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V
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iA'
.<>'
A^
A®
<^'
iP
•p-
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A
THEORETICAL
DESIGN
OP
REINFORCING RINGS FOR
CIRCULAR CUTOUTS
IN
PLAT PLATES
IN
TENSION
A THESIS
SUBMITTED
TO
THE
GRADUATE
FACULTY
of the
UNIVERSITY
OP
MINNESOTA
by
LEWIS WALTER SQUIRES
,
IN PARTIAL FULFILLMENT
OP
THE REQUIREMENTS
for the
DEGREE
OF
MASTER OP
SCIENCE IN AERONAUTICAL ENGINEERING
August,
1950
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PREFACE
In
the field of aircraft
design, the question
of weight
versus
structural
strength
has
always
been
Important.
In recent
years, with the trend
toward high
speed
and
high performance In modern
aircraft. It Is of the utmost Importance
that
the
weight
factor be made as small
as
practicable.
It Is therefore necessary
that
the
structural
engineer
use every means
at his
command
for ob-
taining
the
necessary strength with
the
least
amount
of weight, and this
c
are
must
be
exercised
in
every phase of design, down
to
the
smallest
detail.
The
subject
of
this thesis
was
chosen
with
this
thought
in mind
as
the design
of reinforcing
rings
around
circular
cutouts
is
still largely
a
cut and try matter.
The writer
wishes
to
express
appreciation
to
his
adviser.
Professor
J.
A,
Wise,
whose
advice
and assistance
was
most
valuable.
11
13196
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TABLE
OP
CONTENTS
Section
Page
PREFACE
11
SUMMARY
1
INTRODUCTION
2
THEORETICAL ANALYSIS
1^
I,
General
Discussion
of Method used
•
• •
4
II. Experimental
Check
of Basic Theory
. •
9
III.
Determination
of
radial and
tan-
gential stress
due
to
the
constant
component
|-
S
of
the
normal forces
•
•
17
IV,
Determination
of radial
and tan-
gential
stress
due
to
the
normal
forces
^
S
cos
20
,
and
the shear-
ing
forces
-
i
S
sin
26
28
V,
Summary of
Theoretical Analysis
....
l^$
APPLICATION OF
THEORETICAL
ANALYSIS
TO TEST
SPECIMEN
I4.9
TEST
DATA
AND
EXPERIMENTAL RESULTS
FROM
TEST
SPECIMEN
53
COMPARISON
AND
DISCUSSION
OP
TEST
DATA
AND
THEORY
58
CONCLUSIONS
61
APPENDIX
62
A. References
.....
62
B.
Bibliography
63
C. Symbols
6?
Ill
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SUMMARY
This
is
a
presentation
of
a theoretical
ana-
lysis
conducted
for
the purpose
of
designing
re-
inforcing rings
for
circular cutouts
in flat plates
in
tension.
In
this report,
the general case
of
a
flat
plate
under a uniaxial tension
load
was
assumed,
together
with
a
circular
cutout
symmetri-
cally
placed
and
reinforced
with
a
circular
ring.
Expressions for the radial
and
tangential stress
in
the ring
and
the
flat
plate
were derived
in
terms of the loading on
the
plate,
the
dimensions
of
the
ring
and plate, and
Poisson's
ratio.
Experimental
data
was
taken
from a
test
spe-
cimen which
was
constructed
and loaded so
as to
agree
with
the
ass unptions
made
in the
theoretical
analysis,
A
comparison of
the test
data
and
that
obtained from
the
theoretical
analysis
showed ex-
cellent
agreement
and attests to
the
validity of
the
theoretical
solution.
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INTRODUCTION
The
trend
in modern
aviation is
toward
high
speed
and
maximum
performance
in flight, especi-
ally
with
regard
to
military
aircraft. Along
with
this
trend
go
the stringent demands
for structural
designs
that
reduce the
weight factor
while still
meeting
the
necessary strength
and
space
require-
ments.
The
design of
reinforcing
rings
around
cutouts
in flat
surfaces
for various
loading
conditions
re-
presents
one of
the
many
fields where a simple and
accurate
method
of
design would
save
much
time
and
effort
and would
represent
a
worthwhile savings in
weight.
The
need
for openings in
the
stress
carrying
skin
of
aircraft
for
maintenance
access,
doors,
win-
dows,
lights,
and
retractable
landing
gear have
pre-
sented
many difficult problems
to the
structural
engineer.
Usually,
these
holes
are
reinforced
with
metal
rings--doubler
plates—
riveted
to
the
sheet,
but little is
known
as
to
how well
such a
reinforce-
ment
approaches
the
ideal
in
reducing
the
stress
concentration
around
the
hole
while
at
the
same
time
adding
least
weight
to
the
structure.
2
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with
this
thought
in mind,
a
theoretical
ana-
lysis of a
flat
plate
under a tension
load
was un-
.
dertaken.
The
flat plate was assumed
to
have a
circular cutout,
symmetrically
placed
and
reinforced
with a circular
ring.
Expressions
for
the radial
and
tangential
stress in
the
ring
and
the flat
plate were derived in terms of the
loading
on the
plate, the
dimensions
of
the
ring
and
plate,
and
Poisson's
ratio.
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THEORETICAL
ANALYSIS
I.
GENERAL
DISCUSSION
OP
METHOD USED.
V/hen
a
small
circular
hole
is
made in a plate
submitted
to
a
uniform tensile stress,
a
high
stress concentration
occurs at
the
edges
of
the
hole
located
at
ninety degrees
from
the
direction
of
the
tension load.
If
the
diameter
of the hole
is
less
than
about
one
-fifth the width
of
the
plate,
the
conditions for
a
plate
of ijifinite
width
and
with
the
load applied
at
infinity
are
approached,
and
exact
theory shows
that the tensile
stress
at
the
above mentioned points
is
three
times that
of
the
loading.
Prom
this theory, it can be seen
that
the
stress concentration
is of
a
very
localized
character
and is confined
to
the immediate
vicinity
of
the
hole.
Since
failure
will
first occur
at these points,
it
is
necessary
in
the
design
of
a
ring
to
reduce
the
stress concentration
in
the
plate
in
those
areas.
It
is
therefore
pertinent that expressions
for
the
radial
and
tangential stress in the ring and the
plate
be
developed
in
terms
of the
loading
and the
dimensions
of
the
plate and ring.
This
was
the basis
of
the theoretical
analysis
as
made
in
this
report.
