NASA TECHNICAL
MEMORANDUM
I
X
i NASA TM X2991
BUCKLING OF A CONICAL SHELL
WITH LOCAL IMPERFECTIONS
by Paul A. Cooper and Cornelia B. Dexter
Langley Research Center
Hampton, Va. 23665
"^eisi*
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY 1974
1. Report No.
NASA TM X2991
2. Government Accession No.
3. Recipient's Catalog No.
4. Title and Subtitle
BUCKLING OF A CONICAL SHELL WITH
LOCAL IMPERFECTIONS
5. Report Date
July 1974
6. Performing Organization Code
7. Author(s)
Paul A. Cooper and Cornelia B. Dexter
8. Performing Organization Report No.
L9331
9. Performing Organization Name and Address
NASA Langley Research Center
Hampton, Va. 23665
10. Work Unit No.
743321101
11. Contract or Grant No.
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, D.C. 20546
13. Type of Report and Period Covered
Technical Memorandum
14. Sponspring Agency Code
1 5. Supplementary Notes
16. Abstract
Small geometric imperfections in thinwalled shell structures can cause large reduc
tions in buckling strength. Most imperfections found in structures are neither axiS5mimetric
nor have the shape of buckling modes but rather occur locally. This report presents the
results of a study of the effect of local imperfections on the critical buckling load of a specific
axially compressed thinwalled conical shell. The buckling calculations were performed by
using a twodimensional shell analysis program referred to as the STAGS (STructural Analysis
of General Shells) computer code, which has no axisymmetry restrictions.
Results show that the buckling load found from a bifurcation buckling analysis is highly
dependent on the circumferential arc length of the imperfection type studied. As the cir
cumferential arc length of the imperfection is increased, a reduction of up to 50 percent of
the critical load of the perfect shell can occur. The buckling load of the cone with an
axisymmetric imperfection is nearly equal to the buckling load of imperfections which
extended 60° or more around the circumference, but would give a highly conservative esti
mate of the buckling load of a shell with an imperfection of a more local nature.
17. Key Words (Suggested by Author(s))
Conical shell buckling
Conical shell instability
Shell imperfections
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BUCKLING OF A CONICAL SHELL WITH
LOCAL IMPERFECTIONS
By Paul A. Cooper and Cornelia B. Dexter
Langley Research Center
SUMMARY
Small geometric imperfections in thinwalled shell structures can cause large
reductions in buckling strength. Most imperfections found in structures are neither axi
symmetric nor have the shape of buckling modes but rather occur locally. This report
presents the results of a study of the effect of local imperfections on the critical buckling
load of a specific axially compressed thinwalled conical shell. The buckling calculations
were performed by using a twodimensional shell analysis program referred to as the
STAGS (STructural Analysis of General Shells) computer code, which has no axisymmetry
restrictions. ,
Results show that the buckling load found from a bifurcation buckling analysis is
highly dependent on the circumferential arc length of the imperfection type studied. As
the circumferential arc length of the imperfection is increased, a reduction of up to
50 percent of the critical load of the perfect shell can occur. The buckling load of the
cone with an axisymmetric imperfection is nearly equal to the buckling load of imperfec
tions which extended 60° or more around the circumference, but would give a highly con
servative estimate of the buckling load of a shell with an imperfection of a more local
nature.
INTRODUCTION
Small geometric imperfections in thinwalled shell structures can cause large
reductions in buckling strength. Much work has been done to establish the buckling
imperfection sensitivity of shellofrevolution structures containing small geometric
imperfections either axisymmetric in shape or in the shape of classical buckling modes.
(See, for example, refs. 1 to 3.) In practice, however, most imperfections found in
structures are neither axisymmetric nor have the shape of buckling modes but rather
occur locally. This report presents the results of a study of the effect of local imper
fections on the critical buckling load of a specific axially compressed thinwalled conical
shell. The study was motivated by a need to establish the degradation of the axial load
carrying ability of a thinwalled conical portion of a fielded missile system which had
sustained local damage during routine handling and shipping. The buckling calculations
were performed by using a twodimensional shell analysis program referred to as the
STAGS (STructural Analysis of General Shells) computer code (ref. 4), which has no
axisymmetry restrictions.
