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Stresses in a torispherical head of a pressure vessel by photoelastic coating method Szekessy, Laszlo Imre 1961

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STRESSES IN A TORISPHERICAL HEAD OP A PRESSURE VESSEL BY PHOTOELASTIC COATING METHOD by LASZLO IMRE SZEKESSY D i p l . Mech. Eng. Technical U n i v e r s i t y Budapest, 1950. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n 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 October, 1961 In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It. i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be alloi^ed without my written permission. Department of M f t o l m n i p.a.l F ,ng i r a r i n g The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date Vancouver. October. 1. 1961 i Abstract. The use of the photoelastic coating method i n deter-mining the stresses i n the t o r i s p h e r i c a l head of a pressure ve s s e l was investigated. I t was found that the method i s valuable to obtain the d i s t r i b u t i o n , d i r e c t i o n , and magnitude of stresses on the surface of any structure. The r e s u l t s obtained with the method showed close a-greement with the t h e o r e t i c a l i n v e s t i g a t i o n s . The maximum stresses i n a t o r i s p h e r i c a l head of a pressure v e s s e l occur i n the torus. The same conclusion was drawn from the r e -s u l t s obtained with the method. I t also revealed,that these stresses were compressive on the outer surface. The m o b i l i t y of the instruments, the r e l a t i v e l y simple way of coating the surface of the structure are other fea-tures of the method. ACKNOWLEDGMENT The author wishes to express h i s gratitude to Professor f . 0. Richmond, Head of the Department of Mechanical Engineering, f o r the guidance i n completing h i s work, f o r making the instruments and la b o r a t o r i e s f r e e l y a v a i l a b l e and fo r the help i n obtaining f i n a n c i a l assistance. i i TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. DESCRIPTION OF THE PHOTOELASTIC COATING METHOD 3 3. DESCRIPTION OF THE PRESSURE VESSEL AND EQUIPMENT 12 4. RESULTS 15 5. COMPARISON WITH OTHER WORKS ON PRESSURE VESSEL HEADS 21 6. SUMMARY AND CONCLUSION 25 7. REFERENCES 27 8. APPENDIX 29 I. Mathematical Theory 1. Normal incidence 29 2. Oblique incidence 32 I I . Casting p l a s t i c sheets f o r the head of the vessel 38 I I I . Determination of the s t r a i n o p t i c a l c o e f f i c i e n t by c a l i b r a t i o n 40 LIST OF TABLES TABLE I. MERIDIONAL AND CIRCUMFERENTIAL STRESSES IN THE HEAD OF THE PRESSURE VESSEL AND THE CORRESPONDING STRESS INTENSITY FACTORS i v LIST OF FIGURES FIGURE Page 1. Schematic Drawing of R e f l e c t i o n Polariscope 44 2. Schematic of Oblique Incidence Polariscope 45 3. Schematic of P o l a r i z i n g Microscope 46 4. The Pressure Vessel 47 5. Casting of P l a s t i c 48 6. Pressure Vessel and Dead Weight Tester 49 7. Test Setup f o r the Large F i e l d Polariscope 50 8. Test Setup f o r the Oblique Incidence Polariscope 51 9. Test Setup f o r the P o l a r i z i n g Microscope 52 10. I s o c l i n i c s on the Head of the Pressure Vessel 53 11. Stress D i s t r i b u t i o n on the Head of the Pressure Vessel 54 12. Circumferential and Meridional Stresses along a r a d i a l l i n e 55 13. Stress D i s t r i b u t i o n i n the Torus 56 14. Test Setup f o r C a l i b r a t i o n of P l a s t i c 57 15. C a l i b r a t i o n l i n e of p l a s t i c 58 E ( l b / i n 2 ) V t ( i n ) C ( i n 2 / l b ) °1> K = CE p 1 + L V 6 ( i n ) € 1 ; e2 ( i n / i n ) ; 0~2 ( l b / i n 2 ) cr ( l b / i n 2 ) r ( l b / i n 2 ) R ( i n ) r ( i n ) P C p s i g ) M ( i n / l b ) L ( i n ) F (lb) I ( i n 4 ) L i s t of Symbols Used. Modulus of E l a s t i c i t y Poisson's r a t i o Thickness Stress o p t i c a l c o e f f i c i e n t Correction f a c t o r s O p t i c a l s t r a i n s e n s i t i v i t y f a c t o r of p l a s t i c R e lative r e t a r d a t i o n of p o l a r i z e d l i g h t P r i n c i p a l s t r a i n s P r i n c i p a l stresses Normal stress Shear stress Radius of head or c y l i n d e r Radius of the knuckle Pressure ac t i n g i n the pressure vessel Bending moment Distance Load Moment of i n e r t i a v i ~ Fringe value of the p l a s t i c used X (in) Wave length of l i g h t 0 (degr.) Angle of incidence of l i g h t Angle between a normal to the cf> (degr.) surface of the s h e l l and the s h e l l axis Angle between a normal drawn to the s h e l l axis from the CX (degr.) junction of the c y l i n d e r and head and a l i n e from a point on the surface of the head. m (deer ) Compensator reading on the ^ S w large f i e l d polariscope. The difference of readings tak-en at a point using the oblique incidence polariscope when the structure i s loaded and unloaded. Subscripts. c = c y l i n d e r p = p l a s t i c n = normal o = oblique w = metal, workpiece 1 Introduction. Stress a n a l y s i s by p h o t o e l a s t i c i t y depends upon the property of temporary double r e f r a c t i o n or a r t i f i c i a l b irefringence possessed by c e r t a i n transparent materials. This birefringence i s proportional to stres s and hence i t i s possible to deduce the s t r e s s from the observation of the o p t i c a l properties. Ordinary methods of photoelastic stress a n a l y s i s use a p l a s t i c model of the structure under a n a l y s i s . A newer technique uses a coating of transparent photoelastic material cemented to the metal part under i n v e s t i g a t i o n so that the photoelastic e f f e c t observed i s a function of the s t r a i n on the surface of the structure.In t h i s i n v e s t i g a t i o n the coating, method i s used i n a study of the stres s d i s t r i -bution i n a t o r o i d a l s h e l l . The use of b i r e f r i n g e n t coating i n photoelastic inves-t i g a t i o n was i n i t i a t e d by Mesnager (France) i n 1930 (1), who used a b i r e f r i n g e n t l a y e r of glass. There was no s a t i s -f a c t o r y bond to the structure investigated and p r a c t i c a l r e s u l t s were not produced at that time. Several attempts to develop the coating method by Mabboux (France) 1932 (2), and Oppel (Germany) 1937 (3), were unsuccessful f o r p r a c t i c a l use, because the coating material s t i l l had low s e n s i t i v i t y , and the bonding was i n s u f f i c i e n t . In the United States of America D'Agostino, Drucker, 2 L i n , and Mylonas performed extensive studies of the behaviour of b i r e f r i n g e n t coatings (4»5) and developed i t as a p r a c t i c a l t o o l f o r stres s a n a l y s i s . They presented the r e s u l t s of t h e i r work at the convention of IUTAM i n Brussels i n 1954. P r a c t i c a l r e s u l t s achieved on the i n d u s t r i a l l e v e l i n Prance were published by Zandman (5»6). The coating method was used on various s t r u c t u r a l materials i n both e l a s t i c and p l a s t i c ranges of deformation. The coating material proved to be stable i n time and temperature. The bond was e f f e c t i v e thus providing the necessary t r a c t i o n between the p l a s t i c and the metal. 