EXPERIMZNTAL AND NUMERICAL ANALYSIS
OF PREBUCKLED CYLINDRICAL SHELLS
UNDER UNIFORM EXTERNAL PRESSURE
A THESIS SUBMITTED TO THE GRADUATE DIVISION OF THEUNIVERSITY OF HANAZZ ZN PARTIAL FULFILLMENT
OF THE REQUIRE&KNTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN MECHANICAL ENGINEERING
AUGUST 1980
Christino Dumlao, Jr.
Thesis Committee:
Ronald, H. Knapp, ChairmanNillezn Stuiver
Jorn Larsen-Basse
We certify that we have read this thesis and
that in our opinion it is satisfactory in scope
and quality as a thesis for the degree of
Structural Mechanics in Nechanical Engineering.
THESIS COMNZTTEE
ACKNOWLEDGMENTS
This investigation was sponsored in part by
NQAA, Office of Sea Grant under grant number 04-6-158-44026
with additional support provided by the marines affairs
coordinator, State of Hawaii, I want to express my sin-
cere gratitude to Dr. Ronald, Knapp and Dr. Rudolph
Szilard for the technical guidance and support they
gave during this effort.
I want to also express my appreciation to Rance
Kudo and Linda Yanagihara for their assistance during the .
tests, and to Linda Arnold, Darlene Larimore and Dannette
Okino for their excellent work in typing this thesis.
ABSTRACT
The Prebuckled Cylindrical PC! is a recent struc-
tural concept whose geometry is an idealization of the
buckled surface of a circular cylinder under axial compres-
sion. A major quality of the PC shell over the circular
cylinder is the higher buckling resistance provided by its
undulating corrugated! surface. This investigation is an
effort to understand the structural behavior of the PC
shell. An extensive parametric study is made which provides
thorough evidence of the shell's superior buckling resistance.'
An approximate method for predicting the buckling pressure
of other PC shell configurations in addition to those inves-
tigated is also provided.'
TABLE OF CONTENTS
~Pa e
viii.
iX.
6
82..
143. DESCRIPTION OF MODELS
143. 1 Geometry
18
4. 2 Model Fabrication 19
4.3 Model Assembly 23
5.2 Test Procedure 27
EXPERIMENTAL AND NUMERICAL ANALYSIS OF PREBUCKLED
CYLINDRICAL SHELLS UNDER UNIFORM EXTERNAL PRESSURE
ACKNOWLEDGMENTS
ABSTRACT
LIST OF TABLES
LIST OF ILLUSTRATIONS
l. INTRODUCTION
1.1 Background
1.2 Scope of Thesis
BUCKZING EQUATIONS FOR CIRCULAR CYLINDERS
3.2 Ioad Description
4. FABRICATION OF MODEL/
4.1 Model Configurations and Mold Design
4.4 Accuracy of Fabricated Models
5. DESCRIPTION OF BUCKLING EXPERIMENTS
5.1 Experimental Configurations
16
18
26
26
P acae
6. EXPERIMENTAL RESULTS
6. 1 Buckling Behavior
6.2 Buckling Curves
6.3 Equivalent. Thickness
6.4 Equivalent Axial Stiffness
32
33
35
6.5 Buckling Nodes 37
6.6 Effect of Shell Iength 41
7. NUMERICAI BUCKLING ANALYSIS
7.1 Finite Element Analysis
42
42
7.2 Solution for Nonlinear Problems
7. 3 Description of Buckling Model....... 45
7. 4 Buckling Results ............. 46
8.2 Test Procedure 51
8.3 Description of Finite Element Nodel 52
8. 4 Stress Results 53
9 . CONCLUSIONS 58
10. BIBLIOGRAPHY
APPENDIX A. DESIGN PARAMETERS
A.l Design Curves
A.2 Mold Design
122
123
A.3 Material Stress-strain Data 123
8. EXPERIMENTAL AND NUMERICAL STRESS ANALYSIS... 50
8.1 Description of Test Model......... 50
APPENDIX 9. BUCKI ING ANALYSIS DATA
B. 1 Experimental Data
B.2 Finite Element Data
131
131
131
APPENDIX C. STRESS ANALYSIS DATA
C.l Experimental Data
C. 2 Numerical Data
APPENDIX D. COMPUTER PRINT-OUTS
D.l STRUDL Buckling Results
D.2 STRUDL Stress Results
150
150
150
163
220
LIST OF TABLES
Table Parcae
Nodel descriptions and test data
Properties for Bakelite VSA-3310!rigid vinyl sheet
Heating schedule for thermal vacuum-forming of Bakelite VSA-3310! rigidvinyl sheets 117
Buckling wave numbers for the PCshell N=10! and equivalent circularcylinder 118
Properties of planar framework elementswith rigid joints REF 13! 119
Properties for Plexiglas Gplastic sheet 120
121Strain gage specifications
Tensile test data 130
Axial shortening test data
B«2
149
Strain gage teat data
Finite element stress data
C-1
159C-2
Finite element radial displacements alongmid-length line of symmetry aee Figure3 5!
LIST OF ILLUSTRATIONS
~Fi ure P acae
Comparison of PC shell and buckledcylinder
Axial stress versus unit end shortening
Geometry and loading of PC shell models
Geometry of typical equilateral triangle
66
67
68
Variations of geometric parameters
Finished PC shell molds for N=6 and N=l0
69
70
71Thermal Vacuum-forming
72Plethod of seam. trimming during forming
Method of sealing edges with doubler72strips o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Press fixture for bonding of doublerstr jps ~ 0 ~ ~ ~ ~ ~ ~ ~ 0 ~ ~ I ~ ~
10
Clamping of PC shell to end plates
Completed models for N=6 and N=10
75
78Test configurations14
Typical model deformations under uniformpressure: N=6, R/T=325 79
Buckled configuration for N=616
Typical fracture failure patterns foruni form hydrostatic pressure, N=10 81
Deviation of the basic geometric quantities . 77
~Pi ure Parcae
18 Buckled configurations forX=2 and M=4
N=1082
Buckled configurations forX=5 and M=6
N=10:83
Buckled configurations for N=10:K=7 and N=B
2084
Set-up for radial deflection measurements,model 425
21
Pressure versus radial node deflectonsNodel 0 25 . . . . . . , . . . . . . . . . . 86
22
23
Critical buckling pressure of PC shell andequivalent c ircular true ! cylinder, N=6 see Table 1 for symbols!
24
88
Critical buckling pressure of PC shell andequivalent, circular true! cylinder, N=10 see Table 1 for symbols!
Critical buckling pressure ratio P /P 1!pc cyl see Table 1 for symbols! . . . . . . . . . 90
Equivalent thickness ratio, T /T seee
Table 1 for symbols!
Equivalent thickness ratio as a functionof N symbols represent mean thicknessfor models of Table 1!........... 92
28
Equivalent axial sti f fness ratio, E /E,e
found experimentally see Table 1f o r symbo 1 s ! r ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
29
Equivalent axial stiffness ratio, E /E,e
along line of symmetry A-B Figure 3! see Table 1 for symbols! 94
Deflection pattern prior to buckling,Nodel 25 . . . . . . . . . . . . . . . . . . 87
31 Buckling mode shapes: N=6, R/T=326
Buckling mode shapes: N=10, R/T=131
Buckling mode shapes: N=10, R/T=197
32 96
33 97
Finite element buckling model, analyzedregion,
35 Node numbering for finite element bucklingmodel see Figure 34! ........... 99
Beam elements for finite element bucklingmodel see Figure 34! 100
37 Symmetry boundary conditions see Figure 34! . . . . . . , . ~ ~ ~ ~ 101
38
39 Test configuration of experimental stressmodel ~ ~ ~ ~ ~ ~ 'I ~ ~ ~ ~ ~ ~ 103
Finite element stress model,region
analyzed104
Node numbering for finite element stressmodel see Figure 40! 105
Element numbering of finitemodel see Figure 40!
42 element stress
106
Membrane hoop stress ratio, aB /a B , alongBm Be'
line A-C 107
Membrane hoop stress ratio, aB /ae, alongBm Be'
line B-D 108
Membrane axial stress ratio, a /a , alongam ae
line A-C 109
Membrane axial stress ratio, a ja , alongam ae
line B-D
PC shell finite element buckling results . . . 102
~Pi ure
47 Bending hoop stress ratio, a8b/ab, alongbe'
line A-C 111
48 Bending hoop stress ratio, a /a , alongbe'
line B-D 112
along
113
50
A-1 115
116
A-3 117
A-4 118
Bending axial stress ratio, a b/abline A-C
Bending axial stress ratio, a /aab be'
line B-D
Design curves for N=6
Des'.gn curves for N 10
Nold construction details for M=6
Nold construction details for N=10
along
114
CHAPTER 1
INTRODUCTION
The Prebuckled Cylindrical {PC! Shell is a recent
structural concept resulting from past investigations of
buckled circular cylinders. Tests have shown that thin-
walled circular cylinders under compression loads go through
abrupt. geometric transformations at the onset of buckling.
For cylinders under axial compression buckling is evidenced
by the appearance of an undulating corrugated! polyhedral
pattern often comprised of triangular surfaces {Figure lb!.
The PC shell is an idealization of this buckled geometry as
shown in Figure l.
The buckling characteristics of circular cylinders
have received considerable attention in the past because of
the large discrepancy between experimental and, theoretical
results. The critical buckling loads obtained experimentally
have been lower than those predicted by classical linear
theories. This disagreement has bee6 particularly serious
for axial compression tests where values as low as one-third
the theoretical value have been obtained. Noreover, the8
test values have shown considerable scatter. It has since
been discovered that the disagreement is due primarily to
the nonlinear nature of the buckling process, first described
by Von Karman and Tsien. By including large-displacement31
and geometrically nonlinear terms in their theoretical ana-
lysis, Von Karman and Tsien found secondary stable equili-
brium states involving large deformations existing below the
critical load calculated by linear theories. The signifi-
cance of this discovery is revealed in Figure 2.
Ideally, a perfect cylinder under axial compres-
sion follows a linear, primary equilibrium path line 0-A!
up to the critical load. An additional increase in load
causes the cylinder to become unstable on this path, and thus
is forced to follow a secondary equilibrium path. Classical
linear theories show that an infinitesimal increase beyond
the critical buckling load results in very large lateral
displacements line A-D!. However, Von Karman and Tsien found
the load-displacement path dropping sharply down on to a
secondary stable equilibrium path curve B-C-D! which begins
at a load considerably less Chan the classical buckling load.
This discovery indicates that two equilibrium states are
possible for the load range below the classical buckling load.
Subsequent investigations have shown that because of existing
imperfections in geometry, loads, and boundary condi-
tions ' ' a jump from the primary to the secondary equili-B,9, 10
brium path can occur before the classical buckling load is
reached, as indicated by the arrow in Figure 2. The actual
buckling loads obtained in tests have been associated with
this jump.
Buckling is evidenced by the sudden transformation
of the cylindrical surface into a new geometric configura-
tion. The new geometry of the buckled cylinder was first
described by Kirste and Yoshimura. They found that the12 34
cylindrical surface can be transformed into a developable
polyhedral surface through an inextensional deformation pro-
cess. Thus, the buckled surface Figure lb! is sometimes
referred to as the Kirste-Yoshimura pattern. The most pre-21
valent configuration has been described by Lange and Newell 18
as. a diamond pattern compri sing triangular f aces. Because
of its tacit affiliation with the buckled state, the poly-
hedral configuration has been regarded as a failed form.