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In this
analysis It
was
assumed
that
the re-
inforcement
Is
an
integral
part
of
the sheet
and
that
the
stresses
do
not vary
across the
plate
thickness. Previous
studies Indicate
that
these
assumptions are reasonably valid
and
give good
re-
sults
outside
the
reinforced
area.
It was
found
that
measured and
computed
reinforcement
strains
agreed best
In
the case
of
a
reinforcing
ring
fas-
tened
with
two
concentric rows
of
rivets.
PIG,
1.
Let
Fig.
1
represent
a
plate,
with
a
small
hole
In the middle,
submitted to
a
uniform
tension
of
magnitude
S
as shown.
Prom
Saint-Venant's prin-
ciple,
the
stress
concentration
that
occurs around
the hole
at m
and n
will be
negligible
at
distances
which
are large
compared to
the
diameter
of
the hole.
As developed
in
reference
(a), and considering
the
portion
of
the plate
within
a
concentric circle
of
radius
b, large
with
respect
to
radius
a,
the
stresses
at
radius
b are
essentially
the
same as
in
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a
plate
without
the
hole
and
thus
are
given
by
A,
'I
e
These
forces, acting
on
the outer
circumference
of the
ficticious
ring
at
r
=
b
give a
stress dis-
tribution
within the ring which can be regarded
as
consisting
of two parts.
The
first is
due
to the
constant
component
g-
S
of
the normal
forces. The
remaining
part consists
of
the
normal
forces
\
S
cos
2
0,
together
with the shearing
forces
—
J
S
sin
29
.
As
is
evident later,
a
stress
function
is
need-
ed for
the
solution
of the
stresses
due
to
the
above
mentioned forces
and
it
would
be
very
difficult
to
find
a
single
stress function
for
representing both
parts.
Therefore for
simplification of
the problem,
the
radial
and tangential
stresses
are
worked
out
separately
for the forces
as divided
above
and
then
are
added
to
give
the
final results
for
the
flat plate
with a
reinforcing
ring.
This is
carried out in
Parts
III and
IV
of
this
section.
Part
II of
this
section
consists
mainly
of
a
check
on
the
validity
of the
basic
theory used
in
this
analysis.
It takes
the
equations
for the
stresses
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in
a
flat
plate
with
a
hole in the
middle, as
ob-
tained
from reference (a),
and further
develops
these
equations to obtain expressions
for radial
and
tangential displacement which were then
checked
experimentally
to
substantiate the
general theory.
Briefly the
equations for
the
stresses
in the
flat plate
are developed
as
follows.
For
the
stress
distribution
symmetrical
about
the
center
due
to
constant component
^
S
of
the
normal forces, the shearing
stress
^^
vanishes,
leaving the
radial and
tangential
stress
as
follows:
<=
f-
('-£)
For
the remaining part,
consisting
of
the
normal
forces
^
S
cos
2
B
,
together with
the
shearing
forces
-^
S
sin
29
f
there
is
produced
a
stress which
may
be
derived
from a
stress function of the
form
<p
=
T^)
cos
2
d
As
developed
in detail in Part
IV
of this sec-
tion
this
finally results
in
.^^
>L
^
9
z.
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8
Adding
the
stresses
produced by
the
uniform
tension
^
S
gives
the
final expressions for
the
stresses
in
a
flat
plate
under a tension load, with
a
hole
in
the
middle.
These are the
equations
which
are used
in
Part
II of
this section.
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THEORETICAL
ANALYSIS
II, EXPERIMENTAL CHECK OF
BASIC
THEORY.
Y
•
f
^
r
s
^
Q
R
1
e
—
\
n/
—
-—
X
—
*
Pig.
2.
For
the above
figure,
the
stresses were given
in
Part
I
as,
^
Y
^6
Ke>
/
i
<^
1
f^
1
7-
^^e
X
^e
Fig.
3.
Let
u
=
radial
displacement
V
=
tangential
displacement
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c/
10
For this distribution of
stress,
the correspon-
ding
displacements
are
obtained by applying
the
basic
relationships
given
below:
&
=
^
(/y)
Prom the
above
equations:
TZ^
F(<y-^'
e
'J^
s_
ii,
.1
_
iL //..,)
^/.^.,)r^ 3
«- i i''
c^
xB
Integrating
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11
s
in which
7(d)
is
a
function
of
Q
only.
e«
=
=^
+
d
=
X
V 7^
=
£
I
-e
>^
y^
Substituting
values for
<3^
and
u,
and simplifying
gives
^
y'
-T
Integrating
^
—t
[(/
1^)
t
z
Oy>)
^
^Oy^^f-
^^
___ __
^Bi^B-^^B
111
^—
' MM.
i
^ ^ f#
—
AM
i
^
' \.r
'
y^^
JL
^
where
P
(r) is
a
function
of
yi-
only.
Substituting
the
above expressions
for
u
and
v in
the shear
equation
r.B
o/^
s
Jy^
^
ffe)
e
^^
/
^
,
^
(
'^^
^
^a^
x^j
f
;l
/t-14.
-V
^**-^
^^
'
^^
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12
A.6
J
r-
^
-7-
f
— .<u-^
;2d
/
/ .. 1 , > /.
)
<*
.
./
I
^*^
a.
'^^6
G-
^yte
K
.
^
^
-
-7
^
^*
-
-
f^
^
^6iiu^)'3f^-^)^^
^y^^^j
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13
j
f(Q)
Aq
4
f^{a] -h
yc
FOc)
~
2 E.
-
^
-TH
A*^
since
the
quantity In
parenthesis
=
o,
\
fid)
cle
-h
f
(d)
-^
y^
fU)
-
Fu)
This
equation
Is
satisfied
by
putting
=
O
In
which
A,
B,
and C
are
arbitrary constants
to be
determined
from
the conditions
of restraint.
Therefore:
xa.
-
r/^)
2.
i-{jiyuL)
c^^
2
6 -f-
Z±
C^
^
B
C#«L
Z
^lA'
--ji\S^t^^^^('y^)t-
^0^)
//
The
conditions of
restraint
are:
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Ik
(1.)
At
(9
=
90°
,
V
=
o
(2.)
At
a
-
00
(3.)
At
6
=
00
&
90°,
^
Prom
condition
(1.)
o
r
-c
4-
Ar
C
=
Ar
From
condition
(2.)
o
«
B
4-
Ar
B
= -
Ar
Prom
condition
(3.)