3 Description of the Photoelastic Coating Method. The s t r u c t u r a l part to he analysed i s coated with a l a y e r of b i r e f r i n g e n t material cemented to the surface so that the s t r a i n s are transmitted to the p l a s t i c (7,8). A r e f l e c t i n g surface i s provided between the metal and the p l a s t i c . B i -refringence due to s t r a i n i s observed by a r e f l e c t i o n po-l a r i s c o p e so that the l i g h t ray passes twice through the p l a s t i c l a y e r . Because of t h i s , the basic law of p h o t o e l a s t i -c i t y expressing the r e l a t i v e r e t a r d a t i o n between the ordinary and extraordinary ray and the difference between the p r i n c i p a l stresses i s (9,10) C = i s the stress o p t i c a l c o e f f i c i e n t and OT> are p r i n c i p a l stresses t = i s the thickness of the p l a s t i c l a y e r When the part i s strained, black and coloured f r i n g e s are v i s i b l e through the analyzer plate of the polariscope using white l i g h t . I f the p o l a r i z i n g axes of the p o l a r i z e r and analyzer p l a t e s are crossed, the black f r i n g e s represent i s o c l i n i c s . I s o c l i n i c s are the l o c i of points where the d i -r e c t i o n s of the p r i n c i p a l stresses are constant and the same as the d i r e c t i o n s of the p o l a r i z i n g axes of the polariscope, where i s the r e l a t i v e retardation, 4 The d i r e c t i o n s of the p r i n c i p a l s t r e s s e s t h e r e f o r e can be determined a t every p o i n t because the cros s e d a n a l y z e r and p o l a r i z e r p l a t e s can be r o t a t e d s i m u l t a n e o u s l y . Knowing the d i r e c t i o n s of the p r i n c i p a l s t r e s s e s a t every p o i n t i t i s p o s s i b l e t o plot; the s t r e s s t r a j e c t o r i e s ( i s o s t a t i c s ) (9»10). When the q u a n t i t a t i v e e v a l u a t i o n of the f r i n g e s i s con-s i d e r e d the i s o c l i n i c s should be e l i m i n a t e d . T h i s i s achieved by i n s e r t i n g two quarter wave p l a t e s i n the path of the p o l a r i s e d l i g h t . The qua r t e r wave p l a t e s are made of b i r e -f r i n g e n t m a t e r i a l of such a t h i c k n e s s t h a t the r e t a r d a t i o n between the o r d i n a r y and e x t r a o r d i n a r y r a y s i s one qu a r t e r of the wave l e n g t h of the monochromatic l i g h t used. One qu a r t e r wave p l a t e i s placed between the p o l a r i z e r p l a t e and the p l a s t i c , the other between the p l a s t i c coat and the a n a l y z e r p l a t e ( F i g . l . ) . The two qua r t e r p l a t e s convert the plane p o l a r i z e d l i g h t to c i r c u l a r p o l a r i z e d l i g h t . I n the case of white l i g h t t h i s i s not q u i t e t r u e s i n c e the l i g h t becomes e l l i p t i e a l l y p o l a r i z e d , but the d i f f e r e n c e i s very s m a l l and has l i t t l e e f f e c t on the s c a l e of i n t e r f e r e n c e c o l o u r s (12). The f r i n g e s seen i n the c i r c u l a r p o l a r i z e d white l i g h t are c o l o u r e d . I f the i n c i d e n c e of the c i r c u l a r p o l a r i z e d l i g h t i s normal, these c o l o u r e d f r i n g e s are the l o c i of p o i n t s where the d i f f e r e n c e of the p r i n c i p a l s t r e s s e s i s 5 constant. A black fringe i n the circular polarized li g h t i n -dicates that at those points the difference of the principal stresses i s zero. As shown i n Appendix I, by determining the retardations at points on the plastic the stresses i n the structure are given by the equation (Cn - c r o ) - w ° n 1 w2'w "1+1/ 2 t l w CE where K = 2.+1/ i s ^ t i e OP^--0^ strain sensitivity factor of the p l a s t i c . l t i s obtained P by calibration. E = modulus of e l a s t i c i t y 1/= Poisson 1 s ratio & = relative retardation The half of the principal stress difference so defined, gives the maximum shear stress, as known from the theory of el a s t i c i t y (13). Tmax = I ( ( T 1 " C r 2 ) Therefore the coloured fringes sometimes are called the constant shear stress lines. In particular state of stress, i f the direction of the principal stresses and their difference determined by the 6 f r i n g e s i s known, the i n d i v i d u a l p r i n c i p a l stresses may he determined mathematically.Often the stresses are sought a-long free boundaries such as i n the case of a hole or a notch where the stress normal to the boundary i s zero, and the f r i n g e value gives the tangential stress d i r e c t l y . When t h i s i s not the ©ase a d d i t i o n a l photoelastic measurements are necessary. I f the p o l a r i z e d l i g h t has an oblique angle of incidence the f r i n g e s obtained w i l l represent the difference of the secondary p r i n c i p a l stresses (9) .Secondary p r i n c i p a l stresses are defined as the p r i n c i p a l stresses r e s u l t i n g from the s t r e s s components l y i n g i n a plane normal to the given d i r e c -t i o n . By choosing the plane of incident and r e f l e c t e d l i g h t properly i t i s always possible to include one of the p r i n c i p a l stresses i n t o the difference of the secondary p r i n c i p a l stresses. Having t h i s a d d i t i o n a l reading with the normal reading, (or two oblique readings) the i n d i v i d u a l p r i n c i p a l stresses can be c a l c u l a t e d (14). For p r a c t i c a l reasons the angle of oblique incidence i s chosen as 45° r e s u l t i n g i n great s i m p l i f i c a t i o n s i n the ex-pressions f o r the i n d i v i d u a l p r i n c i p a l stresses. The i n s t r u -ment used i s again a r e f l e c t i o n polariscope, the p o l a r i z e r and analyzer plates being set with t h e i r p o l a r i z i n g axes crossed and at 45° to the plane of the incident and r e f l e c t e d rays ( F i g . 