However, Niura has suggested that it. might, instead, be21
viewed as a new type of 'unfailed structure. He referred to
the new structural form as the Pseudo-Cylindrical Concave
Polyhedral PCCP! shell. Previous studies have identified
the new structure as the PCCP shell; in this investigation
it shall be referred to, simply, as the Prebuckled Cylindrical
PC! shell Figure la!.
Niura has described some of the principal features
inherent to the geometry of the PC shell. Because of the
greater moment of inertia provided by the corrugated surface,
the PC shell's bending rigidity is much higher than that of
a circular cylinder. The PC shell should, therefore, exhibit
higher buckling resistance to externally applied surface
loads. Also, since the tendency for circular cylinders is
to buckle into the PC shell configuration, the PC shell should
represent a more stable configuration, making it less sensi-
tive to geometric imperfections. A negative feature of the
PC shell is its greatly reduced axial rigidity, causing high
axial deformations when compared with the circular cylinder.
This is due to the bellows-like folding effect occuxring along
the intersecting edges of the triangles.
Numerical and experimental investigations conducted
by Knapp ' and by Tanizawa and Miura have confirmed the13, 15 27
above characteristics. Buckling tests performed by the two
parties have shown conclusively the improved buckling resist-
ance of the PC shell compared to that of the circular cylinder
under both hydrostatic and radial pressures. Knapp used
equilateral triangles to model a PC shell having twelve
circumferential N=l2! and four axial M=4! triangles. The
model was subjected to uniform radial pressure, resulting in
a buckling pressure ratio P /P 1! of 1.6 when compared topc cyl
a perfect cylinder. A numerical analysis by the finite element
method verified his results. Tanizawa and Miura conducted
model tests on two PC shell configurations using isosceles
triangular faces. The models vere subjected to uniform
hydrostatic radial and axial! pressure. When compared to
the buckling load of a perfect cylindex, pressure ratios of
5.39 and, 3.82 for six N=6! and eight N=S! circumferential
+triangles, respectively, were obtained. Descriptions of
these tests are provided at the bottom of Table 1. Although
the triangle geometry of the PC shells in the separate studies
vere different, a comparison of the buckling pressure ratios
shovs that the buckling resistance increases with reduced
number of circumferential triangles, N. Accordingly, the
buckling resistance approaches that of a cylinder as N goes
to infinity..
Stress analyses were also performed by both parties.
The experimental and numerical stress analyses performed by
Tanizava and Kiura are detailed in Reference 26. Their com-
parison of the numerical and experimental stress results vere
poor; however, they showed that high bending stresses are
developed which dominate the membrane stresses, particularly
along the intersecting edges of the triangular ' surfaces.
The folding action which takes place along the edges as axial
shortening occurs contributes significantly to the magnitude
The circular cylinder buckling pressures used for thecomparison were calculated from equation 2.3 in Chapter 2.
of these bending stresses. Numerical stress analyses by
15Knapp for PC shells of different geometries have also
revealed the high bending stresses along the edges due to
axial shortening. Knapp has shown that the bending stresses
can be significantly reduced by providing axial restraint.
Practical applications of the PC shell have been
discussed in References 15 and 21. The high buckling resis-
tance is advantageous in an environment where the primary
loads are compressive; thus, the PC shell would be well suited
as a shallow-depth undersea structure. Designing for such
an application requires a more thorough understanding of the
shell's buckling behavior than is presently available. From
earlier investigations, the buckling resistance has been found
to be significantly influenced by the number of circumferen-
tial triangles. However, this is only one of the major geo-
metric parameters which describes the PC shell, and an exam-
ination of the other pertinent geometric parameters is needed
to increase the understanding of its buckling characteristics.
1.2 Ob'ective and Sco e of Thesis
The purpose of this investigation is to conduct an
experimental and numerical buckling analysis of the Prebuckled
Cylindrical PC! shell in order to determine the influence
of its basic geometric parameters. The major effort in this
study has been experimental, involving the design, fabrica-
tion, and testing of PC shell models. A series of twenty-five
models have been tested under hydrostatic and radial compres"
sion to determine the buckling characteristics over a range
of the fundamental geometric quantities, namely: shell length
I!; radius R!; wall thickness T!; and number of circum-
ferential triangles N!. The test results have been compared
with the classical theoretical buckling results of circular
cylinders having the same dimensions. A numerical buckling
analysis by the finite element method has also been performed
for one model to compare with the test results.
In addition to the buckling tests, numerical and
strain-gage stress analyses of a single model have been con-
ducted to ensure that the test models have been designed to
the results of the earlier studies.
CHAPTER 2
BUCKLING EQUATIONS FOR CIRCUIAR CYLINDERS
Various theories for the stability of circular
cylinders have been presented in literature. 1,2,3,7,29,31
The sets of governing differential equations developed from
these theories are, in their original form, highly nonlinear,
making them difficult to solve. However, through simplifying
assumptions the complexity of the expressions can be reduced,
thereby enabling the analytical solution of a range of buckl-
ing problems. By assuming the strains and rotations to be
small higher order differential terms can be neglected, reduc-
ing the equilibrium relationships to a set of linear differen-
tial equations. By further assuming that no imperfection
exists in the geometry and boundary conditions and that the
applied loads are uniform, explicit formulas for the minimum
buckling values of different load conditions have been
derived.
Tests have shown that for some conditions these
linearized solutions often overestimate the buckling load of
real cylinders. For axial compression, it has been shown
that the discrepancy can be very large. However, for cylin-
+ders under radial and hydrostatic pressures, the
+See Chapter 3 for the definition of these loads.
difference has not. been as severe, and, in some cases,
linear solutions have agreed with experiment. 32
Presented are several closed-form linear solu-
tions for calculating the minimum buckling pressure of
perfect circular cylinders under radial and hydrostatic
pressures with clamped and simply-supported edges. The
variables defining the cylinder geometry in these formu-
las, namely: length L!; thickness T!; and radius R!
correspond to the dimension variables which describe the
PC shell in Chapter 3. An important parameter appearing
in the buckling formulas is the buckling wave numer, n.
It, is an integer variable which represents the number of
circumferential half-waves developed at. the onset of
buckling. In each equation the number n must be found for
which the critical pressure value is a minimum.
For a simply supported circular cylinder ends
free to rotate! under radial pressure, the Von Mises'
formula as corrected by Nindenberg and Trilling is33
expressed in nondimensional form as
~rs 1 ~ 1 K~n"-X~n'+X> ] 16 T/R} 'E 3 n~-1 I �-v !z
T/R
Of direct interest to this investigation is the
formula for the critical uniform hydrostatic pressure of a
circular cylinder with clamped ends. Bijlaard has derived
an accurate solution for this case. From Donnell s theory,3 I 7
the simpli f ied equi librium equation for shallow shells
R/T! under hydrostatic pressure is expressed as
ET ~ew+ + R P ! q + � =PET B w w 1 B w
12 �-ve! R Bx BY2
w=casnQ Zw um m �. 3ii!
where
cosh' x cosa xm m
um
cosh' c cosa cm m
Lg C �* 31ii!
in which w and y are the radial and circumferential surface
displacements, respectively. For the clamped shell, where
w=O and Bw/Bx=O at the ends, Bijlaard assumed a shape func-
tion of the form
.Differentiating Equation {2.3iii!, the following relation-
ships are obtained:
�. 3iv!
where
cosha x cosa xu +
mcosha c cosa c
Pl m
{ 2. 3v!
Because u are orthogonal functions, u can be expanded intom m
a series of u m'
um = k, ul+k2 u2+k3 U + �. 3vi!
-12-
a4um
a ux" m m
g2um a 2z m m
Limiting the solution to only the first term m=1! in Equa-
tions �.3ii! and �.3vi!, and substituting these Equations
into Equation �.3i! gives the critical hydrostatic pressure
as
~cl D n 9 +12.3n g +188.4n'S~+386n g +985!+31.4TR'g8 R {n B'+7.7n"g"+62.8n g +48.34!
�. 3!
where
D T12{1-v~!
p~cl 1.38 T/R!
L/R! R/T! '-0.954
Equations �.3! and 2.4! have been used to cal-
culate the hydrostatic buckling pressures of the circulax
cylinder used for comparion with t' he PC shell. The values
for P /E for the circular cylinder corresponding to thecyl
geometry of each PC shell model is presented in Table l.
An approximate solution for P 1/E can be obtainedcyl
by minimizing Equation �.3!; 8 P /E!/8n=O, ignoring thecyl
less significant terms of the resulting expression, a formula
for the curve which envelopes the set of minimum values of
Equation �.3! is obtained:
CHAPTER 3
DESCRIPTION OF PC SHELL MODELS
3.1
= SI ' 1 � COSB/3 SING
�- 1!
Axial compression tests have shown that the buckled
surface of circular cylinders is made up of concave surfaces
arranged in either a diamond or hexagonal pattern. However,
a preference has been found for the simple diamond pattern 18
Figure la!. For this reason, PC shell geometries consisting
of simple equilateral triangle surfaces, like the one examined
.earlier by Knapp, were used in this investigation.13
The principal geometric variables of the PC shell
are shown in Figures 3 and 4. The primary geometric quanti-
ties ar' e: the number of circumferential triangles N!; the
number of axial triangles N! and the corresponding axial
length L!; the shell radius R!; and the shell thickness
T!. Since equilateral triangles are used to model the PC
shell, other pertinent variables are described by the geo-
metric relationships which follow.
The vertical inclination of a triangle Figure 4!
is given by
where
W3 SINS COSyR
with the overall axial length L!
L� N
R R� ~ 3!
The maximum amplitude along the surface is
v 3 SINO SIN/R �.4!
The radius R! of the PC shell is defined by the distance
of the intersecting triangle corners to the central axis
as shown in Figure 3. The triangle vertical height i! of
the PC shell is
The planar dimensions of the individual triangles
as show~ in Figure 4 are calculated from
2 SINGR
and
Equations �.1! through {3.6! reveal that the
geometry of the PC shell is primarily determined by the
number of circumferential triangels. A plot of 0/R, PR,
and 4/R in Figure 5 shows the greatest variations in these
quantities occurring for N less than 20.
3.2 broad Descri tion
The PC shell models were subjected to two kinds of
uniform compression loads: external hydrostatic radial and
axial! pressure and external radial pressure. These two
load conditions were differentiated in the tests by the
presence or absence of axial shortening. The loads are
illustrated in Figure 3. For hydrostatic pressure the ends
-16-
of the models were allowed to displace, permitting unrestric-
ted axial shortening as a result. of the existing axial load.
For external radial pressure the shell was loaded radially
and axially; however, axial restraint was provided by an
internal tube of much higher stiffness see Chapter 5! which
prevented any appreciable axial shortening.
Descriptions of the PC shell geometry and the
loading for each model are given in Table 1. The ends of
all the models were clamped to end plates to provide a
fixed-edge boundary condition. Details of the clamping
method are presented in Chapter 4.
-17-
CHAPTER 4
FABRICATION OF PC SHELL NODELS
4.1 Nodel Confi rations and Nold Desi
The tests conducted by Tanizawa and Miura N=627
and N=8! and by Knapp N=12! have shown that the bucklingl3
resistance of the PC shell increases with decreasing number
of circumferential triangles in a smooth exponential fash-
ion. To provide additional data with which to determine the
trend, configurations of six and ten circumferential triangles
were selected for testing. The results of the earlier
investigations were used in establishing the design parameters
for the models.
