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V393.R46 I
HYDROMECHANICS GENERAL INSTABILITY OF RING-STIFFENED CYLINDRICAL
SHELLS SUBJECT TO EXTERNAL HYDROSTATIC PRESSURE
AERODYNAMICSThomas E. Reynolds and William F. Blumenberg
STRUCTURALMECHANICS
STRUCTURAL MECHANICS LABORATORY
RESEARCH AND DEVELOPMENT REPORT
APPLIEDMATHEMATICS
June 1959 Report 1324
PRNC-TNB-648 (Rev. --58)
- , - ~./e , --
- I _~
GENERAL INSTABILITY OF RING-STIFFENED CYLINDRICAL
SHELLS SUBJECT TO :.XT2~NAL HYDROSTATIC PRESSURE
by
Thomas E. Reynolds and William F. Blumenberg
Report 1324NS 731-038
June 1959
TABLE OF CONTENTS
Page
A B ST R A C T .................................................................................................................................. 1
INT R OD U CT ION ........................................................................................................................ 1
SUMMARY OF EARLIER STUDIES ......................................................................................... 1
RECE NT ST UDIES .................................................................................................................... 2
Tests with Different End Conditions ................................................................................. 2
Tests with Movable Insert Disks ......................................................................................... 7
DISCUSSION AND CONCLUSIONS ........................................................................................ 10
F UT U R E W OR K .......................................................................................................................... 11
RE FERE NC ES ............................................................................................................................ 11
LIST OF FIGURES
Figure 1 - Test Cylinders with Gages Installed ................................................................. 3
Figure 2 - Schematic Diagram Showing Dimensions of Machined Cylinders ................... 4
Figure 3 - Schematic Diagram of End Closure Arrangements ............................................. 5
Figure 4 - Closure Plates used in Tests ............................................................................. 6
Figure 5 - Experimental Pressures for Different End Conditions withTheoretical Pressures for Simple and Fixed Support ............................... .. 7
Figure 6 - Arrangement for Tests with Internal Movable Bulkheads ................................. 8
Figure 7 - Experimental Pressures versus Internal Bulkhead SpacingCompared with Kendrick's Theory for Simple Support ..................................... 9
LIST OF TABLES
Table 1 - Results of Tests with Varied End Conditions ................................................... 6
Table 2 - Results of Tests with Internal Bulkheads........................................................... 8
-r' I I ,_~ II I I 1 II I, I I I I I
ABSTRACT
Hydrostatic pressure tests were conducted on four machined ring-stiffened
cylinders of different lengths to investigate the influence of end fixity on the
elastic general-inktability pressure. Each cylinder was tested nondestructively
with several different end conditions. Pressure variations of as much as 65 per-
cent were observed for a particular cylinder.
Other tests were conducted with one cylinder reinforced by internal mov-
able bulkheads whereby buckling pressures were obtained as a function of
bulkhead spacing. The resulting experimental curve confirms the existence of
discontinuities which occur, according to Kendrick's theory, at the transition
between two circumferential buckling modes.
INTRODUCTION
A project designated "Deep Dish" was instituted at the David Taylor Model Basin forthe study of elastic general instability of ring-stiffened cylindrical shells under external hy-
drostatic pressure. This work has been reported in References 1 through 7.* A primary ob-
jective of the project has been the evaluation of Kendrick's theoretical work8 '9 through tests of
small machined cylinders having systematic variations in geometric parameters. This report
summarizes background information on this project and presents results of recent studies.
SUMMARY OF EARLIER STUDIES
Three series of cylinders have been studied, the first of which (designated DD-8 3 , 4 )had frame size as the single variable. The other two series, DD8-1 and DD8-2, had overall
length as the variable but were geometrically different. Tests of these two series were de-
scribed in a previous progress report. 5 All cylinders were equipped with heavy bolting rings,and usu'ally the bottom ring was bolted to a flat closure plate; the top ring was held against
the head of a vertical pressure tank. With this arrangement it was believed that a condition
of clamped support at the edges of the shell would be approximated.
Although Kendrick's theory assumes the edges to be simply supported, his contention
is that clamped ends will have only a local influence on the buckling pattern and that the re-
sulting buckling pressure will not differ appreciably from that obtained for the case of simple
support. Kaminsky, 1 0 in extending Kendrick's analysis for the clamped condition, found this
difference to be considerable. The buckled shape which Kaminsky assumed is, however,
only one of many possible shapes fulfilling the clamped condition. Moreover, there is no
assurance that by using some other (possibly more correct) shape substantially lower pres-
sures could not be obtained. Although Kaminsky's work thus did not refute Kendrick's
*References are listed on page 11.
