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A Design of a Photo-ilastic

Stress Analysis Machine

7 A Thesis Submitted to
The Faculty of

fiICHIGAN smgws COLLscz
O

f
AGRICULTURJ AND APPLIED selauca

By

Co ‘No giQhOIBS

Candidate for the Degree of

Bachelor of Science

June 1953

r\ r
\f .) Ks)

Introduction

Great proaress has been made in the science of
design in sgite of the severe handicaps under which the
designer works. Heretofore, the designer has had access
to several methods of design but the inherent weaknesses
of most of the methods are readily apparent. For instance,
in judging the safe loads to be applied to a structure
the designer has access to service records or destrdction
tests. The lack of service records of he? structires with
increased loads is a severe handicap. Destruction tests
are inadequate in that they seldom indicate methods of im-
provement. Improvement COuld be made only by constructing
numerous types of details and destroying them under load.
During the constriction of the Mid-Hudson Bridge, there was
some doubt as to the strensth of a certain detail. To
prove its strenrth, an exact replica of the detail had to
be constructed and destroyed under load.

The desijner also uses, in conjunction with the fore-
going methods, current information regarding the stress
fields of the structure, physical data of the materials
used, and the theories of strength best suited to the prob-
lem. The use of these items involves the computation of
stresses by analytical nethods. In problems involving any
bit the most simple stress distribution systems encounters

inedrmountable mathematical difficulties. It is only in a

1031.70

very limited number of cases that strict mathematical fore
mulae may be applied. These are based on theories of
elasticity, temperature stress, etc., and are efficacious
only when the assumptions of load conditions are repro-
duced in the field.

In the final analysis, the failure of the methods
of the desianer is borne out by the fact that the designer
doubts the accuracy of his work and must apply a safety
factor to insure the stability and safe functioning of the
structure.

The study of the effects of loads on structures by
photo-elastic methods provides a practical, complete and
accurate analysis of stress distribution. The following
discussion presents a design and method of use of an

apparatus for stress analysis by photo-elasticity.

C 013 51‘" K) T I OH

In the design of this Photo-Slastic apparatus two
major considerations were kept constantly in mind, namely,
simplicity and economy. Several assumptions concerning

the optical apyaratus were necessary in order to permit

omitting optical apgaratus as sad in more complete and

d in

w
.L

complicated set—ups. These assumptions are uyhe

acne cases by atttal £31€limCLtall0n and in others by

theory.

It was assumed that the effect of temperature
was negligible in comparison with the range of visible
rays. No attempt has been made in this design to elim-
inate heat radiations by means of a water—cooler. The
effect of heat on the polarizing unit is also neglected
due to the type of prism used. Neglect of temperature
consideration is immaterial as borne out by the experi~
ment. See Ref. 1.

Another presumption was necessary in View of the
economical design. This was concerning the Optical ac-
curacy and grade of slass used in the lenses. The con~
clusion reached was that a alass of the grade of eood
watch glass was sufficiently accurate for the purpose of
this apparatus. All results obtained from the apparatus
are purely relative. Jach ray of light passing through
the Optical parts is acted upon within the limits of

accuracy noticeable to the eye. The :rimary considera—

tion in the construction of the apparatus is to keep the
optical axes of the various parts on the same line as
slight variations would cause a leakage which would be

detrimental to the production of a clear image.

The Lamp House — Blate II

The lamp house and source of liaht may need an ex—
planation. In order to secure an accurately controlled

'4

licht source of sufficient intensity, a oC‘ “att bulb and

(‘

parabolic mirror ccmbination V98 utilized. The light

source is fixed, to being; taken to have the source on
the optical axis of the apparatus. .The parabolic mirror
is adjustable only on the horizontal plane. This permits
focusing the light on the first condenser. See Fig. l.
The lamp house is just large enough to admit the mirror.
This eliminates all side or vertical displacement. The
bulb should be silvered on the side opposite the mirror,
the silvered area beirg large enough to prohibit direct
light from the source from enterirg the condensing lens.
The addition of the silvered area also increases the
amount_of light by approximately 50 percent. The interior
of the lamp house is finished in a dull black to eliminate
reflection interference.