A thermal vacuum-forming process was used in manu-
facturing the PC shell models. This method of fabrication
has been found effective in accurately reproducing the cor-
rugated features of the PC shell. In addition, the process
is simple, allowing the economical construction of a quantity
of models.
The thermal vacuum-forming technique requires the
construction of molds on which the desired model geometries
are machined. Hence, the design of the molds is dictated by
the requirements of the test models. Elastic buckling has
been the prime consideration in the model design; the maximum
-1B-
stresses at the onset of buckling must be below the yield
stress and within the elastic range of the selected material.
This is to prevent failure of the material to occur before
actual buckling of the model. Because of the test method-
the external pressure is applied by evacuating air from
within the model - the buckling pressure is limited to less
than fourteen psi gauge pressure. From the results of the
earlier tests, design curves for N=6 and N=10 were developed
which gives the latitudes available in the model parameters
to ensure elastic buckling below fourteen psi. The design
curves are presented in Appendix A-1. These curves were'
used in designing the molds. The molds were machined from
commercially available aluminum tube stock. The size of the
models was determined by the design curves and, in addition,
by the space limitation of the heating oven. The construction
details of the molds are given in Appendix A-2. The finished
mold segments and end plates are shown in Figure 6.
4.2 Nodel Fabrication
The material selected for fabrication of the PC
shell models was a rigid vinyl sheet, BAKEI ITE VSA-3310!,
manufactured by the Union Carbide Company. The vinyl was
selected for its near linear stress-strain curve and for its
good formability at moderately low temperatures. The mate-
-19-
rial properties are given in Table 2. The value of Young's
modulus E=444,000 psi! is the one obtained from the tensile
tests conducted by Knapp and has been used in all the13
analyses for this investigation. Tensile tests Appendix
A-3! conducted later in the experiments gave values for the
modulus in agreement with Knapp's value. The tensile test
specimens were exposed to the thermal cycles of the vacuum-
forming process used in making the models.
In manufacturing the models, the mold is first
assembled by stacking the ring segments to the desired
number of axial triangles as shown in Figure 6. Dowel pins
are used to accurately locate the adjoining segments. Shims
are inserted between segments to provide a one-thousandths
inch gap needed to .draw the sheet onto the mold surface when
vacuum is applied. A vinyl sheet of the required thickness
is cut so that the longitudinal edges are lapped approximately
one inch when wrapped around the mold. The circumferential
edges are cut approximately one-eighth inch, beyond, the end
segments. This extra length allows for shortening of the
sheet as it is drawn onto the mold surface during vacuum
forming. In addition, it also provides a strip to which the
sealing tape can. be applied. The sheet is then secured in
place by a cloth harness which is tensioned around the vinyl
sheet and mold. This assembly is placed on a cradle which
-20-
applies a sufficient compression, load to the end plates to
prevent the mold from sagging as it is placed horizontally
in the heating oven. The complete assembly is shown in
Figure 13a. A vacuum pump can be connected to the mold by
means of a heat resistant air hose attached to fittings on
the top end plate.
During the thermal vacuum-forming process the tem-
perature of the oven is monitored and regulated by means of
a thermal couple suspended near the molding assembly.
Heating schedules for the process have been determined and
are presented in Table 3 for the pertinent sheet, thick-
nesses. The thermal vacuum-forming process is performed in
four steps as described below:
Step l. Initial Heatin and Formin
The molding assembly is heated to and maintained
at the forming temperature see Table 3! until the vinyl
becomes sufficiently pliable. The cloth harness is then
tensioned, causing the vinyl sheet to conform to the surface
of the mold. This reduces the possibility of- trapped air
pockets, particularly along the ridgelines, when vacuum is
applied. This step also serves to bring the vinyl closer to
the mold surface in an inextensible manner so that little
stretching of the material occurs during vacuum forming.
-21-
Step 2. Sealing.
The molding assembly is taken out of the oven and
allowed to cool until the vinyl surface becomes rigid. The
cloth harness is then removed, and, heat resistant. aluminum
tape is used to seal all the edges of the vinyl. The cloth
harness is replaced and. the molding assembly is put back
into the oven with the vacuum hose attached.
Step 3. Vacuum-formin
The vinyl and mold assembly is again heated to
forming temperature. When the vinyl becomes pliable, a vac-
uum is applied. Air is drawn from within the mold by means'
of a vacuum pump. After the maximum vacuum pressure is
obtained the cloth harness is removed and the vinyl is in-
spected for wrinkles. Any wrinkles and' trapped air pockets
that appear can be removed by releasing and reapplying the
vacuum. The assembly is then allowed to cool before again
releasing the vacuum.
Step 4. Cuttin the Seam.
Early buckling experiments have shown that a
lapped joint along the longitudinal seam can cause premature
buckling failure due to secondary be~ding actions. To
prevent this effect, the lapped edge is cut to form a butt-
joint. Figure 8 illustrates the method of forming the
joint. A cut is first made through the lap section Figure
-22-
8a!. The cut edges are then butted and a strip of heat
resistant tape is used to seal the butt-joint. The assembly
is reheated to forming temperature under vacuum in order to
eliminate the step created in the lapping Figure Bb!. Once
the seams are flushed with the mold surface the assembly is
again cooled. The completed seam appears in Figure 8c.
4.3 Nodel Assembl
Before removing the vinyl shell from around the
mold, the ends of the shell are trimmed to where the support
rings are to be mounted on the ends plates Figure lla!.
The shell is then slipped off the mold for final assembly.
Doubler strips one-thousandths inch thick and half
inch wide of the same material are bonded to each side of
the butt-joint to seal and reinforce the seam. This creates
a slight stiffening effect near the region of the seam. The
tests, however, have shown that the additional stiffness has
little influence on the buckling results since in most of
the models buckling occurred away from the seam. The press
fixture shown in Figure 10 is used for bonding the doubler
strips. The edges of the seam are butted and held in place
by strips of tape attached to the inside surface of the
model. The seam region is then slipped over and rested on a
silicon rubber support block see Figure 10!. A doubler
-23-
strip is bonded to the outer surface of the seam using
pressure sensitive cyanoacrylate adhesive. A thin film of
adhesive is applied to the region along the seam. The strip
of material is positioned onto the surface and taped in
place. The plaster press block is then clamped over the
seam region as the adhesive cures. The process is repeated
for the inside doubler strip except that the strip is placed
on the support block first. The adhesive is applied to the
strip before slipping the shell into position. The press
block is then clamped in place while the adhesive cures.
Curing time for each strip is approximately one hour.
The completed shell is mounted on to the same end
plates used in the thermal vacuum-forming process. The
support rings on the bottom end plate are first clamped into
place as shown in Figure lla. The model is then seated on
the support rings. The rings are carefully removed so as
not to disturb the seating of the shell. The edge of the
model is sealed with aluminum tape and silicone sealing
compound. The support rings are then clamped back in place.
The rubber compression ring is then slipped around the
model. The lower ring is positioned onto the bottom end
plate and secured as shown in Figure lib. The compression
ring is clamped so that no portion of it extends beyond the
support edge of the end plate. The top end plate, support
rings, and rubber compression rings are mounted in the same
manner described. The assembled PC shell models for six and
ten circumferential triangles are shown in Figure l2.
4.4 Accurac of Fabricated Models
All the models were measured for manufacturing ac-
curacy. The measured values of the radius R!, thickness
T!, length L!, and triangle edge length X!, have been
compared with the design values and the accuracies in terms
of percent deviation are given in Table l for each model.
Figure l3 illustrates the range of variations in the basic
model parameters. The variations were typical for all the
models. The consistency in the buckling results showed that
the variations were not cr'itical.
CHAPTER 5
DESCRIPTION OF BUCKLING EXPERIMENTS
5, 1 Ex erimental Configurations
Figure 14 shows the test configurations for uni-
form hydrostatic and radial external pressures see Section
3.2!. External pressure is applied by evacuating air from
the interior of the models using a vacuum pump. Hose at-
tachmentss are provided on the top end plate. The pressure
hoses are connected to the vacuum pump and to an electronic
pressure transducer. The rate of air evacuation is regu-
lated by means of a manual gate valve located along the
vacuum hose. Mercury and water filled manometers are used
to monitor the pressure during the tests for purposes of
controlling the rate of evacuation and the taking of inter-
mediate data. The magnitude of the external pressure is
accurately recorded by means of a pre-calibrated chart
recorder electrically connected to the pressure transducer.
Four dial indicators are equally spaced around the
edge of the top end plate to measure axial deformations. A
counterweight is attached to the neutral axis of the top end
plate to eliminate axial loads due to the weight of the top
end plate and pressure hoses.
-26-
5.2 Ex erimental Procedure
The PC shell model is first secured to the base of
the rigid frame Figure 14!. Next, the pressure hoses and
the counter weight are attached to the top end plate. A
small pressure load is then applied in order to seat the end
plates properly. Othervise, erroneous axial deflection
readings may result due to the settling of the end plates
vhen the actual pressure is applied. The dial indicators
are positioned and zeroed. The water or mercury manometer
and the pressure recorders are then adjusted.
The axial deflections measured by the four dial
indicators are recorded as the external pressure is applied.
The uniformity in the axial displacements is revealed by the
difference in the indicator readings. ln all the model
tests the deviation, in readings did not. exceed five percent.
This signifies a reasonably uniform axial loading. The
small deviation also indicates that the additional stiffness
of the seam has little influence on the buckling behavior of
the model.
The axial deflections are monitored until the
model buckles, indicated by a sudden collapse of the shell
wall. A definite drop in pressure is associated vith the
collapse, and the buckle pattern for those models vhich have
not failed by fracture are recorded.
-27-
CHAPTER 6
EXPERINENTAL RESULTS
6.1 Bucklin Behavior
Twenty-five models were tested to examine the PC
shell's buckling characteristics. Results of the tests are
presented in Table 1. PC shells of six circumferential
triangles were represented by models one through nine and
those of ten circumferential triangles by models ten through
twenty-five. Two sets of thickness identified by the radius
to thickness ratio R/T! were examined for each configura-'
tion. The length, identified by the length to radius ratio
L/R!, and the corresponding number of axial triangles N!
were varied for each set'of thicknesses.
Photographs of the observed deformations and
buckled patterns are presented in Figures 15 through 20. As
external pressure was applied, surface deformations developed
which became larger with increasing pressure. The distor-
tions on the concave surfaces of model 41 in Figure 15 are
typical. Because of the high radius to thickness ratio
R/T = 325! the distortions are readily visible. The large
lateral deformations on the triangle surfaces, particularly
near the ridgelines Figure 15a!, give evidence of the high
bending stresses present.
-2 8-
Figure 15b shows model Cl immediately following buckling.
Here, the initial collapse occurred locally, as shown, then
eventually propagated throughout the rest of the surface to
the final buckled form similar to that, shown of model 55 in
Figure 16a. This "slow" buckling process was observed in
models one through five which have the largest radius to
thickness ratio R/T = 325! of all the models. The buckling
process occurred much more rapidly for the other models
having lower R/T ratios. They can, be compared with that of
the circular cylinder which becomes increasingly sensitive
to local imperfections, and hence is more likely to exhibit
initial buckling locally, as wall thickness decreases. The
deviations in the PC shell geometry, i.e., the slight vari-
ations in wall thickness or radius, may have caused the
local failure which initiated the buckling process. In the
circular cylinder, a slight imperfection will also cause a
significant decrease in the buckling resistance. However,
the resistance of 0he PC shell models was consistent when
compared with the perfect circular cylinder see Figure 26!.
