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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. T e c h n i c a l 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 t h e Department of MECHANICAL  ENGINEERING  We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA October, 1961  In p r e s e n t i n g  this thesis i n p a r t i a l fulfilment of  the requirements f o r an advanced degree a t t h e B r i t i s h Columbia, I agree t h a t the a v a i l a b l e f o r reference  and  University  of  L i b r a r y s h a l l make i t f r e e l y  study.  I f u r t h e r agree t h a t  permission  f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may g r a n t e d by  the  Head o f my  Department o r by h i s  be  representatives.  It. i s understood 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 gain  s h a l l not  Department o f M f t o l m n i p.a.l The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada.  be a l l o i ^ e d w i t h o u t my  F,ngi r a r i n g Columbia,  Date Vancouver. October. 1.  1961  written  permission.  i  Abstract.  The  use o f the p h o t o e l a s t i c  c o a t i n g method  in  deter-  m i n i n g the s t r e s s e s i n the t o r i s p h e r i c a l head o f a p r e s s u r e v e s s e l was  investigated.  I t was found t h a t the method i s v a l u a b l e  t o o b t a i n the  d i s t r i b u t i o n , d i r e c t i o n , and magnitude o f s t r e s s e s surface The  on  o f any s t r u c t u r e . r e s u l t s o b t a i n e d w i t h the method  showed  greement w i t h 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 .  c l o s e a-  The  maximum  s t r e s s e s i n a t o r i s p h e r i c a l head o f a p r e s s u r e v e s s e l i n the t o r u s . The same c o n c l u s i o n  was drawn from  s t r e s s e s were compressive on the o u t e r  the s u r f a c e  t u r e s o f the method.  these  surface.  m o b i l i t y o f the i n s t r u m e n t s , the r e l a t i v e l y  way o f c o a t i n g  occur  the r e -  s u l t s o b t a i n e d w i t h the method. I t a l s o r e v e a l e d , t h a t  The  the  o f the s t r u c t u r e a r e o t h e r  simple fea-  ACKNOWLEDGMENT  The author wishes t o e x p r e s s h i s g r a t i t u d e t o P r o f e s s o r f . 0. Richmond,  Head  of  the Department  E n g i n e e r i n g , f o r t h e guidance  i n completing  of  Mechanical  h i s work, f o r  making t h e i n s t r u m e n t s and l a 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 f o r the help i n obtaining f i n a n c i a l  assistance.  ii  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 i n c i d e n c e  29  2. O b l i q u e i n c i d e n c e  32  I I . C a s t i n g p l a s t i c s h e e t s f o r the head o f the v e s s e l I I I . D e t e r m i n a t i o n o f the s t r a i n coefficient  by  calibration  38  optical 40  LIST OF TABLES  TABLE  I . MERIDIONAL AND CIRCUMFERENTIAL STRESSES IN THE HEAD OF THE PRESSURE VESSEL AND THE CORRESPONDING STRESS INTENSITY FACTORS  iv  LIST OF FIGURES  FIGURE  Page  1. Schematic Drawing o f R e f l e c t i o n 2. Schematic o f Oblique  Incidence  Polariscope Polariscope  44 45  3. Schematic o f P o l a r i z i n g M i c r o s c o p e  46  4. The  47  Pressure  Vessel  5. C a s t i n g o f P l a s t i c 6. P r e s s u r e  V e s s e l and  48 Dead Weight T e s t e r  49  7. T e s t Setup f o r the Large F i e l d P o l a r i s c o p e  50  8. T e s t Setup f o r the Oblique  51  Incidence  Polariscope  9. T e s t Setup f o r the P o l a r i z i n g M i c r o s c o p e 10.  I s o c l i n i c s on the Head o f the P r e s s u r e V e s s e l  11.  S t r e s s D i s t r i b u t i o n on the Head o f the Vessel  12.  13.  C i r c u m f e r e n t i a l and M e r i d i o n a l  52 53  Pressure 54  Stresses  along a r a d i a l l i n e  55  S t r e s s D i s t r i b u t i o n i n the Torus  56  14. T e s t Setup f o r C a l i b r a t i o n o f P l a s t i c  57  15.  58  C a l i b r a t i o n l i n e of p l a s t i c  List  E  (lb/in ) 2  o f Symbols Used.  Modulus o f E l a s t i c i t y Poisson's r a t i o  V t  (in)  Thickness  C  (in /lb)  Stress o p t i c a l c o e f f i c i e n t  2  Correction  °1>  K=  CE 1  Optical strain sensitivity factor of p l a s t i c  p  V  +L  factors  (in)  Relative retardation of p o l a r i z e d l i g h t  (in/in)  Principal  strains  (lb/in )  Principal  stresses  (lb/in )  Normal  (lb/in )  Shear s t r e s s  R  (in)  R a d i u s o f head o r c y l i n d e r  r  (in)  R a d i u s o f the k n u c k l e  P  Cpsig)  Pressure a c t i n g i n the p r e s s u r e v e s s e l  M  (in/lb)  Bending moment  L  (in)  Distance  F  (lb)  Load  I  (in )  Moment o f i n e r t i a  6 € 1  ; e  2  ;  cr r  0~  2  2  2  2  4  stress  vi ~  Fringe value the p l a s t i c  of used  X  (in)  Wave l e n g t h o f l i g h t  0  (degr.)  Angle o f i n c i d e n c e o f l i g h t A n g l e between a normal t o the s u r f a c e o f the s h e l l and the shell axis  cf>  (degr.)  CX  (degr.)  A n g l e between a normal drawn to the s h e l l a x i s from the j u n c t i o n o f the c y l i n d e r and head and a l i n e from a p o i n t on the s u r f a c e o f the head.  (deer ) ^ S w  Compensator r e a d i n g on the large f i e l d polariscope. The d i f f e r e n c e o f r e a d i n g s t a k en a t a p o i n t u s i n g the o b l i q u e i n c i d e n c e p o l a r i s c o p e when the s t r u c t u r e i s l o a d e d and u n l o a d e d .  m  Subscripts. c = cylinder p =  plastic  n = normal o = oblique w = m e t a l , workpiece  1 Introduction.  Stress property  of  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 temporary  b i r e f r i n g e n c e possessed  double by  refraction  upon  the  or  artificial  c e r t a i n transparent  materials.  T h i s b i r e f r i n g e n c e i s p r o p o r t i o n a l to s t r e s s and hence i t i s p o s s i b l e t o deduce  the s t r e s s  o p t i c a l p r o p e r t i e s . Ordinary  from the o b s e r v a t i o n  methods o f  of  photoelastic  the  stress  a n a l y s i s use a p l a s t i c model o f the s t r u c t u r e under a n a l y s i s . A newer t e c h n i q u e uses a c o a t i n g o f t r a n s p a r e n t m a t e r i a l cemented  t o the m e t a l  t h a t the p h o t o e l a s t i c e f f e c t s t r a i n on the s u r f a c e  part  under i n v e s t i g a t i o n so  observed  i s a f u n c t i o n of  the  o f the s t r u c t u r e . I n 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 o f the  stress  distri-  The use o f b i r e f r i n g e n t c o a t i n g i n p h o t o e l a s t i c  inves-  bution i n a toroidal  t i g a t i o n was who  photoelastic  shell.  initiated  by Mesnager ( F r a n c e ) i n  1930  used a b i r e f r i n g e n t l a y e r o f g l a s s . There was no  factory  bond t o the  structure investigated  and  r e s u l t s were n o t produced a t t h a t time. S e v e r a l develop the c o a t i n g method and Oppel (Germany) 1937  by  I n the  United  satis-  practical attempts t o  Mabboux (France)  1932  (2),  ( 3 ) , were u n s u c c e s s f u l f o r p r a c t i c a l  use, because the c o a t i n g m a t e r i a l and the bonding was  (1),  still  had low  sensitivity,  insufficient. States  o f America  D'Agostino,  Drucker,  2 L i n , and Mylonas performed  e x t e n s i v e s t u d i e s o f the b e h a v i o u r  o f b i r e f r i n g e n t c o a t i n g s (4»5) and developed i t as a p r a c t i c a l t o o l f o r s t r e s s a n a l y s i s . They p r e s e n t e d the r e s u l t s o f t h e i r work a t the c o n v e n t i o n o f IUTAM i n B r u s s e l s i n 1954. P r a c t i c a l r e s u l t s a c h i e v e d on the i n d u s t r i a l Prance were p u b l i s h e d by Zandman (5»6).  