SYMBOLS
E Young's modulus
m multiple of thickness such that m_t defines axial extent of imperfection
X X
n onehalf of multiple of thickness such that 2n_t defines lineal circumferen
y y
tial extent of imperfection
P critical axial buckling load
Ppj. . classical critical axial buckling load (see eq. (2))
R nominal radius of cone at central location of imperfection
t shell wall thickness
u meridional displacement
V circumferential displacement
w normal displacement
w maximiun amplitude of buckled shell
max ^
W normal measure of imperfection measured from nominal cone surface
W maximum amplitude of imperfection
max ^ ^
X lineal axial distance measured from small radius end of cone
X local axial distance measured from the beginning of the imperfection
y angular circumferential distance measured from center of imperfection, rad
a semivertex angle of cone
/3 circumferential extent of imperfections, deg
coordinate in the circumferential direction measured from center of
imperfection, deg
(i poisson's ratio
(p meridional edge rotation
PROBLEM DEFINITION
Shell Geometry, Edge Condition, and Loading
The shell segment, with dimensions as shown in figure 1, is an idealization of an
unstiffened aluminum truncated conical shell with stiff end rings. The end rings are
assumed to be rigid in the end plane and, thus, are approximated by simply supported
boundary conditions such that the normal displacement w and the circumferential dis
placement V are fixed. At the upper edge (small radius edge), the meridional displace
ment u and the edge rotation (p are free. At the lower edge, the edge rotation is
also free but the inplane displacement u is fixed to support the applied load. A uni
form compressive meridional unit line load is applied at the small radius edge. Any
normal load component which might occur in the actual missile system is assumed to
be equilibrated by the stiff end ring idealization.
Imperfection Geometry
The imperfection sizes of immediate interest in this report have an axial extent of
50t, with a maximum inward depth nomial to the surface of 5t and various circumferen
tial lengths, where t is the shell wall thickness. The imperfection covers a portion of
the shell bounded by two meridians and two parallel circles. The meridional center of
the imperfection is located about two thirds of the axial distance from the small radius
edge of the cone (fig. 2(a)). In this study, the circumferential extent of the imperfection
is varied from 0° to 180°. A highly localized imperfection (i.e., one contained within a
small region of the shell) with a meridional length of 50t, circumferential arc length of
50t, and maximum normal amplitude of St.was studied in detail, and results for this case
are presented in a subsequent section. The imperfection is assumed to have a shape
defined by
w
H^)h^
(1)
To make use of symmetry properties in the analysis, a diametrical plane of symmetry
is assumed so that two imperfections centered 180° apart are assumed to exist for all
studies made in this report. For an extent of 180°, the two diametrical imperfections
meet.
ANALYSIS
All asymmetric imperfection calculations were performed with the STAGS com
puter code. The STAGS code uses a twodimensional finitedifference scheme to approx
imate the shell energy equations which are minimized to obtain the stress distribution
and/or stability of thin general shell structures. As shown in equation (1), the imper
fection is formed by using the normal displacement only, and only the first partial deriv
atives of the imperfection function are used in the STAGS analysis. This method of
representing the imperfection is an approximation to the accurate shell equations which
would strictly define the imperfection and is probably accurate only for imperfections of
shallow amplitudes (less than lOt). The STAGS code is capable of calculating either a
nonlinear collapse load or a bifurcation buckling load away from a linear prebuckled
state. In this study, only a bifurcation buckling analysis is performed and prebuckling
rotations are not taken into account.
To perform the analysis most efficiently, only a 90° portion of the shell (fig. 2(b))
was studied. Thus, symmetry conditions were enforced aloi^ the meridian at 0=0°
and 90 . In this study, the number of finite difference stations along the meridian is 33
and the number of stations along onefourth of the circumference is 51. A preliminary
study showed that the grid size resulting from the use of this number of stations was
sufficiently fine to give an accurate solution. This grid point network contains approxi
mately 5000 degrees of freedom. The locations of the finite difference stations along the
meridian and along the circumference are shown in figures 2(a) and 2(b), respectively,
for the imperfection with a circumferential extent of 50t. Regions of large stress grad
ients have a denser grid spacing. All analyses with the exception of the perfect cone
(i.e., a cone with no imperfection) use the same number of degrees of freedom and same
location of stations along the meridian. The number of stations along the circumference
is the same for all problems, but the spacing of the stations is adjusted for each circum
ferential imperfection length investigated so that close spacing is attained in the region
of the imperfection.