This reveals that the PC shell is not as sensitive to geo-
metric deviations as is the circular cylinder. This may
also be an indication that the pressures at which the local
failures initiated may not be far from the pressures at
which overall buckling should ideally occur.
-29-
The buckling process proceeded very rapidly for
models six through twenty-five, resulting in either the ap-
pearance of the buckled form, or in the fracture failure of
the material immediately following buckling. For models six
through nine N=6, R/T=162! and models ten through twenty-two
K=10, R/T=131!, those that were subjected to external
hydrostatic pressure no axial restraint! failed catastro-
phically by fracture, while those under external radial
pressure axially restrained! buckled intact. Examples of
the fractured surfaces are shown in Figures 16 and l7. The
fracture of the models under hydrostatic pressure can be
explained by the high bending stresses developed along the
edges of the triangles as axial shortening takes place,
coupled 'with the strain rate dependency of the material.
The high strain rates induced as the surface rapidly col-
lapses cause the rigid vinyl to become more brittle. Together
with the high bending stresses already existing, the maximum
stresses then exceeds 0he yield stress of the material,
causing fracture. The failure patterns in Figures 16 and 17
show the fracture propagating along the ridgelines where the
triangles intersect. This is expected since the maximum
bending stresses have been shown to develop along the ridge-
lines. The fact that. these bending stresses can be consi-
derably reduced by restraining the axial displacements is
-30-
supported by non-fracture buckling of the models provided
with axial restraints. The resulting buckle configurations
of the axially restrained models are shown in Figures 18
through 20. Nodels tventy-three through twenty-five M=10,
R/T=197! were subject to hydrostatic pressure. Because of
the higher R/T ratio the maximum stresses vere below the
stress limits of the material so the models buckled intact.
Dial indicators were positioned, along the midlength
of model twenty-five to track the radial deflection of the
nodes Figure 35!. The radial deflection paths are shown in
Figure 22. The large and nonlinear nature of the deflections
are apparent when compared to the linear deflection path of
a perfect cylinder dashed line! calculated by
2PR
6R=ET �. 1!
-31-
Figure 23 shows the radial node deflections dashed
lines! of model 525 just prior to buckling. The model buckled
with three circumferential half-waves, In Figure 23, the
buckle pattern is seen developing. The region near the seam
showed the least variation in deflections due to the addi-
tional stiffness provided by the doubler strip. The regions
around nodes 7 and 4 which are away from the seam buckled
first Figure 21b! . In all the models tested buckling
initiated away from the seam.
6.2 Bucklin Curves
A comparison of the critical buckling pressures of
the PC shell and circular cylinder have been made. The
buckling results of the model tests have been used to con-
struct the graphs of Figure 24 through 30. A consistent set
of symbols are used in reference to the models in Table l.
The buckling resistance as a function of the geo-
metric parameters: length to radius ratio L/R!, radius to
thickness ratio R/T!, and the number of circumferential
triangles N! was examined. Figures 24 and 25 compares the
nondimensional buckling pressure P /E! of the PC shellpc
with that of a perfect true! circular cylinder {P l/E! ascyl
a function of L/R. Equation {2.3! was used to determine the
buckling pressure of an equivalent circular cylinder whose
geometry is defined by the same variables that. define each
PC shell model. Alternatively, the buckling pressure as a
function of the number of axial triangles {N! is shown. The
significantly higher buckling resistance of the PC shell is
immediately apparent for the different R/T ratios and N. A
similar dependency on L/R is seen for the PC shell and
circular cylinder,
-32-
Models ten through twenty-two were used to examine
the effects of pxoviding axial restraint,. The dashed curve
in Figure 25 represents the axially restrained models. The
difference in the buckling resistance between. the restrained
and unrestrained models increases significantly for L/R
values less than 2.0. Evidently, the PC shell becomes
sensitive to load conditions at short lengths. For longer
lengths I./R ! 2.0! the addition of end constraints had
little effect on the buckling resistance.
The buckling pressure ratio P /P l is shown inpc cyl
Figure 26 as a function of I/R. The relationship appears
linear for each R/T ratio. A slight improvement in the
buckling pressure ratio is seen as shell length increases.
However, this improvement diminishes for thicker shells.
The influence of axial restraint is also apparent in Figure
26. For an Z/R ratio close to l.0, the P /P ratio ofpc cyl
the axially constrained shell nearly doubles that of the
freely displaced shell.
6.3 E ivalent Thickness
The most significant influence on the P /Ppc cyl
ratio appears to be the number of circumferential triangles,
N. A considerable increase in pressure is shown in going
from ten. to six circumferential triangles; the buckling
-33-
pressure ratio increases from two to eight. This strong
dependency on a single parameter suggests that the equation
N
6
8
1012
2.32
1.72
1.3?
1.23
These values are plotted in Figure 28. A good
curve fit through these points was obtained. By using the
-34-
for the circular cylinder {equation {2.3!! might be modified
to predict the buckling pressure of the PC shell by assuming
it to be an equivalent circular cylinder. P 1 was set equalcyl
to the experimental buckling pressure, P , and an equivalent,pc
thickness, T , was computed from the approximate bucklinge'
Equation �.4!. All other parameters of the equivalent cyl-
inder remained constant. The resulting T /T xatio for eache
model is plotted in Figure 27 as a function of L/R. The
variation of T /T is seen to be small; thus, for the purposee
of calculation, T /T may be considered constant. The meane
values of the T /T ratio for six and ten circumferentiale
triangles are indicated by the horizontal, lines.
The mean values of the T /T ratio as a function of
N are given below along with the T /T values obtained frome
the result.s o f re f e rene es 13 and 27:
large. Thus, the PC shell behaves more like a circular cyl-
inder as N goes to infinity which supports the geometric
arguments made by Niura.21
6.4 E ivalent Axial Stiffness
The axial deformation was measured by dial indi-
cators as described earlier. The average axial shortening
was compared with that of a circular cylinder in terms of an
equivalent axial modulus, E , of the PC shell, given by15e'
Ee
E 2vrR TbE�.2!
-35-
curve in Figure 28 along with Equation �.3!, an estimate of
the buckling pressure for the PC shell can be made. In the
above calculations the PC shell is considered a circular
cylinder having an apparent wall thickness dependent only on
N. The P /P 1 ratio calculated by this method for six andpc pcl
ten circumferential triangles are represented in Figure 26
by dashed lines. This assumption leads to a constant
P /P 1 ratio against L/R. The lesser variations in thepc cyl
buckling resistance due to L/R and R/T are discussed in
Section 6.5.
In Figure 25 the equivalent thickness decreases
with increasing N, and approaches unity as N becomes very
where
E = actual tensile modulus of elasticity
E /E {mean!
10
106
6
131
197
162
32S
0. 177
0.160
0.062
0.022
The values of E /E are considerably less than unity. Ine
effect, the increase in buckling resistance is accompanied
by an equally significant decrease in axial rigidity. In
going from ten to six circumferential triangles, the buckling
resistance increases nearly 3QQ percent {Figure 26!. However,
the axial rigidity is dropped by more than 60 percent Figure
F = axial compressive force
L = shell length
R = shell radius
T = shell thickness
6 = measured axial shortening
The calculated equivalent modulus for the model tests are
presented in Appendix B-1. The resulting E /E ratios are
given in Table 1.
Figure 29 shows the E /E ratio as a function ofe
K/R. The axial stiffness does not appear to be dependent on
length. The average E /E values as a function of R/T and Ne
are tabulated below:
29! . The axial rigidity also decreases with increasing R/T
ratios.
Knapp has given finite element results for the15
equivalent axial stiffness for different values of N; these
results are presented in Figure 30. The experimental values
of this study show good agreement with those results; the
axial stiffness is seen to increase as N increases. Again,
evidence is shown that the PC shell tends toward the physical
behavior of a circular cylinder as N goes to infinity.
6.5 Buckling Nodes
Buckling modes observed in the tests are shown in
Figures 31 through 33. A buckling mode is characterized by
the number of circumferential half-waves; thus, each "snap-
through" region which usually extends the length of the
shell! appearing on the buckled surface defines a half-wave.
The mode shape refers to the surface pattern made by the out-
line of these regions. Because it was difficult to determine
the buckling modes of the models which failed catastrophi-
cally, only the modes of those which buckled intact are
presented,.
Two regular, symmetric half-waves were observed for
the PC shell models of N=6 and R/T=325 Figure 31!. All the
mode shapes were similar indicating that the range of N and
corresponding L/R ratios! examined was beyond the range
»37»
that would otherwise influence the buckling mode. However,
the models of N=10, with R/T=131 and R/T=197, provided dif-
ferent mode shapes due to the variations in the geometric
parameters. A close examination of these interesting buckl-
ing patterns may provide additional clues to the high buckl-
ing resistance of the PC shell.
The buckling modes for N=10 and R/T=131 are shown
in Figure 32. Regular symmetric mode shapes were observed
on the PC shells with even number of axial triangles, while
regular anti-symmetric patterns were observed for those with
odd numbers. This difference may be caused by the natural
tendency of the mode shapes to conform to the triangle pattern
of the PC shell geometry by following the ridgelines of the
concave surfaces. In order for the half-wave boundaries to
be contiguous an even number of axial triangles may allow
two axis of symmetry {since N is also even in this case!,
whereas an odd number may allow only one axis of symmetry
for the mode shapes.
In Figure 32 the number of circumferential half-
waves is seen to decrease from five to four in going from
M=4 to M=5. Although some distortion is seen in the mode
shapes at the mid-length it is apparent that they follow
closely the original lines of the PC shell surface. This
necessary adaptation to the corrugated surface may contribute
-38-
to the PC shell's high buckling resistance unlike the circu-
lar cylinder whose smooth and continuous surface allows the
natural mode shapes to be easily developed, the corrugated
surface of the PC shell makes it difficult to achieve the
natural mode shapes, thereby increasing its buckling resis-
tance.
The buckling modes observed for R/T=197 Figure
33! were similar to those of R/T=131 except at N=S where
three half-waves appeared. Here the strong influence that the
original lines have on the mode shape is evident. Nore sig-
nificant is the appearance of the three half-waves at K=8
for R/T=197. Comparing with the four half-waves observed for
R/T=131, the number of circumferentiql half-waves is seen
to decrease as the shell thickness decreases increasing R/T!.
Table 4 compares the number of half-waves of the
PC shell and equivalent circular cylinder as a function of
the pertinent geometric parameters. The number of half-waves
decreases for both shells as f/R increases with constant R/T.
However, converse trends occur as R/T increases with constant
L/R; the number of half-waves decreases for the PC shell,
but increases for the cylinder. The reason for this discre-
pancy may be revealed by further examination of the PC shell
geometry.
It has. been shown earlier that the geometry of both
the PC shell and circular cylinder greatly affects their res-
pective buckling modes. Comparing the geometric parameters
which influence the number of buckling half-waves for the
two shell form,
n = n L/R, R/T, 8/T!pc pc
n = n l L/R, R/T!cyl cyl
an additional parameter is seen for n . The o/T term ispc
introduced to account. for the influence of the PC shell ' s
undulating surface. Hence, gr'T may be considered the "effec-
tive corrugation" parameter.