The  level  coating  was u s e d on v a r i o u s s t r u c t u r a l m a t e r i a l s i n b o t h  in  method  elastic  and  p l a s t i c ranges o f d e f o r m a t i o n . The c o a t i n g m a t e r i a l proved t o be s t a b l e i n time and temperature. The bond was e f f e c t i v e thus p r o v i d i n g the n e c e s s a r y t r a c t i o n between metal.  the p l a s t i c and  the  3 D e s c r i p t i o n of the P h o t o e l a s t i c C o a t i n g Method.  The  s t r u c t u r a l p a r t t o he  of b i r e f r i n g e n t material s t r a i n s are surface  transmitted  i s provided  refringence  due  analysed i s coated with a l a y e r  cemented to the to the p l a s t i c  surface  (7,8).  between the metal and  t o s t r a i n i s observed  so t h a t  A  reflecting  the p l a s t i c .  by  a  the  Bi-  reflection  po-  l a r i s c o p e so t h a t the l i g h t r a y passes twice  through  p l a s t i c l a y e r . Because o f t h i s , the b a s i c law  of p h o t o e l a s t i -  c i t y expressing and  the r e l a t i v e r e t a r d a t i o n between the  extraordinary  stresses i s  r a y and  the  ordinary  d i f f e r e n c e between the p r i n c i p a l  (9,10)  i s the r e l a t i v e r e t a r d a t i o n ,  where  C = i s the  stress optical c o e f f i c i e n t  and  are p r i n c i p a l s t r e s s e s  OT>  t = i s the  thickness  o f the p l a s t i c  When the p a r t i s s t r a i n e d , b l a c k are v i s i b l e  and  through the a n a l y z e r p l a t e o f  u s i n g white l i g h t . I f the p o l a r i z i n g axes and  the  analyzer  p l a t e s are c r o s s e d ,  i s o c l i n i c s . I s o c l i n i c s are r e c t i o n s o f the as the  layer coloured the  fringes  polariscope  o f the p o l a r i z e r  the b l a c k f r i n g e s r e p r e s e n t  the l o c i o f p o i n t s where the d i -  p r i n c i p a l s t r e s s e s are  c o n s t a n t and the  d i r e c t i o n s o f the p o l a r i z i n g axes of the  same  polariscope,  4 The  d i r e c t i o n s of the p r i n c i p a l stresses therefore  d e t e r m i n e d a t e v e r y p o i n t because t h e c r o s s e d  can  analyzer  p o l a r i z e r p l a t e s c a n be r o t a t e d s i m u l t a n e o u s l y . 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  be and  Knowing point  the  i ti s  p o s s i b l e t o p l o t ; t h e 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 t h e q u a n t i t a t i v e e v a l u a t i o n o f t h e f r i n g e s i s c o n s i d e r e d t h e i s o c l i n i c s s h o u l d 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 q u a r t e r wave p l a t e s i n t h e p a t h  of  the  p o l a r i s e d l i g h t . The q u a r t e r wave p l a t e s a r e made o f f r i n g e n t m a t e r i a l o f such a t h i c k n e s s t h a t  bire-  the r e t a r d a t i o n  between t h e 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  quarter  o f t h e wave l e n g t h o f t h e monochromatic l i g h t u s e d . One q u a r t e r wave p l a t e i s p l a c e d between  the p o l a r i z e r  p l a t e and t h e p l a s t i c , t h e o t h e r between t h e p l a s t i c and  t h e a n a l y z e r p l a t e ( F i g . l . ) . The two q u a r t e r  convert  coat  plates  the plane p o l a r i z e d l i g h t t o c i r c u l a r p o l a r i z e d l i g h t .  I n t h e case o f white l i g h t t h i s i s n o t q u i t e t r u e  since 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 , b u t t h e d i f f e r e n c e i s v e r y s m a l l and h a s l i t t l e  e f f e c t on t h e s c a l e o f 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 t h e c i r c u l a r p o l a r i z e d  are coloured.  I f the incidence of the c i r c u l a r  white  polarized  l i g h t i s normal, these coloured f r i n g e s a r e the l o c i points  where t h e d i f f e r e n c e o f t h e p r i n c i p a l  light  stresses  of i s  5 constant. A black f r i n g e i n the c i r c u l a r p o l a r i z e d l i g h t i n dicates that at those points the difference of the p r i n c i p a l stresses i s zero. As shown i n Appendix I, by determining the retardations at points on the p l a s t i c the stresses i n the structure  are  given by the equation  (Cn  -cr  1  w  o ) 2'w  -  °  w  "1+1/  n  2 t l  w  where  CE K = 2.+1/ P  i  s  ^  t i e  OP^-- ^ s t r a i n 0  sensitivity  f a c t o r of the p l a s t i c . l t i s obtained by c a l i b r a t i o n .  E = modulus of e l a s t i c i t y 1/= Poisson s r a t i o 1  & = r e l a t i v e retardation The h a l f of the p r i n c i p a l stress difference so defined, gives the maximum shear s t r e s s , as known from the theory elasticity  of  (13). Tmax = I  ( ( T  1 "  C r  2  )  Therefore the coloured f r i n g e s sometimes  are c a l l e d the  constant shear stress l i n e s . In p a r t i c u l a r state of s t r e s s , i f the d i r e c t i o n p r i n c i p a l stresses and t h e i r difference  determined  of the by the  6 f r i n g e s i s known, the  individual p r i n c i p a l stresses  determined m a t h e m a t i c a l l y . O f t e n the  stresses  are  l o n g f r e e b o u n d a r i e s such as i n the  case  n o t c h where the  boundary  the  s t r e s s normal to the  f r i n g e v a l u e g i v e s the  t h i s i s not  the  tangential  ©ase a d d i t i o n a l  of  may  he  sought  a hole is  a-  or  zero,  stress d i r e c t l y .  a and  When  p h o t o e l a s t i c measurements are  necessary. I f the the  fringes  polarized  l i g h t has  o b l i q u e angle o f i n c i d e n c e  o b t a i n e d w i l l r e p r e s e n t the  secondary p r i n c i p a l s t r e s s e s are  an  d e f i n e d as  the  difference  p r i n c i p a l stresses  c h o o s i n g the  resulting  p l a n e of i n c i d e n t  p r o p e r l y i t i s always p o s s i b l e stresses stresses.  i n t o the  ( o r two  stresses  can  chosen as  be  of the  principal the  individual  normal principal  angle of o b l i q u e i n c i d e n c e i s  individual principal stresses.  analyzer plates a t 45°  rays (Fig.2).  light  reading with  ment used i s a g a i n a r e f l e c t i o n p o l a r i s c o p e ,  c r o s s e d and  reflected  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  p r e s s i o n s f o r the  and  direc-  (14).  p r a c t i c a l reasons the 45°  The  being set with t h e i r to the 45°  the  given  secondary  o b l i q u e r e a d i n g s ) the calculated  and  from  to i n c l u d e one of the p r i n c i p a l  Having t h i s a d d i t i o n a l  reading,  For  difference  the  (9) .Secondary p r i n c i p a l s t r e s s e s  s t r e s s components l y i n g i n a plane normal to the t i o n . By  of  p l a n e of the  setting  of  the  the  The  ex-  instrupolarizer  polarizing  axes  i n c i d e n t and r e f l e c t e d  the  p o l a r i z i n g axes  was  7 r e q u i r e d by the q u a r t z wedge compensator. T h i s compensator i s s i t u a t e d ahead o f t h e a n a l y z e r p l a t e i n the p a t h . The  plane of i n c i d e n t  and  reflected  reflected  light  always c o i n c i d e w i t h one o f the p r i n c i p a l  stress  The  to  compensator v a l u e so o b t a i n e d  refers  s t r e s s p e r p e n d i c u l a r t o the plane o f i n c i d e n t  light  should  directions.  