4 1 -rB
contention, it still led investigators at the Model Basin to believe that if clamped ends could
be approximated in the laboratory, the experimental pressures would exceed Kendrick's pres-
sures by a considerable margin. The results of the first (DD-8) series agreed well with Ken-
drick's theory. However, this close correlation did not persist in the second and third series
of tests. 5 Some of these cylinders collapsed at pressures well above the Kendrick pressures,
although not nearly as high as those given by Kaminsky's theory.
The results of the DD-8, DD8-1, and DD8-2 series appeared to be so inconsistent that
no definite conclusions regarding the validity of available theories could be made without
additional investigation. Furthermore, it seemed clear that any intelligent interpretation of
these or future test results would require some knowledge of the influence of the experimental
end conditions on the collapse pressure. In Reference 5 it was proposed that duplicates of
of the three cylinders of the DD8-2 series which showed poorest agreement with Kendrick's
theory be tested with several different end closure arrangements. The buckling pressures
would be determined nondestructively by means of the Southwell method. 6 In this way it was
also hoped that some closure arrangement giving close agreement with Kendrick's theory
could be found, since it was desirable to use such an arrangement with forthcoming studies of
cylinders with intermediate heavy stiffeners.
RECENT STUDIES
So far the three duplicate cylinders have been tested with five different end attachments,
and an additional cylinder (DD8-2-6), the shortest of the series, has also been tested. Fur-
thermore, the longest cylinder, 4-A, had been tested using a technique with movable insert
disks to provide a test section of varying length. Use of this arrangement should more closely
simulate the actual conditions existing in a submarine. Also, since the only rotational re-
straint at the ends of the test section would come from the continuity of the shell, this con-
dition would appear to approximate that of simple supports more closely than the closure
arrangements used previously, thereby providing a more valid test of Kendrick's theory. Since
experimental points could be obtained at small intervals of length, a true experimental curve
of buckling pressure versus length could be established. Of particular interest was the tran-
sition between two and three circumferential lobes, a point where, theoretically, a disconti-
nuity (cusp) should exist.
This method thus providesa rapid and inexpensive means of obtaining much additional
data through the use of one existing cylinder. Thus far, tests have been completed with ten
different compartment lengths. The results of these tests together with those obtained using
the various end closure arrangements are summarized briefly in this third progress report.
TESTS WITH DIFFERENT END CONDITIONS
The four cylinders under consideration designated DD8-2-6, 2A, 3A, and 4A. are pic-
tured in Figure 1. In this series the length-to-diameter ratio varies from 1.91 to 6.85.
I
Figure 1 - Test Cylinders with Gages Installed
Cylinder dimensions are shown in Figure 2. As with the original series of cylinders, these
cylinders had rectangular stiffening rings and were accurately machined from steel tubing
whose high yield strength (85,000 psi) was sufficient to prevent inelastic action prior to the
onset of buckling. The various end closure arrangements are illustrated in Figure 3. Case IV,
the arrangement used with the original set of cylinders, 4 employed a closure plate approxi-
mately 1 in. thick. The 2-in. plates were chosen to find out whether a thicker plate would
affect the collapse pressure significantly. Case II closure was used to determine the effect
of replacing the distributed pressure over the surface of the thinner plate with a concentrated
load around its edge. This was done by means of pressure caps which are shown in Figure 4
along with the /-in. and 2-in. closure plates. All the closure plates were equipped with ma-
chined inserts fitting tightly within the cylinder wall to prevent radial deflections at the ends.
Each cylinder was instrumented with electrical resistance strain gages most of which
were confined to the exterior of the centermost stiffener and oriented circumferentially. The
buckling pressures were obtained by the Southwell method 6 applied to the strain-pressure plots.
The buckling pressures obtained under the various end conditions are given in Table 1
together with the pressures for the original group of cylinders and the theoretical pressures
according to Kendrick Part III* and Kaminsky.
*Since it has been shown3 ' 7 that the theory of Kendrick Part I is often unconservative, and sometimes may be
unsafe to use, attention is confined to the second solution of Kendrick Part III, which is simply a more rigorous
treatment of the same problem.
Figure 2 - Schematic Diagram Showing Dimensions of Machined Cylinders
All dimensions are in inches.
