granted by the Head of my Department or by his representatives. It. is understood ... mining the stresses i n the t o r i s p h e r i c a l head of a pressure vessel was ...
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 .
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 . No. 8. 638-642 [1956] 8., Zandman, P.,and Wood, M.R., " P h o t o - S t r e s s a new t e c h n i q u e f o r p h o t o e l a s t i c s t r e s s a n a l y s i s f o r o b s e r v i n g and measuring surface strains on a c t u a l structures and p a r t s " P r o d u c t E n g i n e e r i n g . 167-178 [1956] 9., Zandman, P., " P h o t o e l a s t i c - C o a t i n g t e c h n i q u e f o r determining stress d i s t r i b u t i o n i n welded s t r u c t u r e s , " Welding J o u r n a l 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 W i l e y & Sons Inc. [1948] V o l . I . and I I . 11., H e t e n y i , H.,[ed] Handbook o f 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 New York. John W i l e y & Sons I n c . [1950]
28 12.,
"INSTRUCTIONS f o r the use o f P h o t o s t r e s s Large Field U n i v e r s a l Meter" T a t n a l l Measuring Systems C o . B u l l e t i n . [ 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. B u l l e t i n . [Phoenixville] No. 8005 [1958]
19.,
" I n s t r u c t i o n s f o r moulding c o n t o u r e d s h e e t s o f photostress plastic" T a t n a l l Measuring System C o . B u l l e t i n . No. IB-8008
20.,
" I n s t r u c t i o n s f o r a p p l y i n g p h o t o s t r e s s sheet plastic" 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. IB-8004 [1959]
21.,
Zandman, P., Redner, S.S., and R i e g n e r , E . I . , " R e i n f o r c i n g e f f e c t o f b i r e f r i n g e n t o o a t i n g s ' T r o c e e d i n g s o f The S o c i e t y f o r 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 No.588.[1959]
22.,
G a l l e t l y , G.D., " I n f l u e n c e coefficirnts and p r e s s u r e v e s s e l a n a l y s i s , " Journal of Engineering f o r Industry ASME Trans, S e r . B. V o l 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 s h e l l s - A c a u t i o n signers." 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
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 " °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
« 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