2 ) . The 45° s e t t i n g of the p o l a r i z i n g axes was 7 required by the quartz wedge compensator. This compensator i s sit u a t e d ahead of the analyzer plate i n the r e f l e c t e d l i g h t path. The plane of incident and r e f l e c t e d l i g h t should always coincide with one of the p r i n c i p a l stress d i r e c t i o n s . The compensator value so obtained r e f e r s to the p r i n c i p a l stress perpendicular to the plane of incident and r e f l e c t e d l i g h t . The magnitude of the p r i n c i p a l stresses i s given by the equations as derived i n Appendix I. <°i>w = 2 ' 3 6 x 1 0 " 7 IT" ^ + T " ) ( C T 2 ) w = 2.36 x IO" 7 -If- ( m2 + ^ ) where m^  and m2 are the compensator reading changes occurring due to load (15). The oblique incidence readings are taken always a f t e r a l l the normal readings are completed and the directions,of the p r i n c i p a l stresses are known. There are d i f f e r e n t ways to evaluate the r e t a r d a t i o n or f r i n g e value and hence the magnitude of p r i n c i p a l stresses. The f i r s t , and l e a s t accurate,is the comparison of the fr i n g e colour to a standard colour scale which i s r e l a t e d to s t r a i n through c a l i b r a t i o n . I f a black f r i n g e i s present, one can count the number of successive " t i n t of passage" fringes.The t i n t of passage i s the colour occurring at the t r a n s i t i o n from 8 red to blue. This d u l l purple colour i s very s e n s i t i v e to small changes i n the difference of the p r i n c i p a l stresses but gives accurate stress readings only when the point under i n v e s t i g a t i o n l i e s on a t i n t of passage. Often the stress i s not high enough to produce s t r a i n s corresponding to one fr i n g e or the point under i n v e s t i g a t i o n l i e s between success-ive f r i n g e s . In t h i s case an o p t i c a l compensator should be used, which produces a t i n t of passage at any point of the b i r e f r i n g e n t p l a s t i c , at any value of s t r a i n . The most accurate method of de f i n i n g f r a c t i o n s of f r i n g e s i s to use photometers. The method of a p p l i c a t i o n of b i r e f r i n g e n t l a y e r s depends on the shape of the structure and also on the accuracy required (16). For plane surfaces, p l a s t i c sheets are a v a i l -able i n various thickness. These sheets are bonded to the surface by means of an adhesive. To provide a r e f l e c t i v e sur-face at the i n t e r f a c e of the structure and coating, the ad-hesive u s u a l l y i s mixed with aluminum powder. In case the surface of the metal i s ground t h i s i s not necessary because the d i f f u s e r e f l e c t i o n provided by the surface, i s s a t i s f a c -tory f o r the observation of the f r i n g e s . When a high degree of accuracy i s not required the p l a s t i c can be applied by brushing l i q u i d p l a s t i c on the surface of the structure, and polymerizing i t by heat applied to the coated area (17). For quantitative a n a l y s i s the thickness of the coat must be also measured at the point under i n v e s t i g a t i o n . When the surface i s complicated i t i s best to use contoured sheets. For contoured sheets the p l a s t i c i s cast on a glass plate and taken from the glass when p a r t i a l l y polymerized (18).. In t h i s state the p l a s t i c sheet i s e a s i l y formed without introducing i n i t i a l b i refringence. The sheet i s moulded over the required surface, and i s l e f t on the surface u n t i l the polymerization i s completed (generally 24 hours ).When hard, the moulded p l a s t i c i s bonded to the surface of the stru c t u -re with a cement, as f o r the sheet p l a s t i c . When casting, a s t r i p i s set aside f o r c a l i b r a t i o n purposes. I t i s bonded to a t e s t bar i n which the surface s t r a i n s may be obtained by c a l c u l a t i o n . Measuring the r e t a r d a t i o n i n the p l a s t i c , the s t r a i n o p t i c a l c o e f f i c i e n t of the p l a s t i c can be determined, as shown i n Appendix I I I , using the equation * " 2 t A £ ± -e2) For high stress gradients or sharp curvatures, a pola-r i z i n g microscope i s used (Fig.3). This c o n s i s t s of an objective, a p a i r of crossed Ni c o l s , one compensator wedge and one ocular. The l i g h t source may be e i t h e r attached to the tube, or to a separate stand. The t o t a l magnification i s about twenty times. The accuracy of the measurements taken with the p o l a r i -10 scope i s defined by the accuracy with which the readings can be made on the compensator. The compensation.''on the large f i e l d polariscope i s a-chieved by the quarter wave plate method; the analyzer i s ro-tated to obtain compensation, with a quarter wave plate be-tween the analyzer and the p l a s t i c . A r o t a t i o n of 180° caus-\ es one wave length r e t a r d a t i o n which corresponds to one f r i n g e . The scale can be read at every two degrees.Thus the smallest change which can be s t i l l observed i s one n i n e t i e t h of a f r i n g e . The oblique incidence polariscope and the p o l a r i z i n g microscope have a quartz wedge compensator. For t h i s a 35 d i v i s i o n scale corresponds to one fring e change. I t can be read at each h a l f graduation and therefore the smallest change to be noticed w i l l be one seventieth of a f r i n g e . Up to t h i s point the r e i n f o r c i n g e f f e c t of the p l a s t i c was not included i n the equations determining the stresses i n the structure. An i n v e s t i g a t i o n by Zandman, Redner and Riegner (21) showed that f o r t h i n coating i n plane stress t h i s r e i n f o r c i n g e f f e c t i s n e g l i g i b l e . In case of bending or combined stresses or when the thickness of the coat i s not small compared to the thickness of metal t h i s r e i n f o r c i n g e f f e c t can not be neglected and must be accounted f o r . The influence of the coat on the s t r a i n s and so on birefringence 11 i s given as a c o r r e c t i o n f a c t o r p l o t t e d as a function of the r a t i o of p l a s t i c thickness to metal thickness i n the above mentioned reference. This c o r r e c t i o n f a c t o r i s e s p e c i a l l y important when t h i n plates are subjected to bending. 12 Description of the Pressure Vessel and Equipment Used. The structure under i n v e s t i g a t i o n was a small pressure v e s s e l obtained commercially ( F i g . 