RESULTS AND DISCUSSION
The results are presented in two parts. First, the results for a series of imper
fections of 50t axial length, 5t depth, and various circumferential lengths up to an
included arc length of 180 are discussed and then results from an indepth study of a
local imperfection with a circumferential arc length of 50t are examined in detail.
Effect of Circumferential Extent of Imperfection
Figure 3 is a plot of the critical buckling ratio Pcr/^cr ^^ *^® circumferential
arc length of the imperfections ^ is increased from 0° to 180°. The classical buckling
load P is approximated by the equivalent cylinder buckling formula
■^ _ 27rEt^cos2a /„x
cr 1 ^ ' ^^f
p{l  m2)
which is given in reference 5 and for the shell of figure 1 would be
Ppj. = 665.9 kN (149 700 lb)
For values of /3 less than 15°, there is essentially no change in buckling strength.
For values of p from 15° to 70°, a rapid drop in buckling load is observed; whereas, for
^ values between 70° and 140°, the critical load remains nearly constant at about 55 per
cent of the critical load of the perfect cone. This behavior is similar to experimental
and analytical results reported for cylindrical shells with cutouts (e.g., ref. 6). As the
edges of the opposing imperfections approach each other, the critical load once again
starts to drop and at /3 = 180°, when the edges of the two imperfections begin to overlap,
the buckling load has reduced to approximately 40 percent of the critical load of the per
fect cone.
The buckling load of the cone with an axisymmetric imperfection defined by equa
tion (1) with y = was determined by using a computer prc^ram for bifurcation
buckling of shells of revolution about an axisymmetric prebuckling state (ref. 7). The
buckling load is 54 percent of the critical load of the perfect cone and the cone buckles
into eight circumferential waves. The buckling load is shown as a horizontal line in
figure 3 for comparison purposes. The axisymmetric results give a fairly accurate
prediction of the critical buckling load for imperfections which extend over 60° or more
of included angle aroxmd the circumference, but the prediction is highly conservative for
imperfections of more local nature. The axisymmetric imperfection results differ from
the results for two diametrically opposed imperfections of 180° extent since the
amplitude of the latter imperfections varies sinusoidally in the circumferential direction,
whereas the amplitude of the axisymmetric imperfection is constant.
Local Imperfection
The prebuckling and buckling results for the specific imperfection shown in fig
ure 2(b) with a circumferential extent of 50t (^3 = 10.5°) are now presented in more detail.
As shown in figure 3, the buckling load for this imperfection is essentially the same as
the classical buckling load for the perfect cone. The axial distribution of meridional
prebuckling stress for the 50t circumferential imperfection is given in figure 4. For a
cone with no imperfection, the meridional prebuckling stress varies linearly along the
meridian. As indicated in figure 4(a), a meridional distribution at 0=1° shows a
rapid reduction in compressive stress in the imperfection to essentially zero stress in
the imperfection center. The stress variation along the meridian at 9 = 38° is pre
sented in figure 4(b) and is seen to be nearly linear and to approximate nominal perfect
shell behavior. Figure 4(c) shows the stress variation along a circumference taken near
the lower edge of the imperfection. The stress is zero at the center of the imperfection
{6 = 0°) with a peak in stress occurring at the edge of the imperfection. The stress then
rapidly damps to the perfect shell value of 0.84 for a unit applied load and remains uni
form from 9 = 20° to 90°. This relatively large stress concentration at the edge of the
imperfection is caused by a stress redistribution around the imperfection. Since the
total axial load must be the same as the applied axial load at the top edge, the stress
peak is expected since the center of the imperfection has zero stress. The maximum
inplane stress occurs at the meridional center of the imperfection just outside the
circumferential edge of the imperfection (x = 54.8 cm (21.57 in.)) and is 1.56; this is a
56percent increase over the maximum inplane stress of the perfect shell.