Due to the variety of shapes and sizes of bulbs, it

was decided to allow sufficient room for the bulb but not

tO hold the builder to a specific design. It would be a
simple matter to insert a socket in the base Of the lamp
house of sufficient height to L‘ing the aource of light

pp to the optical axis of the instrument.

The Polarizing Unit - Plate III

The Condenser Haunt I and the polarizing unit were
node.riaid mith respect to each other for a specific
reason. It was felt that elimination of minor adjustments
would facilitate the Operation of the machine as a whole.
The spacer tube should be constructed Of just sufficient
length to bring the condenser and nicol prism in exact
focus when the polarizing element is in place. The con-
denser lens is held in place by a retaininw ring and the
83805? tube. The polarizing mount Of this unit should be

‘Otatable through ninety derrces in either direction.

LJ

es a snux fit of the riccl prism mounting
in the conderoor tulc. For experimental rurpcees it would
be ’OECleJ to calibrate the tube in degrees of rotation.

Due to the small dimensions of the nicol yriem and
difficulty of machining a part to fit the prism the follow-
ing arrangement was deemed necessary. The prism holder is
a brass asting machined to the specified dimensions. A

hole five eiahths of an inch in diameter should be drilled

through the casting with the center Of the hole exactly on

the Optical axis. The condensing lens should be mounted
in position and light projected through it from the lamp
house. The nicol prism should then be cemented into posi-
tion with plaster of paris. Theoretically, the center of
the prism should be six and one half inches from the face
of the lens. Accurate setting of the nicol prism is es—
sential and care SlOlld be exercised in this phase of con-
struction.

The quarter wave plate attachment needs no adjust~
ment. It should be constructed to fit snugly enouah to
allow no excessive play yet should be capable of removal

without disturbing the adjustment of the various other

units.
Condenser and Analyser

The analyser unit is essentially the same as the
polarizing unit. The respective positions of the quarters
wave plate and the nicol prism being reversed. This
necessitated placing the knurled shoulder on the Opposite
side of the casting from the quarter-wave plate. In this
unit, also, the quarter-wave plate is removable and the
nicol prism mount is rotatable. Provision has been made,
in the condenser tube, for the removal of the quarter~vave

plate.

Condenser Mount III

Condenser mount III was desianed as a single unit
for compactness. The lenses are fired in their respective
tubes yet are adjustable relative to each other. The re~

tainina rinse of all lenses are identical.
The lension Frame

The operation of this Photo«dlastic Apparatus

necessitates using a piece of material cut from the same

.)

sheet as the model as a standard. This Oprration will be

L

t

discussed in detail later. Tie standard must have known
loads a plied, the loads being equivalent tensile loads.
The Tension Frame used in tiis design has been simplified
greatly. To eliminate bulky apparatus and complicated
construction the following method was devised. The load
is applied by a hand wheel and screw arrangement through
a movable chuck to the standard. The lower chuck has no
vertical movement but it can be rotated. The loads are
measured by a strain Pause appliance fastened to the two
chucks, measuring the amount by which the standard elon—
sates under the load. A statement to the effect that,
"Most isotrapic materials, such as celluloid, elonaate in
a straight line ratio nearly to the point of failure",
Ref. II, was the basis of the device. It will be neces-

sary to calibrate the device and if larse differences are

found in the Moduli of jlasticity of the different stan~
dards, a coefficient must be applied to each standard.
The calibration of the device is a simple matter. The
method of procedure would be as follows, the apparatus
could be un~ended, known weight applied to the movable
chuck and the position of the gauge noted for each load.
Thus every divisi n passed over by the indicator would
represent a load of so many pounds, etc. The calibra-
tion unit is placed in a heavy ring to obviate any dis-
placement due to the bending of the frame. The entire
unit is rotatable about 2 axes -- one vertical and one
horizontal. This permits placing of the standard to
correspond with any direction of lines of stress in the

model.