The same information is presented graphically in Figure 5. The close agreement between Cy-
linders 2 and 2-A, 3 and 3-A, and 4 and 4-A under Case IV in Table 1 indicates that the three
cylinders were accurately duplicated. Most striking, perhaps, is the very large increase in
strength realized in progressing from Case I to Case V. For Cylinder 3-A, this amounted to
a pressure increase of 65 percent. Also remarkable is the nearly perfect agreement with
Kendrick's theory for Case I. Another interesting result was that, in those cases (I-IV) where
one or both %-in. plates were used, the appearance of the characteristic lobar strain pattern
prior to buckling was influenced by the orientation of the end plates.. That is, with Case II,
Cylinder 3-A, for example, a rotation through 90 deg of the end plates produced a 90 deg rota-
tion of the circumferential strain pattern, along with a slight change in the buckling pressure,
.........." W 11111 M
Case I
Case I" Case 1Z
Case -2
Figure 3 - Schematic Diagram of End Closure Arrangements
---- ~- ----- MINI1
Figure 4 - Closure Plates used in Tests
TABLE 1
Results of Tests with Varied End Conditions
Theoretical Pressures, psi Experimental Pressures, psi
Cylinder Frame Length - Kendrick III KaminskyNumber Spaces Diameter (Simple Support) (Clamped Support)Case I Case II Case Ill Case IV Case V
(_iI _ _ _ _ _ _ _ (u
1.91
2.61
3.36
4.31
5.16
6.85
499(3)*
415(3)
317(2)
216(2)
180(2)
159(2)
910(3)702(3)
575(3)
551(3)
494(2)
360(2)
504(3) 576(3)
339(2) 1365(2)
229(2) 246(2)238(2)**
178(2) 1189(2)
576(3)
394(3)
298(2)
231(2)
492(3)
408(3)
398(3)
321(2)
315(2)
238(2)
228(2)
173(2)
700(3)
409(3)
378(2)
289(2)
*Numbers in parentheses indicate the number of circumferential lobes.
**End plates rotated 90 deg with a corresponding rotation of lobar strain pattern.
6
1
2
2-A
3
3-A
4
4-A
5
14
19
25
31
37
49I -
- -- 011141161411
1000- 100 Experimental Pressures
o Case I
* Case Il900 A Case I
A Original se 11_V Duplicate Case"
o Case YT800 * End Plates Rotated 90 degrees
700 (3)
Komaminsky Theory/(Clamped Support)
600 Reference 10\ ~(3)
S500 (3) (3)a-
S
__40 ___ _ _ ' _
a 400 --- - --N
S(2) O(2)
-Kendrick Part IMr Theory 0(2)(Simple Support) V(2)
300 Reference 9 A(2) 0(2)
___ ___- -___ *q&2)-(2)1
200 -2) -
100
Cylinder Cylinder Cylinder Cylinder Cylinder CyinderSNo.6- No.I No. 2 N .3 No .45
0 -- 1 1 1
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50Frame Spaces
Figure 5 - Experimental Pressures for Different End Conditions with TheoreticalPressures for Simple and Fixed Support
Numbers in parentheses indicate number (n) of circumferential lobes.
as shown in Table 1. This cannot be attributed to any lack of circularity in the plate inserts,
which were more nearly circular than the cylinders, nor was there any obvious lack of uniform-
ity in the plate cross sections which would indicate a lack of symmetry in the bending rigidity.
Rotation of the 2-in. plates, on the other hand, had no effect on the orientation of the strain
pattern nor on the buckling pressure.
TESTS WITH MOVABLE INSERT DISKS
Cylinder 4-A, the longest of the duplicate group, was tested with movable insert disks
simulating bulkheads. These were circular disks with edges machined to a small radius to
I MINIlmIUM
18"
8.1
I R
Figure 6 - Arrangement for Tests with Internal Movable Bulkheads
TABLE .2
Results of Tests with Internal Bulkheads
Kendrick III ExperimentalFrame Length Pressure, Pressure,Spaces Diameter
psi psi
17 2.40 428(3)* 473(3)
21 2.96 404(3) 422(3)
23 3.25 367(2) 412(3)
25 3.53 305(2) 401(3)
26 3.67 281(2) 398(3)
27 3.81 262(2) 394(3)
28 3.96 246(2) 391(3)
29 4.10 233(2) 383(2)
31 4.38 212(2) 329(2)
33 4.66 197(2) 281(2)
*Numbers in parentheses indicate the number of
circumferential lobes.
minimize the area of contact with the shell.
The disks were held in place by circular wood-
en spacers of various widths as shown in Fig-
ure 6. By separating the disks with different
combinations of the spacers, the length of the
central test section could be varied from one-
third to the full length of the cylinder in incre-
ments of one frame space. In each test the
disks were located directly beneath a stiffener
and the ends of the cylinder were closed as in
Case I of Figure 3.
In addition to the strain gages located
around the centermost stiffener for the previous
tests, additional gages were installed circumfer-
entially along two generators over one half of
the cylinder length to measure the longitudinal
shape of the deflection pattern. As before, the
Southwell method was used to determine the
elastic buckling pressures.