Figures 5(a) and 5(b) show the normal displacement buckling mode normalized
with respect to w„„„ along a circumference which cuts close to the center of the
imperfection (x = 55.1 cm (21.7 in.)) and along a meridian which nearly cuts through the
center of the imperfection {9 = 1 ). The maximum displacement occurs in the vicinity
of the imperfection. However, the buckling amplitudes do not damp appreciably and the
instability may be classified as a general instability rather than a local instability.
The critical load with two diametrically opposed imperfections each of 90° extent
is 52 percent of the classical critical load. Figures 6 and 7 which show the prebuckling
meridional stress resultant and normal displacement buckling mode, respectively, are
included to demonstrate the difference in the behavior of the structure when the circum
ferential extent of the imperfection has been increased from a highly local extent to 90°.
A local stress rise occurs near the edge of the imperfection as shown in figure 6(a).
This large stress rise can be contrasted with the slight stress increase along a meridian
6
at a location outside the imperfection {6 = 70°) as shown in figure 6(b). This stress
distribution character along the circumference is illustrated in figure 6(c) where the
effect on stress of the imperfection rapidly dissipates outside the imperfection. The
local character of the buckling mode in the meridional direction as shown in figure 7(a)
can be contrasted with the more global character of the buckling mode shown in figure
5(b) for the local imperfection. The more local character of the buckling mode along
the circumference can be seen by comparing the modal behavior shown in figure 7(b)
with that of figure 5(a). The buckling displacements remain local to the imperfection
and damp rapidly away from the imperfection in both the meridional and circumferential
directions.
CONCLUDING REMARKS
A brief study was made of the effect of a particular type of local imperfection on
the buckling of an axially compressed thin walled conical shell. Results show that the
buckling load foimd from a bifurcation buckling analysis is highly dependent on the cir
cumferential arc length of the imperfection type studied. As the circumferential arc
length of the imperfection is increased, a reduction of up to 50 percent of the critical
load of the perfect shell can occur. The buckling load of the cone with an axisymmetric
imperfection is nearly equal to the buckling load of imperfections which extended 60° or
more around the circumference, but would give a highly conservative estimate of the
buckling load of a shell with an imperfection of a more local nature.
The bifurcation buckling analysis of a highly localized imperfection shows no sig
nificant drop in buckling load but the linear static stress analysis shows that the imper
fection does cause a local stress rise of over 50 percent above the maximum stress in
the perfect cone. For small imperfections the buckling mode can be classified as a
general shell instability, but the buckled region tends to remain local to the imperfection
as the imperfection size is increased circumferentially.
Langley Research Center,
National Aeronautics and Space Administration,
Hampton, Va., May 2, 1974.
REFERENCES
1. Stein, Manuel: Some Recent Advances in the Investigation of Shell Buckling. AIAA J.,
vol. 6, no. 12, Dec. 1968, pp. 23392345.
2. Amazigo, J. C; and Budiansky, B.: Asymptotic Formulas for the Buckling Stresses
of Axially Compressed Cylinders With Localized or Random Axisymmetric Imper
fections. Trans. ASME, Ser. E: J. Appl. Mech., vol. 39, no. 1, Mar. 1972,
pp. 179184.
3. Narasimhan, K. Y.; and Hoff, N. J.: Snapping of Imperfect ThinWalled Circular
Cylindrical Shells of Finite Length. Trans. ASME, Ser. E: J. Appl, Mech.,
vol. 38, no. 1, Mar. 1971, pp. 162171.
4. Almroth, B. O.; Brogan, F. A.; and Marlowe, M. B.: Collapse Analysis for Shells of
General Shape. Volume I  Analysis. AFFDLTR718, U.S. Air Force,
Aug. 1972.
5. Anon.: Buckling of Thin Walled Truncated Cones. NASA SP8019, 1968.
6. Starnes, James H., Jr.: Effect of a Circular Hole on the Buckling of Cylindrical
Shells Loaded by Axial Compression. AIAA J., vol. 10, no. 11, Nov. 1972,
pp. 14661472.
7. Cohen, Gerald A.: Computer Analysis of RingStiffened Shells of Revolution. NASA
CR2085, 1973.
Unit line load
82.3 cnr;
Figure 1. Shell geometry.