Theory

The results obtained from the apparatus will be more
clearly understood by the operator if he has a knowledae of
the theory underlying this method of analysis.

The fundamental principles upon which the theory of
Photo-Elasticity is based are stated by Prof. Coker as follows;

1. "The distribution of stress through any loaded iso-
trOpic elastic structure is independent of the material of
which the structure is made and depends solely on the form
of the structure and the way in which it is made."

2. "A transparent isotropic material, such as glass or
celluloid, acquires doubly refracting preperties when stress-
ed differently in different directions, and the degree to
which these properties are produced depends on the differ-
ences between the principle stresses in the material."

By "Isotropic", in the first principle, it is meant
that the material has the same physical preperties in every
direction. This class includes most materials used in con-
struction, including, concrete and steel. This property is
also the basis for assuming that the materials used in the
construction of the model is stressed in exactly the same
manner as if the structire were built of steel or concrete.

The doubly refractin; properties, principle 2., of
isotropic materials when stressed have been thoroughly in—

vestigated. Ref. II. The amount of refraction depends solely

on the differences in the principal stresses. Any element
in a stressed material may be thought of as being acted

upon by three principal stresses which are mutually perpen-

P4

dicular. They are referre: to as P Q, and R stresses.

.
The P and Q stresses alone are considered in this work.

The elimination of tee R stress is made possible by using
plate structures and considering only the stresses acting
in the plane of the structure.

A discussion of the phenomenon of polarization of
light is deemed unnecessary except in reaard to the action
of a stressed specimen on the lisht.

The effect of a quarter-wave plate on plane polar-
ized light is to retard one vibration a quarter of a wave
length with respect to the next vibration. This is known
as circularly polarized light and may be thought of as
having a horizontal and a vertical component vibrating a
quarter of a wave length apart in time and phase.

When this circularly polarized light is projected
through a stressed Specimen, the same effect is produced as

with the retardation plate except that the directions of

J

tre'

U)

S

(I

vibrations are the same as the lines of principal
or P and Q for any point in the specimen. If the principal
stresses P and Q differ in intensity, the relative retarda-

tion of the vibrations is proportional to the difference

(P f‘QL

Thus it can be seen that if a mono chromatic light is
used, the color would be produced when the difference between
P and Q is "Test enouch. The color would be totally extin-
guished where P and Q are equal. This is a condition of
zero stress. f the difference between P and Q is sufficient
to produce a retardation of one wave length of the liaht,
black will again result.

To apply this principle to ordinary light, it is first
necessary to state that different colors are retarded differ-
ent amounts by the same stress. Passins ordinary light
through a stressed Specimen will give rise to the typical
"interference colors of the crossed Nicol arrangement". It
will be noticed that with celluloid used as the specimen a
definite series of colors will be obtained.

With the production of the series of colors we have
also obtained a measure of the intensity of the difference
(P - Q). A discussion of the method used to obtain ( P+-Q)
will appear later.

When the quarter wave plates are removed the colors
no lonaer sive a measure of (P - Q) except where the direc-
tion of the principal stresses is at 45 degrees to the plane
of the crossed Nicole. When the directions of stress coin-
cide with this plane of polarization, the light is all cut
out and dard bands are super—imposed on the image. These
dark bands correspond to the locus of points of the same
principal stress direction. Therefore, by rotating the

prisms, keeuins then always in the crossed position, a map

of the directions of the principal stresses may be procured.
For example, with the Nicole in a normal position a set of
isoclinics marked 0 degrees may be drawn connecting the
darkest points on the image. If the prisms are rotated

5 degrees, a new set of lines marked 5 degrees are drawn.

In this connection it may be necessary to enumerate several
fundamental rules governing the charting.

l. The curvature of a stress line varies continuously
if at all.