~ __
r
- ----- -~~ --
~/~//~,~/////~P~
500
450 Experimental Curve
400- -- - .) 13
O(2)
Kendrick Port m Theory50 (Simple Support) - Reference 9350 -
2300 -_
250
200_
1 501 I. _ I __ L _ I __ _ _ I __ _I __ I _ _ I __ __ __ .1 __ ___ __ __ _
16 18 20 22 24Frame Spaces
30 32 ~4
Figure 7 - Experimental Pressures versus Internal Bulkhead SpacingCompared with Kendrick's Theory for Simple Support
Numbers in parentheses indicate number (n) of circumferential lobes.
Tests were conducted with ten different compartment length-liameter ratios ranging
from 2.40 to 4.66. Table 2 lists the compartment length-diameter ratios, the corresponding
experimental pressures, and the pressures given by the Kendrick Part III theory.* These re-
sults appear graphically in Figure 7. It is seen that, although the experimental points lie
somewhat above the theoretical curve, they follow a similar curve, proving the existence of
the discontinuity (cusp) formed at'the intersection of the curves for two and three circumfer-
ential lobes. This fact, not previously verified experimentally for plate and shell structures,
should be of more than mere academic interest, since it indicates that compartment lengths
must be chosen with care if the material is to be used efficiently. For example, the experi-
mental curve shows that a compartment comprising 31 frame spaces can be appreciably strength-
ened if it is shortened by two frame spaces. Beyond that point, however, little benefit is gained
unless a reduction of ten spaces or more is made. In comparing these results with the previous
tests, good agreement is found between the pressures for 25 and 31 frame spaces and those
under Case IV for the cylinders (2 and 3) of corresponding length.
*The calculated pressures for 25 and 31 frame spaces given in Table 2 do not agree exactly with those given
in Table 1 for Cylinders 2 and 3 because, for all cylinders, the end frame spacings are slightly shorter than
typical, whereas the test sections in Table 2 had uniform frame spacing.
9
....... A
4
1
DISCUSSION AND CONCLUSIONS
Probably the most important conclusion that can be drawn from the results of these tests
is that the boundary conditions have an appreciable influence on the buckling pressure. Fur-
thermore, it is significant to note that while Kendrick's theory in some cases seriously under-
estimates the buckling pressure, in no case does his theory appreciably overestimate it. Tests
with Case V, where the ends would seem to be extremely rigid, indicate that the influence of
clamped ends may be overemphasized by Kaminsky.
To draw more specific conclusions regarding the effect of boundary conditions would be
unwarranted on the basis of the limited data available at this time. However, one possible ex-
planation bears mention. In Reference 5, it was suggested that "the introduction of partial
fixity at the ends of a simply supported cylinder is equivalent to shortening its effective
length." It is possible that this effect may depend on the simple relationship,
Le = KL
where L is the actual length of the cylinder,
Le is its effective length, and
K is a constant depending on the degree of fixity, being less than one for all cases
falling between simple support and clamped ends.
Such a relationship is valid for the buckling of a centrally compressed column and has been
utilized with some success by Arnold and Warburton 11 in a study of the effect of boundary con-
ditions on the flexural vibrations of unstiffened tubes.
If this relationship were to hold for the case of general instability, all conditions of
edge fixity could be represented by a single curve of pressure versus effective length, pro-
vided K is known as a function of edge fixity. From this representation comes the interesting
result that the transition from one circumferential mode to another must occur at the same pres-
sure, regardless of the degree of fixity. Figure 7 shows just this sort of correspondence be-
tween Kendrick's solution for simple support and the experimental curve obtained from the
tests with internal bulkheads. The theoretical transition between two and three lobes occurs
at about 401 psi, while experimentally it appears at about 389 psi. This small difference in
pressures could result from variations in cylinder dimensions within the specified tolerances
or from slight inaccuracies arising from the use of nominal values for Young's modulus
(30 x 106 psi) and Poisson's ratio (0.3) in the calculations.
On this basis Kendrick's theory would appear to be very accurate. On the other hand,
Kaminsky's theory for clamped ends (Figure 5) shows the transition from two to three lobes at
about 520 psi, considerably higher than that found experimentally.
It is emphasized that this concept of effective length is presented merely as one inter-
pretation of the results thus far obtained. At the present time there are insufficient data for a
proper evaluation of the idea. It can only be said that none of the tests to date has shown it
to be invalid, and that one of the immediate objectives of Project Deep Dish should be to pro-
vide the additional data necessary for a proper evaluation.
61WM101 0 II 0 I io
~I I I I II I I L I r Illir
, w I AlMlil mla rIYII I I
FUTURE WORK
In order to continue the line of study described in this report and to investigate other
related areas of interest, the following work is anticipated:
1. Further tests employing internal bulkheads to provide a central compartment of
varying length. In particular, tests ire now underway using the shorter cylinder 2-A to inves-
tigate the influence of a shorter end compartment on the buckling pressures already obtained
as well as to provide additional data in the range of shorter central compartments.
