San Fernando Valley State College
Deflection Rings with Holes
A project submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
Harry Shoji Hatanaka
J anuar.r, 1972
I
I
l , ,_. .~--~-------------------·-·---------~---·-·---------~-----.----.------------------------~------J
The project of Harry S. Hatanaka is approved:
San Fernando Valley State College
January, 1972
""''~-····~·-~----·-----~--------·-------------·------··---·----~--~~------·-·-------~-···-
---1
I
I
I
ABSTRACT
Deflection of Rings with Holes
by
Hari~
Shoji Hatanaka
Master of Science 1n Engineering
January, 1972
The project covers the analysis and testing of circular rings with holes&
are considered.
~io
static loading conditions
The first loading condition is diametrical
compression under two opposing concentrated loadse
The
second. loading condition is out-of-plane four-point loading.
Two analytical approaches are presented for
deter~
mining the deflection of these rings under both of the
above loading conditions.
A test program is carried out
to verify the analytical results.
,
I
I
l.------··-·-····----··------·---,.----------------------·-·------··---·---- ________
!
j
Table of Contents
Page
Introduction
1
2.0 Statement of Work
2
1.0
J.O
J.l
0Ut-.pla.11e Load.i.."'lg--Solid Section Ring
3.. 2
4.0
P.?.:oposed
Ar~ysis
• " •• * ••• a • • .,. •
5.0 Theoretical Calculations
6.0
Propos~d A..'1~ysis C?.~culat:l-ons
8.. 0 Test
Data/Res~uts
9.0 Discussion of
T~:;st
~
• o
~
... "'" • •
6
~
., .,
o: • • .JI;. "it
.,. • • • • • ., • •
~
,.
~
It
a ,.
.5
8
10
12
Results
13
9.2
10.0 Concluding Rena:rkG
10.2
Out-pl2.n:e Lending
15
lLO
1J.O
14-.o
'7Q
l ,
Illustrations and. T.ibula:tions
PageFigure 1- A ~~o Degree of Freedom Gimbal System ••••·~·····~·¥··~···· 1
Figure 2 Loading Conditions for InFigure J Test Ring Configuration
Figure 4
Fig-,.u·e·
~~d
Out-plane Deflections •••••• 2
••••"••o••••·~·· .......
Definition of Parameters ...............
4 .... "
& ..
~• ...... ~~~"·"~
••••••
» ••
N •
~G
•
•
•
3
4
5 De.firlition of Section Pararneters ..................... ~ , ., •• ~ .. * ~ 6
Figure 6 Defi...11.ition of saver ......... ~. e ...
a. II~. tt
Fi~1re 7 Stress Concentration Factor, Kt' for a Slotted Bar
IJ. • • • I . . . . . . . . . 4 .
1)
in Bendirlg •••••••••••• ~ • ~ •. " .... ,. u • • ,
Figtlre 8
Test Setups
~
a ••
•
tl' • •
a
l) •
.,
0 . ') •)
!}
o • • -· • • e " • •
••••••••••~•~•o••.,••eJSa•o•~••••'3•"•oe-(t••w•"o"•
8
9
11
Figure 9 In-plane I...oading, DY vs Load 11 P11 , r"" 1.9125
I
Figure 10
~
r "" 2 .. 9125
3.5
I
Figt;re 11
D /2 vs Load "PH, r "" 1.9125
X
Figure 12
37
p
Figure 13
J•...
Figure 14
39
Figure
15 Out.-pla:.1e lo3.di.'1.g, Deflection vs Load
~
17
I
Figure
):' "" 1.9125 ,
~
I
Figure 16
np!!~
8. •
..
;
lj.Q
5.
Table I
Correcti~te
Table II
Theoretical Results for Solid Rings; P = 1 a."1d 0 = rr/2
Table III
Proposed Analysis Results for Rings wit.h·Holes;
p = 1 and e c rr/2 ••••ac~*~···o•e~~··4~0~0·~b~·~·a~~·~f0~ 10
Table IV
Test Results,
Factor-s for Thiel": Rir1gs
~.
,
in~plane Load.ing-~Solid
I
a
lOt, •.•• " •• ,~.
tJ • •
o ••
•to
10
Rbg (r ~ L9125) • 16
Illustrations ~ld Tabulatio~D «cont 1 d)
Page
Table
V
Test Resul·ts,
in~ plane
loading=-.250 Diarnet:;;;r Hole (r
Table VI
.500 Diametl2'er Hole
Table VII
.?50
-
Table VIII
~
Hole
•••••••
Table X
.500
Table XI
.750 Diar1eterr Hole
Test Re.sul ts, out-plane LcB..ding~-Solid Ring
t'
"'=
4ii.3Sit0~·
17
20
21
a~•a~o•
22
.... ., "'
I
19
<'!.,
• • ~., ;&
~
~
23
.r.
25
1. 9125)
.• 250 Diamet<er Hole (r ""'
Table XIII
e • a-. • • •
I
Hole
«
~
18
_L_
.250 Diamet·..=r Hole (r "" 2$902.5)
Diameter:~
..
I
= 2o9125)
Solid Ring (( r
.... 0 •
~
Table LX:
Table XII
(
Dia~.etm~
= L9125)
lo9125)
.ao":!}
~
.. .,
I
t''l
..
Table XIV
.soo
DiamebBr Hole
Table XV
.750
Diamet•.er Hole
Table XVI
Solid RL~g {r ~ 2.9125)
28
Table XVII
.250 Diarnet,:.;r Role
29
Table XVIII
.500
JO
I
~.
-~~'i)~~
Hole
Di~~et~r
26
27
Table XLX:
Table XX
Effect of U3ing Solid Ring Analysis in li'a•Jor of Using P::copcsed
Analysi-s { L~~plane Load.ing) •• ".~~a a ""(I" •• -·.~a
~~II~
a.,
9
Effect of Us:t.11g Solid Ring Analysis in :?aver of Using Proposed
L-oa.d L~g ) • a -a
~~
~
e
a ., • a • a a " ~
-& ;) 1.\ !l ., ..
&
Table XXI
Anal:)tS is (Out~p1.ru1 e
3
..
II 6
G
•
g .a!)$.
0.
0
4
"' 9
.g;, 9
•
1t
tt
i)
••
§I \\
<J !
!'
I
·,
•
0
4
~. ~ ~
'3
~
•
J2
1. 0 Introduction
In the missile guidance field of endeavor, the use of stablized plat-
forms to stablize the missile tracking mechanism during launch is a:l..r1ost
a foregone conclusion.
These stablized platforms usually re.quire tvm, and
sometimes three, degrees of freedom.
degrees of
freedom~
In order to achieve the multiple
the stablized platforms are constructed out of two or
three structural members.
These structural members are connected in such
a w.anner as to allo;.r a pivoting or gimba.lling motion in one direction only.
Each pivot axis is made perpendicular to a prior ·pivot a.,"{iS.
for an illustration of a two degree gimbal
See Fig,. 1
system~
member 2
pivot axis 2
:nember 1
pivot axis 1
Figur~
.1 A
two degree of freedom gimbal systero.
The outer member, member 2, of the ttv-o degree of freedom systeM can
be approximated, if r.ot exactly, as a circular ring in ma1·w
instances~
Theoretical a~P~yses for deflection and stress of solid section circula~
rings are available.
Hm-1ever, no a.na.lyses can be found
to
account for ·
uniform sized holes in circular rings; those holes ·are the result of attachment points for the gi<'7lbal pivot a::\.es·.
This paper will therefore be directed
towards the analysis o:r this type of strc1cture.
2
2. 0 Statement of. \>fork
The
plane
propos~d
de~lections
project is to study the effects of holes on in- and cut-.
o£ circular rlngs.
loading ':lill be considered for
loading will be considered
Only the condition of
out-pl~e
deflection while
fo~r-point
or~y
for in-plane deflections {Sea Fig.
dirunet.r:i.cal
2)o
Loading
will be of a static nature for beth caseso
I
0-
p
p
a). Out...pla.ne loading Condition
/
b)
Figure
,g
In=plane Leading Condition
Loading conditions for
21~ ~"ld ou~pl;;:u1e
deflections
Theoretical expressions are a7ailable i·xhich give deflection rc:.m1 ts
for both the in- a..."ld out= plan~ leading conditions shmm in Fig. 2..
equations,
hOHB"J"er,
apply only to rings tr.at r..ave
perties, iee. no holes or irregu.larities.
which
t~ke
These
C{in.St&i1t sr:;c~ion pro~
No at<alyse5 have b6en found
into acco'.lnt the effect of holes;
the theoretical and prop:>sed analyses will be
therefo~·e a.."'l.
verifi~d
aa::.lysis 1-ri.ll
by .a test prog:ra.rte
3.
Eight test rings 1iill be fabricated out of 606T'-T6 ·a.J.u..'llinum a.11d tested~-_
(
The eight rings will be broken
ring in a given set
r.~11
d~rm
-
into two sets of f.rrur rings each. - Each
be the same size&
The
dif~erence
being that one
ring -vrill have no holes while t.t'le other three in ths same set will h.o.Ye
varyL11g size holes.
The
res~ts
on the solid rL11g
theoretical results and will serve as the standard
ri.11gs vrill be
be compared to the
~gaL11st
which the other
'I\-ro different size rings a'l"'B beLYJ.g used to check
c-cmpa:r~-•
for arJi,r size effects.
~~ill
Both sized rings will have tlne sarne section pro-
perties hence eliminating these as test variables..
F!'igure J depicts the
test ring configt1rations.
.125
L
~
}til1g
If.