the  principal  and  reflected  l i g h t . The magnitude o f t h e p r i n c i p a l s t r e s s e s  i s given  by  the e q u a t i o n s as d e r i v e d i n Appendix I .  <°i>w = ' 2  ( CT ) 2  where m^ and m due  2  3 6  x  1 0  "  7  IT"  = 2.36 x I O "  w  7  -If-  ^  ( m  T " )  +  2  +  a r e the compensator r e a d i n g  t o l o a d (15).  The o b l i q u e i n c i d e n c e  ^  )  changes o c c u r r i n g  r e a d i n g s are  always a f t e r a l l the normal r e a d i n g s a r e completed  and  taken the  d i r e c t i o n s , o f the p r i n c i p a l s t r e s s e s a r e known. There a r e d i f f e r e n t ways t o e v a l u a t e  the r e t a r d a t i o n o r  f r i n g e v a l u e and hence the magnitude o f p r i n c i p a l The  first,  and l e a s t a c c u r a t e , i s the comparison o f the f r i n g e  c o l o u r t o a s t a n d a r d c o l o u r s c a l e which i s r e l a t e d through count  stresses.  calibration.  I f a black  the number o f s u c c e s s i v e  to s t r a i n  f r i n g e i s p r e s e n t , one  can  " t i n t o f passage" f r i n g e s . T h e  t i n t o f passage i s the c o l o u r o c c u r r i n g a t the t r a n s i t i o n from  8 red to blue.  This dull  purple colour i s very  sensitive  s m a l l changes i n the d i f f e r e n c e o f t h e p r i n c i p a l gives  accurate  stress  investigation l i e s  readings  on a t i n t  o n l y when  o f passage.  to  s t r e s s e s but  the p o i n t under  O f t e n the s t r e s s i s  not h i g h enough to produce s t r a i n s c o r r e s p o n d i n g  to  one  f r i n g e o r the p o i n t under i n v e s t i g a t i o n l i e s between s u c c e s s i v e f r i n g e s . I n t h i s case an o p t i c a l compensator  should  used, which produces a t i n t o f passage a t any p o i n t b i r e f r i n g e n t p l a s t i c , a t any v a l u e o f s t r a i n .  o f the  The  a c c u r a t e method o f d e f i n i n g f r a c t i o n s o f f r i n g e s  be  most  i s t o use  photometers. The  method o f a p p l i c a t i o n o f b i r e f r i n g e n t l a y e r s depends  on the shape o f the s t r u c t u r e required  ( 1 6 ) . F o r plane  and  also  on  the  accuracy  s u r f a c e s , p l a s t i c sheets are a v a i l -  a b l e i n v a r i o u s t h i c k n e s s . These s h e e t s a r e bonded s u r f a c e by means o f an a d h e s i v e .  to  the  To p r o v i d e a r e f l e c t i v e s u r -  f a c e a t t h e i n t e r f a c e o f the s t r u c t u r e and c o a t i n g , h e s i v e u s u a l l y i s mixed w i t h aluminum powder.  In  the adcase  s u r f a c e o f the metal i s ground t h i s i s n o t n e c e s s a r y  the  because  the d i f f u s e r e f l e c t i o n p r o v i d e d by the s u r f a c e , i s s a t i s f a c t o r y f o r the o b s e r v a t i o n o f the f r i n g e s . When a h i g h of accuracy  i s n o t r e q u i r e d the p l a s t i c c a n be  brushing l i q u i d p l a s t i c  degree  applied  by  on the s u r f a c e o f the s t r u c t u r e , and  p o l y m e r i z i n g i t by heat a p p l i e d t o the coated a r e a ( 1 7 ) . F o r q u a n t i t a t i v e a n a l y s i s the t h i c k n e s s o f the c o a t must be a l s o  measured a t the p o i n t under i n v e s t i g a t i o n . When the s u r f a c e i s c o m p l i c a t e d i t i s b e s t t o use contoured  contoured  sheets.  For  sheets the p l a s t i c i s c a s t on a g l a s s  plate  and  t a k e n from the g l a s s when p a r t i a l l y p o l y m e r i z e d  (18)..  In  t h i s s t a t e the p l a s t i c  sheet i s e a s i l y formed  without  i n t r o d u c i n g i n i t i a l b i r e f r i n g e n c e . The sheet i s moulded over the r e q u i r e d s u r f a c e , and i s l e f t p o l y m e r i z a t i o n i s completed the moulded p l a s t i c  on the s u r f a c e u n t i l  ( g e n e r a l l y 24 hours ).When h a r d ,  i s bonded t o the s u r f a c e o f the s t r u c t u -  r e w i t h a cement, as f o r the sheet p l a s t i c . When s t r i p i s s e t a s i d e f o r c a l i b r a t i o n purposes.  c a l c u l a t i o n . Measuring  casting, a  I t i s bonded t o  a t e s t b a r i n which the s u r f a c e s t r a i n s may be  obtained  the r e t a r d a t i o n i n the p l a s t i c ,  strain optical coefficient as  o f the p l a s t i c c a n be  by the  determined,  shown i n Appendix I I I , u s i n g the e q u a t i o n  * " 2tA£  ±  -e ) 2  F o r h i g h s t r e s s g r a d i e n t s or sharp c u r v a t u r e s , r i z i n g microscope  i s used  (Fig.3).  objective, a pair of crossed Nicols, and  the  This one  consists  a polaof  compensator wedge  one o c u l a r . The l i g h t source may be e i t h e r a t t a c h e d  the tube,  an  to  o r t o a s e p a r a t e s t a n d . The t o t a l m a g n i f i c a t i o n i s  about twenty t i m e s . The  a c c u r a c y o f the measurements taken w i t h the p o l a r i -  10 scope i s d e f i n e d by the a c c u r a c y  w i t h which the r e a d i n g s c a n  be made on the compensator. The chieved  compensation.''on the l a r g e f i e l d p o l a r i s c o p e  i s a-  by the q u a r t e r wave p l a t e method; the a n a l y z e r  i s ro-  t a t e d t o o b t a i n compensation, w i t h a q u a r t e r wave p l a t e between the a n a l y z e r and the p l a s t i c . A r o t a t i o n o f 180° caus\ es one wave l e n g t h r e t a r d a t i o n which c o r r e s p o n d s t o one f r i n g e . The  s c a l e c a n be r e a d a t every two degrees.Thus the s m a l l e s t  change which c a n be s t i l l  observed i s one  ninetieth  of  a  fringe. The  oblique  i n c i d e n c e p o l a r i s c o p e and the  microscope have a q u a r t z wedge compensator.  polarizing  For  this  d i v i s i o n s c a l e c o r r e s p o n d s t o one f r i n g e change. r e a d a t each h a l f g r a d u a t i o n  It  a 35 c a n be  and t h e r e f o r e the s m a l l e s t change  t o be n o t i c e d w i l l be one s e v e n t i e t h o f a f r i n g e . Up to t h i s p o i n t the r e i n f o r c i n g e f f e c t o f the was n o t i n c l u d e d i n the e q u a t i o n s d e t e r m i n i n g i n the s t r u c t u r e . An i n v e s t i g a t i o n by Zandman,  the  plastic stresses  Redner  R i e g n e r (21) showed t h a t f o r t h i n c o a t i n g i n plane  and 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 . I n case o f bending o r combined s t r e s s e s o r when the t h i c k n e s s s m a l l compared t o t h e t h i c k n e s s e f f e c t c a n n o t be n e g l e c t e d  o f metal  o f the this  coat  i s not  reinforcing  and must be accounted  f o r . The  i n f l u e n c e o f the c o a t on the s t r a i n s and so on b i r e f r i n g e n c e  11 i s g i v e n 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 f u n c t i o n o f the ratio  of p l a s t i c  t h i c k n e s s to metal t h i c k n e s s  mentioned r e f e r e n c e . T h i s c o r r e c t i o n f a c t o r i s important  when t h i n p l a t e s are s u b j e c t e d  in  the above especially  to bending.  12 D e s c r i p t i o n o f the P r e s s u r e  V e s s e l and  Equipment Used.  The s t r u c t u r e under i n v e s t i g a t i o n was a s m a l l v e s s e l obtained  pressure  c o m m e r c i a l l y ( F i g . 