2. Parallel stress lines correspond to uniform stress;
converreut lines to increasing stress and diverging lines
to diminishing stress.

5. No two stress lines of any one system can intersect
or merge into each other.

4. Along any free boundary of a structure, one system
of stress lines is tanyential and the other is at right
angles to the boundary. Ref. III.

A furtner discussion of the charting of stress lines
will be made under the topic of use of the instrument.

In the foreaoing discussion it was stated that a
measure of the intensity of (P -Q) was obtained. It now
becomes necessary to obtain a quantitative measure of the
difference between these principal stresses. An accurate
measure,in pounds, can be obtained by the following method:

Place a standard, out from the same material as the

model, in a calibration member or tension frame. Adjust

the standard to coincide with the imare of the model.
From the nature of the loading of the model it will be
simple to decide which of the principal stresses is ten-
sion. The bar of celluloid is set at right aneles to this
direction of maximum ension. A load is applied to the
standard until the super-imposed imaves of the two pieces
are dark. Note the amount of tension applied to the
standard. The tension applied may be called "T". This
tension T is the force neceSSary to cancel the effect of the
stressed model on the polarized llfht, or, in other words,
is the difference between the principal stresses (P - Q);

or,

r-Q=T. (1)

From the definition of two-dimension stress it can
be seen that the stress perpendi‘ular to an unloaded
boundary of a model is necessarily zero. From this it can
be seen that a principal stress may be obtained directly
from the above ecuation. However, this is a special case
and not applicable to internal areas. It now becomes
necessary to evolve a new equation to be used in conjunction
with the above.

When a load is applied to a material, there is a
chanve of thickness of the material proportional to the sum

of the principal stress

(D

s. By the use of Poisson's Ratio

for the material >nd calibration of the chanae in thickness,

the sum of the orincipal stresses or (P +-Q) could he com-

f“

puted. The exact value of Poisson's Patio mirht be difficult

to obtain so an alternative method is en we

Q:

te

.' C
U)

A simple tension member, the stand»

L“
‘i
pi

previously men-
tioned, is loaded until its chance of thickness is equal to
the chanee of thickness of the model. Therefore (P-fQ) of
the standard equals (P Q)of the desired part of the model
equals T', the tension applied to the standard,

or,

P-rQ = T'. (2)

The P +‘Q terms of these equations are identical

allowinr a simultaneous solution of the equations, from which,

r= :‘g—(T' a T) (5)

Q - em" - r) (4)

(D

Th Quantitative values of P'V“ tosethsr with the
charts showing principal stress directions and concentrations
complete the results for ordinary problems and form the basis

for further experimental studies.

l'/ p '
W; ‘1/ A
‘ ff'k \ "- K '
I ,

rhe above sketcn is an examcle of the types
of imapes obtained With the Photo-Slastic Stress
Analysis Kachine. The colors are the interference

colors of the crissed Nicol arranwenent.

Procedure

The following procedure, or use of the apparatus, is
enerally used only when the fullest infornrtion concerning
the stresses in the structure is desired. In many instances.

modifications of these suggestions will suffice for the
problem at hand. For instance, if a model is to be examined
for the purpose of finding the weak points in the design, all
that is necessary is to examine the model with circularly
polarized liaht. The highly stressed areas are clearly re—
cornizable by the colors of the imase. The weak portions of
the design will be immediately apparent.

To obtain complete information it is necessary first
to determine the directions of the Principal Stresses P Q.
A model, cut from a sheet of celluloid or Eyralin %" thick,
to the exact reduced scale dimensions of the structure to be
eramined, is mounted in the beam of olane LGlHTiZHd liyht as
shown on Plate I. This means that the quarter wave plates
are removed from the apparatu‘. The polarizing axes of the
polarizing and analyzing units are mutually perpendicular,
preferably, one horizontal and one vertical.