Niura has shown that the high circumferential2
bending rigidity of the PC shell is due largely to the
increased moment of intertia provided by its undulating geo-
metry: the shell behaves as though it were a thicker shell.
An increase in u/T would indicate an increased moment of
inertia. From Equation �.4!, in going from N= to N=6,
the effective corrugation parameter, g/T, would increase.
Associated with this is a significant increase of the buckling
resistance. To a lesser degree, and if N is held constant,
it can be seen that +/T can also be increased by reducing
the wall thickness as Table 4 shows. This increase in PT
implies that, from a buckling standpoint, the PC shell would
-40-
behave as a thicker shell the thinner it physically becomes.
This is supported by the T /T ratio discussed in Sectione
6.3. In Table 4 the T /T ration is seen to increase as R/T
increases. This thicker shell behavior may explain the
improved buckling resistance of the PC shell, especially at
higher R/T ratios Figure 26!.
6.6 Effect of Shell Len th
In Figure 26 the buckling resistance is seen to
improve with increasing shell length, particularly for the
higher R/T ratios. This improvement may be due to the dimi-
nishing influence of the shell ends as length increases.
Test results of circular cylinders under hydrostatic pressure
showed poor agreement with theoretical results at short
lengths because of increased sensitivity to edge effects. 1
However, agreement became good as length increased. Similar
effects may have occurred during the PC shell model tests,
which would explain the improved buckling resistance as
length increases. The PC shell is more stable than the cir-
cular cylinder so this improvement is seen to be slight.
CHAPTER 7
NUNERICAL BUCKLINC ANALYSIS
7.1 Finite Element Anal sis
To verify the experimental buckling results of this
investigation, a numerical buckling analysis was performed
on a single model. Comput: er modeling using finite elements 35
has become a very effective method for analyzing complex
structural problems due to the method's simplicity and its
availability in existing large program libraries. The dis-
13,14,15placement method was used successfully by Knapp ' ' in
analyzing the PC shell. A similar analysis has been conducted
in this study.
Basically, the finite element method assumes the
real structure or continuum to be an assemblage of elements
interconnected at a discrete number of nodal points' 1, 34
The properties of these elements are derived such that they
will approximate the stress and deformation states of the
original structure. In general, these derivations are based
on satisfying the equations of elasticity, i.e., the equili-
brium and compatibility equations as well as the relevant
constitutive relationships along the boundaries of the element.
The result is a set of equilibrium equations the coefficients
of which constitute the stiffness matrix, [K] , of the ele-
ment. Detailed discussions on the methods of deriving stiff-
ness matrices can be found in references 1, 5, l8 and 34.
The individual stiffnesses are compiled to form a set of
equilibrium equations for the total structure represented in
matrix form by
<P! = fKj d! + PG] �.1!
where P represents the external nodal forces; d , the
nodal displacements; and [K], the stiffness matrix of the
total structure. Any terms due to nonlinearities are included
in PG . If .nonlinear effects are neglected, then the vector
PG is ignored and the structure is assumed to behave
linearly. The set of equations represented by equation �.1}
are solved for the unknown displacement states in d
-43-
7.2 Solution for Nonlinear E ations
The PC shell carries load by both membrane and
bending actions, and for large displacements these actions
become coupled and dependent upon the displacement state; 14
thus, the stiffness properties can vary with the applied
loads. Equation �.1! represents a set of nonlinear equa-
tions which can be solved by several available methods. ' The
Newton-Raphson iterative technique has proven to be one of
the best methods of solution available and has been incor-
porated in every existing structural programs.
An important characteristic of the Newton-Raphson
method is its ability to converge with extreme accuracy for
higly nonlinear behavior. The technique involves succes-28
sive incremental load stepping with an updating of the stiff-
ness matrix on each increment. Intermediate corrective cycl-
ing may be performed in each load step until equilibrium
between forces is achieved. This enables tracking of the
load-deflection paths up to the point of structural insta-
bility. Details of the Newton-Raphson method of solution
are given in references 6, 13, and 28.
One major drawback to the incremental solution is
its cost, particularly for structures with large degrees of
freedom. For this analysis a PC shell model having a total
98 degrees of freedom was analyzed. Eight load steps were
carried out amounting to an average of $5.00 per load step.
This can be compared to the analysis performed by Knapp 13
for a PC shell model with 311 degrees of freedom in which
the cost per load step was approximately $75.00.
-44-
7.3 Descri tion of' Bucklin Model
The PC shell model selected for the finite element
stability analysis is shown in Figure 34. The geometry and
properties are identical to that of test model twenty-three
see Table 1!. Because of the 1/5th symmetry exhibited by
the buckling mode of model twenty-three, only the shaded
region of the PC she11 of Figure 34 was modeled. Thirty
node points Figure 35! interconnected by 79 beam elements
Figure 36! represent the analyzed surface.
The framework type of finite element was used to
model the PC shell. The advantages of this type of element
for stability analysis have been presented by Knapp and14
include the following:
l! The derivation of the properties are based, on
the same principles used to drive properties of a con-
tinuous element;
{2! Framework elements inherently possess edge
continuity, so consequently, they .show good monatonic
convergence characteristics over continuous elements in
large deflection and instability problems; and,25
�! Framework elements can easily be used in
existing space frame programs.
Framework elements are composed of an assemblage
of bars arranged in a definite geometric pattern. The prop-
erties of the equi lateral and right triangular patterns used
to model the PC shell are given in Table 5. The membrane
properties for the right triangle elements are not available;
however, the membrane properties for the equilateral triangle
have been found to be applicable. Properties of the beam14
elements for the model Figure 36! were calculated {Appendix
B-2! from the equations in Table 5. Each interior node point
of Figure 35 contains six degrees of freedom--three rotational
and three translational. Translations and rotations on the
edges of the model not excluded by symmetry conditions are
illustrated in Figure 37. The model contains a total of 98
degrees of freedom.
7.4 Buckling Results
The computer program utilized was the STRUctural
Design Language STRUDL! program package developed at the
Massachusetts Institute of Technology, Civil Engineering
Department. Subroutines for finite element analysis havell
been incorporated in the program which includes subprograms
for nonlinear space frame analysis. No modifications of the
program were required for the framework element model.
Descriptions of the structural program and the required input
parameters are given in references ll and 22. The input data
for the buckling analysis is presented in Appendix D-1. All
-46-
computations were carried out on the IBN 370 computer at the
University of Hawaii. The PC shell model was subjected to
uniform hydrostatic pressure, similar to test model twenty-
three, from 0.0 to 3.0 psi at load increments of 0.2 psi.
Because of the small load increment, the Newton-Raphson method
was used with no corrective cycling applied to the load steps.
The complete computer output of the analysis is presented in
Appendix D. The total cost of the analysis was less than
$50.00.
Results of the nonlinear analysis are shown in
Figure 38. The radial load deflection paths of node 7, 15,
and 11, located on the line of symmetry see Figure 35!, have
been plotted. It is seen that the radial displacements were
nearly linear up to 1.2 psi. The displacements then became
very large from 1.2 to 1.6 psi. At load increment 9 the
analysis terminated automatically when structural instability
was detected by the STRUDEL program. Thus, buckling occurred
between 1.6 and 1.8 psi shaded region!. Comparing this to
the buckling pressure of test model 523 �.18 psi!, the nume-
rical buckling pressure is seen to be 17 to 27 percent below
the test buckling pressure. The result is still higher than
for the equivalent cylinder by approximately 1.5 times.
The lower buckling load obtained by the finite
element analysis may be explained by the convergence char-
-47-
acteristics of the framework element. Szilard has shown26
that framework elements have excellent monotonic convergence
characteristics over that of surface i.e., plate-shell type!
elements that are currently available. However, the conver-
gence path of the element types differ. Continuous elements
tend to give upper bound values of the true solution, whereas
framevork element tend to give lower bound values. As a
structure is modeled with finer subdivisions, the values given
by both element types approaches that of the true solution.
The lover bound approach indicates that structures modeled
with framework elements may be more flexible than the actual
structure if the subdivisions taken are not sufficiently fine.
Consequently., for coarse subdivision , the resulting stresses
and displacements tend to be larger than for finer subdi-
visions. This suggests that the subdivisions used in our
buckling analysis may have been too coarse. Subsequently,
the resulting displacements may be larger than what would
othervise be obtained using a finer subdivision, causing the
structure to become unstable at a lower load. Modeling the
PC shell vith finer subdivisions may increase the buckling
region shovn in Figure 38, bringing it closer to the actual
test value. Unfortunately, the additional analysis required
to determine the effect of a finer subdivision vas not per-
formed, and should be considered in later investigations.
-48-
lf the buckling results obtained for the model of
Figure 35 holds for other PC shell configurations, then an
important aspect of using framework elements would be its
conservative estimate of the buckling load. This feature is
important especially when analyzing full scale configurations
where testing of an actual model would be impractical. Thus,
a conservative estimate of the buckling load to within
twenty-five percent can be obtained cheaply using a coarse
subdivision such as that shown in Figures 36 or 37. A bet-
ter estimate of the buckling load can, then be made using a
more refined model. The results of the first analysis can
be used to make the second analysis more efficient, mini-
mizing the cost.
CHAPTER 8
EXPERINENTAL AND 1WXERZCAL STRESS ANALYSIS
B.l Descri tion of Test Nodel
To ensure that the test models buckled elastically,
experimental and numerical stress analyses were performed.
The test model constructed for the analyses is shown in Figure
39. A plastic sheet of "Plexiglas G", O.OSS8 inch thick,
manufactured by the Rohm-Haas Company was used. Properties
of Plexiglas are given in Table 6, The same vacumm-forming
process described in section 4.2 was used in fabricating th4
model with ten circumferential and two axial triangles. The
temperature for the forming process was 160 centigrade.
The seam construction was modified because of the
different material and the large sheet thickness. The sheet
was accurately cut before the fabrication process so that
the butt-joint was formed on the initial 'heating and vacuum-
forming. After the thermal vacuum-forming process, shrink-
age of the material left a one-eighth inch gap along the seam.
The shell was removed from the mold, and the gap filled with
a liguid polymer bonding agent. Once hardened the fillet
has nearly the same mechanical properties as Plexiglas. The
completed model was then heated to 50' centigrade for 8 hours
to reduce the residual stresses along the seam edges. The
-50-
model was then assembled in the same manner as those of the
buckling tests see Section 4.3!
set gl - hoop strains on concave surface
set g2 - hoop strains on convex surface
set, 53 - axial strains on concave surface
set g4 - axial strains on convex surface
A-C!
B-D!
A-C!
B-D!
Gage sets 05 through g7 were positioned at equal intervals
along the height of the convex surface B-D! to measure the
hoop strains along the axis.
8.2 Test Procedure
The PC shell model was tested in procedure described.
in Chapter 5. The model was subjected uniform hydrostatic
pressure. The strain gages vere electrically connected to a
-51-
Before assembly, sets of strain gages vere mounted
on the triangle surfaces farthest away from the seam as shovn
in Figure 39b. The gages were specially manufactured for
use on Plexiglas. Specifications of the strain gage are given
in Table 7. Each set is comprised of two strain gages, one
mounted on the inside, and the other on the outside surface
of the PC shell. Both are aligned in the same direction.