2. Efforts to design a test arrangement that will simulate the simple support condition,
assumed by Kendrick in his theoretical treatment.
3. Further tests of cylinders having one intermediate heavy stiffener, toinvestigate the
efficiency of deep frames in breaking up the overall length of a submarine compartment. The
size of the stiffener will be systematically reduced, and the collapse pressure will be deter-
mined at each stage by nondestructive testing.
4. An investigation of the range of boundary conditions existing in an actual submarine
so that future structural model tests of submarine designs would incorporate more realistic end
closure bulkheads.
5. An examination of the clamped-end case to find a more realistic theoretical buckling
shape which would lead to buckling pressures lower than those given by the Kaminsky analysis.
REFERENCES
1. Slankard, R.C., et al, "An Experimental Investigation of the Effect of Radial Excitation
on the General-Instability Strength of Stiffened Cylindrical Shells Subjected to Hydrostatic
Pressure (Models 1A and 1K)," David Taylor Model Basin Report C-724 (Jan 1956)
CONFIDENTIAL.
2. David Taylor Model Basin CONFIDENTIAL letter C-SS/S11 Serial 0171 of 28 Feb 1955
to Bureau of Ships.
3. Slankard, R.C. and Galletly, G.D., "The Effect of Reinforcing Rings on the General-
Instability Strength of Machined Cylindrical Shells under External Hydrostatic Pressure,"
David Taylor Model Basin Report C-822 (Jun 1957) CONFIDENTIAL.
4. Galletly, G.D., et al, "General Instability of Ring-Stiffened Cylindrical Shells Subject
to External Hydrostatic Pressure-A Comparison of Theory and Experiment," Journal of
Applied Mechanics, Vol. 25, Trans. ASME, Vol. 80, pp. 259-266 (Jun 1958).
5. Reynolds, T.E., "Progress Report. General Instability of Ring-Stiffened Cylindrical
Shells Subject to External Hydrostatic Pressure," David Taylor Model Basin Report C-841
(Jun 1957) CONFIDENTIAL.
I_ _ 111=l1 1
ilwl~ ,,srrmneha* *rrn~~**~Cw*riuruuorr~i~*ao f3Cr~-
11,
6. Galletly, G.D. and Reynolds, T.E., "A Simple Extension of Southwell's Method for
Determining the Elastic General Instability Pressure of Ring-Stiffened Cylinders Subject to
External Hydrostatic Pressure," Proceedings for the Society of Experimental Stress Analysis,
Vol. XIII, No. 2, p. 141 (1956).
7. Reynolds, T.E., "A Graphical Method for Determining the General Instability Strength
of Stiffened Cylindrical Shells," David Taylor Model Basin Report 1106 (Sep 1957).
8. Kendrick, S., "The Buckling under External Pressure of Circular Cylindrical Shells
with Evenly Spaced Equal Strength Circular Ring Frames-Part I," Naval Construction Research
Establishment Report NCREI/R.211 (Feb 1953).
9. Kendrick, S., "The Buckling under External Pressure of Circular Cylindrical Shells
with Evenly Spaced Equal Strength Circular Ring Frames-Part III," Naval Construction
Research Establishment Report NCRE/R.244 (Sept 1953).
10. Kaminsky, E.L., "General Instability of Ring-Stiffened Cylinders with Clamped Ends
under External Pressure by Kendrick's Method," David Taylor Model Basin Report 855
(Jul 1954).
11. Arnold, R.N. and Warburton, G.B., "The Flexural Vibrations of Thin Cylinders,"
Proceedings (A) for the Institution of Mechanical Engineers, Vol. 167, No. 1, pp. 62-80 (1953).
,, I a
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, 1611, 1 W I 6111116 11
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1 CHONR, Mech Br (Code 438)
1 OPNAV, Op 373
1 Prof. J. Kempner, Dept of Aero Engin & ApplMech, Polytechnic Institute of Brooklyn,N.Y.
1 Pressure Vessel Research Committee,New York, N.Y.
1 Dir of Def Res & EnginAttn: Tech Library
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1 Dr. E. Wenk, Jr., ChairmanDept of Engin Mech, SW Res Inst,San Antonio, Texas
1 Dr. G.D. Galletly, Emeryville Res Ct,Shell Dev Co, Emeryville, Calif.
1 S. Kendrick, Naval Construction ResearchEstablishment, St. Leonard's Hill,Dunfermline, Scotland
1 H. Becker, College of Engin, New YorkUniv, University Heights, N.Y., N.Y.
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9.