4) • Two t o r i s p h e r i c a l d i s h e d  heads were welded to a c y l i n d e r , which was r o l l e d s t e e l p l a t e and welded a l o n g  the g e n e r a t i n g  was complete; the o t h e r c o n t a i n e d  from  l i n e . One  two tapped h o l e s  a head  for 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 p h 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 t o  a v o i d r e i n f o r c i n g a t the j o i n t o f t h i s head t o  the c y l i n d e r , the weld was ground f l u s h to the p a r e n t m a t e r i a l s o u t s i d e and i n s i d e . Care was t a k e n a l s o t h a t the c y l i n d e r was  s u f f i c i e n t l y l o n g so t h a t the o t h e r  end c l o s u r e  had no  e f f e c t on the s t r e s s e s i n the head under i n v e s t i g a t i o n . The m a t e r i a l o f the v e s s e l was m i l d s t e e l w i t h a s t r e n g t h o f 30000 p s i .  The  modulus  of  elasticity  yield was  30 x 1 0 p s i . 6  The w a l l t h i c k n e s s , one e i g h t h o f an i n c h , was i n the heads and the c y l i n d e r , and s i n c e torus  ( k n u c k l e ) p a r t o f the head was  v e s s e l c o u l d n o t be c o n s i d e r e d the r a d i u s o f c u r v a t u r e  uniform  the r a d i u s o f the  s m a l l , t h i s p a r t o f the  as a t h i n s h e l l . The r a t i o o f  t o the w a l l t h i c k n e s s was  |.84  the r a t i o f o r t h i n s h e l l s i s d e f i n e d as above i s t e n .  and The  13 other  p a r t s , the s p h e r i c a l cap and the c y l i n d e r ,  s h e l l s having a radius of curvature and  to thickness  were  r a t i o o f 64  32 r e s p e c t i v e l y . To make t h e p h o t o e l a s t i c measurements e a s i e r , t h e  was  thin  mounted on a stand and p r o v i s i o n s was made so  that  c o u l d he r o t a t e d about an a x i s a t the mid l e n g t h cylinder Due  vessel  of the  (Pig.6,7). t o the double c u r v a t u r e  o f t h e head o f t h e p r e s s u r e  v e s s e l i t was n e c e s s a r y t o use the moulding  technique  to c o v e r i t w i t h a p l a s t i c l a y e r , as d e s c r i b e d A p a r t i a l l y p o l y m e r i z e d p l a s t i c sheet was  showed no r e s i s t a n c e a g a i n s t  obtained  forming and was formed  head w i t h o u t i n t r o d u c i n g i n i t i a l b i r e f r i n g e n c e . p l a s t i c hardened, i t was cemented t o t h e head r e s i n cement  on the  When by an  the'y epoxy  (19).  s t r a i n o p t i c a l c o e f f i c i e n t o f the  plastic  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 o f the c a l i b r a t i o n  was obcast  i s given  t h e Appendix. The  in  by  sheet  t a i n e d by c a l i b r a t i o n u s i n g a c a l i b r a t i o n s t r i p from the  in  (17)  i n Appendix I I .  c a s t i n g l i q u i d p l a s t i c on a l e v e l g l a s s s u r f a c e . T h i s  The  i t  p r e s s u r e v e s s e l was f i l l e d w i t h water w h i c h h a d been  t h e open a i r b e f o r e  f i l l i n g to release  A h i g h p r e s s u r e rubber hose p r o v i d e d  the absorbed a i r .  the connection  the p r e s s u r e v e s s e l and a dead weight t e s t e r ( P i g . 6 ) , produced t h e r e q u i r e d p r e s s u r e .  Before applying  between which  any p r e s s u r e ,  14 the system was "bled o f a i r a t the h i g h e s t The  point.  dead weight t e s t e r remained connected t o the p r e s s -  ure v e s s e l throughout the t e s t s and i t s p i s t o n to secure u n i f o r m p r e s s u r e  while  was  rotated  the p h o t o e l a s t i c  measure-  ments were taken. A l l measurements were taken a t 500 p s i g . The  p h o t o e l a s t i c p a r t o f the experiment  the i n v e s t i g a t i o n o f a s e t o f p o i n t s a l o n g  consisted a  radial  from the c e n t e r o f the head t o the c y l i n d e r , d e f i n e d CA (Pig.4).  angle from  a  Measurements were t a k e n a t every  ( X = 0° t o  0(= 20°  = 20°  a = 90°  to  and a t every f o u r  Instruments used were the l a r g e f i e l d  line by the degree  degrees  from  polariscope,which  was used t o determine the d i r e c t i o n s and the d i f f e r e n c e the p r i n c i p a l s t r e s s e s (Fig.7)>  the o b l i q u e i n c i d e n c e  r i s c o p e , which s u p p l i e d t h e v a l u e s cipal  of  of pola-  f o r the i n d i v i d u a l  prin-  s t r e s s e s ( F i g . 8 ) , and t h e p o l a r i z i n g microscope, which  provided  control values  large f i e l d polariscope  t o check the v a l u e s  obtained  by t h e  (Fig.9).  When c a l c u l a t i n g the s t r e s s e s bending had t o be i n t r o d u c e d ,  a  correction factor f o r  because bending moment e x i s t e d  i n the t o r u s due t o the i n t e r n a l p r e s s u r e .  The  f a c t o r was o b t a i n e d  (21)  from F i g 2 i n r e f e r e n c e  e x i s t i n g r a t i o o f p l a s t i c t o metal  thickness.  correction for  the  15 Results.  The  d i r e c t i o n s o f t h e p r i n c i p a l s t r e s s e s were e s t a b l i s h -  ed f i r s t w i t h t h e use o f the l a r g e the  f i e l d polariscope  q u a r t e r wave p l a t e s . I t was found t h a t w i t h i n  without  a  circle  drawn a t (X = 32° around the a x i s o f t h e s h e l l on the s p h e r i cal  c a p , every p o i n t 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 e q u a l i n a l l d i r e c t i o n s . A n o t h e r dark r i n g appeared i n the f i e l d riscope  a t (X = 10° where the p o i n t s  of  proved t o  be  which meant t h a t b o t h p r i n c i p a l s t r e s s e s became At t h e o t h e r p a r t s  o f t h e head  a  stresses  that  the  were m e r i d i o n a l  (Fig.10).  direction  pola-  singular,  zero.  r a d i a l dark  c o i n c i d i n g always w i t h t h e p o l a r i z i n g a x i s o f plate indicated  the  of  the  line polarizer  the p r i n c i p a l  and c i r c u m f e r e n t i a l  respectively  The m a t h e m a t i c a l l y 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= 1 0 ° ; and changed to m e r i d i o n a l  between  CX= 1 0 ° and  d  = 32°  These o b s e r v a t i o n s l e d t o the c o n c l u s i o n d i o n a l and c i r c u m f e r e n t i a l the  t h a t the  meri-  stresses along a r a d i a l l i n e  were  p r i n c i p a l s t r e s s e s , and so the maximum o f m e r i d i o n a l  circumferential stresses As  s t r e s s e s were determined, g i v i n g  and  the maximum  a t the p o i n t . part  o f the q u a n t i t a t i v e  a n a l y s i s normal and  oblique  16 i n c i d e n c e measurements were  taken.  When t h e p l a s t i c was viewed w i t h t h e l a r g e f i e l d  pola-  riscope set f o r c i r c u l a r l y polarized l i g h t field,three  areas  showed dark p a t t e r n s . One a r e a was w i t h i n t h e c i r c l e  drawn  at  (X = 32° around t h e a x i s o f the s h e l l on  cap.  the s p h e r i c a l  The o t h e r was a r i n g a t OC = 1 0 ° , and the t h i r d a n o t h e r  ring at field,  oi = 0 ° .  these  principal  Due t o t h e c i r c u l a r l y p o l a r i z e d  dark areas  light  i n d i c a t e d t h a t the d i f f e r e n c e o f the  s t r e s s e s was zero t h e r e . F a r t h e r i n v e s t i g a t i o n how-  ever showed t h a t the p o i n t s on the r i n g a t  ct  = 10°  s i n g u l a r , and t h e p o i n t s on the s p h e r i c a l cap w i t h i n c i r c l e a t the angle Oi  = 0° were The  Oi = 32° and the p o i n t s  f r i n g e v a l u e i n c r e a s e d from  on the r i n g a t  Oi = 0 ° , t o zero  reached i t s  at  i n c r e a s e d a g a i n from  to  zero a t Oi = 3 2 ° . The d i f f e r e n c e o f the p r i n c i p a l  Oi  oi = 10° t o Oi =17°  OC = 1 0 ° .  It  not reach  the  isotropic.  