V
=
o
^
z
O
/\^
#=
°
=
-h
[Ot-)
~-('y^)£-^(T)i]-''
'^
A
o
B
=
o
C
=
o
The final
equations
for
u
and
v are
<•
r
«-
^
At
^
=
0°
and r
=
a
3
So^
u
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15
At
5
=
90°
and
r
»
a
u
=
-
^
To
find
B
for
u
=
o
at
r
=
a
cos
Z& ~
~\
cos
120°
=
-i
a
=
60°
Using
the above
equation
for
radial
displacement
(u), a
plot
of
the
displacement
was
made
as
shown
in
Fig, Ij.
for a
loading
of
eight hundred
pounds
on
the
test specimen as indicated
in
the same figure.
An
experimental
check using the Huggenberger Tensometer
was
also
made on
the test
specimen
and
gave the
ex-
perimental
values
as
shown. The
results indicated
that
the
basic
theory used
is sufficiently
valid for
purposes
of
this
analysis.
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ns'
\^M,.
TE$f
'sPEc'^il^ tiirotR
A iENsip
LOib
dF
i
Boo
iBs.
1
'
.
' 1
J
.
. .
-^Jf^
\
\^
y
/
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17
THEORETICAL
ANALYSIS
III. DETERMINATION
OP
RADIAL AND
TANGENTIAL
STRESS
DUE
TO
THE
CONSTANT
COMPONENT
J
S
OF
THE NOR-
MAL FORCES.
Let
^^'
=
thickness in
region
of
ring,
^jj
=
thickness of flat
plate
(outside ring).
To
obtain
the radial stress
(fe
)
and
the
tan-
gential
stress
(
c^
)
in
both
regions
{
u
.
V*
t^^)
due
to
the
constant
component
5
S of
the normal forces,
the
procedure
is
as follows:
6\
g
*
0,
since
the
stress
distribution
is
sym-
metrical,
the
stress
components
do
not
depend
on
and
are
functions
of
r
only.
This
is
also
evident
from the polar
equation of
equilibrium,
^
_
_L
^
^
-
-^
^
The
displacement
(u)
is
a
function of
the
radius
only,
because
of symmetry.
Since
.
€
=
-^
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18
Then
<^.
^
E
y
-
£
Considering
the
equilibrium of
a
small
element
cutout
from
the plate
by the
radial
sections
C?c
^^^
1
perpendicular
to
the
plate,
and
by
two
cylindrical
surfaces
a
d
and
b
c
with radii
r
and
r
4-
dr,
normal
to
the
plate
,
O
A^
Fig.
5.
^oj
(^
remains
constant
because
of
symmetry,
C^Q~
O
(from
previous
considerations)
Summing
the
forces
in
the
radial
direction
®
2
Neglecting
second
order
quantities
and
cancel-
ling
out
dr
dd
gives
dU.
=
o
Vu
Substituting
/r^
and
^
in
the
above
equation.
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19
Multiplying through
by r.
To
solve
this ordinary
differential
equation.
Let r
=
e
where
D
=
.^ ^
t-
^^u.
X
>u
-
^
-h
«
•
—
cL±^
'L.
dj'.u^
JLjuu
_
c^-
-**-
A^
,
-f
^^O
dL^
^
^
^
t:
z
y^
d^
..-*4-.
<X.
-
«<.
/u -
^
-e
•
^
t:
-
1_- ^
^
d1-
D.
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20
Therefore
may
be
written as
[_D
(
D-0
i-
D-J
=r
O
—
O
Lei
^
=
e-^
-6
Since r
=
C
To
check
on the correctness of
this
formula,
consider
a
flat plate
as
follows
with
a
constant
thickness:
with
boundary conditions as
follows:
When
r
a,
cr^
»
o
When
r
a
C
,
<5^
=
%'
Using the
original
equation for
^^,
-A.
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21
3
equation:
i
2
Substituting boundary conditions
In
the
above
A
s
c
For
c
=
«*0
'or a flat plate of constant thick-
ness
with
a
hole
at
the
center.
On
page
fifty-five
of reference
(a),
the
follow-
ing
formula
Is
given
for a
hollow
cylinder
submitted
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22
to
uniform
pressure
on
the
inner
and
outer
surfaces:
oyb-if.-p:)
J^
^
fZ^
a,-
-^„
h^
In the
above case:
f.-
:
O
P'-
s
(pos
lU-
^^a)
/
(positive
for
compression)
g1
=
'
Multiplying
numerator
and
denominator by
—
:
/-
-.
)^
^
-
^
At
b
=
oO
,
s
[-^^7
This gives an exact
check
on
the
formula
pre-
viously
developed;
B
For
the case of a
flat plate with a different
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23
thickness
around
the
hole
as
follows:
Fig.
7
At
the
boundary
(b),
the
designation
Is
1
-
Inside
o
-
outside
as
indicated
in
the
above
figure.
^
±
''y
Boundary
conditions
are
as
follows:
(1.)
At
r
=
a,
(si^=
<^
.
_
5
(2.)
At r
=
C,
ff^
-
-^
(3.)
At
r
»
b,
^^^
^
^
o
(l^..)
At r
=
b,
G^
^
-
^
^
Using
the
formulas
previously
developed
in
this
section
-L
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2k
and
Q
u
=
Ar
+
-^
the above
boundary
conditions give
the
following
equations, assuming
the modulus of elasticity
(E)
to
be
a
constant.
—
(1.)
=
e£(/^>.*)
t\^
-
(
'-^)
-X--
(2.)
f'E[Oy-)
A.-f,^)
1=,
J
(3.)
/I_,i, ^
:^
=
/\,l
r
i^
These
equations
may be solved by the method
of
determinates for the
constants
A
•
Q.
/)_
s
Uid
l^
h\
(numerator)
=
—
—
^
•
(numerator)
«
—
J^^J^-A^
f\
(numerator)
=:
_£i^
A
V^
^Z^^-
'
^
J
^^^t^i^y^)-i>^t.M
f\
(numerator)
=^
—
<-
n
^
A,
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Denominator
a.
25
__ 0;^
/
The
validity
of
considering
the
load
applied
at
infinity
has
previously been proven
in
section
II,
and this
is now
done
for
the value
of
c
as it
appears
in
the denominator
of the four
constants
listed above.
Denominate.
-^
L[
£^.
,)t
^1^^
(
,y.)
.