4) • Two t o r i s p h e r i c a l dished heads were welded to a c y l i n d e r , which was r o l l e d from a s t e e l plate and welded along the generating l i n e . One head was complete; the other contained two tapped holes f o r f i l l -i n g and p r e s s u r i z i n g the v e s s e l . The ph o t o e l a s t i c i n v e s t i g a t i o n was made on the complete head, and to avoid r e i n f o r c i n g at the j o i n t of t h i s head to the c y l i n d e r , the weld was ground f l u s h to the parent materi-a l s outside and i n s i d e . Care was taken also that the c y l i n d e r was s u f f i c i e n t l y long so that the other end closure had no e f f e c t on the stresses i n the head under i n v e s t i g a t i o n . The material of the v e s s e l was mild s t e e l with a y i e l d strength of 30000 p s i . The modulus of e l a s t i c i t y was 30 x 10 6 p s i . The wall thickness, one eighth of an inch, was uniform i n the heads and the c y l i n d e r , and since the radius of the torus (knuckle) part of the head was small, t h i s part of the ve s s e l could not be considered as a t h i n s h e l l . The r a t i o of the radius of curvature to the wall thickness was |.84 and the r a t i o f o r t h i n s h e l l s i s defined as above i s ten. The 13 other parts, the sphe r i c a l cap and the c y l i n d e r , were t h i n s h e l l s having a radius of curvature to thickness r a t i o of 64 and 32 r e s p e c t i v e l y . To make the photoelastic measurements easier,the v e s s e l was mounted on a stand and provisions was made so that i t could he rotated about an axis at the mid length of the cy l i n d e r (Pig.6,7). Due to the double curvature of the head of the pressure v e s s e l i t was necessary to use the moulding technique (17) to cover i t with a p l a s t i c l a y e r , as described i n Appendix I I . A p a r t i a l l y polymerized p l a s t i c sheet was obtained by cas t i n g l i q u i d p l a s t i c on a l e v e l glass surface. This sheet showed no resistance against forming and was formed on the head without introducing i n i t i a l b i r e f r i n g e n c e . When the 'y p l a s t i c hardened, i t was cemented to the head by an epoxy r e s i n cement (19). The s t r a i n o p t i c a l c o e f f i c i e n t of the p l a s t i c was ob-tained by c a l i b r a t i o n using a c a l i b r a t i o n s t r i p from the cast p l a s t i c . A d e t a i l e d d e s c r i p t i o n of the c a l i b r a t i o n i s given i n the Appendix. The pressure vessel was f i l l e d with water whichhad been i n the open a i r before f i l l i n g to release the absorbed a i r . A high pressure rubber hose provided the connection between the pressure v e s s e l and a dead weight t e s t e r (Pig.6), which produced the required pressure. Before applying any pressure, 14 the system was "bled of a i r at the highest point. The dead weight t e s t e r remained connected to the press-ure v e s s e l throughout the t e s t s and i t s p i s t o n was rotated to secure uniform pressure while the photoelastic measure-ments were taken. A l l measurements were taken at 500 ps i g . The photoelastic part of the experiment consisted of the i n v e s t i g a t i o n of a set of points along a r a d i a l l i n e from the center of the head to the c y l i n d e r , defined by the angle CA (Pig.4). Measurements were taken at every degree from ( X = 0° to 0(= 20° and at every four degrees from a = 20° to a = 90° Instruments used were the large f i e l d polariscope,which was used to determine the d i r e c t i o n s and the difference of the p r i n c i p a l stresses (Fig.7)> the oblique incidence pola-riscope, which supplied the values f o r the i n d i v i d u a l p r i n -c i p a l stresses (Fig.8), and the p o l a r i z i n g microscope, which provided c o n t r o l values to check the values obtained by the large f i e l d polariscope (Fig.9). When c a l c u l a t i n g the stresses a c o r r e c t i o n f a c t o r f o r bending had to be introduced, because bending moment existed i n the torus due to the i n t e r n a l pressure. The c o r r e c t i o n f a c t o r was obtained from F i g 2 i n reference (21) f o r the e x i s t i n g r a t i o of p l a s t i c to metal thickness. 15 Results. The d i r e c t i o n s of the p r i n c i p a l stresses were e s t a b l i s h -ed f i r s t with the use of the large f i e l d polariscope without the quarter wave plat e s . I t was found that within a c i r c l e drawn at (X = 32° around the axis of the s h e l l on the spheri-c a l cap, every point was i s o t r o p i c , i . e . the p r i n c i p a l s t r e s s -es were equal i n a l l d i r e c t i o n s . Another dark r i n g appeared i n the f i e l d of the pola-riscope at (X = 10° where the points proved to be singular, which meant that both p r i n c i p a l stresses became zero. At the other parts of the head a r a d i a l dark l i n e c o i n c i d i n g always with the p o l a r i z i n g axis of the p o l a r i z e r plate i n d i c a t e d that the d i r e c t i o n of the p r i n c i p a l stresses were meridional and ci r c u m f e r e n t i a l r e s p e c t i v e l y (Fig.10). The mathematically l a r g e r p r i n c i p a l s t r e s s , CJ]_ was c i r c u m f e r e n t i a l between CX = 0° and OC= 10°; and changed to meridional between CX= 10° and d = 3 2 ° These observations l e d to the conclusion that the meri-dional and ci r c u m f e r e n t i a l stresses along a r a d i a l l i n e were the p r i n c i p a l stresses, and so the maximum of meridional and cir c u m f e r e n t i a l stresses were determined, g i v i n g the maximum stresses at the point. As part of the quantitative a n a l y s i s normal and oblique 16 incidence measurements were taken. When the p l a s t i c was viewed with the large f i e l d pola-riscope set f o r c i r c u l a r l y p o l a r i z e d l i g h t f i e l d , t h r e e areas showed dark patterns. One area was within the c i r c l e drawn at (X = 32° around the axis of the s h e l l on the sp h e r i c a l cap. The other was a r i n g at OC = 10°, and the t h i r d another r i n g at oi = 0°. Due to the c i r c u l a r l y p o l a r i z e d l i g h t f i e l d , these dark areas indicated that the difference of the p r i n c i p a l stresses was zero there. Farther i n v e s t i g a t i o n how-ever showed that the points on the r i n g at ct = 10° were singular, and the points on the sp h e r i c a l cap within the c i r c l e at the angle Oi = 32° and the points on the r i n g at Oi = 0° were i s o t r o p i c . The f r i n g e value increased from Oi = 0°, reached i t s maximum at OK= 5»5°» then decreased to zero at OC = 10°. I t increased again from oi = 10° to Oi =17° and decreased to zero at Oi = 32°. The difference of the p r i n c i p a l s t r a i n s did not reach one fr i n g e at any point from Oi = 0° to Oi = 90° which means that the re t a r d a t i o n measured was a l -ways l e s s than one wave length of,the white l i g h t . Oblique incidence measurements were made a f t e r the d i -r e c t i o n of the p r i n c i p a l stresses was determined with the large f i e l d polariscope. The plane of incident and r e f l e c t e d l i g h t of the oblique incidence polariscope was f i r s t aligned to coincide with a meridional l i n e and the readings so ob-17 tained were used to determine the i n d i v i d u a l c i r c u m f e r e n t i a l s t r e s s . S i m i l a r l y i t was aligned perpendicular to the pre-vious d i r e c t i o n to supply the values f o r the c a l c u l a t i o n of the meridional stresses. Each measurement was taken when the pressure v e s s e l was loaded and again when i t was unloaded. The sign of the compensator values was defined by t h e i r l o c -a t i o n from the zero f r i n g e i n the compensator wedge and by the a l g e b r a i c .subtraction of the no load reading from the reading taken under load. A dimensionless stress i n t e n s i t y f a c t o r was defined as the r a t i o of the stress measured to the c i r c u m f e r e n t i a l s t r e s s i n the c y l i n d e r remote from the end closures Stress Intensity = — where CT = stress measured p = pressure i n the ves s e l R Q= radius of the c y l i n d e r t Q = wall thickness of the c y l i n d e r This stress i n t e n s i t y f a c t o r i s given with the stress values at each point on the head i n Table I. The points on the head were defined by the angle OC , but the corresponding values f o r the angle (£> were also included, because i t was found that t h i s angle was more often used i n the l i t e r a t u r e . 