maximum a t OK= 5»5°» t h e n decreased  did  were  one f r i n g e a t any p o i n t from  and  Oi  decreased strains  = 0°  to  = 90° which means t h a t the r e t a r d a t i o n measured was a l -  ways l e s s than one wave l e n g t h o f , t h e white Oblique  i n c i d e n c e measurements were made a f t e r the d i -  r e c t i o n o f the p r i n c i p a l large f i e l d  light.  s t r e s s e s was determined  p o l a r i s c o p e . The plane  with  the  o f i n c i d e n t and r e f l e c t e d  l i g h t o f the o b l i q u e i n c i d e n c e p o l a r i s c o p e was f i r s t a l i g n e d to  c o i n c i d e w i t h a m e r i d i o n a l l i n e and the  readings  so ob-  17 t a i n e d were u s e d t o 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 a l i g n e d vious  perpendicular  pre-  d i r e c t i o n t o s u p p l y the v a l u e s f o r the c a l c u l a t i o n  the m e r i d i o n a l  of  s t r e s s e s . Each measurement was t a k e n when the  p r e s s u r e v e s s e l was l o a d e d and a g a i n when i t The  t o the  was  unloaded.  s i g n o f t h e compensator v a l u e s was d e f i n e d 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 the a l g e b r a i c reading  . s u b t r a c t i o n o f the no l o a d r e a d i n g  and from  by the  t a k e n under l o a d .  A d i m e n s i o n l e s s s t r e s s i n t e n s i t y f a c t o r was  d e f i n e d as  the r a t i o o f t h e s t r e s s measured t o 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 c l o s u r e s Stress Intensity = —  where  C T = s t r e s s measured p = p r e s s u r e i n the v e s s e l R=  radius  t =  wall thickness  Q  Q  This  o f the c y l i n d e r o f the c y l i n d e r  s t r e s s i n t e n s i t y f a c t o r i s g i v e n w i t h the s t r e s s v a l u e s  a t each p o i n t were d e f i n e d  on the head i n Table I . The p o i n t s by the angle  on the head  OC , but the c o r r e s p o n d i n g  f o r the a n g l e (£> were a l s o i n c l u d e d ,  because i t was  t h a t t h i s a n g l e was more o f t e n used i n the l i t e r a t u r e .  values found  18 The  s t r e s s e s were a l s o p l o t t e d on l i n e s  t o the head a t each p o i n t tem  o f angle OC v e r s u s The  maximum  perpendicular  ( P i g . 1 1 ) , and i n a c o o r d i n a t e  sys-  the stresses (Pig.12).  meridional  stress  of  o c c u r r e d i n the t o r u s p a r t o f the head a t  O^j CK  = -48500 p s i =5.5°,  while  the maximum c i r c u m f e r e n t i a l s t r e s s r e a c h e d i t s maximum v a l u e of  Q~  6  = -33500 p s i a t  OC = 5 ° . Both  p r e s s i v e on t h e o u t e r s u r f a c e o f the head.  s t r e s s e s were com-  19 Table I.  M e r i d i o n a l and  C i r c u m f e r e n t i a l S t r e s s e s i n the Head  o f the P r e s s u r e V e s s e l and  the  Corresponding  Stress Intensity Factors.  Points by  a 0.0  defined angles  4> 90  1.0 1.2  80  2.0 2.4  70  3.0 3.5  60  4.0 4.5  50  5.0 5.5  40  6.0  S t r e s s e s 10 Circumferential  psi  Meridional  Stress Intensity Factors Circumferential  Meridional  -2.00  -2.00  -1.25  -1.25  -2.65  -2.90  -1.66  -1.81  -2.70  -3.05  -1.69  -1.91  -2.95  -3.50  -1.84  -2.19  -3.00  -3.75  -1.88  -2.34  -3.20  -4.05  -2.00  -2.53  -3.25  -4.30  -2.03  -2.69  -3.30  -4.50  -2.06  -2.81  -3.30  -4.70  -2.06  -2.94  -3.35  -4.78  -2.10  -2.98  -3.30  -4.85  -2.06  -3.03  -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  -2.50  -3.65  -1.56  -2.28  -2.35  -3.30  -1.47  -2.06  -2.10  -2.65  -1.31  -1.66  -1.50  -1.50  -0.94  -0.94  8.0 8.4  27  9.0 10.0  26  20 Table I .  Points by  defined angles  a 11.0  25 24  13.0 14.0  23  15.0 15.4  22  16.0 17.0  21  18.0 18.6  20  19.0  Stress Intensity  4 S t r e s s e s 10 Circumferential  12.0 12.6  (continued)  psi  Meridional  Factors CircumMeridional ferential  -0.85  0.00  -0.53  0.00  -0.20  1.00  -0.13  0.63  0.10  1.50  0.06  0.94  0.55  1.80  0.34  1.12  0.70  2.60  0.44  1.63  0.95  2.75  0.59  1.72  1.05  2,85  0.66  1.78  1.20  2.95  0.75  1.84  1.30  3.10  0.81  1.94  1.40  3.15  0.88  1.97  1.41  3.10  0.88  1.94  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  1.30  1.65  0.81  1.03  28.0 30.0  15  1.15  1.40  0.72  0.88  32.0  14  0.97  1.18  0.61  0.74  0.86 • • • 0.71 • • • 0.57  0.94 •  0.54 • •  0.59 • • • 0.48 • • • 0.37  36.0 • • • 54.0 • • • 90.0  • • • • 7 • • • • 0  *  • 0.76 t • *  0.59  *  0.44 • • • 0.36  21 Comparison w i t h o t h e r works on p r e s s u r e v e s s e l heads.  I n o r d e r t o a s s e s s the a c c u r a c y o f the p r e s e n t  stress  i n v e s t i g a t i o n , the r e s u l t s were compared to v a l u e s  obtained  i n the l i t e r a t u r e . These v a l u e s were a r r i v e d a t by  calcula-  t i o n s , u s i n g the t h i n s h e l l t h e o r y , and  the dimensions  of  the v e s s e l s a t i s f i e d the c o n d i t i o n s o f t h i n s h e l l s . T h e v e s s e l i n the p r e s e n t 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 .  f o r e the d i s t r i b u t i o n o f the s t r e s s e s and  There-  the maximum s t r e s s  i n t e n s i t y f a c t o r s were compared. The heads i n two  o f the work  t o r i s p h e r i c a l and one was The  used as comparisons  were  ellipsoidal.  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  Kraus, B i l o d e a u and Langer ( 2 4 ) .  published  by  R e s u l t s were p r e s e n t e d  t h i s paper f o r an e l l i p s o i d a l head h a v i n g a two  t o one  in  ratio  o f the major and minor axes. S t r e s s i n t e n s i t y i n d e x e s were t a b u l a t e d f o r v e s s e l s determined  by parameters  T and  r a t i o o f the major a x i s to the minor  Where  the e l l i p s o i d a l head D  diameter  t  cylinder thickness  T  head t h i c k n e s s  of c y l i n d e r  a s e r i e s of . axis  of  22 For  comparison j] i n t h e p r e s e n t i n v e s t i g a t i o n was t a k -  en as t h e r a t i o o f t h e r a d i u s o f the head.  o f the c y l i n d e r t o  The maximum s t r e s s 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 k n u c k l e and 2.15 cap.  the h e i g h t  The parameters were  /) /j = 3  D r£ = 50  i n the s p h e r i c a l T ^ = 1  p r e s e n t i n v e s t i g a t i o n these parameters were  In the  ^3=3  ^ = 64  T ^  = 1  3.00  and t h e maximum s t r e s s i n t e n s i t y  i n t h e knuckle was  and i n t h e s p h e r i c a l cap 2.28. The  papers d e a l i n g w i t h t h e t o r i s p h e r i c a l  heads  were  p u b l i s h e d by G a l l e t l y (22,23). One o f these papers ( 2 3 ) , shows t h a t t h e ASME Code f o r Unfired torus.  P r e s s u r e V e s s e l s g i v e s v e r y low s t r e s s v a l u e s i n the 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  o f 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 revolution,  The r e s u l t s show the d i s t r i b u t i o n o f the m e r i d i o n a l  bending s t r e s s e s the  torus,  the  material. The  and the c i r c u m f e r e n t i a l  and t h a t  both s t r e s s e s  direct stresses  in  exceeded the y i e l d p o i n t o f  comparison o f t h e s e r e s u l t s t o the r e s u l t s o b t a i n e d  i n the present i n v e s t i g a t i o n  (Pig.13) a r e made on the  base  o f the s t 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 s t r e s s i n t e n s i t y f a c t o r , and f o r t h e s i g n o f the stresses. The  dimensions o f t h e head were  23 Radius of s p h e r i c a l cap  227.5  in  Radius o f the c y l i n d e r  138.23  in  T h i c k n e s s o f the head  0.625 i n  T h i c k n e s s o f the c y l i n d e r  0.460 i n  Pressure The  second paper (22)  ure v e s s e l s of influence  t o r i . The  stresses torus  dimensions o f the head were: Radius of the  s p h e r i c a l cap  281.25  in  Radius o f the  cylinder  150  in  T h i c k n e s s o f the head  0.625 i n  T h i c k n e s s of the  0.5  cylinder  s t r e s s due  psig.  to i n t e r n a l p r e s s u r e i n a  by u s i n g  the ASME Code f o r U n f i r e d e q u a t i o n used CT=  Q~=  in  60  a l s o determined i n the  where  use  (Pig.13).  