^
(
,y)\^
Denominator
=
~i^7T
j(^~^
J
'^
-^
(
^^J
A,
b
-
t
^
[h-
f/^)
^
c^
{
i-^]]
Denominator
^^-^
—
-^
T
**'
i^'y^
Combining
the
numerators
and denominator give
the
constants:
8
C-b^t^S
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26
/)
-
-7
Considering
previous
formulas
developed;
By
proper substitution,
the above gives:
(1.)
<
=-
£
/7/^j
-^
-
(^y^):z'-J
(2.)
(T^^
-
e[(/^)
^+( y^')^-]
(3.)
^
=
^--
^
X
Therefore
for
the
radial and
tangential
stresses
due to
the constant
component
(i-
S)
of the normal
forces,
the
following
equations apply
in which
the
above
constants
A.
S>
r\
^
and
O^
are
to
be
used
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27
(o^
-a:
8.
Also,
B
AX
-
A
-X
-<-
So
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28
THEORETICAL
ANALYSIS
IV,
DETERMINATION
OP
RADIAL
AND
TANGENTIAL
STRESS
DUE
TO
THE
NORMAL
FORCES
(|
S
cos
2
6
)
AND
THE
SHEARING
FORCES
(-^
S
sin
2
^
)
Neglecting
body
forces,
the equations
of
equilibrium
for an
element,
expressed
in
polar
coordinates are:
,.^ ,^
t
These
equations are
satisfied
by
taking:
where
M
is
a
function
o/
-/u
sm</ ^
a
m
«/
is
knovm
as
a
stress function.
As
developed
in
reference
(a),
the
stresses
developed
from the normal forces (|^
S
cos
2d
)
and
the shearing forces
(-J
S sin
2©)
maybe
derived
from
a stress
function of the form
Substituting
this into
the
compatibility
equation
in
polar
co-ordinates
,
/ i.
/
\
gives
the
ordinary differential equation,
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29
J^v
+^
J^'
^-
-^-0'^--
Solving
this
equation in
the
same
manner as
used in Section
III
gives
The
roots
are
fif,
4-2,
and
-2
Therefore _
and
^
=
(/^.^-
^.0^''
^i:
^^je^se
The
corresponding
s tress components
are:
<-
-(-^'^
^^
2d
^l'^
^
^
^.
]
C^
SLB
To obtain general
expressions
for
radial dis-
placement (u)
and
tangential
displacement (v)
for
the
above conditions,
the
procedure
is
as
follows.
For
this
distribution of
stress, the
corres-
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30
ponding displacements are obtained
by
applying
the
relationship
as
follows:
G
=
('y^'>
)r
—
T^
T
~i
—
zr
G-
^^
£e£^
^,^;^
z
B^-
/^
r<^y
/
'^y
ie
^
i
Integrating,
In
which
fCd^
is
a
function
of
Q
only.
fe
-
£
^7^
•
i
r^-
-.-
<)
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31
i^'f(^.-^<]-
Substituting in the
above
equation gives:
2
A.
<X
-/-
£
Integrating,
2>^ -<u^
Z
^lA
^^
where
P
(r) is
a function
of r only,
n
-J
he)
Ji^
-^
F^)
_
3.
><u;^^
;eg
Substituting
the
above
expressions
f
or u and v
in
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32
ir^^'H^^^'^'-y-^fTy^f]^^^
4-^
f(e) dd
-
^
'^A.6
G-
^^9
'^e
6-
.
-±u^
t^
^^^
-
^-^
-
^
^^45
<iC
_
;?
/>
)
y
_
-2
P
('^y^U
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33
J
f(e)
JLg
i
il
6
)
i-yi
F(^)
-
F(^]
-
Z
A.
-^-c^
2
:zC
Q}^:>
VP
-
^(^/j^;
-B^'^i^y^)
_
-S_
(^z-^)
^_^^
|<,^j
Since the quantity in brackets
is equal
to zero,
-h
yx.
F(^)
-
F(^)
=
O
fcd;
i^
t
flo)
This
equation
is
satisfied by
putting
f
(^)
=
V
>o
<9
^
c^
5
in
which
X,
Y,
and
Z
ai^e
arbitrary
constants
to be
determined
from the conditions
of
restraint.
Substituting these values
in the
equations
for
=
-
u
and
V
^ivos
V
gives
~1
\y^
-y
itWAO*^)-z
d^y
^
J;
(,p^)
^^y
Y^6
^Z^e
X
For
the
purpose
of determining
the
value
of
X,
Y,
and
Z,
the conditions of
restraint are:
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3h
(1.)
At
^
-
90°
,
V
=
(2.)
At
^
= 0°
,
V
=
(3.)
At
e
=
90^
or
0^,
~
=
O
Prom
condition
(1.)
«
-Z
4
X
r
Z
=
X
r
Prom
condition
(2,)
=
Y
4
X r
Y
= -
X
r
dA.
Prom
condition
(3.)
X
»
Y
=
Z
=
3
0^-(jy.)-^(<-^)
^J
Oy-ll^
X
Therefore
.>u_
—
—
i^[-A(,y^)-z
B->
^;5r/^;^
^J
^
^
^^.uUiej-^
f^^j
^
^^-(y^^
y^
^''^^J
A^^
These
equations are
the
general expressions
for
radial displacements
(u)
and
tangential
displacement
(v)
for
the
normal forces
\
S
cos
2B
together with
the
shearing
forces
-
J
S
sin
2^.
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35
As
a
check
on
the
equations
as derived
above,
the equations
for
c^
<al
and
'^^
on
page
29
can
bo
used
to determine the values of the constants of
integration from
conditions
for the
outer boundary
and
from
the
condition that the
edge
of the hole is
free from
external
forces.
The boundary
conditions
are:
(1.)
At r
=
b,
c^^
=
X
S
<^^^
^^
(2.)
At
r
-
a,
G^
=
o
(3.)
At r
=
b,
^^=
-
i:
s
-**^
^^
(I4..)
At
r
=
a,
^^e^
^
These
conditions
give
—
^^
-
^
^
/\
-t
;iA
-^
6C
f
aJ^
^^
^
.a^--
f.
-
^
--^^
2A
+
^
^^--
>^
-
^^
4^
*-
—
a
Solving
these
equations
and putting a/b
=
0,
ie.,
assuming
an
infinitely large
plate,
gives:
A
=
-sA
B
»
C
—
-TT-
«->
¥
¥
D
5
_f^
3
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36
Substituting
the above
values
for
the
con-
stants
A,
B,
C,
and D
in the equation
for
u
gives
the
following
result:
5^
/(/^^l
_
^
1,^^)
-^
t±l
]^^/)-^('t^)^
^i]'^-'
This expression for
u contains the last three
terms
of the
general
eauation developed for u on
page
llf
of
section
II,
The
other
two
terms
in
the
general equation ere not dependent
on
B
and are due
to
the
constant
component
g-
S
of
the
normal
forces.