18 The stresses were also p l o t t e d on l i n e s perpendicular to the head at each point (Pig.11), and i n a coordinate sys-tem of angle OC versus the stresses (Pig.12). The maximum meridional stress of O^j = -48500 p s i occurred i n the torus part of the head at CK = 5 . 5 ° , while the maximum c i r c u m f e r e n t i a l stress reached i t s maximum value of Q~ = -33500 p s i at OC = 5°. Both stresses were com-6 pressive on the outer surface of the head. 19 Table I. Meridional and Circumferential Stresses i n the Head of the Pressure Vessel and the Corresponding Stress Intensity Factors. Points defined by angles Stresses 10 p s i Stress I n t e n s i t y Factors a 4> Circum-f e r e n t i a l Meridional Circum-f e r e n t i a l Meridional 0.0 90 -2.00 -2.00 -1.25 -1.25 1.0 -2.65 -2.90 -1.66 -1.81 1.2 80 -2.70 -3.05 -1.69 -1.91 2.0 -2.95 -3.50 -1.84 -2.19 2.4 70 -3.00 -3.75 -1.88 -2.34 3.0 -3.20 -4.05 -2.00 -2.53 3.5 60 -3.25 -4.30 -2.03 -2.69 4.0 -3.30 -4.50 -2.06 -2.81 4.5 50 -3.30 -4.70 -2.06 -2.94 5.0 -3.35 -4.78 -2.10 -2.98 5.5 40 -3.30 -4.85 -2.06 -3.03 6.0 -3.18 -4.75 -1.99 -2.97 6.3 30 -3.10 -4.70 -1.94 -2.94 7.0 28 -2.85 -4.45 -1.78 -2.78 8.0 -2.50 -3.65 -1.56 -2.28 8.4 27 -2.35 -3.30 -1.47 -2.06 9.0 -2.10 -2.65 -1.31 -1.66 10.0 26 -1.50 -1.50 -0.94 -0.94 20 Table I. (continued) Points defined by angles 4 Stresses 10 p s i Stress I n t e n s i t y Factors a Circum-f e r e n t i a l Meridional Circum-f e r e n t i a l Meridional 11.0 25 -0.85 0.00 -0.53 0.00 12.0 -0.20 1.00 -0.13 0.63 12.6 24 0.10 1.50 0.06 0.94 13.0 0.55 1.80 0.34 1.12 14.0 23 0.70 2.60 0.44 1.63 15.0 0.95 2.75 0.59 1.72 15.4 22 1.05 2,85 0.66 1.78 16.0 1.20 2.95 0.75 1.84 17.0 21 1.30 3.10 0.81 1.94 18.0 1.40 3.15 0.88 1.97 18.6 20 1.41 3.10 0.88 1.94 19.0 1.42 3.07 0.84 1.92 20.0 19 1.45 2.95 0.90 1.84 22.4 18 1.50 2.60 0.94 1.63 24.0 17 1.47 2.35 0.92 1.47 27.0 16 1.35 1.80 0.84 1.12 28.0 1.30 1.65 0.81 1.03 30.0 15 1.15 1.40 0.72 0.88 32.0 14 0.97 1.18 0.61 0.74 36.0 • • • • 0.86 • • 0.94 • * 0.54 • • 0.59 • • • 54.0 • • • • 7 • • • 0.71 • • • 0.76 t • * 0.44 • • • 0.48 • • • 90.0 • • 0 • 0.57 * 0.59 • 0.36 • 0.37 21 Comparison with other works on pressure v e s s e l heads. In order to assess the accuracy of the present stres s i n v e s t i g a t i o n , the r e s u l t s were compared to values obtained i n the l i t e r a t u r e . These values were a r r i v e d at by c a l c u l a -t i o n s , using the t h i n s h e l l theory, and the dimensions of the v e s s e l s a t i s f i e d the conditions of t h i n shells.The vess-e l i n the present i n v e s t i g a t i o n was not a t h i n s h e l l . There-fore the d i s t r i b u t i o n of the stresses and the maximum stress i n t e n s i t y f a c t o r s were compared. The heads i n two of the work used as comparisons were t o r i s p h e r i c a l and one was e l l i p s o i d a l . The e l l i p s o i d a l head i n v e s t i g a t i o n was published by Kraus, Bilodeau and Langer (24). Results were presented i n t h i s paper f o r an e l l i p s o i d a l head having a two to one r a t i o of the major and minor axes. Stress i n t e n s i t y indexes were tabulated f o r a ser i e s of vessels determined by parameters T and . Where r a t i o of the major axis to the minor axis of the e l l i p s o i d a l head D diameter of c y l i n d e r t c y l i n d e r thickness T head thickness 22 For comparison j] i n the present i n v e s t i g a t i o n was tak-en as the r a t i o of the radius of the c y l i n d e r to the height of the head. The maximum stress i n t e n s i t y f a c t o r i n r e f e r -ence (24) was 3.00 i n the knuckle and 2.15 i n the sph e r i c a l /) D T cap. The parameters were /j = 3 r£ = 50 ^ = 1 In the present i n v e s t i g a t i o n these parameters were ^3=3 ^ = 64 T ^ = 1 and the maximum str e s s i n t e n s i t y i n the knuckle was 3.00 and i n the sphe r i c a l cap 2.28. The papers dealing with the t o r i s p h e r i c a l heads were published by G a l l e t l y (22,23). One of these papers (23), shows that the ASME Code f o r Unfired Pressure Vessels gives very low stress values i n the torus. The c a l c u l a t i o n s were based on the s o l u t i o n of the d i f f e r e n t i a l equations f o r constant thickness s h e l l s of rev-o l u t i o n , The r e s u l t s show the d i s t r i b u t i o n of the meridional bending stresses and the ci r c u m f e r e n t i a l d i r e c t stresses i n the torus, and that both stresses exceeded the y i e l d point of the m a t e r i a l . The comparison of these r e s u l t s to the r e s u l t s obtained i n the present i n v e s t i g a t i o n (Pig.13) are made on the base of the st r e s s i n t e n s i t y f a c t o r . Good agreement p r e v a i l s f o r the maximum stress i n t e n s i t y f a c t o r , and f o r the sign of the stresses. The dimensions of the head were 23 Radius of spherical cap 227.5 i n Radius of the c y l i n d e r 138.23 i n Thickness of the head 0.625 i n Thickness of the c y l i n d e r 0.460 i n Pressure 60 psig. The second paper (22) gives two examples i n which press-ure vessels of d i f f e r e n t shapes were designed with the use of influence c o e f f i c i e n t s . I t showed also that high stresses occurred i n the t o r i . The s t r e s s d i s t r i b u t i o n i n the torus was presented and t h i s was also compared to the values pres-ently obtained i n (Pig.13). The dimensions of the head were: Radius of the s p h e r i c a l cap 281.25 i n Radius of the c y l i n d e r 150 i n Thickness of the head 0.625 i n Thickness of the c y l i n d e r 0.5 i n Pressure 60 p s i g . The s t r e s s due to i n t e r n a l pressure i n a t o r i s p h e r i c a l head was also determined i n the presently investigated v e s s e l by using the ASME Code f o r Unfired Pressure Vessels. The equation used was CT= £ ( 0.5 M f- + 0.1 ) where Q~= stress i n the wall p = i n t e r n a l pressure 24 R = i n s i d e radius of the s p h e r i c a l cap t = thickness of the head 7| = j o i n t e f f i c i e n c y (unity) M = 1.77, when the radius of the knuckle i s 6 % of the radius of sph e r i c a l cap. The stress was CT= 28400 p s i and the s t r e s s i n t e n s i t y f a c t o r was 1.78. The d i s t r i b u t i o n curve was f l a t t e r f o r the present i n -v e s t i g a t i o n which was a t t r i b u t e d to the wall thickness of the torus. 