D
0
D
_!_j
t-1.000
D
r
2
no holes
.2.50
.3
.sao
4
.?50
5.930
5.9.)0
5.930
5
no holes
.). 950
6
.250
7
.500
-750
).950
.3.950
1
8
Figu.re
2
r
D
r
5~950
3~950
Test ring configuration
4
J.l Out-Pla..'1e Loading-=Solid Section Ring
The deflection, y • and bending moment, 1·1, for the loading condtion
shmm in Fig. 2a can be e..xpressed as:
y =
~{2(0 - l}~inS
1
+ (29 + 2
rr)cosS
+
TT -
(1)
2
w + A/(6 + 20 - IT)cosO - 2(.3- e)sine + rr + 4e -
fl} -
M = Pr(sinO - cos9)/2
(2)
= ring deflection
M = bending moment
P ... applied concentrated load
r ""' mean ra.1.ius of curta.ture
0 ~ angle at which y arrl M are measured
where:
y
E - modulus of elasticity
G = modulus of rigidity
I = moment of inertia o£ ring cross-section abo1J.t
w
.
the lv=ax:ts
K = torsion~~ shape i'actor
>.. - EI't-1'/GK
TT
A,
D.."'ld
th
-· +
• e mce:nen,.
.fJ'
o.~-
.
by
:l.!.i.ert':ta, I. , are g1.ven
•
w
.!.1..
~.>1li:l
f' , 1
.
, -+.
h.l.P"'•
-~. 2
_o.u,m.n.ng
re,uJ,..,J..Ol.'lS.
{ 3)
1mere:
a and b are defined belo-tT
•p
2a.
.
J....
h-a.,'(iS
·~---r-'
j
3
~-_J
~-~r
.--{
~2b-~
Def~1ition
---•: .. a.x:is
y
of parameters
i --
J. Z
In-Pl~me
Loading--Solid Section
Ri~g
·-
-----
a ring loaded in diametrical compression,
the following relationships: 3
The deflections for
Fig. 2b, -are given by
see
(5)
(6)
= change in diameter in the y-direction
~D
where:
y
= change in diameter in the
P = applied conc~ntrat~j load
E = modulus of elasticity
~D
X
Ih
~
moment of inertia
about ~he h=2..Xis
... 2a(2b)--'/12
of
x~direction
ring cross-section
(7)
Kx
If the ring is relatively thick, Dx must be multiplied by
o.nd Dy
by K where K · and K are ccrrective factors th.:1t depend on the :r:-c;ttio of
y
X
y
.
_,·
outer radius, r
0
Tablf:l 1
it1..Y1er
radius, r i' and have the foll-:,.;qing values:
4
Corrective factors for thick rings
1.4
1.5
1.6
1.7
L8
L9
K
1.050
1.115
1.175
1.225
1.27.?
LJ25
1.360
K
y
1.030
L055
1.090
1.114
1.155
1.180
1.225
X
11
to
1.3
ro/ri
4.0
,
Prooosed" .:1\.nalysis
The following analysis is prop? sed to accour.1t .fpr the possible effects
I
o.f
holes in circular rings.
The pointd of rr:ax:inu.m stress coincides with
the points of maximu.fl deflection for both loa.d conditions considered in
this reporL
The holes being located at these· f,'-cints of rr.axim;.ua stress
have a tendency to w..crease the stress levzl above that of a ·solid ring.
6
If we assume:
1) that the stress·concentration due to the hoies varies
..
linearly 'vith dista.."'l.oe from t.h.e holes, 2) the stress concentration becomes
d.ist~•ce
neglible, i.e. ·stress concentration factor equals 1.0, at a
of
ltD 1-rhere D is the dia.rr!.eter of the holes i."1 the ring, a.nd 3) that the de-
flections in the rings are directly proportional to the stress leyels in
the ring.
Hav:i.."lg made the above assumptions, we are no>-r ready to attempt
to explain hmv to account for the holes.
The effect of holes ha.s a tendency to increase the deflections since
the stress levels are higher in these r;ngs.
Equations (1),
(5)
~~d
{6)
shm-r that the applied load, P; the radius, r; and the section properties,
I w· t I,n 1 2.nd K are directly related to deflectiono
radius of curva.ture do not change_just because
The applied load and
sc~e
holes have been added.s
However, I , Ih' and K all do cha.'1ge ..
'tf
Let us a:!ccunt for the changes j_n the section properties by arriving
(
at expressions f'or equ.ivalent section
The section properties
prcperties~
for the circwlferential length of {2rrr ~ I1-D) are dei'ined in Eqr1.!s (3), (4)
a.'id (?)a
If one assu.:~e:s that the cross section height varies line:rrly ·
over the dia.."lleter of the holes, the section properties a.t the hol·:s
be defir-1ed. as Eqn 1s {8), (9) ard (lO){See Fig" 5).
These
ties apply for only 1:-D of the c:i.rcun1.ferential length&
T
2a.,., 1.000
2b ,
FigJ.::-e
2
Definiti.on of section
p,:~.ra.;1.eters
~125
C<!..:~
7
I
h
.
.
3·
net= (2b).LZa-
2~7/12
. (8}·
....
(9)
K
.
net = ab31l6/J ~ 3.36(b/a)(l - b4/12a4l7
~ ~b31)L6/J - J.J6(b/~)(l - b4{12ai17
(10)
If we n<Y...r arrive at equivalent section properties by weighi.t'lg the
different properties in proportion to their applicable circu.ntferential
lengths, Eqnws (ll). (12) and (13) result.
Equations (11), (12) and {13)
are to be used in place of Eqn•.s (J), (4) a."1d (7) respectively 1men de=
termL~L~g
the deflections of rings through the use of
E~n's
(1), (5)
and (6).
K
eq
I
= (2n'r
- 4D)K + 4DKne..,
+
(11)
2rrr
w e-q ,. (2rrr ... 4D)Ivl. + 4DIvr net
(12)
2Tir
I h · eq
~
r~
~~~~~-
-
hn\1 T
-~
~h
+ 4DT-h
(1.3)
2rrr
In order to account i'or the higher stresses <t.t the holBs,
2.
correction
, will be forn.ulated. This corrct~tion factor will be. based
cor
upon the stress concentraticn factor at the holes.
factor, F
F
cor
= 2ITr
+ 4(2 x Saver x ltD)
The terms 2rrr represent the mean
in the second. term
L~
the ntwierator
circu•·~feren~$
dasign~tes
of the
rin~.
The4
the number of holas in the
ring 'HhUe the 2 stands for the fuct th:at the stress on either. side of
The p2.ra.'noter
Saver is the avsra!!e
.
"" stress at-ove the
nominal stress over the affGcte.d area due t6 t.he holes (See Fig~ 6)$
the hole is highero
8
(
smav--------------1~--------­
-sa.vers _
nom
Definition of S
aver
The value of S
Turning to Fig.
7,
aver
can be determined i f the value of S
max
one can find S
max
by letting t
is knmm.
= ra
Equa:tions (1), (5) a.'l1.d (6) are to be mu.ltipiied by tht:l above correction
factor to arrive at the final results.
Iri su..rmnary,
(14)
~D
X
=
.lqr:~p
A
PrJ
cor
(15)
------~----
_EIh eq
PrF
/.
cor,z{e - l)sin9 + (29 + 2 = rr}cose + rr - 2
8EI
•
1'1 eq
+
+ 29 - TT)cos9 - 2(J- G)sinB + TT + 4B
(16)
y - .
t..oc/(6
where:
- 9.7)
= EIw eq /GKeq
Ih eq = eqn (13)
I
· = eqn (12)
. w eq
A. eq
Keq
= eq.n (ll)
Using Equations (1), (5) and (6) the defle•::tions for our 11 solid" test
rings arc
sun:.T.a,.,~ized
e ""
To- ohto..ii1 the defJ.ection values for "P" o:i:.her tl:x: unit.y, 1rr0l-
rr/2o
iil Table II..
The m:.t-pl.a:!e re::.uJ.,ts are for P ., 1 a.nd
:
tiply the values for ~:ft 6Dy and 6D
. X in Table II bv the valuo of P ..
vr
v
tl:l
~IJ:Effi•·················~
W;:\4~
,_j~-
1:.::.:-
.iii:f !:ll.rF'"
,2-4
I
''
tf:i:
i+t'
~·-
2:2
~;::
-~· -~·::<
.tm2a-.
.. -,....
~·-.4
,~
2.0
H:±:
'·i
~
·c-0~
.. i+t, . H
,,
-
+++trr
+++' '1±1-l~,__.1_]_-l u
H4-H·
j
T..-;-T
r
f+l~~
-/,':·
i'TT
1.8
,-t''
ii1Jll+-
-~~~
"}!''lflmt
''+ ' ',
-ri-t·
·p"
H-''
-I
·1+++1:HH'
~~~~
.....
r-··
,.~~;
.
+++trn:i
...
~: .....~.~-
')
J..
St::---ess
Cor~c~ntratin:: Fact.,~r, K.a..e
"'
1:;
fer a Slot·t,ed Bar in Bend:l.ng,..f
~r
·:..:..::
i2~
'•:J
10
Table II
Theoretical Results for Solid R:L-1gs; P "" 1 and
Test Ring #
1
(.5.950
0.835 X 10=.3
0.2)6 X 10~3
2 (.3.950 O.D.)
6& 0 Proposed
P.~'1alysis
~D
~D
y
o.D . )
e = rr/2
2.262
X
X
10- J ~
_-:t
0 .. 640 X 10
.J
2~080
X
10-3
0 .. .589
X
10 --'
_":\
Calculations
The results for the test rings with holes are sur.::rr..a.rizcd in Table IIL.
Equations ( 14), ( 15) and ( 16), the proposed a.naJ.ys is equations, ·vrere used
to arrhre o.t these resultse
11
P11 >;~m.s taken as unity and
e ""
rr/2.