head was  The  the  the  press-  a l s o compared to the v a l u e s p r e s -  Pressure The  examples i n which  stress distribution i n  t h i s was  obtained i n The  two  c o e f f i c i e n t s . I t showed a l s o t h a t h i g h  p r e s e n t e d and  ently  gives  psig.  o f d i f f e r e n t shapes were d e s i g n e d w i t h  o c c u r r e d i n the was  60  presently  investigated vessel  Pressure  Vessels.  was £  ( 0.5  s t r e s s i n the  M f- + 0.1 wall  p = i n t e r n a l pressure  torispherical  )  24 R = inside radius  o f the s p h e r i c a l  cap  t = t h i c k n e s s o f the head 7| = j o i n t e f f i c i e n c y M = 1.77, is The s t r e s s  (unity)  when the r a d i u s  6 % o f the r a d i u s  of  the k n u c k l e  of spherical  cap.  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 The d i s t r i b u t i o n curve was  1.78.  f l a t t e r f o r the p r e s e n t  v e s t i g a t i o n which was a t t r i b u t e d t o the w a l l t h i c k n e s s the  torus.  inof  25 Summary  The  and  Conclusions.  p h o t o e l a s t i c c o a t i n g method p r o v i d e d  for evaluating pressure  a new approach  the s t r e s s e s i n the t o r i s p h e r i c a l  head o f a  v e s s e l . The coat a p p l i e d t o the s u r f a c e made p o s s i -  b l e the d e t e r m i n a t i o n  o f the s t r a i n s  on the  s t r u c t u r e . I t a l s o supplied the d i r e c t i o n s  surface  o f the  o f the p r i n c i p a l  s t r e s s e s . I n t h i s c a s e they were c i r c u m f e r e n t i a l  and  me-  ridional. The  maximum v a l u e  and the l o c a t i o n  of  t h e s t r e s s e s as  w e l l as t h e i r s i g n was determined and these c l o s e l y w i t h the v a l u e s  obtained  i n other  values  agreed  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 p r e s e n t i n v e s t i g a t i o n t h a n t h e d i s t r i b u t i o n i n t h e compared T h i s was a t t r i b u t e d t o the t h i c k e r t o r u s i n  the  works. pressure  vessel investigated. The  maximum v a l u e  o f the s t r e s s e s exceeded  point o f the m a t e r i a l , although during  no  yielding  the  yield  was n o t i c e d  the t e s t . I t i s b e l i e v e d that previous  unsuccessful  t e s t i n g caused work h a r d e n i n g i n the m a t e r i a l . The  thickness  o f the p l a s t i c  determined the s e n s i t i v i t y  o f the measurements and i n the p r e s e n t A thicker coating increases  case i t was q u i t e low.  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 a l s o i n c r e a s e s . The c a s t i n g and f o r m i n g the t h i c k e r sheet i s a l s o more  difficult.  of  When l i q u i d p l a s t i c  i s east,  quite large  quantities  s h o u l d he mixed w i t h the a c c e l e r a t o r to o b t a i n polymerization. tal; non  a little  The  a  amount o f a c c e l e r a t o r added  uniform i s very v i -  d e v i a t i o n from the r e q u i r e d p r o p o r t i o n  causes  u n i f o r m sheet f o r m a t i o n . T h i s , and  f a c e s c o u l d he  the  forming of t h i c k coatings  on c u r v e d  i n v e s t i g a t e d i n a s e p a r a t e work. P h o t o e l a s t i c  measurements on a s h e l l o f r e v o l u t i o n i n which were o b t a i n e d a n a l y t i c a l l y would be a l s o In conclusion  the  stresses  i n s t r u m e n t s used can be  the  to  in  be a  the  v i d e d an e l e c t r i c o u t l e t i s a v a i l a b l e . The equipment and  very  pressure  e a s i l y c a r r i e d and  p h o t o e l a s t i c i n v e s t i g a t i o n i s p o s s i b l e even on the  structure requires l i t t l e  stresses  valuable.  the c a s t i n g method proved  u s e f u l method f o r e v a l u a t i n g v e s s e l . The  sur-  thus  site,pro-  coating  a s h o r t time  of  the  only.  27 References.  1., Mesnager, M., "Sur l a d e t e r m i n a t i o n o p t i q u e des t e n s i o n s i n t e r v e n u e r 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., " A p p l i c a t i o n s d e 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 P o l a r i z a t i o n s o p t i s c h e S c h i c h t v e r f a h r e n zur Messung d e r Oberflachenspannung am beanspruchten Baut e i l ohne M o d e l l , " 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 a n a l y s i s o f p l a s t i c b e h a v i o u r o f m e t a l s w i t h bonded birefringent plastic," P r o c e e d i n g s o f The S o c i e t y for E x p e r i m e n t a l S t r e s s A n a l y s i s . V o l . X I I . 2, 115-122.(1955) 5., D'Agostino, J . , Drucker, D.C, L i u , O.K. and Mylonas, C, "Epoxy a d h e s i v e s and c a s t i n g resins as Photoelastic plastics" Proceedings o f The S o c i e t y f o r Experimental S t r e s s A n a l y s i s . V o l . X I I . 2, 123-128. (1955) 6., Zandman. P., "Analyse des c o n t r a i n t e s par v e m i s photoelastiques," Groupment 1'Advancement Methodes d'AnalC o n t r a i n t e s V o l . 2, No. 6 1-12 (1956) 7., Zandman, P., "Mesures p h o t o e l a s t i q u e s des deformations e l a s t i q u e s e t p l a s t i q u e s e t des f r a g m e n t a t i o n s c r i s t a l l i nes dans l e s metaux," Revue de M e t a l l u r g i e V o l . L I I . 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( P h o e n i x v i l l e ) No, 8005.(1958)  13.,  M i n d l i n , R.D., " D i s t o r t i o n o f the p h o t o e l a s t i c fringep a t t e r n i n an o p t i c a l l y u n b a l a n c e d polariscope." ASME T r a n s . V o l . 59. No. A 170 (1937)  14.,  Timoshenko, S., G-oodier, J.N., Theory of Elasticity E n g i n e e r i n g S o c i e t i e s Monograph., McGraw-Hill Book Co. I n c . New York. Second ed. (1951)  15.,  McMaster, R.,(ed) N o n d e s t r u c t i v e T e s t i n g Handbook. York. The Ronald P r e s s Co., (1959) V o l I I .  16.,  " O p e r a t i n g i n s t r u c t i o n s f o r t h e Oblique I n c i d e n c e 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 ( P h o e n i x v i l l e ) No. BN-8022 (1959)  18.,  "Instructions f o r Applying Photostress L i q u i d Plastic" T a t n a l l Measuring System Co. 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Journal of Engineering f o r Industry. T r a n s . Ser. B. V o l . 81. 51-66 (1959)  24.,  K r a u s , H., B i l o d e a u , G.G., Danger, B.F., " S t r e s s e s 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 T r a n s . S e r . B. V o l . 83. 29-42 (1961)  New  t o deASME  29 Appendix I .  The M a t h e m a t i c a l Theory.  Normal I n c i d e n c e .  The c o n n e c t i o n between t h e p r i n c i p a l refringent material ordinary  and t h e r e t a r d a t i o n o b t a i n e d between the  and e x t r a o r d i n a r y  rays of a polarized  p r e s s e d i n Neumann's law f o r plane  <5 «0 CT^;  G~2  a  £>  n  G t Rewriting  r  e  '^  xe  P i r  relative  n  c  i  P l  l i g h t i s ex-  stresses.  ( 0 ^ - 0 ^ 2  a  Where  stresses i n a b i -  1.  t  s t r e s s e s i n the p l a s t i c  a  retardation  stress optical  coefficient  thickness  to give  the d i f f e r e n c e o f the p r i n c i p a l  stresses 2.  