Gage sets one through four were positioned to measure the
hoop circumferential! and axial strains at the centroid of
the triangle faces as follows:
BLH D. C. Model strain indicator through a multiple bridge-
balancing unit. Each strain gage vas balanced before the
test. A mercury manometer was used to monitor the pressure
as vacuum vas applied. At regular intervals of the applied
pressure, strain indicator readings were recorded. The
resistance of the gages vas translated directly into strains
by the strain indicator. The driving voltage {6 volts! to
the gages were higher than the recommended voltage �.5 volt!,
thus the duration of' the measuring times were kept as short.
as possible to minimize the effect of localized heating of
the surface area around the gages. The measured strains for
the range of the pressure load were linear. The resulting
data are given iq Appendix C.l, and are discussed along with
the numerical stress results in Section 8.4.
8.3 Descri tion of Finite Element Model
A finite element stress analysis was carried out
for the PC shell experimental stress model Figure 39!. The
finite element representation is shown in Figure 40. A linear
stress analysis vas performed. Because of geometric and
loading symmetry, only the shaded region A-B-C-D shown in
Figure 40 was needed to be analyzed.
The analysis was performed with the STRUctual Design
Language STRUDL! program on the IBM 370 computer at the
-52-
University of Hawaii. The SBCT plate-shell type element
contained in 0he STRUDEL element library was used. A conver-
gence study was first carried out in order to insure reason-
able accuracy of the results. The resulting mesh in Figure
41 has more than 200 degrees of freedom. Node and element
numbering are shown in Figures 41 and 42, respectively.
Boundary condition are similar to the symmetry boundary con-
dition of the finite element buckling model in Figure 37.
The model was subjected to uniform hydrostatic
pressure similar to the experimental stress model. The com-
puter output for the analysis is listed in Appendix D-2.
The stress results are presented in Appendix C-2 and are
discussed in the followi'ng section.
8.4 Stress Results
The experimental and numerical stress results are
presented graphically in Figures 43 through 50 in nondimen-
sional form. Stress distributions are shown for the lines
of symmetry A-C concave surface! and B-D convex surface!
of Figure 40.
The membrane stresses of the PC shell have been
compared to the stresses of the equivalent circular cylin-
-53-
der. The equivalent axial stress of the circular cylinder
is given by
Fa
aae 2mRT
8 ~ 1!
where F is the axial load on the PC shell. For hydrostatica
pressure
F = PNR SING COSBa
8.2!
Thus,
-PNR SINe COSe2%T
The equivalent hoop circumferential! stress of the circular
cylinder is
PR
Be T S. 4!
ob = 0. 1071P T 8. >!
-54-
The maximum stress of a clamped, equilateral triangular plate
has been used for the equivalent bending stress: 26
The bending stress is positive if the compressive component.
occurs on the pressure side of the plate.
The membrane hoop stress distribution obtained nu-
merically is compared with the experimental stress results
in Figures 43 and 44. Good agreement can be seen for the
hoop stress at the centroid of the concave surface {Figure
43!. The numerical result is approximately 25 percent higher
than the test result. Along the convex surface Figure 44!,
the numerical and experimental results show similar trends
for the hoop stress distribution. However, the numerical
stress values are considerably larger than the test values;
from l.9 to as much as 2.5 times. The hoop stress for the
concave and convex surfaces are primarily compressive. The
membrane. axial stress distributions are shown in Figures 45
and 46, Like the membrane hoop stress, reasonable agreement
is seen for the numerical and experimental stress on the
centroid of the concave surface . Figure 45!, whereas the
numerical stress is approximately twice the value of the
experimental at the centroid of the convex surface {Figure
46!. The membrane axial stress changed from compressive to
tensile along line A-C, while the stress along line B-D
remained compressive.
The bending hoop stresses are shown in Figures 47
and 48. The stress predicted numerically is seen to be
approximately twice the experimental value at the centroid
of the concave surface Figure 47!. The numerical stresses
along the convex surface Figure 4B!, however, show excel-
lent agreement with the experimental results. The maximum
bending hoop stresses are seen near the line of symmetry C-D
corresponding to the midlength of 0he PC shell. The magni-
tude of the maximum stress ratio is less than 2.0. The bend-
ing axial stress distributions are shown in Figures 49 and
50. The numerical stress value at the centroid of the con-
cave surface Figure 49! is approximately 4.0 times the
experimental stress value, whereas good agreement is seen
for the stresses at the centroid of the convex surface
Figure 50!. The maximum axial bending stress ratio is less
than l. 0 in magnitude.
The discrepancies between the experimental and nu-
merical stress results seem dependent on the location. Re-
ferring to Figure 40, the membrane results were in good agree-
ment along the concave surface line A-C!, but disagreed by
a factor of approximately 2.0 along the convex surface line
B-D!. On the other hand, the bending stress results were in
agreement along the convex surface, but in disagreemet along
the concave surface by a factor of 2.0 to 4.0. The numerical
values were consistently higher in all cases. The cause of
the discrepencies described are not known. However, several
-56-
explanations are possible. For the experimental stress model,
the ends of the shell were held in place by rubber compres-
sion rings. Thus, they were not perfectly clamped; the rubber
allowed some rotations to occur, however slight. This may
have the effect of reducing the bending stresses along the
surface. Other effects that may have also affected the stress
values are: misalignment of the strain-gages; variations in
the shell geometry; and local heating of the surface during
strain measurements. For the finite element model, the node
displacements along A-B were not uniform see displacement
print-out in Appendix D-2!. The axial shortening at point A
node 1! was 63 percent greater than at point B node 6! see
Figure 41!. The variation in the axial Road indicated by th
difference in axial shortening may perhaps be significant
enough to create the discrepancies.
The excellent agreement between the experimenta.l
and numerical bending stresses along line B-D indicates that
the maximum bending stress shown in Figure 48 is accurate.
In designing the models for the buckling experiment, a maxi-
mum bending stress ratio of 2.0 was assumed. From these
results the maximum bending stress ratio is seen to be less
than 2.0, which supports the assumption.
CHAPTER 9
CONCI US I ONS
In this investigation the buckling behavior of the
PC she11 was ezamined. Twenty-five models were tested in
order to determine the influence of the PC shell's basic
geometric parameters. The results give conclusive support
to earlier claims ' ' ' ' ' of increased buckling13,14,15,16,21,27
resistance of the PC shell to that of the circular cylinder.
The ezperimental test results have been summarized in refer-
ence 16.
Zn summary, the following PC shell characteristics
have been revealed:
�! The buckling resistance is primarily dependent on
the number of circumferential triangles, N. As N
decreases the buckling resistance increases, and,
correspondingly, as N increases the buckling resist-
ance approaches that of the circular cylinder.
�! The buckling resistance when compared to the perfect
circular cylinder improves for the PC shell as the
thickness decreases because of the increase in the
effective corrugation, 6/T, which makes the shell
seem thicker from a buckling stand point.
�! The buckling resistance also improves with increas-
ing L/R, due probably to the reduced influence of
the shell ends.
�! The PC shell is considerably more stable than a
circular cylinder, thus is less sensitive to geo-
metric and load imperfections.
�! The axial stiffness of the PC shell decreases as N
increases, and as shell thickness decreases.
�! The addition of an axial restraint increases the
buckling resistance of the PC shell for low L/R
ratios, but its effect diminishes as L/R increases.
�! The addition of an axial restraint considerably
reduces the bending stresses in the PC shell.
By assuming the buckling resistance to be depen-
dent only on N, an approximate estimate of the buckling pres-
sure for PC shells with a given number of circumferential
triangles can be obtained by using Figure 27 to determine
the equivalent cylinder thickness, T , which is then substi-
tuted for T in Equation 2.4. Further investigations may
provide a more accurate method of predicting the buckling
resistance by including the effects of the other geometric
parameters.
A finite element buckling analysis was performed
to verify the model tests. The buckling pressure found for
-59-
a particular shell configuration! was lower than the corres-
ponding experimental value due to the coarse mesh size of
the finite element model used. It is argued that for analyz-
ing large size PC shells, such a coarse model can provide an
inexpensive and conservative estimate of the buckling pres-
sures to be expected.
An experimental and numerical stress analysis showed
that the bending stresses clearly dominate the membrane stres-
ses, and must be the first consideration in design. In
addition, the analysis confirmed that the experimental buckl-
ing models were designed within the elastic range; thus, thh
buckling results are valid.
-6G-
BIBI IOGRAPHY
l. Almroth, B.O. and Brush, D,O., Buckling of Bars, Plates,and Shells, McGray-Hill, New York, 1975.
2. Batdorf, S.B., "A Simplified Method of Elastic-StabilityAnalysis for Thin Cylindrical Sells, 1-Donnell'sEquation," NACA TN 1341, 1947.
3. Bijlaard, P.P., "Buckling Stress of' Thin CylindricalClamped Shells Subject to Hydrostatis Pressure,Journal of Aeronautical Science, Vol. 21, Dec.1954, pp. 852-853.
4. "Buckling of Thin-Walled Circular Cylinders," NASA TNSP-8007, Aug. 1968.
5. Coexter, H.S.M., Introduction to Geometr , Wiley,New York, 1974.
6. Cook, R.D., Conce ts and A lication of Finite ElementA~nal sis, John Wiley and Sons, Inc., New Yock,].974.
7. Donnell, L.H., "A New Theory for the Buckling of ThinCylinders Under Axial Compression and Bending,"Transaction, ASME, Vol. 56, 1934, pp. 795-806.
S. Donnell, L.H. and Wan, C.C., "Effects of Imperfectionson Buckling of Thin Cylinders and Columns UnderAxial Compression," Journal of A lied Mechanics,Vol. 17, No. 1, '1950, p. 73.
9. Dym, C.L., "On the Buckling of Cylinders in Axial Com-pression," Journal of A lied Mechanics, ASME PaperNo. 73-APM-BBB.
10. Hoff, N.J., Madsen, W.A., and Mayers, J., "PostbucklingEquilibrium of Axially Compressed Circular Cylin-drical Shells," AIAA Journal, Vol. 4, No. 1, Jan.1966, pp. 126-133.
11. Jordan, J.C., ed., ICES: Pro rammers Ref'erence Manual,Report R67-50, Dept. of Civil Engineering, M.I.T.,Oct. 1967.
12. Kirste, L., "A Bwickelbare Verformung diinnwandigerKreiszylender," Oesterreichisches IngenieurArchiv, Vol. 8, May 1954, pg. 149.
13. Knapp, R.H., Finite Element Nonlinear Bucklin Anal sisof a Pseudo-C lindrical Concave Pol hedral ShellUnder External Pressure, PhD dissertation, Dept. ofOcean. Engineering, University of Hawaii, Honolulu,Hawaii, Aug. 1973.
14. Knapp, R. H., "Numerical and Experimental Analysis of aPseudo-Cylindrical Shell," IASS/CISM Symposium onFolded Plates and Spatial Panel Structures, Udine,Italy, Sept. 1974, also ulletin of the IASS, No. 59,Vol. XVI-3, Dec. 1975.
15. Knapp, R.H., "Pseudo-Cylindrical Shells, A New Conceptfor Undersea Structures," Trans. ASME, Journal ofEn ineerin for Industr , Vol. 99, No. 2, May 1977,pp. 485-492.
16. Knapp, R.H. and Dumlao, C., "Experimental Investigationsof Prebuckled Cylinders Under External Pressure,"Trans. ASME, Journal of En ineering for IndustrVol. 101, No. 2, May 1979, pp. 178-184.