A cor:-
plate set of sample calcn.llatio:J.s is presented in App:3ndb: I.
Table III
(
Propoeed Analysis
Test Ring
Holes;
,,
Jl,
6Dy
y
6DX
2 (5.930 O.D., .250 Dia Hole)
0.829 :< 10~3
2~2.5'+ ~: 10~3 2~093 ,.
3 {5.930 O.D.,
()500
Di~t
Hole)
0.814-2
2 • .302
2o1l6
4 (5.930 O.D.,
~750
Dia Hole)
L084
2. 971~
2 .• ?35
6 (J.950 o._D'",
~250
Dia Hole)
0.218
0.,647
Oa595
7 (3.950 O.D., .,500 Dia Hole)
0.243
Oe668
0~614
8 (Je950 O.D., .. ?50 Dia. Hole)
O,J51
X
1o=J 0,.971
.....
(1- .J
X -1 v
":1
10=..)
OG89J :( 1o=J
I
by Fig. 8.
An Instron ten3il.:a
Dillon re<1dout meter ind:..~at.•.:d tho applied 1cw.s while a di.:a.1 ir.d:ic~:tcr
11
t.
a)
In-plane Test Setup
(
f
777
b)
(
Ou~ plane
Test Setup
G) .Instron Tensile Tester, Hodel
@ Bro~,m and Sha.."'Pe Differential
Full
FT M~L
A.ilplifier, Hodel 992
"" .020 inches, 1 Divis1on "" .0005 inches
.
h
·Full Scale ""' .006 1nc~.es,
1 Division "" .0001 inches
Full ScaJ~e - .002 inche:>, 1 Division = .00005 inches
S::z~e
Full Scale
Q)
=
.0006 inches, 1 Division
Lufkin Dia~ Indicator
Full Scale "" • 004- inches • 1 Division
·Total Re.:ading """
@
= .00001 L"lches
= ~ 0001
7 rotaticns or • 028 inches
Dillon Lo:'.d Reada-.tt Heter, Hodel ACNlE20li
Full Scale "': 100 lbs, 1 Division
G)
t
®
Test Rir.,s
hlChes
=1
r
.
lb
Inch Di;;u-::.'?ter Steel Rod
Figu.re
~
Test
S~t~tJS
I
.
12
8.0 Test Dat~/Results
(
Three test runs for each of the loud. conditions were
eight test ri.'1gs..
for all
Com~
These tests were conducted at the· Hughes Aircraft
pany, Missile Syst'?ms Division, Ca..'1oga P3!k 1
data for these tests are presented in
California~
T'.ae rat-; test
AppendL~ II~
Tables IV through XIX compare the test data w·ith the
sults.
w~e
pr~,;dicted
re-
Percent de"<;;·iations from the calculated and test results are also
given in these tabJ.es as well as the average deviation for any given tests
In arriving at the percent deviations, Eqlliltions (1),
(5) and (6) were
used as the theoretical stfu""ldard for t~he solid rings vThile Equations (14),
(15) a.11d (16) '\-/ere used for the rings with holes.
Fig-u.res 9 through 17
present the above results in graphic form.
(
9.0 Discussion of Test Results
Referring b.ack to Tables IV and VIII, ·He see tha:l:. the percsnt er.<'q:r•
for both the solid test rL'1gs is virtually neglible 8.nd Hell
test
variations~
adequate to
obt~in
~-ri thin noTI'~~
'E"'"lis fact indicates th<?.t tJ'l.e in=plane test setup is
the accuracy
~equired
for our
tests~
The error for the rir.1gs 1-vith h0las i.."'lcreases in proportion to hole
size as depicted in Tables V, VI, VII, LX, X and XI.
The error;
< 22~~
I
for the rir!gs •nith holes up to • 500 inches in dia..111etor are hoH-sver. accer;t.able for most engin8ering a:!'"..l alyses~
A.s the hole si.ze increasqs to
'
the percent error becorr.c3 quite high, lJ..0-50;1;, and una.ccep~;,bl€~
e
7.50,
1tlhat
appears to be happe.-nir:.--.; is as the remaining ··ring section is reduc<::dt the
ring no longer acts as one continuous structure but rather as
tw"J
str-:..1c-
1.3
tures hinged at two of the four holes.
This
w~uld ~£co~~t
for the fact
that we are getting more deflection than predicted by the analysis.
Let us
no~..r
investigate T.tihat kind of error that would have resulted
if' the effects of holes were neglected and the solid ring analysis Ttras
used instead of the proposed .c:u"1alysis for test rings Hith holes$
X.\ suin.marizes the average percent error
usi..~g
the proposed analysis vs
the percent error using the solid ring analysis.
rings with holes up to .250
results
th~"1
the proposed
di~~eter,
Table
Here we seB that for
the solid ring analysis gives better
the hole size increases,
a~alysis.
the proposed analysis yields better results.
In· fact, for the rings ·with
• 750 holes this inp:t.•ovement ~ras b•atter than lOQ1t~
Out-plane L-oading
Tables XII
thro~gh
/
XL\ summarizes the
resu~ts
for. Otlt-plane loa:hng.
Tables XII ar:d XVI are the r<::sults for the solid rings&
Here \>ie See errors
of appl"'oximately 10'{; uhich are higher than normal test variations..
Tb.es•3
resul-ts -vmuld indicate t.hB.t the test setup for out-pl2..i'1e lo:J.d.ing could ·
and/or should be imp~oved~
One difficulty 'jrith the present setup vras trying to ba1:.,nce the test
r:Ll'lgs bet-vreen the tHo -!- L1ch dia..-·neter rods such that equal d<.:;flection
readings could be obtai.'led from the tHo deflection Lndicators (See Fig~
8b).
For all test r..n1s, there 1.;as some unbalance behreen t'he two re<J4:'t.p.&;.'.i.
This indicates that t11e fc'ur load points were not equally
in the analysis.
spar~ed
as assuned
It is felt that ;_.P equal four-poi.nt loadin.; could. be
achieved, the test results for the solid rings \iOuld closely match the
pr;edicted results.
14
The results for t.'!Je rings with holes tended to ;f:oll<Wr the sa;.Yfle trend
as for in•plane loading; the larger the holes in. the rings, the larger
the percent devia·tion from predicted values.
from 2.5-.50%.
The percent deYiation v-aried·
If we w·ere able to el:Lrni..'tla.te the approximately 10% built~j-"1
error due to m1equaJ. load spacing, the error 1wuld still be 15-Ll-C%.
the error tended to increase with hole size, one might suspect "tli.e
phenomenon as the
culprit~
F~~er
point the source{s) of error.
investigation is
necessa~J
Since
11
hinge 11
to pin
It IDE\Y" be one of tHo things--the test pro-
cedure/ setup or the a..'l.a.lysiso
Let us now turn to Table
L~I
to see the effects of using the solid
ring a."lalysis i...Tl favor of the proposed analysis for
loading.
other~
Here we can see no clear cut advantage of one analysis over the
This r:w,y be because of the large errors
used~
rega.....-;(l~ess
c£ the analysis
I£ the test setup errors could be reduct:d. (see discuss:i.on ::,bove) a
trend may dt;;·velop.
Hm·mver t at this time it appears that
th~~
propos,-x.l
anr:lysis yields better results for the smaller test rings, rin6s
artd
6, 7,
8, l"lhile the solid ring .analysis yields better re.sults .for the larger
rings.
10. 0 Concluding Rem?_rks
10.1
In-plane Loadir.:g
the effects of holes u.p to and including • 500 ir..ches :i.n diameter
engineering applications i f the hei~}}t of the ring is
>1. ooiJ
fOl"
most
inches,.
error 'rrom this simpH.fication arnoun.tcd to less tha.."1 22% :for all rings
tested.·
For rings 'jd.th holes greater than • 500 dia:."?.eter, neither the solid
1'4~
15
. ring nor proposed analysis yielded accurate in-ph. .-'1e res:,1ts; errors o:f
34· to
132% >..rere recorded.
The proposed analysis did however give consid-
erably better results than the solid ring analysis for the • 7.50 dia.'7leter
hole t.est ringso
This :iJnprovement was better than 100'{6 for both test rings
havi:."'lg • 750 dia."lleter holes.
10.2
O~t-plane
Lo<?.ding
The results for the out-plane loading ccnditicn vrere rather inconelusive.
Neither analysis predicte-d the perfonnance of the test rings
very accurately.
the test ri.'<!gs.
The error ra:.1ged from a lm'i of 8 to a high of 90% for
This error, as a whole, increased with ring hole size.
It is .felt that approximately 10·% of the error recorded for out-plane
lo?nirig can be
(
attritr~ted
If this is the case~
to a faulty test setup.
the errors for ri."1gs -vii th • 250 holes become sw.EJ..l and one could ignore
the effects of holes up to about .250 inches for
o~t-plane loading~
'Regard.less of the a..Ylaiytical apprr~a.cl1 used in thiS repor~l:,~ the !"esu_l-ts
for the • 750 hole rL"lgs -vmre in considerable disagree:::1ent •rrith predicted
results for both in- a.."'ld
out~plane
loading.
Further testing or
1-rork w-ill be required to resolve this discrepancy.
~.naJ..yt.ical
,.. ~ ......,
-~-
Table IV
TEST RESULTS
!
In-plane Loading--Solid Ring (r
Load
(lbs)
Dy*
(inches)
1
0.67
2
L33
3'
10-3
= 1.9125)
n;:theo
%Er:rox•
(inches)
0.640
10-3
D:/2
(inches)
%Error
+ lj..?