To obtain the
other
two terms, in the general
equation
of
section
II,
which
are due to
the
constant
component
^
S
of
the
normal
forces,
proceed
as
fol-
lows :
Prom
section III
8
u
=
A r
+
—
_
S
Lt^
^
^
ZE(>y^)(^-f^^
^
ly-
^
'^
('y^('-^)
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37
For
C
—
^
<^^
u
=
^^
u
~z
/
,
--a.
[t.
^^
I
^^
^
_eb
.
^^/^
Since
'Z^
iH
L
z^'-
-^'^
y^'
,-M.^^
I
u
=
I-
/;^';-J
'
;g:'
('^-Jj
Therefore,
using
the
principle
of
superposition,
and
adding the radial
displacement
due
to
the
con-
stant component
|- S
of
the
normal
forces,
ie
u
=
[i -/)
^
±:
('ty^
and
that
due
to
the
normal forces
^
S
cos
Z9
toge-
ther
with the shearing
forces
-
i
S
sin
2^
,
ie.;
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38
gives
the
final
expression
and
is
identical to
that
on
page
1I4.
which
is
the
for-
mula for u due
to all of the
loading.
This gives
a
complete
check
on the validity
of
the equation
for
u.
In
the same manner,
the expression
for v
can
be checked
as
follows:
v
=
~
[ ^
i,.^)
f
3^^
i^y.)
^
J
(/^)
-J
0;^
Again using
A
=
-
SA
B
a
C
=
-
CL.
3
D
=
4^
^
V
=
—
if
|('^)
^f.V(^^
^^y^y^n
^^
^^
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39
Since
there
is no
tangential
displacement
(v)
due
to
the
symmetrical loading by the
constant
com-
ponent
1^
S
of
the normal
forces,
this
is
also
the
expression
for the
total
tangential
displacement,
and as
such,
it agrees with
the expression for v
on
page ll}.
of
section II.
To determine e^
and
<o^
in
the
ring and
plate
due
to
the
normal
forces
(^
S
co3^0
)
and the
shearing
forces
(
-
i-
S
sin
S&
),
the
constants
A,
B,
C, and
D must
be
determined in both
regions,
i
and
0,
as
indicated
in
Pig.
8,
Pig.
8.
At the
boundary
b,
a stress
flow is assumed
as
in
Pig.
9
(b)
instead of
the
actual
stress
flow shown
as it would
be
in
Fig.
9
(a).
f^.
(h.)
E.
Pig.
9.
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1^0
Since there will
be
eight unknowns.
A-
B.
C-
A
a
C^o
C,
and
L^
,
there must
be
eisrht
equations.
The
boundary
conditions
for these eight
equa-
tions
are
as
follows:
At
r
=
a
(1.
(2.
At r
=
C
(3.
At
r
=
b
(5.
(6.
(7.
(8.
^^&
=
^
'^o
~
-2-
The
general
equations
to be used
with
these
boundary
conditions
are:
^.=
(^ ^
-h
6>
aJ~
B
~
^'
:id
^^
o^ ;td
/l^
=
2
^U
.dA^
id
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Using
these
formulas
and
the above
boundary
conditions,
the following
equations are
set up
for
determination
of
the
constants:
The
subscripts
are
as
indicated in
Pig.
8
with
relation
to
the radius
b,
i
-
inside
-
outside
These
equations
also assume that
It
should be
noted
that
these
constants
are
not
the same
as those which are
evaluated
in
section
III.
(1.
(2.
(3.
(5.
(6.
J
tt
.•
^
^At
'-
o.-t
A.-
il^C-
il'
n.
-L.
A.
+3-t.h'^
P^-
l^
e,-
:L
p^
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h2
(7.)f/+^)
A_,
t
z
by
3.
-
.^
C.
-
^
D-
Solving these equations
by
the
method
of
determinants
for
the
constants,
and
imposing
the
condition
of
C
—
^
oO
gives the results
listed below.
Because
of
the
length of the terms,
the
numera-
tors
are
listed
separately,
as
is
the denominator.
They are
not combined
to give
the
complete
constants
as
it
is more
convenient to
work them out separately
for
a
specific
case.
The numerators
of
the constants are:
Num
A
^
/o
8
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k3
Nun.
C-
-ff^^f.^.
Ot^Xl^.^}}^
T^T^M-^.^.^^
Nun
/^^^
:f-
]
D^t46u4H^^f4rj
Num
5=0
Mum
.0
=
2^
Num
D
^
_/
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kit
^J'
For
the
stresses and
displacement due
to
the
normal
forces
(^
S
cos
?5
)
and
the
shearing
forces
(-^
S
slni?^),
the above constants
are
to
be
placed
In the equations
as
listed
below,
using
either the
subscripts
1
or as
appropriate.
<=
-(^^
^^ ^
^i^)
<^^^
U
s
3.
A.
-^4^
V
-
'/:M^-^->5-^c
t^.oj
^^[O^)
^
^/^i^U'Bi-
-^
^-^^
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ks
THEORETICAL
ANALYSIS
V.
SUMMARY
OP THEORETICAL
ANALYSIS
The
radial
and
tangential
stresses
due
to
both
the
constant
component
(i-
S)
of
the
normal
forces
and the
normal
forces
(^
S
cos
^6)
together
with
the
shearing
forces
(
-^
S
sin
2
6
)
must
be
added
toge-
ther
to
give
the
total
radial
and
tangential
stresses
in the
ring
and
plate.
The
radial
and
tangential
stresses
due to
the
constant
component
(J
S),
together
with
the proper
constants
are:
-
~
TTi^)
L-V/;^-;
(t^-t^)
-L^i^
(/f^)-i,^t^
Oy.)l
Z
E
(lyT)
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46
The
radial
and
tangential
8
tresses
due to
the
normal
forces
{}
S
cob^B)
and the
shearlne;
forces
(-
i
S
sin
2©),
together
with
the
numerators
and
denominator
of
the constants
are:
^^
=
(
2
A^ ^>z
0^^^
-t
±^
C,)
^^
a«
o
Nun. /i^
=
-
1
|C-^.f
;
i^T^
^
t
^.