The imare of the model will appear on the screen with
black lines and areas. These lines and areas represent por-
tions of the model in which the condition of stress is such

that no effect is made on the polarized light. The principal

stresses may be parallel and perpendicular to the polarizing
axis or they may he eqial to each other or zero. With the

tion the

U]
(0
H.

axes of the polar riser and analgser in thi yo ,
darkest areas are outlined and the central part of the dark
bands are marked. The polarizer and analyser are then ro-
tated to a new position and a new set of lines drawn on the
above chart. iach line is ;arkeu with its deare e of rota-
tion, such as 0 deg., 5 deg., 10 dea., etc. These lines are
the so-called isoclinics previously mentioned. In inter-
pretina the information of the charts, the general case is
‘ssumed in which one of the principal stresses is perpendicu-
lar to the axis of the polarizer. If at any point on a line
marked 5 deg., another line is drawn inclined at 5 dea. to

the horizontal, this second line will represent the direction
of one of the principal stresses fo1 the corresponding point

on the model. These second lines may be placed so as to form
smooth curves. It is necessary to adhere strictly to the rules

stated in the discussion of the theory of the apparatus. The

stress line diagram now obtained is a reproduction of the

‘w

m .+ . 1:. ..
i063 SUTSSS (Liesram

(D
7"}
(n
0

directions of only one principal s tr

of the other principal t-ess is obtained by drawing another

(Q

system of lines Which intersect the first system perpendicularly.
This complete diagram is the stress system of the structure.
Kn wing the stress directions it is now necessary to

find the (iffe erence in principal stress s ( ? - Q). The model

is set up as before with the polarizer and analyser in their
original positions. The quarter wave plates are then set in
with their optic axes mutually perpendicular and inclined at
forty-five deg. to the horizontal. An estimate as to the in-
tensity of streso ma” oe obtained by a study of the image.
The color of the fringes depends on the difference (P - Q).
To find the quantitative value of (P - Q}, mount the stan-

dard in the calibration unit, adjust to the desired position,

apply the load of the imase of the portion of the model under

*9

1

observation is dark, and note the deflection of the scale.

From this deflection compute the te sile load aoplied. This

J.

is the value of "T."

To compute the value of (P‘tQ), measure the thickness
-of the portion of the model under consideration, Without load.
Apply the desired load and again measure the same portion of
the model. Afgly a load on the calibration unit such that the

standard changes in thickness to corresoond exactly with the

‘

1'.

U)

chanre in the model. The tensile load wpolied i
From the equations (5) and (a),
1
P - :— m» T)

H
M“
A

H
I

F3
V

With the values of P, and Q and the directions in which
they act the stresses of the structure due to the known load

have been completely analysed.

l

‘ " Q , “ ‘ o ‘. ~‘1‘ " -(‘ ‘1 —. v, I'jlr ',—~ ~ w--— >'\ \ : -- 1- ‘
n Culflfinght recOLd or the .na es Mt; L8 ottained oy

-. - _ '1: —- ,-. w 11 — 1. c v 'V ’T" ."1 5‘ 1 T" . ‘ u -- fr C
uEle Finley 5.2tjs C. J. 9301 “do .. JJiTtn ct., L. Y. .,
3
I

for tie colored images,and ordinary photOgraphic neyatives

for the dark band images.

Bibliography

Pef.
'v a " ‘7 " a. l‘ -~ ’ ' 1““ -\ A A... r« . (‘1
I 837119 :Jil {)deiztiue: - II Cucfl s L,rgsit. —
"‘ .V—/\ .‘,,_..J--, ‘0 1~(“~-\ ,.
U111 m 2,? ;.J 1.5 Ci 111..)thilv‘Z-‘o

II Prof. Coker - General ilectric Review.

III Prof. Coker - Photo-Elastic Apparatus

Adam Hilner,~Limited — London.

General.

n

D. V. Baud — Westinghouse, Jlectric & M25.

‘-

Bastman Kodak 00., Rochester, H. Y.
Baush & Lomb Optical 00., Rochester, N. Y.

Du Eont Viscoloid 00., Wilmington, Del.

 

 

 

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