0.27
1.280
+ ).9
0.57
0• .588
1..97
1..920
+ 2.6
0.84-
0,.882
. - 4.8
4
2.62
2.. 560
+ 2 .. .3
1.19
1.176
+ 1.2
5
3~22
;3.200
+ 0 .. 6
1.46
1.470
- 0.7
6
3.87
J.Sl+O
+ 0.. 8
1.79
1 .. 764
+ 1.5
7
4.. 55
4 .. 480
+
2.,10.
2 .. 0.58
+ 2 .. 0
8
5Gl6
2.,.35 X 10-3
2.352
X
X
10-3
5,.120
X
X
10-J
Aver***
1~6
+ 0.. 8
X
10-J
**
2
Dx,theo/
(inches)
== + 2.0
0.294
X
10-3
- 8.2
- 3.1"
X
10-3
- 0.. 1
Aver*** = + 0.?
·*These values represent the average of all test runs ..
** These
values are based on the formulas fou:nd in 11 F'ormulas f'or Stress and Strain 11 by Ro J. Roark.
*** The high and low values have been excluded from this average.
~
0'\
,,,
Table V
TEST RESULTS
In-plru1e
Loading-~.250
n*y
Load
,(lbs)
Diameter Hole (r
n**
(inches)
y.theo
= 1.9125)
(inches)
1
0~57
2
1.27
1..294
3
1 .. 90
4
n;;z
n**
t2
x.theo
(inches)
(inches)
%Error
%Error
0.26 x lo-3
0.298 x lo-3
- 12.7
- 1.9
0 .. 51
0.596
- 14.4
1 .. 941
- 2.1
0.80
0.894
~
2 • .59
2 • .588
+ 0.,1
1 .. 13
1.192
- .5.2
5
3.. 26
3.. 235
+ 0.8
1 .. 39
1.490
- 6.?
6
3.. 78
3.882
- 0.3
1.63
1 .. ?88
~
7
4.. 49
5.06 x.lO-3
4.529
- 0.9
1.9.5
2.086
- 2.3
2.21
8
X
10-J
0.. 647
5.176
X
X
10-3
10""'3
- 11.9
Jl:
10-3
Aver*** .., . - 1.1
*These
.
** These
I
10.5
8.8
·- 6.. .5
2,.)84 X 10-3
Aver***
' - ?.3
lilt!
-
8.4
. ,,
values represent the average of all test runs.
.
-
values are base-d on the "proposed analysis. 11
*** The .high
and low values have been excluded from this average.
1-'
""'l
' !~
'f(::··
\
Tab~e
VI
TEST RESULTS
..,
I
'
In-plane Loading--.500 Diameter Hole (r
•r."
'.
= 1.9125)
l
.,'·
Load
n*y
n;:theo
(lbs)
(inches)
(inch.es)
%Error
lf'*
Dx,theo/2
Dif./2
X
(inches)
'% Error
(inches)
I
.+ 15.3
0. y +X 10-3
0.307 x 10"'" 3
1,.336
+ 16 .. 0
0.68
0.614
+ 10.8
2.36
2.004
+ 17.8.
1~05
0.921
+. 14.0
4
3.. 17
2.672
+ 18.6
1.45
1 .. 228
+ 18.1
5
J .. 99
J,.340
+ 19.5
1..82
1.535
+ 18.5
6
4.90
4 .. 008
+ 22.2
2.13
1.842
+ 15.6
7
5.68
4.. 676
+ 21.5
2.55
2.. 149
+ 18.6
8
· 6.50
s~J!W x Io- 3
+ 21.7
2,.96
1
0. 77
2
L55
3
X
X
10-J
10-3
0.668
X
lO.. J.
Aver***
*These
** These
*** The
"
X
10-J
= + 19.2
2.4,56
+ 10.?
!
'
X
10-3
+ 20.5
Aver.*** "" + 15.9
values represent the average o.f all test runs.
values are based on the 11 proposed analysis."
high and low values have been excluded from this average.
:1,.
1-'
co
'
Table VII
TEST RESULTS
In-plane Loading--.7.50 Diameter Hole (r,.;,
n*
Load
y
(inches)
· (lbs)
n**
y,theo
1.912.5)
n*;z
X
%Error·
(inches)
(inches)
/2
n**
0.4_56
10-3
%Error
+
30 .. 8
0.6.5
1.842
+
47.6
1.. 35
0.912
+ 48.0
4.04-
2.. 813
+ 4) .. 6
2~·os
1.)68
'+ .52.1
4
5.50
) .. 684
+ 49.3
2.. 78
1.824
+ 52 .. 4
5
6.88
4.. 655
+
47.8
3.. 4.5
2"280
+ .51.4
6
8.. 33
5.626
+ 48 .. 1
4.. 17
2.736
+ 52 .. 5
7
9.. 69
6.497
+ 49.1
4.81.J.
J.l92
+ 51.7.
50~3
,5 .. 47
1
1.27
2
2.72
3
X
10-3
11.08 ~ 10-3
8
0.971
?.J68
X
X
10-J
10-3
+
X
X
10-J
x,theo
(inches)
10-3
Aver*** ... + 47.6
*These
** These
*** The
3.648
X
X
10-J
Aver***
+
+
42o5'
49.9
= + 50.9
'',
1Jhl..ues represent the average oi' all test runs.
values are based on the
11 proposed
mlalysis. •l
high and lovr .values have been e:x:cluded. from this average.
f-1
\.0
Table VIII
TEST RESULTS
In-plane
Loading~So1id
Ring (r ... 2.9125)
:~ ' '
*
Load:
D·
y
(inches)
(lbs)
1
2.18
2
1h38
J
6 .. 77
4
9.18
X
10-3.
X
10-J .
---------
5
6
7
8
---
9
n**
y,theo
(i.'1ches)
2 .. 262
10-3·
n*/2
X
n**
· /2
x,theo
(inches)
(inches)
%Error
1.oa x lo-3
1. 040
4 .. 524
- J.2
2.0J
2.080
- 2.4
6 .. 786
- 0.2
3.17
).120
+ 1.6
9 .. 048 X 10-J
+ 1.5
4 .. 08.
4.160
- L9
5.J2
5.200
+ 2.J
6.. 25
6.240
+ 0.2
7.38
7.280
+ 1.4
-
8.42
. 8.320
+ 1.2
...
9.90
------
--IJOIJ--
-
-
X
lO""J
Aver*** ""- 0.9
*These
** These
Error
- 3.6
X
9. 360
X
10-3
fa
x. 1o·J
I
t
+ J.B
+
5.8
Aver*** ... + 0.9
values represent the average of all test runs.
values are based· on the fonr.rulas .found in 11 Formula..o:; for Stress and Strain" by R. J. Roark.
***. The. high
. values ha,.re been excluded
'
and lot..r
.from this average.
N
0
· Table IX.
·
In-plane Loading-.250 Diameter Hole (r
Load
D*y
(inches)
(lbs)
n**
TES~RESULTS
= 2.9,025)
n;._/2
**t.heo/ 2
Dx,
(iriehes)
(inches)
%Error
Yttheo
(inches)
1.86
2.073
- 10.1
- 7.9
2.75
).109
-11.5
- 6.7
).80
4.145
..
11.270
- 5.. 0 ....
~.69
.5 .. 181
- 9e5
13.524
- 4.8
.5o48
6.218
- l1.8
- 5.8
6.. 24
7.254
- 13 .. 9
-
7.. .37
8.290
- ll.O
-
8.30
2.2,54
2
4al7
3
10-J
.... 5.9
1. 01
J+.5o8
- 7.5
6.23
6.762
4
8.41
9.016
5
10.74
6
12.87
7
14.. 87
---
8
---
9
Error
-
2.12 X 10-3
10-J
<fo
1.0)6 X 10-J
1
X
c
15.. 778
X
10-J
X
-
---
X
:x:
10-J
1o-.3
Aver*** ""- 6o2
9.,326
X
10-J
2 .. 5
8 .. ,3
- 11.0
Aver*** ... - 10 • .5
.
·,
*These values represent the average of all test
** These
*** The
nL~s.
valu.es are based on the "proposed analysis. 11
high a.nd lovT values have been
'
~.X.(;luded.
'
from this
average~
1',
N
f-1
Table X
TEST RESULTS
"'
In-plane Loading--.500 Diameter Hole (r • 2.9025)
**
Dy,theo
(inches)
Dy*
(inches)
Load
(lbs)
1
2 • Lr-:~
r.,; ....
"',.,
2
1!-. 94
3
lo-3
D*/2
:JC:
n•*
x,theo/2
(inches)
(inches)
~Error·
1.058 x lo-3
+ 12.5
2.40
2.116
+ 13.4
J.6J
J.1?4
+14.4
4.71
ll-.232
+ ll.J
+ 13.3
6.0.3
5.290
+ 14.0
·13.810
+ 13.5
7.24
6~348
+ 13•7
16.111
+ 12.6
8.44
?.406
+ 1.).7
+ 12.9
9.58
8.464
+ 13.. 2.
2,..)02
10-J
5.6
1.19
4.603
+ ?.J
?.48
6.905
+ 8.J
4
10.17
9a206
5
13. ()/+
11.508
6
15.68
?
18.15
8
20.78
X-
10-3
X
18.412
_.....
9
+
+
X
---
10-J
10~5.
-
X
10-J
10.,88 X 10""J
Aver*** = + 10.8
.*T.hese values represent
** These
.% Error
9.522
X.
10-J
+ 14• .3
,I
!.
!
,,
Aver*** = + 13.5
the average of all test runs~
'ralues .r:.re based on the "proposed analysis. 11
*** The. high
.
and: lmv- values have been excluded
from this average.
N
N
' ~ '
"·
·r
Ta'ble X+
TEST RESULTS
.--
\:.