17. Knapp, R.H. and Szilard, Rudolph, "Non-linear StabilityAnalysis of Pseudo-Cylindrical Shells," ACTA Tech-nica Academiae Scientiarum Hungaricae, Vol. 86,1978, pp. 9-41.
18. Lange, C.G. and Newell, A.C., "Spherical Shells LikeHexagons: Cylinders Prefer Diamonds," Journal ofA lied Mechanics, TRANS. ASME, June 1973.
19. Martin, H.C. and Carety, G.F., Introduction to FiniteElement Anal sis, McGray-Hill, New York, 1973.
20. Meyers, Holm, McAllister, Handbook of Ocean and Under-water En ineerin , North American Rockwell Corp.,1969, pp. 9-3 to 9-15.
21. Niura, Koryo, "Proposition of Pseudo-Cylindrical ConcavePolyhedral Shells," ISAS Report No. 442, Vc 1. 34,No. 9, 1969.
-62-
22. Roos, Daniel, ed. ICES S stem: General Descri tion,R67-49, Dept. of Civil Engineering, N. I . T., Sept.1967.
23. Rubinstein, N.F., Natrix Com uter Anal sis of Struc-tures, Prentice-Hall, New Jersey, 1966,pp. 143-144.
24. Salonen, Eero-Matti, "Triangular Framework Nodel forPlate Bending," Journal of the Engineering Mechan-ics Division, ASCE, Vol. 97, No. ENl, 1971,pp. 149-153.
25. Salonen, Eero-Natti, "A Gridwork Nethod for Plates inBending," Acta Pol tech. Scand. Civil Eng., Build-ing Const. Serv., No. 59, 1969.
26. Szilard, R., Theor and Anal sis of Plates: Classicaland Numerical Methods, Prentice-Hall, Inc.,Englewood-CLiffs, 1974.
27. Tanizawa� K. and Niura, K., "Stress Analysis of a Con-cave Polyhedral Shell," Report No. 523. Instituteof Space and Aeronautical Science, University ofTokyo, 1975.
28. Tillerson, J.R., Stricklin, J.A., and Haisler, W.E.,"Numerical Methods for the Solution of Non-linearProblems in Structural Analyses," Numerical Solu-tion of Non-linear Problems, AMD-Vol. 6, 1973,pp. 67-101.
29. Timoshenko, S.P. and Gere, J.M., Theor of Elastic Sta-b~il.it , McGraw-Hill, New York, 1961.
30. Timoshenko, S.P. and Goodier, J.N., Theor of Elasti-~cit , McGraw-Hill, New York, 1951.
31. Von Karman, T. and Tsien, Hsue-Shen, "The Buckling ofThin Cylindrical Shells Under Axial Compression,"Journal of the Aeronautical Science, Vol. 8,June 1941, p, 303.
32. Weingarten, V.I. and Seide, Paul, "Elastic Stability ofThin-Walled Cylindrical and Conical Shells UnderCombined External Pressure and Axial Compression,"AIAA Journal, Vol. 3, No. 5, Nay, 1965.
-63-
33. Windenburg, Dwight F. and Tri lling, Charles, "Collapseby Instability of Thin Cylindrical Shells UnderExternal Pressure," ASME Trans., Vol. 56, PaperAFM-56-20.
34. Yoshimura, Y., "On the Mechanism of Buckling of a Circu-lar Cylindrical Shell Under Axial Compression,"NACA TM 1390, July l955.
35. Zienkiewicz, O.C., The Finite Element Method in Struc-tural and Continuum Mechanics, McGraw-Hill, NewYork, 1967.
-64-
Radial
Pressure
TOP VIEW
Axial Pressure l l l l RadialPressure
l l l I l l l 1SIDE VIEW
NOTE: SHELL EDGES FULLY CONSTRAINED FOR RADIAL PRESSURE.AXIAL DISPLACEMENT PERMITTED FOR UNIFORM PRESSURE.
GEOMETRY AND LOADING OP PC SHELL MODELS
PI GURE
-67-
a! CONFIGURATION DURING HEATING
b! CONFIGURATION AFTER COOLING
THERMAL VACUUM-FORMING
FIGURE 7.
-71-
a! LAPPED SEAM AFTER INITIAL HEATING
b! TRIMMED SEAM BEFORE SECOND HEATING
o! FINISHED BUTT JOINT FOLLOWING FINAL FORMING
METHOD OF SEAM TRIMMING DURING FORMING
FIGURE 8.
-72-
a! EDGES SEALED WITH ALUNIMUN TAPE
b! RUBBER EDGE CLAMPS IN PLACE
CLAMPING OP PC SHELL TO END PLATES
FIGURE 11.
M=2
L/R= 1. 07!
M~7
L/a=3.73!
BUCKLING NODE SHAPES: N=10, R/T=131
FICURE 32.
-96-
@=4
L/R=2. 13!
M=5
L/R=2.67!
N=6
L/R=3. 20!
X=8
L/R=4.26!
DEVELOPED
IRCUtFZIEZCE
/ /4 /
M=2
L/R=l.07!
K=8
LyR=4.26!Pf= Q
L jR;-2. 67}
BUCKLING NODE SHAPES: N=10, R/T=19 7
FIGURE 3 3.
-97-
g
C3
~ coa
o I-IM
M
O CQ W W Cu O tQQ 4 4
Gl R N w w w w w O O
led! ZHflSSZH8 'IVNHZZXZ-102-
R 0
0 0 0
a! NODLL CONFIGURATION
b! STR~XINJ GAGE ORIENTATIONS'J'
TEST CON~I<URATIO'0 QF E:CPERI&<mNTAL STRESS NODEL
FIGURE 39.
-103-
te~ e
0 M ' ~~ I le
04 440
4 o0 D
M ~te teN ~ ~~ I
tl0 00 0
0 C92 ElD
t 0Cl
~ ICt Eta
DM 0~ I
M SlaEt
PlPlCt tlCl0e a
~ II M o
~ 0Plel~ I
I ~
~ I 0 te4 IM
0 tlM te pl tepl te o~ IM tePl tt M Pl te ItM~ ~ I~ e
Cl Ie 0 'Cl0 ~IClaD IS tea ~0
~ C
Ct~ I
tt tlI Sl
CICI
0
~ I92Et
ItD
~ 0teCl'I ~
a a0 0N ~0
IteCl 0alt
IS' ~ ~0 0
Ik ISM Ie
te sl Cl4 aIS IO tlM
I~ I
00 oo M 0 44 0 tlClN
00 tl Cl
DI 0 M 4
Pl ~~ 92 ISC92
~ 0 elM~ eCl tleltl ~ ~ ~
o a 0 0
el
Clte 00 tICl Cl r te0 a
'M Sl
~ I ~0 M tltto0I
Cl ~ 92Pl ~tt
Eeel~ ICt
~ IICIIPl
00 M t0 ~ P Pl ~ I le
~ I 40 0~ C4
jf
0 0 0 0
<Std 0 4 M tl O
~ VIC
Pl0 a1tl
10 tetl 0N
0 It 00 0 CItl M
0a aM
O ttl
leel0 sl ISo
0 Va a
0t I92 ~ ClCt I~ I ~CI
~ l~ e IS
I~ '92e
~ IPl
~ 92'ltPt
te
IC
~ 0PlIPl o~ e
~ I~ I~ e
0 aIatlM I
M lePIEl m 0 te ~ 0~ 'MIe tlte
0
A
o I
8 !
!
A0 @
!
w .!N Qa
D O! s ZSdl
OT 4 !sd!
axle
D 0! ~ ZOd!
D�D! 4 Ztd!al e
!ZOd! e
a 0 o
IS M 0a M 00 0 o 40 M a
o o a
t 0 r ~ ~
0 a I ~
It IS ~
Ie Cl ISM Pllt tl
M te Ie
0~ ~ ~CII
~ ~ ~
!! Ce!
444,000 PSI
166,917 PSI
TENSILE MODULUS:
SFXAR MODULUS:
POISSON'S RATIO:
YIELD STRESS!
0 33
PSIXO, 000
-116-
TABLE 2.
PROPERTIES FOR BAKELITE VSA-+310!RIGID VINYL SHEET.
a
Q H IXC4
U 'gM
Q Q U
0 ZQ H
+ MQ M
5
VJ
z
O H
I
Q N XQ HtQ A
U WR 4H R
LA0 0
I I0
Ch 0
0 0 0
0 0 0
ahCILa
0 0~ J 0a rS
Cp4a0 L0 0CI.~ II0 CII
g-
tP A0IM q5CI LalCI CI
oa
0I
4r 0CU I~ CCCr MIa 0CL IA
7+ ~
CI
TABLE 6.
PROPERTIES POR
PLEXIGLAS t PLASTIC SHEET,
TENSILE MODULUS: 450,000 PSI
SHEAR MODULUS' 1,67,000 PSI
POISSON'S RATIONS 0.35
YIELD STRESS' 10,500
-120-
TABLE 7.
STRAIN GAGE SPECIFICATIONS
MANUFACTURER: MICRO-MEASUREMENTS INC.
GAGE FACTOR: 2 ' 035 + 0 ' 5% �75 F!
-121-
GAGE TYPE:
RESISTANCE:
EA-41-062AQ-350
350.0 + 0.15% OHMS
APPENDIX A
DESIGN PA%V%TERS
A.l Design Curves
The model design curves are presented in Fig-
gure A-1 and A-2. The lower bound curves dashed lines!
give the buckling pressure as a function of R/T for theA
clamped circular cylinder, calculated by Equation �.4!
using the axial length L! that corresponds to each N.
The upper bound curves solid line! are those of the PC
shell obtained by multiplying Equation �.4! with the
P /P 1 ratios given in Reference 15. Thus,pc cyl
= P /P ! xPpc pc cyl cyl
A. 1!
Pressure as a function of R/T at which the maximum allow-
able stress would be expected to develop. Because the
maximum stress was found to occur in bending, the curve
was obtained by using equation 8.5!, modified to
limit 1
R z X zF 0 1071 A. 2!
where P /P != 8.0 for N=6 and P /P 1!=3.5 for M=10.pc cyl pc cyl
The shaded curve in the two figures gives the
where al. . =cr . ld/2.0 to be conservative. The factor,limit yield
F g i s the maximum a eb/ab ! ratio that wou 1 d be expectedbe
From the earlier investigations it was found that the
stress ratio did not exceed 2.0; thus, F=2.0 for both M=6
and M=10.
Construction details of the mold segments and,
end plates are shown in Figures A-3 and A-4.
A.3 Material Stress-strain Data
Results of the tensile tests conducted on rigid
vinyl specimens are presented in Table A-1. Values of the
Young's modulus compared with that obtained by Knapp 13
8=444,000 psi!, whose value was used throughout. the tests.
Z%54 P
I ALL II»»RC»i%»O»IS IJIR. IH IIA'CIICSe,g. Uaa1AC CIT»»CATii»A. %PC If»CO» ALL .
II»VC»A>R»»i i» AIIL $,0oh,g IA»ATCI»»AI » TFOC TI»» AL ~ ' '
..C 4'TY RO.O ~ 8 AC@'CS't~ J IAIOICL APE'4 VRC'eillIII4
»
PC SEIELL MOLD FOR N=10
FlGURE A-4.