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bles
, re
fs.
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pr
essu
re t
ests
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on f
our
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truc
tive
ly
with
sev
eral
dif
fere
nten
d co
ndit
ions
. P
ress
ure
vari
atio
ns o
f as
muc
h as
65
perc
ent
wer
e ob
serv
ed f
or a
par
ticu
lar
cyli
nder
.O
ther
tes
ts w
ere
cond
ucte
d w
ith o
ne c
ylin
der
rein
forc
ed b
y in
-te
rnal
mov
able
bul
khea
ds
whe
reby
buc
klin
g pr
essu
res
wer
e ob
-ta
ined
as
a fu
ncti
on o
f bu
lkhe
ad s
paci
ng.
The
res
ulti
ng e
xper
i-m
enta
l cu
rve
conf
irm
s th
e ex
iste
nce
of d
isco
ntin
uiti
es w
hich
occu
r, a
ccor
ding
to
Ken
dric
k's
theo
ry,
at t
he t
rans
itio
n be
twee
ntw
o ci
rcum
fere
ntia
l bu
ckli
ng m
odes
.
1.
Cyl
indr
ical
she
lls
(Sti
ffen
ed)
-B
uck
ling
-M
odel
tes
ts.
2.
Cyl
indr
ical
she
lls
(Sti
ffen
ed)
-B
uck
lin
g
-G
raph
ical
an
alys
is.
3.
Cyl
indr
ical
she
lls
(Sti
ffen
ed)
-B
uck
ling
-T
est
met
hods
(N
on-
dest
ruct
ive)
4.
Subm
arin
e hu
lls
-C
hara
cter
isti
cs -
Mod
elte
sts.
I. R
eyno
lds,
T
hom
as E
.II
. B
lum
enbe
rg,
Will
iam
F.
III.
N
S 73
1 03
8
1.
Cyl
indr
ical
she
lls
(Sti
ffen
ed)
-B
uck
ling
-M
odel
tes
ts.
2.
Cyl
indr
ical
she
lls
(Sti
ffen
ed)
-B
uck
ling
-G
raph
ical
an
alys
is.
3.
Cyl
indr
ical
she
lls
(Sti
ffen
ed)
-B
uck
ling
-T
est
met
hods
(N
on-
dest
ruct
ive)
4.
Subm
arin
e hu
lls
-C
hara
cter
isti
cs
-M
odel
test
s.I.
Rey
nold
s,
Tho
mas
E.
II.
Blu
men
berg
, W
illia
m F
.II
I.
NS
731
038
Dav
id T
aylo
r M
odel
Bas
in.
Rep
ort
1324
.G
EN
ER
AL
IN
STA
BIL
ITY
OF
RIN
G-S
TIF
FEN
ED
CY
LIN
DR
I-C
AL
SH
EL
LS
SU
BJE
CT
TO
EX
TE
RN
AL
H
YD
RO
STA
TIC
P
RE
S-
SUR
E,
by T
hom
as E
. R
eyno
lds
and
Will
iam
F.
Blu
men
berg
.Ju
ne 1
959.
ii,
13
p. p
hoto
s.,
tabl
es,
refs
. U
NC
LA
SSIF
IED
Hyd
rost
atic
pre
ssur
e te
sts
wer
e co
nduc
ted
on f
our
mac
hine
dri
ng-s
tiff
ened
cyl
inde
rs o
f di
ffer
ent
leng
ths
to i
nves
tiga
te t
he i
n-fl
uenc
e of
end
fi
xity
on
the
elas
tic
gene
ral-
inst
abil
ity
pres
sure
.E
ach
cyli
nder
was
tes
ted
nond
estr
ucti
vely
with
sev
eral
dif
fere
nten
d co
ndit
ions
. P
ress
ure
vari
atio
ns o
f as
muc
h as
65
perc
ent
wer
e ob
serv
ed
for
a pa
rtic
ular
cyl
inde
r.O
ther
tes
ts w
ere
cond
ucte
d w
ith o
ne c
ylin
der
rein
forc
ed b
y in
-te
rnal
mov
able
bul
khea
ds w
here
by b
uckl
ing
pres
sure
s w
ere
ob-
tain
ed a
s a
func
tion
of
bulk
head
spa
cing
. T
he r
esul
ting
exp
eri-
men
tal
curv
e co
nfir
ms
the
exis
tenc
e of
dis
cont
inui
ties
w
hich
occu
r, a
ccor
ding
to
Ken
dric
k's
theo
ry,
at t
he t
rans
itio
n be
twee
ntw
o ci
rcum
fere
ntia
l bu
ckli
ng m
odes
.
Dav
id T
aylo
r M
odel
Bas
in.