In-plane
Load
(lbs)
Loading--.750 Diameter Hole (r
D*y
n**
·yttheo
(inches)
(inches)
10""' 3
2~974
+
Y+.3
5.81
4.102
+ J4;o3
+ )4.8
7.80
5.. 470
+ 4·2.6
___._.
-
9.95
6 .. 837
+ 45.5
---
-
l1.98
8o92.J
5
6
16.04
X
10-;;
-
lla 898
.
---
10-J
+ 42.2
3
4
1.')67
2.735
X
X
10- J
X
10-.J
(inches)
3-89
5.949
':l
(inches)
%Error
33~5
7.94
'
**
2
0x,theo/
D~:/2
X ..
%Error
+
2
'
10-3
2.9025)
2.0,5
3 .. 92
X
~
.·.
'+ 31.9
l
.
11.99 x lo-3
Aver*** .., + :33.9
X
·a. 204 x
1o-3
\
+ 50.0
I.
+ 46 .. 2
Aver*** .., + 44.1
*These values. represent the average of all test runs.
**These values are based on the "proposed ant:U.ysis .. "
***' The
high and
lO'I-1
-:
values have been excluded from this average •
.'
1\)
\..I)
I"'.
Table XII
TES'X RESULTS
Out-plane Loading-Solid Ring (r = 1.9125)
Max Deflection**
(inches)
'load*
(lbs)
***
Hax Defleotioh.theq
cgt Error
..
(inches)
o.2363 x 10... 3
'4
0.2.5
8
0.53
o.J~,726
+ 12.1
12
0.74
0.7089
+ 4.4
16'
1.05
0.94.52
+ 11.1
20
l..JO
Ll815
+ 10.0
24
1 • .50
1 .. 41713
+ 5.. 8
28
1.78
1.6541
+ ?.. 6
32
2.06 X 10-J
*The
Jt
10-3
..
1. 890!~
X
10- J.
value of npn for use in the analysis is (loail/2).
**.These
+ 5.8
+ 9.. 0
Aver****
a
+ 8.2
value represent the average of all test runs.
*** These
values are based on the formula found in 11 How to Find Deflection and Moment of Rings .·.
a.'l'ld Arcuat,e Beams, 11 _!'rod.uct Engineerj..ng, ,Jan 3 63 9 p. 69. ·
**** The
high and low values have been excluded in this average.
N
-{:"
·.r·,
"
\.
'!' :'··
·..
Table XIII
TEST RESULTS .
Ou~plane
Loading--.250
Load*
·(lbs)
Di~~eter
Hole {r
Max Deflection**
(inches)
10-3
= 1.9125)
.
***
%Error
10-J
- 28..4
Max Deflectiontheo
(inches)
0.2.374
4
0.17
8
0.52
o.lt-7.1.:.8
+ 9.5
12
0.91
o. 7122
+ 27.8
16
1.29
o.9496
+ 35.8
20
1.66
1.1870
+ 39.9
24
2.. 03
1.42l.J.4
+ 42.5
28
2.40
1.6618
+ 44.• 4
32
2. 76
-·
X
X
10-3
l. 8992
X
X
10-J
-
value of
-.
** These
*** 'I'hese
11
,,
'
'
.
values are based on the
and
= + 25.3
pu for t.tsa in the a.11aly$is is (load/2).
. represent the average of all test runs.
values
**"'* The. high
.!
+ 45.4.
·Aver****
*'The
I
I
I
lo~r
11
propQsed
..
a.11alysis~ 11
values have been excluded in this average.
N
\..1\
'.
I'
.......,
Table XIV
TEST RESULTS
Out-plane Loading-.500 Diameter Hole (r ... 1.9125)
Load*
Max Deflection**
(inches)
(lbs)
10-3
***
Max Deflectiontheo
(inches)
o.z4J4
4
0.,2J
8
0.64
0.4868
12
1..09
.0 .. 7302
16
1.51
0 .. 97.36
20
1.93
loll70
24
2.33
1 .. 4604
28
2.72
1.7038
32
.3 .. 08 x 10-J
l. 9472
*The
value of
** These
*** These
**** The
npn
X
x
lo-3
X
10-3
%Error
.
for use in the analysis
is (loarl/2).
values represent the average of all test runs.
;ralues are based on the 11_proposed analysis."
high ' and.
l~;
values have been excluded from this average.
1\)
0'\
-,"1>,·,
I
Table XV
·I
I.
TEST RESUJ..TS
Out-plane Load.i.ng--. 7.50 Diameter Hole (r:Ma.x Deflection**
Load*
(J.bs)
(inches)
-'l
..1
= 1.9125)
}1ax
·
Deflection.***
h
~ eo
(inches)
%Error
Oo 3513
+ 1~.6
4
O.Ji.2
8
0 .. 90
0.7026
+ 28.1
12
1.36
·le05J9
+ 29.0
16
1 .. 82
1 .. 4052
+ 29.5
20
2o27
1 .. 7565
+ 29.2
24
2. 67
*The
value of
** These
*** These
11
X
.X:
10
10-3
2,.1078
X
X
10-.3
_-:a
10
P11 .for use in the analysis is (load/2).
1
+ 26 .. 6
J
Aver****
= + 28o2
values _r.epresent the average of all test runs.
values are based on. the
11
i
propcsed analysis .. "
''
****.The
. lm·r values have been excluded from this average.
high and
l\)
"'-.J
·..... ·:
Table XVI
TEST RESULTS
Out-plane Loading-Solid Ring (r
Load*
'(lbsY
Max
Defl~ction
= 2.9125)
** .
***
Hruc Deflectiontheo
(inches)
%Er:ror
(inches)
2
0.54 X 10-~_,
0.8)47
4·
1.. 28
1 .. 6694·
... 23.3
6
2.14
2.5041
8
2.98
J.J388
- 14• .5
_. 10.7
10
. J.91
4.1735
- 6.)
12
l+-..72
.5.0082
5.7
14
. 5 • .50
5.8429
5.9
16
6.,28
6.. 6776
- 6.0
18
6.91+
7.. 5123
7.6
20
7o ?1
*The
X
10"" 3
B.J470 x lo-3
10-J
value of 11 P1' for use in the analysis is (load/2).
**
· These
X
~
... 35.3
- ?.6
Aver**** = - 10.2
values represent the average of all .test runs.
***·These values are
·. based on the formula found in 11 HOi·r tc Find Deflection and Moment of' Rings
.and-Arcuate Beams, 11 Pror.iuct
**** The
Engineering~
Jan
8 63~
p.
69~
high and low values. have been excluded in th:i.s average ..
1.\)
co
•'•
Table XVII
TEST RESULTS
I'··
Out-plane Loading--.2.50 Diameter Hole (r
.
*
Load
(lbs)
Max Deflection**
= 2.902.5)
***
Hrut~ Def'lectiontheo
(inches)
2
0.39
l}
10-3
tfo Error·
(inches)
o.8287 x lo-3
- 53.0
0.. 9.5
1.6571~>
- 42.7
6
1.73
2.4861
- 30.4
8
2s51
). Jllt8
~
10
3.25
4.143.5
- 21.6
12
).97
l}.9722
- 20.2
1!~
4.72
.5 .. 8009
- 18.6
16
5.,35
6.6296
- 19.3
18
6.. 07
7 .. 4583
- 18.6
20
6.?7
*The
X
X
10-3
8., 2870
X
10- J
value of .11 P11 for use in the analysis is {load/2).
** These
'
24.3
... 18.3
Aver****
=-
24 • .5
values represent the average of all test runs •
.'
*** These values are based .<)n t.l-J.e upropo:sed analysis;, 11
'
**** The high and lovr values
.
have been excluded from this average.
II,
N
'-0
'
'
,•""~
Table XVIII
!.~§.±RESULTS
Out-plane Loading--.500 Diameter Hole (r = 2.. 9025)
Max Deflection**
(inches)
Load*
(1bs)
Ha.x
.
%Error
0., 8421
- 50.2
10-3
2
0.,1+2
4
0.95
1.6842
- 43.5 .
6
1.48.
.2.5263
- 41.4 .
8
2 ..1+7
3.3684
... 26.6
10
J .. l4
4o2105
-
12
) .. 90
,5.0526
- 22.9
. 14
li-.1~7
5.8947
-· 24.. 2
16
5.20
6o 7Jf)fJ
- 22o8
.18
. 5.98
7 .. 5789
- 21 .. 1
X
10-3
***
Deflectionth
eo
(inches)
6 .. 65 x lo-3
20
8,)1210
X
X
10-3
2.5~4
- 21.0
,,
*The
vaJ.ue of
** These
*** These
~alues
11
P11 .for use in the anaJ.ysis is (loa.ii/2).
represent the average of all test
Aver****
=-
28.5
~~s.
values are· based on the "proposed analys:i..so"
~
0
**** The
high and
l~Y.v
values have bee11 excluded from t.his average. ·
t •,
I
I'
./
Table XIX
TEST RESULTS
Out-plane Loading--. 750 Diruneter Hole (r
. **
Max DefleotJ.on
(inches)
Load*
{lbs)
I·
= 2.9025)
***
Max
· Deflectionth.eo
(inches)
10-J
1 10 08J7
%Error
2
0)¥8
4
1.04
2.1674·
- .51.9
6
1.81
J .. 25ll
- 44.4 '
8
2 .. .q.a
4.JJ1+8
- 42.8
10
3.28
5 .. 4185
- J9.5
l2
3.99
6.5022
- 38.6
14
11-.76
7.5859
- 37.3
16
5.. 1+8
8.6696
- 36.8
18
6.22
9 .. 7533
- 36.2
20
7.. 00
*The
.....