C 2 t  The d i f f e r e n c e o f the p r i n c i p a l s t r e s s e s c a n a l s o  be  e x p r e s s e d as  (CT-,1  " ^ p " 1 +V. XT  (  €  1  "€ 2  }  p  3.  30. Where  E = Modulus o f 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 l  Strains  a  Combining e q u a t i o n s 2 and 3  5  E  C 2 t  1 + \/ ^  1  c  2 p ;  4  *  or &  (1 + V-)  n  Denoting CE„ 1  e q u a t i o n 4/a becomes  (€ Where  -  x  &  n  2 t K  =  5  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 s u a l l y determined  *  and i s u -  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 c o a t and s t r u c ture  (E and the d i f f e r e n c e  1  -€ ) 2  p  = ( £  x  o f the p r i n c i p a l  - €  2  )  w  strains  5/a  i n the s t r u c t u r e  31 i s g i v e n by  (Gi -  £  2  K  " 2 t K  6  *  To f i n d t h e d i f f e r e n c e o f the p r i n c i p a l s t r e s s e s Hooke's law i s u s e d  (crl -cr) ^ p l  U  cS  E w  7  n  1 + V  w  2 t K  '*  T h e r e f o r e t h e d e t e r m i n a t i o n o f the r e l a t i v e r e t a r d a t i o n provided  the d i f f e r e n c e o f the  principal  s t r u c t u r e . From t h i s the maximum shear  f  max  stress  is  i n the obtained  2  The i n d i v i d u a l p r i n c i p a l s t r e s s c a n w i t h one normal r e a d i n g  stresses  be o b t a i n e d  however, when t h e p o i n t 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. these p o i n t s t h e normal s t r e s s must be zero and gives  also  At  equation  7  the t a n g e n t i a l s t r e s s . Two e q u a t i o n s a r e n e c e s s a r y t o d e f i n e  t h e two i n d i v i d u a l  s t r e s s e s a t p o i n t s n o t on a f r e e s u r f a c e . These two  equa-  t i o n s c a n be o b t a i n e d by t a k i n g two o b l i q u e  o r one  normal  and one o b l i q u e  reading.  readings  32  The  Oblique Incidence.  As  seen from t h e p r e v i o u s d i s c u s s i o n , when the i n c i d e n c e  o f the l i g h t i s normal, o n l y the d i f f e r e n c e o f the p r i n c i p a l s t r e s s e s c a n be o b t a i n e d . lique incidence,  Taking another  reading  under ob-  the r e t a r d a t i o n observed w i l l be the d i f f e r -  ence o f the secondary p r i n c i p a l s t r e s s e s , thus g i v i n g a n o t h e r e q u a t i o n f o r s o l v i n g the two p r i n c i p a l s t r e s s e s i n d i v i d u a l l y .  As  the c o o r d i n a t e  systems show, i t c a n be always a r r a n g -  ed t h a t . o n e o f the p r i n c i p a l s t r e s s e s c o i n c i d e s w i t h one a x i s o f t h e secondary p r i n c i p a l s t r e s s e s . S i n c e the d i r e c t i o n s o f the p r i n c i p a l s t r e s s e s c a n be o b t a i n e d w i t h a dence p o l a r i s c o p e ,  the  oblique  incidence  a l i g n e d w i t h one o f these d i r e c t i o n s , thus  normal  inci-  instrument can including  be  one  p r i n c i p a l s t r e s s i n the d i f f e r e n c e o f the secondary p r i n c i p a l  33 stresses.  S t a r t i n g w i t h Neumann's e q u a t i o n a g a i n and u s i n g  Mohr's c i r c l e  drawn f o r a t h r e e dimension  case we can w r i t e  the e q u a t i o n s f o r the r a t a r d a t i o n t a k i n g two o b l i q u e  inci-  dence measurements  (CT-, - O V )„ = 1 " 2 p ~ 2 t / cos u  (  where  (S  q1  °~2  " °I  (SO2  and  ;  }  ol  Q±  5 o2  p ~ 2 t / cos 9  10,  C  2  11,  C  a r e the r e t a r d a t i o n s measured i n  l i q u e i n c i d e n c e the instrument b e i n g a l i g n e d w i t h r e c t i o n o f the p r o p e r p r i n c i p a l s t r e s s .  ob  the d i 0~ '  o  2  be d e f i n e d from Mohr's c i r c l e as shown  (CT ') 2  p  = (CT  2  cos  2 Q  l  )  p  C O  l'>p  =  (G  ~1  G  o  s  2  e  2>p  a  n  34 t h e r e f o r e e q u a t i o n 10, and 11 can be expressed i n terms  of  the p r i n c i p a l s t r e s s e s as f o l l o w s  (cr -ar  2  oos  (CTg-C^OOS  S o l v i n g these simultaneous  (CT)  (  C  *  r ) 2  * *  2  2  s  S  9 ) 2  °  p  t  c o s  c  c  o  s  2  0  and  CT"  2  e  14.  2  ^  S cos 9 2 t / cos 9, C 2  Ql  C  0  9  +  2  2  g  ±  c o s * 9_  15.  ]  Great s i m p l i f i c a t i o n can be i n t r o d u c e d by c h o o s i n g angle o f i n c i d e n c e f o r b o t h cases a t 9-^ = 9 which e q u a t i o n s 14 and 15 become reduced  ^1 P ;  T T T  -  13.  ^02 i C 2 t / cos 9-, C ^ ± 9 cos  ^o2 2 t / cos Q 1 - cos  2  O"^  ,  Q  o2  °e  1 2  +  1 - cos  =  2  /  equations f o r  ^ol 2 t / cos  =  c  i>p = 2 t / co3°i  (  °ol  +  T~>  2  = 45°  the  with  to  1 6  *  35  6\  ( CT ) ~= 3 t C (o&2 '2'p v  0  w  + ^ )  Q  17,  Knowing from the normal incidence d e r i v a t i o n that C  K =  1  E. + V /  P  and from this. K (1 + l/^)  C =  E  P  S u b s t i t u t i n g t h i s value of C i n t o equations 16 and 17 assuming that ( £  1  ) = C £ p  1  )  w  and  V  =  Prom  and  Hooke's  law \[2  <°"2>» "  3  t  E  c5  ~  V J  (  5  02  «•  +  As stated e a r l i e r i t i s not necessary that the oblique incidence reading be taken to determine the i n d i v i d u a l p r i n c i p a l stresses. One oblique reading with the normal reading would be s u f f i c i e n t . I f an oblique  reading  i s taken f o r a  r e t a r d a t i o n value i n c l u d i n g the p r i n c i p a l stress equation become  CT-^, the  36  "° P p = 2~ro (  c  r  l  " cr ' ) 2  p  - ( O i -cr  2  cos  2 S  l  )  Solving these equations f o r determining  p  =  -  2o  2  j f ^  t  and  the  c  2 1  -  equations  the stresses i n the structure w i l l be t cr ^ °i'w  " t  ( a  2 w  " t K (1 + v > " 2 " °ol "  }  w ,Vg (l + vJ) ~  £ j§*u °oi ~ ~  c  E  K  {  P 9 2 2  «  }  °n  (  S i m i l a r l y the equations determining  the  }  '  2 3  stresses  in  the structure when the oblique reading "includes the p r i n c i p a l stress  are  < ° i > w " t K (1 T v/J ^  ^02 " n>  E  frrM (  G^'w  E  w  ~ t K ( l + V,)  §  ,V2£ {  —  °o2  determined from the equation  *'  &ns _  ~  -  ?(  2 5  }  The r e t a r d a t i o n i n the case of normal incidence l i g h t was  2  of  '  where  and  k  i s any p o s i t i v e number (0,1,2,...)  y5  i s the compensator r e a d i n g  /\  i s the wave l e n g t h o f l i g h t used  i n the case o f o b l i q u e  incidence  ^ol where  =  ~33~  m^ i s the d i f f e r e n c e o f the compensator r e a d i n g s under l o a d and no Equations  18 and 19 a r e expressed  i n t r o d u c i n g the n u m e r i c a l  values  white l i g h t and P o i s s o n ' s  r a t i o as  A=  2.27 x 1 0 "  V/ =  0.3  5  i n a s i m p l e r form by  f o r the wave l e n g t h o f the  in  -7 w 2.36 x l O ' ^  ,„  E  <Oi> « w  (CT ) 2  = 2.36 x I O " 7  W  load  (  J  m  . 2, - f ) m  i  +  - (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 c o v e r the head w i t h the p h o t o e l a s t i c p l a s t i c , a t o u r e d sheet had  to he used. The  sheet was  by/ u s i n g a p a r t i a l l y p o l y m e r i z e d p l a s t i c t i a l l y polymerized  sheet was  formed on the head  sheet.  