Rep
ort
1324
.G
EN
ER
AL
IN
STA
BIL
ITY
OF
RIN
G-S
TIF
FEN
ED
CY
LIN
DR
I-C
AL
SH
EL
LS
SU
BJE
CT
TO
EX
TE
RN
AL
HY
DR
OST
AT
IC
PR
ES
-SU
RE
, by
Tho
mas
E.
Rey
nold
s an
d W
illia
m F
. B
lum
enbe
rg.
June
195
9.
ii,
13p.
pho
tos.
, ta
bles
, re
fs.
UN
CL
ASS
IFIE
D
Hyd
rost
atic
pre
ssur
e te
sts
wer
e co
nduc
ted
on f
our
mac
hine
dri
ng-s
tiff
ened
cyl
inde
rs o
f di
ffer
ent
leng
ths
to i
nves
tiga
te t
he i
n-fl
uenc
e of
end
fix
ity
on t
he e
last
ic g
ener
al-i
nsta
bili
ty p
ress
ure.
Eac
h cy
lind
er w
as t
este
d no
ndes
truc
tive
ly w
ith s
ever
al d
iffe
rent
end
cond
itio
ns.
Pre
ssur
e va
riat
ions
of
as m
uch
as 6
5 pe
rcen
tw
ere
obse
rved
fo
r a
part
icul
ar c
ylin
der.
Oth
er t
ests
wer
e co
nduc
ted
with
one
cyl
inde
r re
info
rced
by
in-
tern
al m
ovab
le b
ulkh
eads
whe
reby
buc
klin
g pr
essu
res
wer
e ob
-ta
ined
as
a fu
ncti
on o
f bu
lkhe
ad s
paci
ng.
The
res
ulti
ng e
xper
i-m
enta
l cu
rve
conf
irm
s th
e ex
iste
nce
of d
isco
ntin
uiti
es
whi
choc
cur,
ac
cord
ing
to K
endr
ick'
s th
eory
, at
the
tra
n ;t
ion
betw
een
two
circ
umfe
rent
ial
buck
ling
mod
es.
1.
Cyl
indr
ical
shel
ls(S
tiff
ened
) -
Buck
ling -
Mod
el t
ests
.2.
C
ylin
dric
al s
hel
ls(S
tiff
ened
) -
Buck
ling
-G
raph
ical
ana
lysi
s.3.
C
ylin
dric
al s
hel
ls(S
tiff
ened
) -
Buck
ling
-T
est
met
hods
(N
on-
dest
ruct
ive)
4.
Subm
arin
e hu
lls
-C
hara
cter
isti
cs -
Mod
elte
sts.
I. R
eyno
lds,
Tho
mas
E.
II.
Blu
men
berg
, W
illia
m F
.II
I.
NS
731
038
1.
Cyl
indr
ical
shel
ls(S
tiff
ened
) -
Buck
ling
-M
odel
tes
ts.
2.
Cyl
indr
ical
shel
ls(S
tiff
ened
) -
Buck
ling
-G
raph
ical
an
alys
is.
3.
Cyl
indr
ical
shel
ls(S
tiff
ened
) -
Buck
ling -
Tes
t m
etho
ds (
Non
-de
stru
ctiv
e)4.
Su
bmar
ine
hull
s -
Cha
ract
eris
tics
-M
odel
test
s.I.
Rey
nold
s, T
hom
as E
.II
. B
lum
enbe
rg,
Will
iani
F.
III.
N
S 73
1 03
8
I
Dav
id T
aylo
r M
odel
Bas
in.
Rep
ort
1324
.G
EN
ER
AL
IN
STA
BIL
ITY
OF
RIN
G-S
TIF
FE
NE
D C
YL
IND
RI-
CA
L S
HE
LL
S SU
BJE
CT
TO
EX
TE
RN
AL
H
YD
RO
STA
TIC
PR
ES
-SU
RE
, by
Tho
mas
E.
Rey
nold
s an
d W
illia
m F
. B
lum
enbe
rg.
June
19
59.
ii,
13p.
pho
tos.
, ta
bles
, re
fs.