X
10-J
l0,.8J70
X
X
10-3
10-3
55.8
... 3.5.4
Aver**** • -
value of 11 P11 :for use in the ana~ysis is (loai!/2),.
** These
L~1.14·.
values represent the average of all test runs ..
**:0:' These values -are
**** The
X
basf~d
on the "proposed. anltiysis .. 11
. have
high and lm.-r values
beo:~m
ext;lucled from this
):l
a.verage~
!'
I
i.. ,
Table XX
Effect of Using Solid Ring Ane.lysis in Favor of Using Proposed Analysis
(In-plane Loading)
Ring #
Aver
%Error
(proposed analysis).
~D.y
.6Dx
2
- 6.2
- 10.5
J
+ 10.8
+ l)e5
4
+ 33.9
+ 44.1
6
- lel
7
+ 19.2
+ 15.9
8
+ 47.6
+ 50.9
-
Aver
~ Error (solid ring analysis}
.6.Dy
- 5. 5
.6.DX
~- -~- ~-··~.:9~9~
8.l~
'
I
o
~
'
Table XXI
.• ·:,.1•
.\,
i ";·,·~·
i
I
Effect o.f Using Solid. Ring Analysis in Favor of Us.ing Proposed Analysis
{Out-plane Loading)
Aver
Ring#
~
Error (proposed analysis)·
24.5.
2
Aver ~ Error (solid ring analysis)
- 24e2
J
- 28.5
27~1
4
- 41.4
22.5
6
+ 25 .. 3
+ 25.9
7
+ 50.0
+ 56.6
8
+ 28.2
+ 90.2
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11.0 APPE!iDIX I
Sample Calculations Using Propose-1. Analysis
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(.
--- - -
.
3>
~ '3 1.:;-
~
/o.OXtcJ
2.&30
2 ( / f - .3/:)-)
/-~ ~~----- ~
iI 7
P.
- ------ -- -------....... -.........,__
/
3.
t1o2 2 .x t 0 ' ·.)
---------
/\ -
I C>. 0 X ( 0
le-t.(
c. (
I. 0
3 t3 9. X
I
0
-
.. 10.3 t35 >(tO/2 2. o972
.,
C:_~~~=~----
-
47. o;s- __
(;. 9 9.:>-.3
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2\0
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2
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lU
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48
1
l2c 0
APPENDIX II
Rm<t Test Data
49
(
In~ plane Loading-~So1id
Tare Load.
Run #
~
5 lbs
Load
D
y
DX /2
(lbs)
(inches)
(inches)
1
(
---
Ring ( r ""' 1. 9125)
\
1
0.66
2
1.)6
0.61
J
2$01
Oe87
4
2. 7lt
1.19
5
3.21~
1.4-J
6
3-93
1,77
7
4.60
2.08
8
.5.25
1
o.68 x 10=3
0.,26
2
1.31
0• .52
3
1..91
0.80
4
2.51
1..18
5
3.25
I.l}B
6
3.84
1.78
7
4.57
2.08
8
5.10
X
X
10-3
10-3
_..,
0.27 X 10
..1
2.29 X J.O-3
'~-~
2
·,
X
10-3
2. J6
X
10-3
X
10-3
50
(
In-plane Loading=-Solid Ring (r
Run #
'\
r···-
~
lo9125) Cont'd
Load
D·
y
D /2
(lbs)
(inches)
(inches)
X
1
0.68
2
1.30
Oo57
3
2.00
0.85
4
2.60
1.20
5
3.18
1.48
6
.).84
1.81
7
4eli-8
2.1)
8
5.1,) X 10-3
X
10-3
0.28
- 2.40
X
10=3
X 10""
3
51
TEST DATA
---
In~pln.ne
Loading=-.250 Diameter Hole {r""" le9125)
Tare Load
Run #
1
(-
2
~
5 lbs
Load
D
{lbs)
{inches)
y
.,.'.:I
1
0.59
2
1 .. 25
Oa52
3
1.89
0.81
4
2.59
1.1.3.
5
3.25
lo41
6
).79
1G62
7
l~-o47
_1.96
8
5.05
10~3
2e21
1
0,58 x lo=3
0.27
2
1.28
Oo51
3
1.89
0.80
4
2.56
L15
5
3a28
1.40
6
3.78
1..65
7
8
X
X
10-3
DX /2
(inches)
.
.L•9
--lj.
~
5.04 x 10- 3
0.27
X
10 ..1
X
10-_..
X
10""..1
X
10-3
-
'.:I
'.:I
1,9L~
2,20
52
TES1' DATA
-----
Run
If
Load
(1bs)
3
"'·
.
~·-
..
D
y
(inches)
(inches)
1
0.55
2
1.28
Oe49
3
1 .. 92
0.80
lt-
2.61
1 .. 11
5
3.26
1.37
6
3-77
1.63
7
4.. 50
1 .. 95
8
5.08
X
X
10-J
DX /2
10... 3
0.25
2.,21
X
10-3
X
10-3
.53
In~plane Lo.ading=~.500 Diameter Hole (r "" 1 .. 9125)
Tare Load .., 5 lbs
Run#
1
Load
D
(lbs)
(inches)
DX /2
(inches)
y
0.33 x 10-3
1
0.7 8
2
1.59
0.74
3
2.37
1.08
4
3.. 2.5
l.JB
5
4.08
1.80
6
.5.02
2.12
7
5.77
-2 .. .50
8
6.,76
x
10=3
2.85
X
10-3
1
0. 76
X
10~3
0o35
X
10.,;, 3
2
1.57
0.73
3
2.)0
lelO
4
3~11
1.51
5
3-93
1 .. 81
6
4o83
2.12
7
5.62
2 • .58
8
6.40
X
10""3
X
10-3
)
~
2
X
':l
10-_,
3. 00
In~plane
Loading==.500 Di.runeter Hole {r "" 1.9125) Cont'd
Run If
D
y
DX /2
(inches}
(inches).
Load
(lbs}
l
0.76
2
lo50
0.56
3
2.42
0.98
4
3ol5
le46
5
3.,95
1.84-
6
4.,84
2.,15
7
5e66
2.56
8
6.,43
X
10=3
0.35
X
10- 3
X
10~ 3
'-.,_
. . -·--·
-----
..
~·--
.,_---·~-
------·- ------ ---
.
--
-
--------
-- ------
---
X
10~3
- 3~ QL~
55
In-plfi!"1e Lo0\.ding~a•.7.50 Diauneter Hole (r
Tare Load
Run#
1
"'
2
~
:g
L9125)
.5 lbs
Load
D
DX /2
(lbs)
y
(inches)
1
1.31 x lo-3
0.61+
2
2.. 75
1.33
3
4.10
2.02
4
5• .51
2.7)
5
6.95
3.. 42
6
8.,)6
!-hl2
7
9.66
4.78
(inches)
X
10""3
-
8
1Ll4 x 10=3
5.,1~-Q
X
10~ 3
1
1 .. 25 x 1o=J
0.67
X
10-~..;
2
2.. 72
1.39
3
4.. 01
2.16
4
5.55
2.82
5
6.84
. 3.. 49
6
8.30
4.20
7
9.. 79
4.89
8
11.09
-·--~-·-·
.·
,..
X
10-3
5.58 x 1o= 3
-
~------
-- -
In=plane Lorv:iing=··e750 Diameter Hole (r""' 1.9125)
Run #
Load
(lbs)
J
.
(
'.,
Dx/2
(inches)
y
(inches)
'l
1,.25
2
2 .. 69
1 .. 34
3
4.00
2o07
4
5.,Ljl~
2 .. 79
5
6.86
J.45
6
8.,JJ
4.18
7
9 .. 6J
4.86
X
10*"-'
0.,64
1
8
..
D
Cont'd
11,.00 X 10~3
- 5.,J.:.2
X
10~ 3
X
10... 3
57
In-plane Loading-=Solid Ring (r "" 2.9125)
Tare Load "" 5 lbs
Run #
Load
(lbs)
1
.,
.•
2
D
D
y
(inches)
X
~3
10 .
0.95
2 .. 15
2
4.1}0
1 .. 90
3
6.80
2.9.5
4
9.25
X
·)
(inches)
1
X
/2
~3
10
X
10~3
X
1Q-3_
3 .. 60
5
5.05
6
6.10
7
?. 10
8
8.35
9
9.50
1
2.25 X 10-3
1.20 X 10-3
2
4.40
2 .. JO
3
6.85
4
9.15
3.35
':I
X
10-..J
4.40
5
5 .. 45
6
6 .. 55
7
7-55
8
8.75
9
10 .. 20 x lo-3
58
(
Run#
3
=-,
Load
D
y
(1bs)
DX /2
(inches)
(inches)
10-3
1.10
1
2.15
2
4·.35
1.90
3
6~65
3o20
4
9.15
X
X
10~3
X
10-3
X
10-3
4.25
5
5.. 45
6
6.10
7
7.50
8
8el5
9
10.,00
--
/r-- ..
59
In-plane Loading=-.250 Diameter Hole (r
Tare Load = .5 lbs
Run#
1
Load
Dy
(lbs)
(inches)
·~
·/
-,
2.12
2
4.13
1.89
3
6.24
2.79
4
8.41
3.. 8J
.5
10~88
lt-.68
1)o00.
