of the plane values'. However, among themselves, models N o « S F - 3 , SF-4 and SF-5 ( a l l of which had a f i l l e t radius equal to ^) show a re la t ive relationship similar to the equivalent plane models. Model No.SF-3 had a maximum shear stress higher than that of SF-2, (which had a f i l l e t radius of ^ ) , This re la t ive relationship i s also i n agreement with the results obtained by the plane experiments. Pr inc ipa l Stresses i n Three Dimensions A s l ice was taken out of the stress-freezing models at 9 0 ° to the axis of each pressurized hole and half-way along i t . The rad ia l and tangential stresses were assumed to be maximum at this point, since the hole opening and the v i c i n i t y of the plug both had restraining effects which reduced these stresses. The difference i n pr inc ipa l stresses was calculated for each l a te ra l s l i ce and compared to the difference i n pr inc ipa l stresses at the point of maximum fringe order on the s l i ce along the hole axis . In every case (except for Np *SF-1, which could not be determined) the stress difference at the point of maximum fringe order was greater by 6 to 20$. Two conclusions can be drawn from th i s : 75 1* Since the hydrostatic pressure equals one of the pr inc ipa l stresses on the wall of the hole:, and i t was the same at both points examined, then the pr inc ipa l tensi le stress at the f i l l e t must be greater than the-tangential stress-around the hole, 2. If the maximum shear stress i s considered the c r i t e r ion for f a i lu re , then the plug would f a i l before the elast ic body containing the hole would burst * I soc l inics and Stress Trajectories There was an important difference between the stress tra jectories of the plane and three-dimensional models:-there was np isotropic point observed on any of the three-dimensional models* A comparison of the i s o c l i n i c patterns for two equivalent models i s shown i n Figure 52* The reason for this 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, I I I . ESTIMATE OF POSSIBLE ERROR The accuracy of the three-dimensional experiments was- not quite as good as the plane invest igat ion. .The Hysol DISTANCE: A L O N G E.DQE OF PLUG (IN HOLE- D I A M E T E R S ) (a) h = 1.5 (b) h = 1.0 d d ' 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 inclusions and inconsistencies, as shown by the irregular fringes i n the photographs. The i s o c l i n i c l ines were par t icu lar ly d i f f i c u l t to trace. They were not- l ines , but rea l ly vague, shadowy areas. In the heating phase of the test', the temperature control was very accurate, since an electronic control ler with a thermocouple sensor was used, which could detect + o fluctuations of — 3 F i n the a i r temperature almost instant-aneously. The possible error in temperature control was consequently negl ig ib le , 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 application i s considered negligible because of the accurate control provided by the dead-weight gauge tester* The greatest possible source of error , however, was the value of the fringe constant. On the actual calcu-l a t ion of i t from results of a test beam* there i s an estimated possible error of - 7$« However, i t s high value compared to the manufacturer's specif icat ion i s unexplained. .CHAPTER VIII SUMMARY OP RESULTS I . COMPARISON OF PLANE AND THREE-DIMENSIONAL RESULTS The s tress- freezing experiments confirmed some of the plane results and disagreed with others. The most Obvious difference was the missing isotropic point i n the i s o c l i n i c pattern. This was a result of the stress t r a -jectories i n the model, which, i n turn, are subject to the method of supporting the model. Since both approaches were approximations of the actual case, the actual i s o c l i n i c pattern may be a compromise between the two* This discrepancy did not, however, affect the agreement on more important points., l i s t e d below: Uniformity of Shear Stress In both cases 1 the shear stress- (7xtj:) along the edge of the plug rose to a peak value and then dropped off quickly, and i t i s , therefore, not v a l i d to consider i t Uniform^. • • Maximum Values of Shear Stress The maximum value of the shear stress (7$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) for a thick plug i n a plane modelw The three-dimensional case indicated that the stress was actually much less , possibly 1/2 or 1/3 of the plane values. 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 for the square-cornered plug and at about the mid-point of the f i l l e t for other plugs. The value of this 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 rapidly as the plug thickness was decreased below l)£d.. These conclusions were made i n the plane experiments, and the results of the five stress-freezing experiments agreed, although the actual values were 1/3 to 1/2 of the plane values. •Diffusion of Shear Stress The shear stress, (TXLJ ) w a s found to decrease tb zero within one hole diameter i n the plane experiments. The three-dimensional tests showed a similar rapid drop-off , but the value s tab i l i zed above zero. In either case,-i t may be concluded that the shear stress ( 7 x u ) drops off 81 to a re la t ive ly 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 stress, decreases to a small percentage i n the distance of one hole diameter. II.. CONCLUSION The preceding section has summarized the experi-mental answers to the problems stated i n Chapter I. The.se conclusions apply to a plug bonded integra l ly i n a hole, through a semi-infinite elast ic body, subject, of course, to the simplifying assumptions made i n that chapter-. 82 LITERATURE CITED 1. S. Timoshenko, Theory of .Elasticity•,• McGraw-Hill Book Co . , New York, 1934. 2. N i l . Muskhel i shvi l i , Some Basic Problems of the Mathematical Theory of E l a s t i c i t y , translated 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 . Clark, "Static and Dynamic Calibration of a Photoelastic Model Material-, CR-39.", Proceedings of the Society for Experimental Stress Analysis , Vol,XIV, No* ! , 1956, p.195-5* A . J . D u r e l l i , R.L* Lake and C.H. Tsao, "Device for Applying Uniform Loading to Boundaries of Complicated Shape", Proceedings of the Society for .Experimental Stress Analysis , Vo l .XI , N o . l , 1953* p.55. 6 . T . J . Dolan and W„M« Murray, "Photoelasticity -Fundamentals and Two-Dimensional Applicat ions" , 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. Taylor, "The Photoelastic Laboratory at the Newport News Ship-building and Dry Dock Company", Proceedings of the Society for Experimental Stress Analysis , 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 Analysis , Vol .VI , N o . l , 1948, p*74. 9. M.M. Frocht, R. Guernsey and D. Landsberg, "Photo-e l a s t i c i t y .- A Precision Instrument of Stress Analysis" , Proceedings of the Society for Experimental Stress Analysis , 1953, Vol .XI , N o . l , p.,105* 10. R.B. Heywood, Designing by Photoelast icity, Chapman and H a l l , L t d . , .London, 1952. 

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