UN
CL
ASS
IFIE
D
Hyd
rost
atic
pr
essu
re t
ests
wer
e co
nduc
ted
on f
our
mac
hine
dri
ng-s
tiff
ened
cyl
inde
rs o
f di
ffer
ent
leng
ths
to i
nves
tiga
te t
he i
n-fl
uenc
e of
end
fix
ity
on t
he e
last
ic
gene
ral-
inst
abil
ity
pres
sure
.E
ach
cyli
nder
was
tes
ted
nond
estr
ucti
vely
with
sev
eral
dif
fere
nten
d co
ndit
ions
. P
ress
ure
vari
atio
ns o
f as
muc
h as
65
perc
ent
wer
e ob
serv
ed f
or a
par
ticu
lar
cyli
nder
.O
ther
tes
ts w
ere
cond
ucte
d w
ith
one
cyli
nder
rei
nfor
ced
by i
n-te
rnal
mov
able
bul
khea
ds w
here
by b
uckl
ing
pres
sure
s w
ere
ob-
tain
ed a
s a
func
tion
of
bulk
head
spa
cing
. T
he r
esul
ting
exp
eri-
men
tal
curv
e co
nfir
ms
the
exis
tenc
e of
dis
cont
inui
ties
whi
choc
cur,
acc
ordi
ng t
o K
endr
ick'
s th
eory
, at
the
tra
nsit
ion
betw
een
two
circ
umfe
rent
ial
buck
ling
mod
es.
1.
Cyl
indr
ical
shel
ls(S
tiff
ened
) -
Bu
ckli
ng
-
Mod
el t
ests
.2.
C
ylin
dric
al s
hel
ls(S
tiff
ened
) -
Bu
ckli
ng
-
Gra
phic
al a
naly
sis.
3.
Cyl
indr
ical
sh
ells
(Sti
ffen
ed)
-B
uck
lin
g
-T
est
met
hods
(N
on-
dest
ruct
ive)
4.
Subm
arin
e hu
lls
-C
hara
cter
isti
cs -
Mod
elte
sts.
I. R
eyno
lds,
Tho
mas
E.
II.
Blu
men
berg
, W
illia
m F
.II
I.
NS
731
038
Dav
id T
aylo
r M
odel
Bas
in.
Rep
ort
1324
.G
EN
ER
AL
IN
STA
BIL
ITY
O
F R
ING
-ST
IFF
EN
ED
C
YL
IND
RI-
CA
L S
HE
LL
S SU
BJE
CT
TO
EX
TE
RN
AL
HY
DR
OST
AT
IC
PR
ES
-SU
RE
, by
Tho
mas
E.
Rey
nold
s an
d W
illia
m F
. B
lum
enbe
rg.
June
195
9.
ii,
13p.
pho
tos.
, ta
bles
, re
fs.
UN
CL
ASS
IFIE
D
Hyd
rost
atic
pre
ssur
e te
sts
wer
e co
nduc
ted
on f
our
mac
hine
dL
ing-
stif
fene
d cy
lind
ers
of d
iffe
rent
len
gths
to
inve
stig
ate
the
in-
flue
nce
of e
nd f
ixit
y on
the
ela
stic
gen
eral
-ins
tabi
lity
pr
essu
re.
Eac
h cy
lind
er w
as t
este
d no
ndes
truc
tive
ly w
ith
sev
eral
dif
fere
nten
d co
ndit
ions
. P
ress
ure
vari
atio
ns
of a
s m
uch
as
65 p
erce
ntw
ere
obse
rved
for
a p
arti
cula
r cy
lind
er.
Oth
er t
ests
wer
e co
nduc
ted
wit
h on
e cy
lind
er r
einf
orce
d by
in-
tern
al m
ovab
le b
ulkh
eads
whe
reby
buc
klin
g pr
essu
res
wer
e ob
-ta
ined
as
a fu
ncti
on o
f bu
lkhe
ad s
paci
ng.
The
res
ulti
ng e
xper
i-m
enta
l cu
rve
conf
irm
s th
e ex
iste
nce
of d
isco
ntin
uiti
es w
hich
occu
r, a
ccor
ding
to
Ken
dric
k's
theo
ry,
at t
he t
rans
itio
n be
twee
ntw
o ci
rcum
fere
ntia
l bu
ckli
ng m
odes
.
1.
Cyl
indr
ical
she
lls
(Sti
ffen
ed)
-B
uck
ling
-M
odel
tes
ts.
2.
Cyl
indr
ical
sh
ells
(Sti
ffen
ed)
-B
uck
lin
g
-G
raph
ical
ana
lysi
s.3.
C
ylin
dric
al
shel
ls(S
tiff
ened
) -
Buck
ling -
Tes
t m
etho
ds
(Non
-de
stru
ctiv
e)4.
Su
bmar
ine
hull
s -
Cha
ract
eris
tics
-
Mod
elte
sts.
I. R
eyno
lds,
T
hom
as E
.II
. B
lum
enbe
rg,
Will
iam
F.
III.
N
S 73
1 03
8
.___
_
3 9080 02754 3237
- 9~
I I -L
MAR 2 '
A 2 :1977APE 21 off
DEC 2 3 I 7
U L 3 1979
MAb-Y
oc T 71g98
~JI I II r r rclbl C~ r ~ ~-
r- I