.594.5
7
2
10-3
Dx/2
(inches)
1
-6
L-;
= 2.902.5)
14.99
X
X
10~_3
1.00
X
10-3
6e20
8
7.31
9
8.27
X
10~3
1.00
X
10=3
10~3
1
2olJ
2
4.27
1.82
3
6.28
2.72
4
8.62
3.. 74
5
10.92
4.70
6
13.19
,5.45
7
1,5.05
X
X
10-3
6.20
8
7.38
9
8.28
60
In=plane Loading-=.250 Diameter Hole (r "" 2e9025) ContDd
Run#
3
Load
Dy
Dx/2
(lbs)
(inches)
(inches)
1
z·~lO
2
4.12
1.86
J
6~16
2 .. 74
4
8 .. 21
J .. 83
5
10 .. 4·1
4.68
6
12 .. 41
5.54
7
14.56 x 1o=J
6.31·
.,
8
,r·-·
·"'~.--
9
...
X
=~
10-J
1.0J
X
10-3
,
X
10- 3
-·
-----·----·-
7.42
8.J5
61
In~pl~~e Load~1g~=.500 Dia~eter
Tare Load
Run#
~
.,
r
~
2.9025)
5 lbs
Load
(lbs)
1
Hole (r
D
y
(inches)
lQ~J
X
10-3
10.90
X
10-J
X
10 -'
X
10-J
2.,55
2
4.91
2e50
3
7.. 59
3.70
4
10.20
4.90
5
13e18
6.11
6
15.. 66
7.. 20
7
18.29
-8.45
8
20.94·
X
X
10~3
9 .. 70
1
2.. 43 x 1o=J
1$20
2
4.94
2.46
3
7.46
3.65
4
10.19
4.84
5
12.90
6.08
6
15.74
7.3?
7
18.04
8._54
8
20.71
9
·,
1.26
1
9
2
DX /2
(inches)
X
10-J
~.,
9.68
10;,98
62
In=plane
Run If
Loa1ing~-.500
Load
{lbs)
J
Diameter Hole (r
~
2.9025) Cont 1d
D
y
{inches)
Dx/2
{inches)
1
2.)2 X 10-J
1.10
2
4.96
2Q24
J
7.)9
3.51J.
4
10.12
4.58
5
1.).. 05
5.90
6
15.63
7.15
7
18.11
8.)2
8
20., 61}
X
10-3
X
lO~J
,,
X
10-3
- 9.. 35
(---
----'""
9
10.77
In=p1ane Loading=-.750
Tare Load "" 5 lbs
Run#
1
Di~~eter
Hole (r = 2.9025)
Load
D
(lbs)
(inches)
y
10~3
lO~J
11.92
:X
10-3
2.04-
X
10=3
4.11
2
8.00
3.88
3
12.01
5,.79
4
16.06
X
10~3
X
2.08
7.82
9.83
6
2
X
1
5
I
DX /2
(inches) )
~..n.~
1
3.68
2
7.83
).87
3
11.86
5.79
4
15.99
X
10=3
/
3
X
10"'"3
7.77
5
9.97
6
11.,92
X
10-3
X
10~3
1
3.96 x lo-3
2.04
2
?.99
3.92
3
12.07
5.8l+
4
16.,07
X
10~3
7.81
5
10.o4
6
J:2ol2 X 10-3
64
(
Ol..lt-plane Loading--Solid Ring (r ""' 1. 9125)
Tare Load "" 10 lbs
Run #
Loe.d
(1bs)
1
.,
(
2
--
,;r.J
Deflection 11 A11
(inches)
=3
Deflection 11 B11
(inches)
0.20
10-3
(A .,.. B)/2
>(inches)
0.24
X
10""3
X
10=J
0,21~ X
10 ..;
4
0.28
8
0.39
0~64
0.52
12
0.14-7
1.00
0.74
16
0.74
1.36
1.05
20
0.81
1.76
1.29
24
0.89
2.11
1.50
28
1.00
2.52
1.76
.32
1.16
X
10-J
2.95
X
1-0~ 3
4
0.29
X
lO~J
0.18
X
10 ..;
8
o.~-2
0.6)
0.53 .·
12
o. 51+
0.9.3
o. 71}
16
0.69
1..38
1.04
20
0.85
1.75
1..30
24
0.95
2.05
L50
28
1.1.3
2.40
1.77
32
1.)0
X 1.0
X
10-.3
2.80
X
X
~")
10-3
2.06
2.05
X
~~
10-J
65 .
(
Out.:.plane Loading~~Solid Ring (r ., lo9125)
Run#
3
{
.·
Cont 1 d
Load
Deflection 11 A11
(lbs)
(inches)
4
0.23 x 1o-3
0~29
8
OcJ9
0.68
0.54
12
0.55
0.92
o. 74-
16
o. 71t
1.40
1.07
20
0.90
1..72
1 .. 31
24
1.00
2~00
1.50
28
1.21
2.40
L82
32
l.Jl
2.80 ~ 10~3
2o06
X
10-J
Deflection 11 Bu
(inches)
X
10~3
(A 4' B)/2
·(inches)
0.26
X
10-3
X
10=;;
')
66
TEST DATA
---
Out~plane Loading~-.250
Tare load
Run_#
~
10 lbs
Load
(lbs)
1
.,
2
Diameter Hole (r "" 1 .. 9125)
Deflection 11 A"
(inches)
J;,.
0.16
8
10=3
Deflection 11 B11
(inches)
(A + B)/2
>(inches)
Oo22 x 10-3
0.19
o.43
0.66
0.55
12
0.7)
1.14
0.. 94
16
1.02
1.62
1.J2
20
1.)2
2.06
1 .. 69
24
1.61
2.52
2.07
28
1.89
2.97
2 .. 43
32
2.21
X
10-3 ·.
3.. )9
X
10
4
0.. 17
X
10-3
0.17
X
10-3
8
0.. 45
0• .58
0.52
12
0.76
loO?
0.92
16
1.08
L56
1.32
20
1.)8
L96
1.67
24
1.65
2.41
2 .. 03
28
1.94
2.86
2.40
32
2.2)
X
X
10-J
3.26
X
=':I
_.~
10~3
X
10-3
2.. 80 x io~3
- 0.17
2. 75
~':I
X
10
X
10-3
._.~
67
TEST DATA
--·--
(
Ott't=plane Loading~=.250 Dia.))leter Hole (r "" 1.9125)
Run #
3
·~
Load
Cont•d
(lbs)
Deflection 11 A11
(inches)
4
0.17 'x 10- 3
8
0.44
0.54
0.49
12
0.7.5
1.00
0.88
16
.1.05
Ll.~l
1.23
20
loJ6
1 .. 86
1.61
24
1 .. 64
2.)6
2.00
28
1.97
2.78
2.38
32
2.2)
X
10=3
Deflection 11 B11
(inches)
0.14 X 10-3
3.21
X
10""3
(A + B)/2
(inches)
"Del6
X
10~3
z~ 72 x 10=3
68
TEST DATA
----
(
OJ.t...plane Load.ing=~.500 Diameter Hole (r ,.. 1.9125)
Tare Load ""
10 1bs
Run #
Lof!,d
Deflection 11 A11
Deflection 11 B11
(lbs)
(inches)
(inches)
1
"
(
,.
• r~-
•• .,. - - - ._..,. ---·---·- ""·- - ....•
.• ·--;,._-
O~ )6
)
(inches)
4
0.12
8
0.40
0.91
0.66
12
0.71
1.1}6
1.09
16
1.02
2.04-
1.53
20
1 • .34
2~55
1.95
24
1.62
.3-$07
2 • .35
28
1.90
3.. 58
2 .. 74
.32
2~18
X
X
.
2
10~.3
(A + B)/2
10~3
':1
X
10-.3
0.24
X
10~3
) .. 10
X
10=3
X
10=J
0.,26
X
10=__.
X
10-J
Oell X 10=-'
Osl~O
8
0.,38
0.. 94
Oo66
12
0.69
L53
l.ll.
16
0.. 99
2.09
1.54
20
1.,31
2.59
1.95
24
1.58
,3.14
2.,36
28
1.85
3.68
2.77
.32
2.1,3
4.01 x lo-3
,3. 07
10-3
lO~J
4.01
4
X
X
~
69
(
TEST DATA
----
.
Out~plane I~ading=~e500 Dia~eter
Run #
'\
. i" ""·· ... ~-
·-.
-~-···~ ~- ~----
•.
--
····.l';:.
~-
.
Hole {r
~
1.9125) Cont 6d
Load
Deflection 111!. 11
Deflection "B 11
(lbs)
(inches)
(inches)
10... 3
~3
{A
+ B)/2
(inches)
D.20
4
0.13
8
0.39
0.83
Oe61
12
0.69
1 ..·42
1.06
16
0.99
1.92
1.46
20
1.31
2.4.5
L88
24
1.57
.).00
2,29
28
1.89
.).4J
2.66
32
2.15 x 1o=J
3.9 6
X
0.27
X
X
10
19~':1-'
J~06
X
10-3
X
10~3
70
Out-plane
Tare Load
Run#
1
"
2
'"·~-·-
!.
-·
Lc~ding=-.750
c
Diameter Hole (r
~
1.9125)
10 lbs
Load
(lbs)
Deflection 11 A11
(inches)
Deflectin 11 B11
(inches)
(A + B)/2
(inches)
4
0.44
X
10-3
0.38
X
10-3
O.lJ.l
X
10-3
8
0.92
X
10=3
0.91
X
10-3
0.92
X
10- 3
12
1.39
1.35
1.37
16
1.85
1.80
1.8J
20
2 .. 27
2.29
2.28
2lJ.
2e68
X
10-3
2.65
X
10~3
2. 67
X
10~3
-4-
0.46
X
10""3
O.J9
X
10-3
0.4J
X
10~ 3
8
0.94
0.85
0.90
12
1.39
1.30
1.35
16
1.85
1.78
1.82 ..