25 Summary and Conclusions. The photoelastic coating method provided a new approach f o r evaluating the stresses i n the t o r i s p h e r i c a l head of a pressure v e s s e l . The coat applied to the surface made p o s s i -ble the determination of the strains on the surface of the structure. I t also supplied the d i r e c t i o n s of the p r i n c i p a l stresses. In t h i s case they were c i r c u m f e r e n t i a l and me-r i d i o n a l . The maximum value and the l o c a t i o n of the stresses as well as t h e i r sign was determined and these values agreed c l o s e l y with the values obtained i n other investigations.The c i r c u m f e r e n t i a l s t r e s s d i s t r i b u t i o n was lower i n the present i n v e s t i g a t i o n than the d i s t r i b u t i o n i n the compared works. This was a t t r i b u t e d to the t h i c k e r torus i n the pressure vessel investigated. The maximum value of the stresses exceeded the y i e l d point of the material, although no y i e l d i n g was noticed during the t e s t . I t i s believed that previous unsuccessful t e s t i n g caused work hardening i n the material. The thickness of the p l a s t i c determined the s e n s i t i v i t y of the measurements and i n the present case i t was quite low. A t h i c k e r coating increases the s e n s i t i v i t y , but the r e i n -f o r c i n g e f f e c t also increases. The casting and forming of the t h i c k e r sheet i s also more d i f f i c u l t . When l i q u i d p l a s t i c i s east, quite large quantities should he mixed with the accelerator to obtain a uniform polymerization. The amount of accelerator added i s very v i -t a l ; a l i t t l e d eviation from the required proportion causes non uniform sheet formation. This, and the forming of thick coatings on curved sur-faces could he investigated i n a separate work. Photoelastic measurements on a s h e l l of r e v o l u t i o n i n which the stresses were obtained a n a l y t i c a l l y would be also valuable. In conclusion the casting method proved to be a very u s e f u l method f o r evaluating the stresses i n the pressure v e s s e l . The instruments used can be e a s i l y c a r r i e d and thus photoelastic i n v e s t i g a t i o n i s possible even on the s i t e , p r o -vided an e l e c t r i c outlet i s a v a i l a b l e . The coating of the structure requires l i t t l e equipment and a short time only. 27 References. 1., Mesnager, M., "Sur l a determination optique des tensions intervenuer dans l e s s o l i d e s a t r o i s dimensions"Comptes  Rendus l'Aoad. S c i . , 190, 1249, P a r i s , (1930) 2., Mabboux, G., "Applicationsde l a P h o t o e l a s t i c i t e dans l e s ouvrages en beton," E d i t o r s : Delmar, Chapon Grounouihou, Prance (1933) 3., Oppel, G., "Das Polar i z a t i o n s o p t i s c h e Schichtverfahren zur Messung der Oberflachenspannung am beanspruchten Bau-t e i l ohne Modell," V D I - Z i t s c h r i f t Bd. 81 Nr. 27, (1937) 4., D'Agostino, J . , Drucker, D.C, L i u , O.K. and Mylonas, C , "An analysis of p l a s t i c behaviour of metals with bonded b i r e f r i n g e n t p l a s t i c , " Proceedings of The Society f o r  Experimental Stress A n a l y s i s . V o l . XII. 2, 115-122.(1955) 5., D'Agostino, J . , Drucker, D.C, L i u , O.K. and Mylonas, C , "Epoxy adhesives and casting r e s i n s as Photoelastic p l a s t i c s " Proceedings of The Society f o r Experimental  Stress Analysis. V o l . XII. 2, 123-128. (1955) 6., Zandman. P., "Analyse des contraintes par v e m i s photo-e l a s t i q u e s , " Groupment 1'Advancement Methodes d'Anal-Contraintes V o l . 2, No. 6 1-12 (1956) 7., Zandman, P., "Mesures photoelastiques des deformations elastiques et plastiques et des fragmentations c r i s t a l l i -nes dans l e s metaux," Revue de Metallurgie V o l . L I I . No. 8. 638-642 (1956) 8., Zandman, P.,and Wood, M.R., "Photo-Stress a new technique f o r p h otoelastic stress a nalysis f o r observing and measur-ing surface s t r a i n s on actual structures and parts" Product Engineering. 167-178 (1956) 9., Zandman, P., "Photoelastic-Coating technique f o r determin-ing stress d i s t r i b u t i o n i n welded structures," Welding  Journal Research Supplement (May I960) 10., Frocht.,M.M., P h o t o e l a s t i c i t y . New York. John Wiley & Sons Inc. (1948) V o l . I. and I I . 11., Hetenyi, H.,(ed) Handbook of Experimental Stress Analysis New York. John Wiley & Sons Inc. (1950) 28 12., "INSTRUCTIONS f o r the use of Photostress Large F i e l d Universal Meter" T a t n a l l Measuring Systems C o . B u l l e t i n . (Phoenixville) No, 8005.(1958) 13., Mindlin, R.D., " D i s t o r t i o n of the photoelastic f r i n g e -pattern i n an o p t i c a l l y unbalanced polariscope." ASME Trans. V o l . 59. No. A 170 (1937) 14., Timoshenko, S., G-oodier, J.N., Theory of E l a s t i c i t y Engineering S o c i e t i e s Monograph., McGraw-Hill Book Co. Inc. New York. Second ed. (1951) 15., McMaster, R.,(ed) Nondestructive Testing Handbook. New York. The Ronald Press Co., (1959) Vol I I . 16., "Operating i n s t r u c t i o n s f o r the Oblique Incidence Meter" T a t n a l l Measuring System Co. B u l l e t i n (Phoenixville) No. BN-8003 (1958) 17., "Suggestions f o r p l a s t i c s e l e c t i o n . " T a t n a l l Measuring  System Co. B u l l e t i n (Phoenixville) No. BN-8022 (1959) 18., "Instructions f o r Applying Photostress L i q u i d P l a s t i c " T a t n a l l Measuring System Co. B u l l e t i n . (Phoenixville) No. 8005 (1958) 19., "Instructions f o r moulding contoured sheets of photo-s t r e s s p l a s t i c " T a t n a l l Measuring System Co . B u l l e t i n . No. IB-8008 20., "Instructions f o r applying photostress sheet p l a s t i c " T a t n a l l Measuring System Co. B u l l e t i n . (Phoenixville) No. IB-8004 (1959) 21., Zandman, P., Redner, S.S., and Riegner, E.I., "Reinfor-cing e f f e c t of b i r e f r i n g e n t ooatings'Troceedings of The Society f o r Experimental Stress Analysis No.588.(1959) 22., G a l l e t l y , G.D., "Influence c o e f f i c i r n t s and pressure v e s s e l a n a l y s i s , " Journal of Engineering f o r Industry ASME Trans, Ser. B. Vol 82 259-269 (I960) 23., G a l l e t l y , G.D., " T o r i s p h e r i c a l shells-A caution to de-signers." Journal of Engineering f o r Industry. ASME Trans. Ser. B. V o l . 81. 51-66 (1959) 24., Kraus, H., Bilodeau, G.G., Danger, B.F., "Stresses i n thin-walled pressure vessels with e l l i p s o i d a l heads." Journal of Engineering f o r Industry. ASME Trans. Ser. B. V o l . 