38.86
cm
1
»i< t =0.16 cm
1 50.8 cm
82
r
) 8.0 cm
/ ( 50 t )
1 t
'ir 0.8 cm
1 ' ( 5 t )
\
46.35 cm
1
■
Finitedifference
stations
(a) Axial grid at = 0°. (b) Circumferential grid at x = 54.8 cm. t = 0.16 cm.
Figure 2. Typical finite difference representation.
JQ"
1.0
0^ .8
o
T3
a
o
c
.6 
o
o
.y .2
o
X
{ V >
\
\
.
'r
_
\^_
Ay3^
"T
^^ — • Axisymmetric
^^^
imperfection
1
1 1 1
I 1
1
1
20
40
60
80
100
120
140
160
180
Circumferential arc length of imperfections, /9, deg
Figure 3. Effect of the circumferential extent of local imperfection on the critical
buckling load of a smallangle cone. P = 665.9 kN.
n
o
o
4)
c
c
3
4)
a.
o
3
C
o
(A
«
(A
c
o
c
o
a>
Center of
imperfection
Perfect cone
( no imperfections)
40 60
Axial distance, x, cm
80
100
(a) 6=1 (near center of imperfection).
Figure 4. Prebuckling meridional inplane stress resultant distribution due to unit
loading for a cone with highly localized imperfection. (Circumferential arc
length of imperfection equals 50t.)
12
T3
D
O
<1>
C
c
.2
"5.
a
o
3
C
O
"S
V>
i_
c/>
(A
c
o
Q.
I
C
o
c
o
a>
40 60
Axial distance, x, cm
(b) = 38°.
Figure 4. Continued,
J
00
13
c
o
(A
•o —
<u o
w. O
(A
a>
a> — _ 4 J
.^ Q.
O o
c ••
O
.2
.4
.6

.8
—
u
.0
r
"?
1
1
1
1
1
1
1
1
1
10 20 30 40 50 60 70
Circumferential angle, d, deg
(c) X = 58.4 cm (lower edge of imperfection).
Figure 4. Concluded.
80 90
14
o
E
c
a>
E
V
u
o
Q.
.52
o
E
o
z
20 30 40 50 60 70 80
Circumferential angle, 6^ deg
(a) X = 55.1 cm (near center of imperfection).
Figure 5. Normal displacement buckling mode of a cone with highly
localized imperfection. (Circumferential arc length of
imperfection equals 50t.)
90
15
40 60
Axial distance, x, cm
(b) 0=1 (near center of imperfection).
Figure 5. Concluded.
100
16
.9 ^
o
o
T3
a
o.
o
3
•o
c
o
Q>
w
</>
a>
c
o
Cl
I
c
o
c
o
2
20
80
40 60
Axial distance , x , cm
(a) = 41.1° (near edge of imperfection).
Figure 6. Prebuckling meridional inplane stress resultant distribution due to unit
loading for a cone with imperfection extending 90° along the circumference.
17
,99 r
T3
O
o
^^
1.00
Q)
C
,^
"c
 I.OI
3
T3
Q)
Q.
1.02
O
O
<D
 1.03
3
T3
^_
C
o
1.04
"5
(A
0)
w
 1.05
CO
<A
a>
k.
*■
(/>
1.06
0>
c
o
Q.
1
c
1.07
o
c
q
1.08
*;o
^
<i>
S
1.09
40 60
Axial distance, x, cm
(b) 9 = 70°.
Figure 6.  Continued.
100
18
.3 r
in
ID
CO
.£ f
'i5 'S
10 20 30 40 50 60 70
Circumferential angle, 0, deg
(c) X = 58,4 cm (lower edge of imperfection).
Figure 6. Concluded.
80 90
19
.2r
K
e
E
e
E
«
Q.
o
E
40 60
Axial distance , x, cm
(a) 9
41.1°.
80
100
Figure 7. Normal displacement buckling mode for a cone with imperfection
extending 90 along the circumference.
20
o
E
c
E
a>
u
_o
ex
o
£
o _
20 30 40 50 60 70
Circumferential angle, 9, cleg
(b) X = 55,1 cm (near center of imperfection).
Figure 7. Concluded.
80 90
NASALangley, 1974 L9331
21
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