('A-J^'
[^y^^^
-
^-.-
''
('Ml
''-
'r-f[-M'^^^
('^^-
':('^}^M'^-
'-
('y^y'y'4
Hum
Ao=
-
-f-
f
/5e«.«..MaforJ
Nun
g
=
O
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^7
Num
h
Num
U
=
-
'
of
-1^
-h
i-
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w
It Is not
practicable
to combine
the
above
nuunerators
and
denominator
because
of
their
length.
At
this
point,
it le
beet
to substitute
actual
values for
a specific
case
when
using
the
above
expressions to deteinine
the
radial and
tangential
stresses.
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APPLICATION
OP
THEORETICAL
ANALYSIS
TO
TEST SPECIMEN
The
results
of the
theoretical analysis are
now
applied to
a
flat
plate under
a
tension
load,
with
a circular cutout symmetric
ally placed
and
re-
inforced
with a
circular
ring.
The dimensions
of
the test specimen
are
taken
from
a
structure
which
was tested experimentally
for
verification
of
this
analysis.
The
dimensions
and
loadin;^
are:
a
2.5
inches
b
a
3*5
inches
^0=
0.0l|.0
inches
t
s
0.120
Inches
yt^ =
0.3
E
=
10.3
X
10^
Ibs./sq.in.
See
the
section
TEST
DATA
AND
EXPERIMENTAL
RESULTS
for
details
of
the
test
specimen.
Applying
the
equations
of
Part
V
of
the
theo-
retical
analysis:
For
the
constant
component
(i
S),
(o^
.
=
Sf
O.JOS'-
IlIJ.
1^9
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50
(^
^
S
(
O.S-
4-
^'^^^
)
For
J-
S
cos
3d
component of normal forces and
-
^
S
sin
2©
shearing
forces,
Combining
the above expressions
to
give the
equations
for
radial
and tangential stresses
due
to
the
total
loading:
In
the
experimental testing,
the data
was
taken
at
8
= 90°
oc
a
^
-
-
y.
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51
For
this
location,
the
above
equations
reduce
to:
<.
=
S(-0.,3J
7- if
_
^2.6,
^^
^
s
i
^i=^ii
+
i.
Z^)
<^e
^
s(/.o -I-
3
:i.(o^
Since
the
experimental data taken
was
for
G
at
=
90°,
and for
r
=
2.6,
3.0,
3.^^,
and
3.6,
the
above
equations
were
used
to
obtain
the
follow-
ing
table.
TABLE
I
^v-
<^..
'^0^
2.5
s(i.900)
2.6
S(1.701)
3.0
S(i.i35)
34
S(o.78l)
3.5
S(0.710)
3(0.833)
3.6
S(0.85l4.)
Since
the
cross
sectional
area
of
the
test
specimen
at the
point
where
the
loading
is
applied
is
1.2
square
inches.
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52
s
r
^
— 733
-2
—
r
Tor
e'&cA
P
=
1000 lbs,
which
is
applied
to the
structiire.
See Fig.
12,
page
6o
for a
plot of the theore-
tical
data
as
determined
above.
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TEST
DATA
AND
EXPERIMENTAL
RESULTS
PROM
TEST
SPECIMEN
The
details
of
the
test
specimen
are
shown
in
Pig.
10,
page
55.
The
reinforcing
rings
are fas-
tened to
both sides
of the
flat
plate
by
hammered
rivets,
spaced
as
shown.
Past
experience
has
shown
that
the
reinforcement
acts
very
much
as
an
inte-
gral part
of
the
sheet
when
it
is reinforced
with
two
concentric
rows
of rivets.
Figure
11
contains
photographs
of the test
section
and
test
equipment,
made while
the
tests
were being made.
Table
II
is
made
up
of the
data as
taken during
the
experiment
and
it is
used
to
make the
plot of
experimental data
as shown
in
Pig.
12,
page
6o.
Strains
were
taken
by
use
of
the
SR-i4.
strain
indicator
together
with
a multiple
switch
box for
ease
of taking the
readings.
The gage
factor set-
ting
on
the
strain indicator
was
1.77
and the
gage
factor of
the
strain gages
was
1.68.
This
necessi-
tated
that
all
strain
measurements
be
multiplied
by
the
factor of
T^^
•
In
the
determination
of
C5l
it
was assumed that
G
53
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51^
e^^O
f
an assumption which
Is obviously
quite
accurate. This
allows a calculation
of
6^
by the
simple
relationship:
where
6-,
Is the strain as measured
by
the strain
gages on
the test
specimen.
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55
DETAILS
OP
TEST
SPECIMEN
f
s
t
t
t
T
OOfO
\0./20
Mill
fljK^
<3K(^
Plah
-
E ^
ST
A
i4
it^
I
K
U
il^
A
Hoy,
i:5
35
it\C£D
//V
FA(f^5
ON
CTH
S/P£$
OF
f^eS.
IN
OHN\S
6/.
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TXO»
II*
S:qperimental Test
Set<<xp
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57
TABLE
II
EXPERIMENTAL
DATA
BX
:}ap:e
strain indicator reading
for
'
P
equ£
il
to:
12^000
2.000
Il.OOO
6.000 8.000
10.000
14.000
1
4,^82
13,902
lk,175
li^,if30
li;,6k8
111,
742
14,535
14,675
14,905
15,120
15.290
2
14,915
14,996
15,052
15,070
^
9,030
9,140
9,193
13,165
13,305
15,288
15,420
9,233
9,265 9,287
9,309
9,245
12,975
13,438 13,570
13,703
13,836
14,020
16,210
?
15,220
15,560
15,712
15,868
16,010
6
10,012
9,950
9,975
10,023 10,083
10,130 10,130 10,015
7
11,830
11,860
11,952
12,050
12,168
12,285
12,402
12,548
8
1
9,158
9,250
9,434
9,620
9,830
10,038
10,255
10,535
Strain
(inches
per
inch)
for
'
P
equ£il
to:
12.000
.000
4.000
6,000 8,000 IC.OOO
14.000
351 637
905
1,162
1,405
1,630
1,810
2
174 273
371
4f5
247
540
598
618
^
115
172
214
270
284
226
190
330
463
595
728
648
861
1,045
5
68
200
340
492
790
990
6
-65
-39
12
76
124 124
3
7
32
128
231
355
478
601
755
l,W
97
290
485
705
924
1,155
Average
Strain
for
P
equal
t
0:
and
8
Radius
2.000
a.