This  from s t i c k i n g  toelastic hardener  baked  inside d i -  put on the g l a s s and s e a l e d w i t h masking  p l a s t i c was  tape  grams o f l i q u i d  mixed w i t h f i f t e e n per c e n t by  phoweight  and a l l o w e d t o r e a c h an exotherm temperature  110° j?. I t was  then poured  to  hardboard  q u a r t e r of an i n c h t h i c k , and 9" x 9"  around i t s e x t e r n a l p e r i m e t e r . Seventy  was  was  t o the g l a s s , a n d was  on the g l a s s i n an oven a t 450°F f o r f o u r h o u r s . A  mensions was  on  the s u r f a c e o f which  p r o t e c t e d by a l a y e r of s i l i c o n e v a r n i s h . T h i s l a y e r  frame, one  par-  o b t a i n e d by c a s t i n g p l a s t i c  an a c c u r a t e l y l e v e l l e d g l a s s p l a t e ,  p r e v e n t the p l a s t i c  con-  onto the prepared g l a s s  of  surface,  where the p o l y m e r i z a t i o n began. A f t e r t h r e e and a h a l f hours a t room temperature t h i c k n e s s , which was  the p l a s t i c  v e r y s o f t . The  the head, c o v e r i n g more than one gence was  formed a sheet of sheet was  then formed  t h i r d of i t .  i n t r o d u c e d w h i l e f o r m i n g . The  uniform  edges  No  birefrin-  of  the  were h e l d i n p o s i t i o n w i t h l i g h t l y a p p l i e d s c o t c h tape, left  f o r c o m p l e t i o n o f the p o l y m e r i z a t i o n which  on  sheet and  required  39 about  24 hours.. When the p o l y m e r i z a t i o n o f the p l a s t i c was  the c o n t o u r e d sheet was ness measured t o one micrometer.  The  completed,  removed from the head and i t s t h i c k -  t e n thousandth  o f an i n c h ,  with  a  edges were c u t and the s u r f a c e c l e a n e d  with  acetone. A reflective tic  used t o bond the p h o t o e l a s -  sheet t o the s u r f a c e o f the head. The  w i t h the hardener, set  type cement was  cement was  t e n p e r c e n t by weight,  f o r t e n minutes or more. A l a y e r o f about  o f an i n c h was  a p p l i e d . A i r bubbles were  a p p l y i n g the sheet a t an a n g l e to the  the o t h e r end. The  one s i x t e e n t h  p r e s s e d out  used  b r a t i o n bar.  by  out w i t h the a i r  edges o f the p l a s t i c were s e a l e d  the r e m a i n i n g cement. The bond hardened i n about  dure was  to  s u r f a c e and g r a d u a l l y  l o w e r i n g so t h a t the excess cement squeezed  the coat: was  allowed  then spread on the head from the mixed cement  and the sheet was  at  and  mixed  a day,  with and  ready f o r the p h o t o e l a s t i c t e s t . T h e same p r o c e to bond the c a l i b r a t i o n s t r i p  to  the  cali-  40 Appendix I I I .  D e t e r m i n a t i o n o f 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 t h i c k n e s s  o f 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 h a r d e n i n g o f the bond took p l a c e  bar.  a t room temperature and com-  p l e t e d i n twenty f o u r h o u r s . B e f o r e  the  calibration  s t a r t e d , the p l a s t i c was examined w i t h a p o l a r i s c o p e was observed t h a t no i n i t i a l b i r e f r i n g e n c e i n g the t e s t bar.  e x i s t e d before load-  One end o f the b a r was then  A f t e r the l a r g e f i e l d p o l a r i s c o p e  f o r the r e a d i n g s ,  clamped  was  to  a  end  positioned  the t e s t b a r was l o a d e d i n two pound i n c r e -  ments and the r e t a r d a t i o n s n o t e d f o r each l o a d .  The  were p l o t t e d i n a compensator r e a d i n g s v e r s u s l o a d system ( P i g . 1 5 ) .  was andl i t  bench and p r o v i s i o n was made t o l o a d i t a t the o t h e r (Pig.14).  The  points  coordinate  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  l o a d and r e t a r d a t i o n , the l o a d c a u s i n g l e n t t o one wave l e n g t h  a retardation  o f the white l i g h t used  mined. The d i f f e r e n c e o f the p r i n c i p a l s t r a i n s  was  equivadeter-  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 f r i n g e v a l u e o f the p l a s t i c . T h i s f r i n g e v a l u e i s n o t e q u a l to the a c t u a l s t r a i n d i f f e r e n c e when the s t r u c t u r e i s bent, because o f the r e i n f o r c i n g e f f e c t o f 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 a p p l i e d .  41 The mined by t h e o r y of  1= The  cipal  difference  of the  p r i n c i p a l s t r a i n s was  also  deter-  c a l c u l a t i o n , u s i n g the w e l l known e q u a t i o n s from  the  elasticity.  LF t e n s i l e s t r e s s r e s u l t i n g from the  s t r e s s a t the  top  s u r f a c e of the  bar,  l o a d w i l l he the  other  prin-  princi-  pal stress i s zero.  The  difference  er, i s given  by  But  above  As  from the  stated  o~2  =  =  02  = 0  p r i n c i p a l s t r a i n s as d e r i v e d e a r l i -  0  earlier, a correction  c a r e o f the i n t o the  of the  birefringence  equation  z  °i  and  f a c t o r was  n e c e s s a r y to  take  caused by bending. I n t r o d u c i n g t h i s  42  This equation g i v e s the f r i n g e value from the s t r a i n s occurri n g a t the surface of the t e s t  bar. Since the s t r a i n s  the same i n the p l a s t i c as i n the metal  at  the  interface,  t h i s f r i n g e value was made equal t o the f r i n g e value by equation 9 i n Appendix I .  e x p r e s s i n g 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  I n the present case the values were *& = t =  2.27 x 1 0 "  5  0.052  in in  P =  25.5  lb  L =  6.0  in  Z =  1.04 x 1 0 " i n 3  E =  30.0  V=  0.3  C=  1.16  2  x 10  6  3  lb/in  2  are  given  43  and  so  b =  1.0  in  h =  0.25  in  K =  0.0905  with, t h i s v a l u e o f K the f r i n g e v a l u e o f the p l a s t i o f =2400 x 1 0 ~  6  in/in  was  44  1-  Observer  Analyzer plate. Q u a r t e r wave p l a t e  Elliptioally l i g h t beam  Light  source  Polarizer J-^r  wX.  plate  Q u a r t e r wave p l a t e  Circularly polarized l i g h t beam  polarized  i i i i i  Birefringent plajs_tic_ Structure  H  ,,  I  ~~] x  W-.\\\\\w\\\;x\  .Reflective  -^> W W N ^ W W ^  P i g . 1. Schematic drawing o f R e f l e c t i o n polarisoope  surface  45  0  = Observer  1  = Light  source  Oo = Ooular  p  = Polarizer plate  a  = Analyzer plate  pr = P r i s m  q  = Quartz wedge compensator  w  = Structure  pc = P l a s t i c  P i g . 2 . Schematic  of Oblique Incidence  Polariscope  coat  46  V  Observer  Ooular Analyzer  plate  Quartz wedge compensator  i  Semi t r a n s p a r e n t mirror  Polarizer plate  !  i Objective  I  1  Light source  1  i i  !  )  Plastic Structure  Pig.3.  Schematic o f P o l a r i z i n g Microsoope  coat  47  R  = =  c L = r = t =  R  in in 8.0 in 0.48 i n 0.125 i n  8.0 4.0  Pig.4. The Pressure Vessel  48  F i g . 5 . C a s t i n g of  Plastic  P i g . 6 . P r e s s u r e V e s s e l and  Dead Weight T e s t e r  50  P i g . 7 . T e s t Setup f o r the Large F i e l d  Polariscope  F i g . 8 . T e s t Setup f o r the Oblique Incidence  Polariscope  F i g . 9 . T e s t Setup f o r t h e P o l a r i z i n g M i c r o s c o p e  53  F i g . 1 0 . I s o c l i n i c s on the Head o f the P r e s s u r e  Vessel  Pig.11. S t r e s s  distribution  o f the p r e s s u r e  on the head  vessel  Stress (l0 psi) 5  Spherical  Torus  Cap  40  30  _ Circumferential Stress  20  _  10  _  15  20  25  30  35  40  50  60  80  90 angle c  10 _  20  30  40  50  70  _  P i g . 12.  C i r c u m f e r e n t i a l and Meridional along a r a d i a l l i n e .  Stresses  Stress Intensity, Factor  56 P r e s e n t work Reference (22)  3-1  —  Reference  (23)  \  30  4 0  50  60  Meridional  70  80  90  angle  Stress  Stress Intensity Factor  P r e s e n t work  3-  30  40  50  60  Circumferential  Fig.13.  7 0 8 0  Referenoe  (22)  Reference  (23)  90  Stress  S t r e s s D i s t r i b u t i o n i n the Torus  angle  Pig.14. T e s t Setup f o r C a l i b r a t i o n o f P l a s t i o  

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