20
2.25
2.27
24
2.65 x 1o=_J
2.64
X
10-__,
2~ 6
4
0.45
0.39
X
1{)-3
Oe 42
8
0.93
0~83
0.88
12
1.)6
1.33
l.J5
16
1.82
1.78
1.80
20
2.22
2.29
2.26
24
2.66
X
10=3
~
X
10-__,
2.70
.,
X
10-3
2.26
5 X 10=3
X
10~3
2.68 _x 10-J
TES1' DATA
-----
(
0J.t-p1ane Loading--Solid Ring (r "" 2. 9125)
Tare Lead
Run#
~
10 1bs
Load
(lbs)
1
....._
2
Deflection 11 A11
(inches)
10-~./
Deflection 11 B11
(inches)
(A + B)/2
(inches)
o.27 x .lo-3
0.51
2s00
0.63
1.32
6
3.15
1.26
2.21
8
4.05
1.90
2.96
10
5.40
2.. 45
3.96
12
6.10
3.29
4-.70
14·
6.95
3.90
5o J.,t.£3
16
?.?5
4.70
6.23
18
8.15
5.4l~
6.80
20
8. 70
2
0, 75
4
X
X
10-3
10-J
6.29
X
10-3
7.50 x.10=3
2
o.45 x 10=3
0,71
X
10~3
0.)8
4
1.45
1.11
1.28
6
2.65
1.61
2.13
8
3.80
2,19
3.00
10
5.00
2.79
).90
l2
5.95
3.59
4,.77
14
?.00
4.00
5.50
16
8.05
4.60
6.. 33
18
8.90
5.20
7.05
20
9.80
X
X
10-3
5.9]
=':I
X
10 ./
7.87
X
10-3
X
10=3
72
Out-plane toading=u·Solid Ring (r "" 2. 9125)
Run#
(-- ·.
...
_.~-
·;
......
-~-=···--~- --.-..-------~
---- ·--·- -·--
Load
Deflection 11 A11
(lbs)
(inches)
Deflection 11 Bu
(inches)
(A + B)/2
(inches)
2
Oo65
4
1.85
0.65
le25
6
2.. 90
1.24
2.07
8
4.05
1.89
2.97
10
5.. .30
2.43
).87
12
6.45
2.9.3
4.69
14
7.55
3$50
5.5J
16
8e90
3.68
6.29
18
9.50
4·.43
6e97
20
10~55
X
X
10-3
Cont'd
10-J
0.38
4.99
X
X
10""3
lO~J
0.52
7. 77
X
10""3
X
10=J
73
Out...pla."'le Loading= .... 250 Diameter Hole (r ""
2$9025)
· Tare Load "" 10 lbs
·-
~---~~--·-
. .
~-
.. -·
(A"+ B)/2·
(inches)
1
2
Oo65
4
1.35
0<>52
0.94
6
2.. 25
1.17
1.71
8
3.20
1 .. 78
2.49
10
4.15
2.36
J,26
12
5.05
2~92
3o98
14
5.90
3e56
4.73
16
6.-7o
3.99
5.35
18
7.50
4.. 68
6.. 09
20
8.30 x 10=3
5.,21
X
10-3
6. 76 x lo=3
2
Oe 70
0.02
X
10-3
0.)6
4
1.45
O~Jll-
0.90
6
2.45
0~87
1.66
8
3.35
~.52
2.,44
10
4.25
2.06
).16
12
5.15
2.65
).90
14
6.00
3.25
4.. 6)
16
6.80
3.. 71
5.26
18
7.55
4.. 40
5.98
20
8,.40
.;
........,.,...,.. ,...-:
Deflection 11 B11
Load
2
")
Deflection 11 A11
(inches)
Run #
_,..,.__
(1bs)
X
X
X
10=3
10-3
10-3
(inches)
0.25
4•95
X
X
10-3
10""3
0. 45 X 10~3
6.68
X
10=3
X
10""3
74
Out..plane Loading==.2.50 Diameter Hole (r ""' 2.9025) Ccnt'd
Run #
3
Load
Deflection 11 AII
(lbs)
(inches)
0~07
(A + B)/2
(inches)
2
0,.65
4
1.50
Oa51
1 .. 01
6
2~55
1.06
1 .. 81
8
; .. 45
1.76
2.61
10
4.. 4o
2,25
3.. 3)
12
.5 .. 20
2e84
4.02
14
6el5
).1}!:.
4.80
16
6.95
3.92
5.44
18
7.65
4.6)
6.14
20
8• .55
X
10-3
Deflection 11 B11
(inches)
X
10-3
'J.J6
X
10... 3
X
10""3
'}
/
.,
. (~
X
10-3
5e 17
X
10=3
6.86
75
!
•
Out-plane Loading=-.500 Diameter Hole (r
m
2.9025)
Tare Load ,.. 10 lbs
Run
1
'+
2
If
Load
(lbs)
·n~flection 11A11
(inches)
10=J
Defled.ion "B"
(inches)
O~J1
10-3
(A + B)~2
>(inches
0.. 51
X
10-J
2
0. 70
4
1.15
0.85
1 .. 00
6
1.,80
1 .. 65
1.7.3
8
2.65
2.60
2~6.3
10
,3 .. 15
3.43
3e29
'12
1}.00
4:.. 17
4.. 09
14
4.55
4,92
4.74
16
5.. 05
5.62
5.. 34-
18
6.00
6.. 39
6.20
20
6.90
X
10-3
6.,87
X
10=3
6.89 x·10-3
2
0.55
X
10~3
0~20
X
10-3
O.,J8 x 10-3
4
1 .. 05
0.76
0.91
6
1.50
1~Ci6
1.58
8
1.65
2~84
2o25
10
1.85
lt~04
2.95
12
2.. 35
4.96
3.. 66
14
2.9:5
5.65
4.30
16
3.. 45
6.46
4.96
18
4.15
7.24
5.70
20
5.00
X
X
10=3
7. 72
X
X
10-J
6.)6
X
10"".3
76
*·
Out-plane Loading==e500 Diameter Hole (r
Tare Load ~ 10 lbs
Run If
3
\
'
·~·
~
2.9025) Contftd
)(A + B)/2
. (inches)
Lofl.d
Deflection 11 AII
Deflection 11 B"
(lbs)
(inches)
(inches)
2
0,70 x 10=3
o.o4 x
4
1.45
o.4J
6
2.30
0.96
1.. 63
8
3.40
1.. 64·
2.. 52
10
4.oo
2.)8
3.19
12
4.70
3.• 22
3.. 96
14
5.05
3a69
4.37
16
6.. 15
4.JJ.2
5.29
18
6.8.5
5.09
-5.97
20
7.70
X
10- 3
5,.68
10~3
0.37
-
X
10-3
X
10-3
0.9lt-
6.. 69 x ·1o"" 3
77
Out-plane Loading==· 750 piameter Hole (r ""
2.9025)
Tare Load"" 10 lbs
Run#
1
.,
--·-
2
Load
(lbs)
Deflection 11 A11
(inches)
(inches)
(A + B)/2
(inches)
2
0.,40
4
0.75
1.1}0
1.08
6
1.50
2o14
1.82
8
2.25
2.86
2.56
10
2.. 90
3.71
3.31
12
3a55
4.50
4.03
14
4.30
5.25
4.78
16
5.. 00
6.,02
5o 51
18
5.80
6.. 67
20
6.,45
X
10~3
7.53
X
10-3
6.24
6----. =3
,99 x·lo
2
0.40
X
10=3
0.,5_3
X
10-3
0.47
4
0.. 65
X
10=3
Deflection 11 B11
0.61
X
10-3
1.32
0.,5].
X
X
10=3
10~3
0.99
-----~-
6
1.75
2&02
1~89
8
1 .. 95
4.. 92
2~44
10
2o70
3.71
3.21
12
J.45
4.53
3.99
14
4.40 .
.5.22
4.81
16
4.80
6.08
- . .5 .. 44
18
5.70
6a62
6.16
20
6.55
x
lo=3
7.,30
X
10""3
6.9J
X
10-J
78
Ottt=plane Loading=-.750 Diameter Hole (r
Run
II
Load
(lbs)
3
Deflection 11 A11
(inches)
,..'l
2.9025) Cont'd
Deflection 11 Bil
(inches)
0.49
10-3
(A + B)/2
(inches)
0.47
2
Oe45
4
0.. 80
1~27
1 .. Olt-
6
1.25
2.17
1.71
8
1.60
3.25
2.. 4)
10
2.. 45
4.01}
3.25
12
3.10
4.79
3.95
14
4.05
5<>35
4e70
16
4.95
6.05
5.50
18
6.00
6.50
6.25
20
7.20
X
10
~
J
X
X
10~3
X
10- 3
'\
·.•
X
10-3
6e97
X
10-3
7e09
79
13.0
Footnotes
1
Alex~r~er Blake. "How to Find Deflection and Moment of Rings
and Arcuate
Beams,*' Product Engineering, January 196J, p. 71.
2
Ibid.
t
p. 70.
~'1nond J.
4rbid., p.
Roark, Formu1as for
Stress and Strain, Nev1 York, 1965t p. 172.
172 •
.\. E. Peterson, Stress Concentration Design Factors, Nevi York, 1962,
p. 100.
80
14.0 Bibliography
\.
(
(
1.
Alexander, 11 Hm.; to Find Deflection and !-foment of" Rings and
Arcuate Beams~ 11 Product Engineering, Jai-ru.ary 196).
2.
Peterson, R~ E., Stress Concentration Design Factors, Nm>t Yorkt 1962,
John Wiley ru~d Sons, Inc.
3.
Roark. Raymond J., Fora:ulas
HcGrGr...T-Hill Book Company.
4.
Sha..'1l.ey • F. R., Strengtl! of Haterials, New York, 19 57, HcGravl=Hill
Book Company.
Blake~
f9£
Stress and Strain, Nevr York. 1965,