000
6.000
T.ooo
10.000
12.000
14.000
1
2.6
224
464
695
934
1,165
1,393
1,629
2
and
7
3.0
103
201
24
67
301
405
50Q
600
687
i
and
6
3
4
3.6
113
162
204
826
115
and
5
129
265
402
Sl^k
6^8
1,018
<s<
for
Radius
'
P
equal
to:
and
8
2,000 li.OOO 6.000
8,000
10.000 12,000
14.000
1
2.6
2,310
4,780
7,160
9,620
12,000
Ik,
340
6,180
16,800
2
and
7
3.0
1,060
2,070 3,100
4,160
5,240
7,080
^
and
5
34
3.6
260
690
1,160
1,670
2,030
2,100
1,150
and
5
1,330
2,730
4,130
5,600
7,090
8,510
io,4Bo
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COMPAHISON
AND
DISCUSSION
OP TEST
DATA
AND
THEORY
A
plot
of
experimental
and
test
data
for
tan-
gential
stress
versus
radius
for
varying
total
loads is
shown
in
Pig.
12.
In
general,
the
agree-
ment
is
excellent,
especially
at
the
inner radius
where
the
stress
concentration
is
critical.
At
this
point,
the
per
cent
variance
of the theore-
tical tangential
stress
from the
experimental
tan-
gential
stress
at
the
varying
total
loads
is:
At
P
5
12,000
lbs.,
10.5^
P
-
10,000
lbs.,
9.7/^
P
=
8,000
lbs.,
8.7/^
P
=
6,000
lbs.,
11.8^
P
=
ij.,000 lbs,,
ll4..3^
P
=
2,000
lbs.,
10.7/^
Observation of
the curves of
Pig.
12,
indicates
that
the
original sheet
between
the rings
is
under
a
slightly higher
stress
than
the
rings,
perhaps
on
the
order of
ten
per
cent,
since the experimental
stresses
(taken
on
the
rings)
are somewhat lower
than the theory indicates.
This
is
probably
due
to
the
fact
that
the rings
are not
acting in complete
accord
with
the
assumption
that they
are
integral
58
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59
with
the
sheet.
It is
likely
that
an
Increased
number
of rivets
would
result
In
better agree-
ment.
Non-agreement
between
the
theoretical
data
and
the
experimental data
Is quite
pronounced
in
the rin^^ area
towards
the outer ed^e. This
is
probably
due
to
the
fact
that the
ring does
not
act
completely
as
an
integral
part
of
the
sheet,
and
if
a
greater
nuniber
of
rivets
were
used, or
if the outer
row
of
rivets were
placed
nearer
the
outer
boundary of the ring,
this part
of
the
ring
would
tend to carry more of the load
and
give
better
agreement
with
the
theory.
The
assumption
of
an
abrupt change
of
stress flow
at the
outer
boundary
of
the
ring
as
explained
on
page
39
is
also
responsible
for
making
the
theoretical
values
high
at
that point.
However,
the
stresses
at
the outer
edge
of
the
ring
are
not
critical,
so
that
the
variance in
this
area
is
inconsequential.
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A
PLOT OP
lEXPEI
-']
h
--'
—
:]
T ':TI'
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CONCLUSIONS
As
discussed
in
the
previous
section, the
comparison
of
the
theoretical
computations
and
the
experimental
data
for the
tanf^ential stress
showed excellent agreement,
and
indicates
that
this
method might
be
used
to
solve problems
of
reinforcement
design
in the case
of a
flat
plate in
tension
with
a circular
cutout.'
Sots
of curves
could
be
drawn
up
for various
thick-
nesses
of
plate
and
ring
and
also
for varying
widths
of
the
ring.
It
would
also
be
useful
to
apply
this
type
of
analysis
to
investigate
the
possible optimum widths of
rings
and
thicknesses
of
rings
for reducing
the
stress
concentration
by a
predetermined
amount,
A
study of
the
effect
of
rivet
placement
on
the
stress
distribution
across
the ring is
sug-
gested
as
a
worthwhile
topic
for
experimental
research
in
this
field.
6X
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APPENDIX
A
REFERENCES
(a.)
Timoshenko:
Theory
of
Elasticity ,
193i|.,
First
Edition, Eighth
Impression,
pp.
52-79,
McGraw-Hill
Book Company, Inc. New York
and
London,
(b.)
Tiraoshenko:
Strength
of Materials , Part
II,
Advanced
Theory
and
Problems,
1930»
PP»
k-^k-
14-57*
D*
Van Nostrand
Company,
Inc.
New
York.
62
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APPENDIX
B
BIBLIOGRAPHY
1,
Kuhn, Paul
and
Moggio, Edwin
M.:
Stresses
Around
Rectangular
Cutouts
in
Skin-Stringer
Panels
under
Axial
Loads,
NACA
ARR
June,
19k^.
2,
Kuhn, Paul;
Duberg,
John
E,
and
Diskin, Simon
H.i
Stresses around
Rectangular
Cutouts
in Skin-
Stringer
Panels
under Axial
Loads
-
II,
NACA
ARR
3J02, October,
I9I4.3,
3,
Parb,
Daniel: Experimental
Investigation
of
the
Stress Distribution
around Reinforced Circular
Cutouts
in Skin-Stringer
Panels
under
Axial Loads.
NACA
TN
12l|l, March,
191^.7.
1^..
Kuhn, Paul; Rafel, Norman and
Griffith,
George
E,
:
Stresses around
Rectangular
Cutouts with
Reinforced
Coaming
Stringers.
NACA TN
1176,
January,
19[|-7.
5,
Dumont,
C:
Stress
Concentration
around an
Open
Circular
Hole in a Plate
Subjected
to
Binding
Normal
to the
Plane
of
the
Plate.
TN
7l|.0,
December,
1939.
6,
Podorazhny,
A.
A,:
Investigation
of
the Behavior
of
Thin-Walled
Panels
with
Cutouts,
TM
IO9I1.,
September,
191^6.
7,
Kuhn,
Paul
and
Moggio,
Edwin
M.:
Stresses
around
Large
Cutouts
in
Torsion
Boxes.
TN
1066,
May,
191^6.
8,
Moore,
R.L.J
Observations
on
the
Behavior
of
Some
Non-Circular
Aluminum Alloy
Sections
Loaded
to
Failure
in
Torsion,
TN
1097,
February,
19^7.