-12 8-
APPENDIX 8
BUCKLING ANALYSIS DATA
B.l Ex erimental Data
The axial shortening test data are presented in
Table B-l. Because the results are analogous to the gene-
ral stress-strain relations, a=Em, the data have been
presented in similar form, where
F
02xRT
Thus, the equivalent axial stiffness cd be expressed. by
E e 0'
E e,E
The resulting values of E /E are given in Table 1.
8.2 Finite Element Data
Table 8-2 gives the radial displacements of the
nodes located on the lines of symmetry of the finite ele-
ment model shown in Figure 35. The displacements are
plotted in Figure 38. These values were obtained by
transforming the global node displacements of the STRUDL
output.
-131-
TABLE B-l. CONT.}
AXXAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 42: N=6, M 8
-133-
TABLE B- 1. CONT. !
AXIAL SHORTENING TEST DATA SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 53: N=6, M=9
-l34-
TABLE B-1- CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL fj4: N=6i M=10
-135-
TABLE 8- l. CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL $5: N=6, N=ll
-136-
TABLE B-1. CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 46: N=6, M=8
-137-
TABLE B-1. CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 47: N~6, M~9
-138-
TABLE B-1. CONT. !
AXj:AL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 48: N=6, M 10
-139-
TABLE B-l. CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 0 11: N= 10, M~2
TABLE 8- l. CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 413: N=la, M=4
141-
TABLE B-l. CONT. !
AXIAL SHORTEN1NG TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL e4 15: N= 10, M= 5
-142-
TABLE B- 1. CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE l FOR SHELL GEOMETRY!
MODEL 016: N=10, M=5
-143-
TABLE B-1. CONT.!
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODEL 017:N=10, M=6
-144-
TABLE B- 1. CONT. !
AXIAL SHORTENING TEST DATA
SEE TABLE 1 FOR SHELL GEOMETRY!
MODLE 419: N=10, M=7
-145-
TABLE B-l. CONT.}
AXIAL SHORTENING TEST DATA
SEE TABLE l FOR SHELL GEOMETRY!
MODEL 423: N=l0, M 2
-147-
TABLE B-2.
FINITE ELEMENT RADIAL DISPLACEMENTS ALONG
NXD-LENGTH LINE OF SYMMETRY SEE FIGURE 35!
POSITIVE INWARD
-149-
APPENDIX C
STRESS ANALYSIS DATA
C. 1 Experimental Data
Strain gage data are presented in Table C-l.
The surface stresses are defined by
o =Esn n
where
a -surface stressn
E -Young's modulus
c -surface strainn
In the axial direction, the aver'age membrane and bend,ing
stresses are defined as
+!-tension
-!-compression
C. 2! +!-concave inward
-!-concave outward
In the hoop direction, the stresses are defined by C.2!
by replacing subscript a with 8.
-l50-
a.+ai 0
am 2
a. a1 0
ab
~i-inner surfaceo-outer surface
C.2 Numerical Data
Stress data from the finite elment analysis are
presented, in Table C-2. The membrane stresses were ob-
tained directly from the STRUDL output, Appendix D-2.
The bending stresses were obtained from the bending moments
of the STRUDL output by the relation
6Na n n a-axial C 3!nb 3 e-hoop
where N is the average bending moment at, the node. ae'
ve , and ab in the table was calculated from Equationsbe
�. 3!, �. 4!, and �. 5! respectively.
Ul LA R O R 0LA CI Cfl CD P4
LA lA LA LA
0 0 0 0 0 0 0
0 Cl & W e4Cl Ul f4 LO LOCl Ch Ch Ch Ch OL Ch
4rl 0 0 0 0 0 0
0 HUl Ul tfl Ul LA Ul Lfl4 ' ~ ' ~
0 W P4 M ~ LA LO0 0 0 0 0 0 0
Pl Lfl ~ OL W Pl
P4 W P4 LO1
P4 P40 W N A CO LA
I W W f4 Pl P!I I I I I I
a oL a' ~ 1
UlCl Fl 0 W Pl
I W & CA PlI I I I I
Ul Q P4 f4CO W W Ch P LO rV
CglI I I I I
I
0
Ul lfl Lfl Ul Ul Ul 0f4 0 CFi P > CXL tD
Ch 0 EO W f4 0 CO
40 H P4 % N Ih
8
'0
Pt4!
H
M
Q4l
0 C!
I
LO CO W Ul4 e
LO CO > 0 OE LALA W LXL 0
P4 Fl
LA CI lfl lA lA lflFl
tA Ch Pl W W UlP4
I I I I
LO
0 Ul0
YlLO
0
C4
0 H
IO LO CFl CFI W N CDLA LA M LA Ul LOh4 P4 C4 h4 P4 C4
4 00 0 0 0 0 0 0
00 Pl 0 Pl K W C!LA LO
LA0
0 0 0 0 0 0 O
LA LA LA lA LA LA Ul
P4 Pl W LA0 0 O O 0 0 O
Pl LA W Ch W Pl
CO Pl Ch Pl Che
P4 W PlCO Pl Ch LA
P4 t W LO1
Pl P40 C W A CCI LA
I W W h4I I I I I I
Ch IA W W LA
CO W Ol NCQ
I I I H W W P4I I I I
Ul 0 LO Pl IRC4 N t CO 0
IA LA CD Pl LAO Ch CO
I W P4 W Pl ~ LAI I I I I I
Ul 0 0 Ul O LA UlLA Ol
Ul O LA LA Ul O lACh P LA LA W LA lA
CO CD C4I I I I
I I
Ch I LO eP P4 0 00'CI' 'CP W W CP CP
0 A P4 Pl cP LA LO
LA
0
155
IA
0
P4I
IACD
0 a w IAIA IA W W W m
P4 Al hl C4 C44 4
o a a a 0 0
LA W W cP Ch LDm IA
IA IA IA IA IA
0 0 0 0 0 0
IA IA IA IA IA0
CDO 0 O O 0 0
m IA C
m
O Chcn
P4 I CV
mm W ~ t
I I II I I
0 O R CO p4~ f
IA W m CqI P4 M ICI t Ch
I I I I I
co o o m w p4IA Ch W lCI
e4
0 AI N ICI0 m Ul H Ch
II I I I I
0 0 0 0 0 00 ChP4 M & cF
0 0 0 0 0 0m m 0 m n mI P4 W LA I m
I I I I I
Ch W LO ~ P4
0 A P4 H N IA
IA
0
0 m 0 I
m ECI
Ch
IA
0
156
Al
'0
RH
0
CII
ID
Fl
ChF4
LA W V LA CZ! P4 LAPl P! & P!
P4 & P4 P4 P4 P40
0 0 Q 0 0 0 0
ChCh 0 & W Al CO LA
Pl & Pl Fl Pl
0 0 0 0 0 0 0
N LCl LA LA LCl Lfl
AI hl W Lll0 0 0 0 0 0 0
OO W W Ch CO Ch
CPi Ch LA M W LllAl 'LCL W 4! W LLD Ch
P4 M C4
P4 W P4 W W CO
Pl Al N0 W N A OO
hlI I I I I I
Lfl
0 CO0 r4
I II I I
CO K CO K 0 ChYl ELD CO W 'cV 'CI'
I0 W W H Ch
h4I I I I I I
0 0 0 0 0 0 0Ch W 0 m
P4 Pl
0 0 0 0 0 0 0CO W Ch CP W CO ChI A F! LA f CO Ch
I I I I I I
Ch W % N P4 CD ChCP W CCL
40 H P4 Yl CI ~ LA
CZ!
0
157
0
MCQ
l4b
0b
H0
Q
8
b b5
b
LLL
M R
0 O OO Cll CLL Ql LXL Cl CO
0 0 0 O O 0
0 0 0 0 0 0 0
Ch N CDLh O Cll M R Pl LALA LC! LA LO LCL LO
0 0 0 0 0 0 0
LA LA LA LA LA LA CD
O 6 hl K 4 LA0 0 0 0 0 0 O
LA W Ch M C4
O O OO W u!I
0 t LA W COhl Cl' 4 I 0 0
P4 W N LCL W CO' ~
0 I ~ + CO
I I I I I I
0 LA LA LA Ch
Ch H R CO0 ~ m W r
II I I I
Ol LCL W 'LO LA ChP7 LA CO W ~ LCLI I I
I I I
0 m h4
CVI I I I I
0 0 0 0 0 0 0n4 CO r! CLL LCL Pl l
Yl P!I I I I I
0 0 0 0 a 0 0Pl P H LA
hd Pl N LCL W COI I I I I I
Ch P LCL N N 0ct' W M M 'cV W CO
0 H h4
0 Cll0 0
Ch
0 'LO0
'LD 0LA 'LD 00
0 0 0 0I I I I
Pl
0
rh W lIlLll
Pl0
0 O O Q
0 H0 LDIA Lll
00 O 0 0
Lll
Ch H 0
PlI
I I
IllUlI
CDPlI
0
I I I I
ChLA 'LD DlI
I I IM LA I
0
0 0 0 0Pl Co
'LD ChI I I I
0 0 0 0CD
P4 'cV Ul
H0 0 Ch
0
Ch I LD
0 rE P4
8
b
8
8
b
CD
H M
Q
z Z
LD
I I
Lll
LD LDLAI I
CD
0 I
LD
0 0Ul
0 Ch
r I
cPI
LD' ~
LtlCh I
0 I
0 0 LA0 I
Ch
0
lll
LA0
0 I
0Lfl
0 I
ChP4
0
m
LD I
CO
I
h4
I
0'CP
I
LD
0 I
TABLE C-2.
FINITE ELEMENT STRESS DATA
MEMBHANE STRESSES ALONG LINE
A-C SEE FIGURES 43 AND 45!
* 0 = -174 PSXQe
** a = -82 PSXae
-159-
TABLE C-2. CONT.!
FINITE ELEMENT STRESS DATA
MEMBRANE STRESSES ALONG LINE
B-D SEE FIGURES 44 AND 46!
a = -174 PSIOe
+* g = -82 PSIae
-160-
TABLE C-2. CQMT.!
FINITE ELEMENT STRESS DATA
AVERAGE BEMDIMQ STRESSES ALONGLIME A-C SEE FIGURES 47 AND 49}
'eb~'bNODE "a 'eb LB IN IM! PSI!
Ne LB IN IN!
a~ PSI}NUMBER
-116 -0. 102
-67 -0.007
-130 0.089
-89 0.371
Q. 207
0.222
0.128
* ab = 503 PSIbe
1
7
13
19
25
31
37
43
49
55
-0.027
-0.002
0.023
0 ' 097
0.130
0.228
0.263
0.294
0 ' 285
0.205
-0.060
-0.035
-0.067
-0.046
-0 ' 022
0.070
0.137
186
251
439
507
567 .
550
394
-43
135
264
399
427
247
0.499
0.873
1.001
1.128
1.094
0.785
-0.230
-0.134
-0.258
-0.177
-0.086
0.270
0.525
0 ' 794
0.849
0.491
TABLE C-2. CONT.!
FINITE ELEMENT STRESS DATA
AVERAGE BENDING STRESSES ALONG
LINE B-D SEE FIGURES 48 AND 50!
* g = 503 PSI
-162-
APPENDIX D
COMPUTER PRINT-OUTS
D.l STRUDL Buckling Results
The following pages contain the computer out-
put for the finite element buckling analysis.
-163-
164
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APPENDIX D
COMPUTER PRINT-OUTS
D.2 STRUDL Stress Results
The following pages contain the computer out-
put for the finite element stress analysis.
20-
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