83. 29-42 (1961) 29 Appendix I. The Mathematical Theory. Normal Incidence. The connection between the p r i n c i p a l stresses i n a b i -re f r i n g e n t material and the r e t a r d a t i o n obtained between the ordinary and extraordinary rays of a p o l a r i z e d l i g h t i s ex-pressed i n Neumann's law f o r plane stresses. Where CT^; G~2 a r e '^xe P r i n c i P a l stresses i n the p l a s t i c £> r e l a t i v e r e t a r d a t i o n n G stress o p t i c a l c o e f f i c i e n t t thickness Rewriting to give the diffe r e n c e of the p r i n c i p a l stresses The difference of the p r i n c i p a l stresses can also be expressed as < 5 a « 0 ( 0 ^ - 0 ^ 2 t 1 . 2 . C 2 t (CT-, 1 " ^ p " 1 +V. ( € 1 " € 2 }p XT 3. 30. Where E = Modulus of E l a s t i c i t y V = Poissdn's r a t i o ^ 1 ' ^2 = ^ r i n c i P a l S t r a i n s Combining equations 2 and 3 5 E C 2 t 1 + \/ ^ 1 c 2 ; p 4 * or Denoting & n (1 + V-) CE„ 1 equation 4/a becomes &n ( € x - = 2 t K 5* Where K = i s the s t r a i n o p t i c a l c o e f f i c i e n t and i s u-s u a l l y determined by c a l i b r a t i o n Assuming a good bond between the p l a s t i c coat and struc-ture (E1 - € 2 ) p = ( £ x - € 2 ) w 5 / a and the difference of the p r i n c i p a l s t r a i n s i n the structure 31 i s given by ( G i - £ 2 K " 2 t K 6* To f i n d the difference of the p r i n c i p a l stresses Hooke's law i s used E cS (cr -cr) w n 7 l U l ^ p 1 + V w 2 t K '* Therefore the determination of the r e l a t i v e r e t a r d a t i o n provided the difference of the p r i n c i p a l stresses i n the structure. From t h i s the maximum shear stress i s obtained f max 2 The i n d i v i d u a l p r i n c i p a l stress can be obtained also with one normal reading however, when the point under i n v e s t i -gation i s located at a d i s c o n t i n u i t y or free surface. At these points the normal s t r e s s must be zero and equation 7 gives the tangen t i a l s t r e s s . Two equations are necessary to define the two i n d i v i d u a l stresses at points not on a fre e surface. These two equa-ti o n s can be obtained by taking two oblique readings or one normal and one oblique reading. 32 The Oblique Incidence. As seen from the previous discussion, when the incidence of the l i g h t i s normal, only the difference of the p r i n c i p a l stresses can be obtained. Taking another reading under ob-l i q u e incidence, the re t a r d a t i o n observed w i l l be the d i f f e r -ence of the secondary p r i n c i p a l stresses, thus g i v i n g another equation f o r so l v i n g the two p r i n c i p a l stresses i n d i v i d u a l l y . As the coordinate systems show, i t can be always arrang-ed that.one of the p r i n c i p a l stresses coincides with one axis of the secondary p r i n c i p a l stresses. Since the d i r e c t i o n s of the p r i n c i p a l stresses can be obtained with a normal i n c i -dence polariscope, the oblique incidence instrument can be aligned with one of these d i r e c t i o n s , thus i n c l u d i n g one p r i n c i p a l stress i n the difference of the secondary p r i n c i p a l 33 stresses. S t a r t i n g with Neumann's equation again and using Mohr's c i r c l e drawn f o r a three dimension case we can write the equations f o r the r a t a r d a t i o n taking two oblique i n c i -dence measurements (CT-, - O V )„ = o l 1 " u 2 ;p ~ 2 t / cos Q± C 10, (°~2 " ° I } 5 o2 where (Sq1 and (SO2 11, p ~ 2 t / cos 9 2 C are the retardations measured i n ob l i q u e incidence the instrument being aligned with the d i -r e c t i o n of the proper p r i n c i p a l s t r e s s . 0~2 ' o a n be defined from Mohr's c i r c l e as shown ( C T 2 ' ) p = (CT 2 c o s 2 Q l ) p C O l ' > p = ( G ~ 1 G o s 2 e2>p 34 therefore equation 10, and 11 can be expressed i n terms of the p r i n c i p a l stresses as follows ( c r r - a - 2 oos 2 si>p = 2 t / co 3°i c 1 2 -2 S o 2 (CTg-C ^ O O S 9 2) p ° 2 t / c o s ° e 2 c 13. Solving these simultaneous equations f o r O"^  and CT"2 ^ o l , ^02 c o s 2 e i 2 t / cos Q 0 C + 2 t / cos 9-, C ( C T ) = ^ ± 14. * 1 - cos 9 2 cos ^o2 S Q l c o s 2 9 2 2 t / cos Q0 C + 2 t / cos 9, C ( C r 2 ) = ^ g ± 15. * * 1 - cos 9 2 cos* 9]_ Great s i m p l i f i c a t i o n can be introduced by choosing the angle of incidence f o r both cases at 9-^  = 9 2 = 45° with which equations 14 and 15 become reduced to ^ 1 ; P T T T ( °ol + T ~ > 1 6 * 35 6\ '2'p ~ 3 t C v wo2 ( CT0) = ( & Q + ^ ) 17, Knowing from the normal incidence derivation that C E. K = 1 + V / P and from this. C = K (1 + l/^) E P Substituting this value of C into equations 16 and 17 and assuming that ( £ 1 ) p = C £ 1 ) w and V = Prom Hooke's law \[2 E ~ c5 <°"2>» " 3 t V J ( 5 0 2 + « • As stated earlier i t i s not necessary that the oblique incidence reading be taken to determine the individual prin-cipal stresses. One oblique reading with the normal reading would be sufficient. If an oblique reading i s taken for a retardation value including the principal stress CT-^ , the equation become 36 " ° P p = 2 ~ r o 2o-( c r l " cr2' ) p - ( O i - c r 2 cos 2 S l ) p = 2 t j f ^ c 2 1 -Solving these equations for and the equations determining the stresses i n the structure w i l l be t cr ^ - Ew ,Vg £ j§*u P 9 c °i'w " t K ( l + v J ) { ~ °oi ~ ~ } 2 2 « ( a 2 } w " t K (1 + v > ( " 2 " °ol " °n } 2 3 ' Similarly the equations determining the stresses i n the structure when the oblique reading "includes the principal stress are < ° i > w " t K (1 E T v/J ^ ^02 " §n> 2*' f r r M Ew ,V2£ &ns ?(-( G^'w ~ t K ( l + V,) { — °o2 _ ~ } 2 5 ' The retardation i n the case of normal incidence of ligh t was determined from the equation where k i s any p o s i t i v e number (0,1,2,...) y5 i s the compensator reading /\ i s the wave length of l i g h t used and i n the case of oblique incidence ^ o l = ~33~ where m^  i s the difference of the compensator readings under load and no load Equations 18 and 19 are expressed i n a simpler form by introducing the numerical values f o r the wave length of the white l i g h t and Poisson's r a t i o as A = 2.27 x 10" 5 i n V/ = 0.3 -7 Ew ,„ . m2, < O i > w « 2.36 x l O ' ^ ( m i + - f ) ( C T 2 ) W = 2.36 x IO" 7 - J - (m 2 + ^ ) 38 Appendix I I . Casting p l a s t i c sheets f o r the head of the v e s s e l . To cover the head with the photoelastic p l a s t i c , a con-toured sheet had to he used. The sheet was formed on the head by/ using a p a r t i a l l y polymerized p l a s t i c sheet. This par-t i a l l y polymerized sheet was obtained by casting p l a s t i c on an accurately l e v e l l e d glass plate, the surface of which was protected by a l a y e r of s i l i c o n e varnish. This l a y e r was to prevent the p l a s t i c from s t i c k i n g to the glass,and was baked on the glass i n an oven at 