9,
Ruffner,
Benjamin
F,
and
Schmidt;
Calvin
L,
Stresses
at
Cutouts
in
Shear
Resistant
Webs
as
Determined
by
the
Photoelastic
Method,
NACA TN
9814.,
October,
19^+5.
63
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6k
10.
Kuhn,
Paul:
Skin
Stresses
around
Inspection
Cut-
outs
•
NACA
ARR,
December,
19lfl.
11.
Kuhn,
Paul:
The
Strength
and
Stiffness
of
Shear
Webs with
and
without Lightening
Holes.
NACA
ARR,
June,
I9I4-2.
12.
Kuhn,
Paul:
The
Strength
and
Stiffness
of Shear
Webs
with Round
Lightening
Holes
Having
14.5°
Flanges.
NACA
ARR,
December,
191^-2.
13.
Kuhn,
Paul and
Levin,
L,
Ross: Tests
of 10-inch
2i|.ST
Aluminum Alloy
Shear
Panels
with
l| inch
Holes.
NACA
RB,
3F29, June, 19^3.
li).,
Kuhn,
Paul and
Levin,
L.
Ross: Tests
of
10-inch
2I4.ST Aluminum
Alloy Shear Panels with
l|
inch
Holes.
II
Panels Having
Holes
with
Notched Edges.
NACA
RB
Ll|D01, April,
19y4-.
15.
Moggio,
Edwin M, and Brilmyer,
Harold
G,:
A
Method for
Estimation of Maximum
Stresses
around
a
Small
Rectangular Cutout in
a
Sheet--Stringer
Panel
in
Shear.
NACA ARR
Li|D27,
April,
l^kk-.
16.
Hoff
, N.J.
&
Kempner, Joseph:
Numerical
Proce-
dures for the
Calculation of
the
Stresses
in
Monocoques.
II
Diffusion of
Tensile Stringer
Loads in
Rein-
forced
Flat
Panels with
Cutouts.
NACA
TN
950,
November, 19i+4.
17.
Kuhn,
Paul
and
Diskin,
Simon H.
:
On the
Shear
Strength
of
Skin-Stif fener
Panels
with
Inspection
Cutouts
ARR
L^COla,
March,
19i4-5.
13.
Hill,
H.N.
and
Barker,
R.S.:
Effect
of Open
Cir-
cular
Holes
on
Tensile
Strength
and
Elongation of
Sheet
Specimens
of
some
Aluminum
Alloys,
TN
197ij.,
October,
19^9.
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65
19#
Relssner,
H,
and
Morduchow,
M,
:
Reinforced
Cir-
cular
Cutouts
In Plane
Sheets.
TN
1852
April,
1914.9.
20,
Hoff,
Boley
and
Klein:
Stresses
in
and
General
Instability
of
Monocoque
Cylinders
with
Cutouts:
I.
Experimental
Investigation
of
Cylinders
with
a
Symmetric
Cutout
Subjected
to
Pure
Bending.
NACA
TN
1013,
1946.
II.
Calculation
of the Stresses
in
Cylinders
with
Symmetric
Cutout.
NACA
TN
10li|.,
1914-6.
III.
Calculation
of the Buckling
Load
of
Cylinders
with
Symmetric
Cutout
Subjected
to
Pure
Bend-
ing.
NACA
TN
1263,
1914.7.
IV. Pure
Bending
Tests
of
Cylinders
with
Side
Cutout,
NACA TN
I26I4.,
I9I4.8.
V. Calculation
of
the Stresses
in
Cylinders
with
Side
Cutout.
NACA
TN
II4-35,
I9W.
VI.
Calculation
of
the
Buckling
Load of
Cylinders
with
Side
Cutout
Subjected
to
Pure
Bending.
NACA
TN II4.36,
March, I9I4.8.
21.
Journal
of the
Aeronautical
Sciences
(a.)
Vol.
15,
pp.
171,
I9I+8.
Effects
of
Cutouts in
Seml-raonocoque
Struc-
tures
by
P.
Cicala,
(not
assuming
a
rigid frame)
(b.) Vol.
12,
pp.
1|7,
1914-5.
Graphical
Solution for
Strains
and
Stresses
from Strain
Rosette Data
by Continu.
pp.
235
Stress in a
ring
loaded
Normal
to
its
Plane
by E.R.
Walters.
(c.)
Vol.
6,
pp.
I4.60,
1938-39.
Analysis of
Circular
Rings for
Monocoque
Fuselages
by
J.A.
Wise.
(d.)
Vol.
7,
pp.
I4-38,
1939-i|-0.
Circles of
Strain by
J.
A.
Wise.
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66
22.
Lundquist, E.E, &
Burke,
W.F,:
General
Equa-
tions for
Stress
Analysis of
Rinc^s,
NACA
Tech.
Report
No.
509,
193k'
23.
Niles, & Newell,
Airplane
Structures,
Vol.
II.
Second
Edition,
pp.
265-281.
Circular
Rings.
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APPENDIX
C
SYMBOLS
(1.)
a
-
Internal
radius
of
reinforcing
ring
-
radius
of
cutout,
external
radius
of
reinforcing
ring,
modulus
of
elasticity,
radial
strain,
tangential
strain,
modulus
of
shear,
shearing
strain,
total
load
on plate
-
lbs.
radius
in inches,
stress function,
loading
on
plate
-
lbs. per
sq. in.
radial stress in
ring
-
lbs, per
sq.
in.
radial
stress outside
ring
-
lbs. per
sq.
in.
tangential stress
inside
ring
-
lbs.
per
sq.
in.
tangential
stress
outside
ring
-
lbs.
per sq .
in.
thickness
of
ring and
sheet,
thickness
of
sheet,
shearing stress
-
lbs.
per
sq.
in.
67
(2.)
b
(3.)
E
(i|-.) 6
(5.)
e.
(6.)
G
(7.)
^.*
(8.)
P
(9.)
r
(10.)
^
(11.)
S
(12.)
(13.)
<-.
(1I4-.)
(15.)
(16.)
(17.)
^
(18.)
^.
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68
(19.)
6
-
angle
measured
from direction
of
load
(see
Pig.
1,)
(20.)
u
-
radial
displacement,
(21.)
J^
-
Poisson's Ratio.
(22.)
V
-
tangential displacement.
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S66<
pla
Thesis
S669
13196
Squi
res
Theoretical
design
of
reinforcing
rings
for
circular
cutouts
in
flat
plates
in
ten-
sion.
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