450°F f o r four hours. A hardboard frame, one quarter of an inch thick, and 9" x 9" insi d e d i -mensions was put on the glass and sealed with masking tape around i t s external perimeter. Seventy grams of l i q u i d pho-t o e l a s t i c p l a s t i c was mixed with f i f t e e n per cent by weight hardener and allowed to reach an exotherm temperature of 110° j?. I t was then poured onto the prepared glass surface, where the polymerization began. A f t e r three and a h a l f hours at room temperature the p l a s t i c formed a sheet of uniform thickness, which was very s o f t . The sheet was then formed on the head, covering more than one t h i r d of i t . No b i r e f r i n -gence was introduced while forming. The edges of the sheet were held i n p o s i t i o n with l i g h t l y applied scotch tape, and l e f t f o r completion of the polymerization which required 39 about 24 hours.. When the polymerization of the p l a s t i c was completed, the contoured sheet was removed from the head and i t s t h i c k -ness measured to one ten thousandth of an inch, with a micrometer. The edges were cut and the surface cleaned with acetone. A r e f l e c t i v e type cement was used to bond the photoelas-t i c sheet to the surface of the head. The cement was mixed with the hardener, ten per cent by weight, and allowed to set f o r ten minutes or more. A l a y e r of about one sixteenth of an inch was then spread on the head from the mixed cement and the sheet was applied. A i r bubbles were pressed out by applying the sheet at an angle to the surface and gradually lowering so that the excess cement squeezed out with the a i r at the other end. The edges of the p l a s t i c were sealed with the remaining cement. The bond hardened i n about a day, and the coat: was ready f o r the photoelastic test.The same proce-dure was used to bond the c a l i b r a t i o n s t r i p to the c a l i -b r a t i o n bar. 40 Appendix I I I . Determination of the s t r a i n o p t i c a l c o e f f i c i e n t by c a l i b r a t i o n . A f t e r the thickness of the p l a s t i c c a l i b r a t i o n s t r i p had been measured the s t r i p was bonded to an aluminum bar. The hardening of the bond took place at room temperature and com-pleted i n twenty four hours. Before the c a l i b r a t i o n was started, the p l a s t i c was examined with a polariscope andl i t was observed that no i n i t i a l birefringence existed before load-ing the tes t bar. One end of the bar was then clamped to a bench and pr o v i s i o n was made to load i t at the other end (Pig.14). A f t e r the large f i e l d polariscope was positioned f o r the readings, the tes t bar was loaded i n two pound i n c r e -ments and the retardations noted f o r each load. The points were p l o t t e d i n a compensator readings versus load coordinate system (Pig.15). Prom the s t r a i g h t l i n e r e l a t i o n s h i p between load and ret a r d a t i o n , the load causing a re t a r d a t i o n equiva-l e n t to one wave length of the white l i g h t used was deter-mined. The difference of the p r i n c i p a l s t r a i n s corresponding to t h i s r e t a r d a t i o n i s c a l l e d the fr i n g e value of the p l a s t i c . This f r i n g e value i s not equal to the actual s t r a i n difference when the structure i s bent, because of the r e i n f o r c i n g e f f e c t of the p l a s t i c (20), and a c o r r e c t i o n f a c t o r must be applied. 41 The difference of the p r i n c i p a l s t r a i n s was also deter-mined by c a l c u l a t i o n , using the well known equations from the theory of e l a s t i c i t y . 1 = L-F The t e n s i l e stress r e s u l t i n g from the load w i l l he p r i n -c i p a l stress at the top surface of the bar, the other p r i n c i -pal s t r e s s i s zero. °i = z 02 = 0 The difference of the p r i n c i p a l s t r a i n s as derived e a r l i -er, i s given by But from the above o~2 = 0 and As stated e a r l i e r , a c o r r e c t i o n f a c t o r was necessary to take care of the birefringence caused by bending. Introducing t h i s i n t o the equation 42 This equation gives the fringe value from the s t r a i n s occurr-ing at the surface of the te s t bar. Since the s t r a i n s are the same i n the p l a s t i c as i n the metal at the i n t e r f a c e , t h i s f r i n g e value was made equal to the fringe value given by equation 9 i n Appendix I . expressing the s t r a i n o p t i c a l c o e f f i c i e n t from here 6 K = 2 t y ( l + l / ) C 2 In the present case the values were * & = 2.27 x 10" 5 i n t = 0.052 i n P = 25.5 l b L = 6.0 i n Z = 1.04 x 10" 3 i n 3 E = 30.0 x 10 6 l b / i n 2 V = 0.3 C2= 1.16 43 b = 1.0 i n h = 0.25 i n and so K = 0.0905 with, t h i s value of K the f r i n g e value of the p l a s t i o was f =2400 x 10~ 6 i n / i n 44 Observer Analyzer plate. Quarter wave plate wX. 1- Light source P o l a r i z e r plate J - ^ r Quarter wave plate E l l i p t i o a l l y p o l a r i z e d l i g h t beam C i r c u l a r l y p o l a r i z e d l i g h t beam i i i i i B i r e f r i n g e n t plajs_tic_ H , , I ~~] - .Reflective surface W - . \ \ \ \ \ w \ \ \ ; x \ x Structure - ^ > W W N ^ W W ^ Pig. 1. Schematic drawing of R e f l e c t i o n polarisoope 45 0 = Observer Oo = Ooular a = Analyzer plate q = Quartz wedge compensator 1 = Light source p = P o l a r i z e r plate pr = Prism w = Structure pc = P l a s t i c coat Pig.2. Schematic of Oblique Incidence Polariscope 46 Observer V Ooular Analyzer plate Quartz wedge compensator i Semi transparent mirror P o l a r i z e r Light ! plate source i Objective I 1 1 i i ! ) P l a s t i c coat Structure P i g .3 . Schematic of P o l a r i z i n g Microsoope 47 R = 8.0 i n R c = 4.0 in L = 8.0 i n r = 0.48 i n t = 0.125 i n Pig.4. The Pressure Vessel 48 F i g . 5 . C a s t i n g of P l a s t i c Pig.6. Pressure Vessel and Dead Weight Tester 50 Pig.7. Test Setup f o r the Large F i e l d Polariscope Fig.8. Test Setup f o r the Oblique Incidence Polariscope Fig.9. Test Setup f o r the P o l a r i z i n g Microscope 53 F i g . 1 0 . I s o c l i n i c s on the Head of the Pressure V e s s e l Pig.11. Stress d i s t r i b u t i o n on the head of the pressure v e s s e l Stress ( l 0 5 p s i ) Torus 40 30 _ 20 _ 10 _ Spherical Cap Circumferential Stress 10 _ 20 30 40 15 20 25 30 35 40 50 60 70 80 90 angle c 50 _ Pig. 12. Circumferential and Meridional Stresses along a r a d i a l l i n e . Stress Intensity, Factor 3-1 56 Present work Reference (22) — Reference (23) \ 30 4 0 50 60 70 80 90 angle Meridional Stress Stress I n t e n s i t y Factor 3-Present work Referenoe (22) Reference (23) 30 40 50 60 7 0 8 0 90 angle Circumferential Stress Fig.13. Stress D i s t r i b u t i o n i n the Torus Pig.14. Test Setup f o r C a l i b r a t i o n of P l a s t i o 

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