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-

O A

a

0 u RQ

-66-

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-

H ~

W 0R

CD

C4

O

R &4

8 M

O R 0

M C"

CD

CD

FINISHED PC SHZLL MOLDS FOR N=6 AND N=10

FIGURE 6.

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-

METHOD OF SEALING EDGES WITH DQUBX ER STRIPS

FIGURE 9.

-73-

PRESS FXXTURE FOR BONDXNG OF DOUBLER STRXPS

FIGURE 10.

a! EDGES SEALED WITH ALUNIMUN TAPE

b! RUBBER EDGE CLAMPS IN PLACE

CLAMPING OP PC SHELL TO END PLATES

FIGURE 11.

COMPLETED MODELS FOR N=6 AND N=l0

FIGURE l2.

O C! O

I

C!

I

N 0 I ld I A 3 G 3 O0'I N 333d

5

0 0 0 R O H H CI-77-

50

5

R0H

Q R UO

l

O 0K

ll

~ a

li

5 CI

R O Q

-79-

II

F4

4 0 F

4

W M

W U

-80-

rvlII

4 0

II

R 0g 00

~ 5U UH HR 0 V 4 U

C!

II

O O O

i

A

A

-82-

li

M W a 0

~ ~

CD II

R 0 A 0 U H R OIO

A

ss

4 0 R ll4

-83-

ll

A

R

P7

Il

A O X

II

k llZ

~ ~C7

1I

R IX0 4 0i8M

R 0 A

ID

C U

R

R

R 0 H A

Io

O

I

C4I

O

I

~ ggO W

Q

o.BnO

0

O W AW

M

Pl

fv

I I I I I

RH

O &I4

4 0 4O

Q UlO 41

B 0O B

A W Q

5 A I I I I I.I Q g H O 0 0 A

-87-

O O88

C!O

C!

O ' O

g Ol>3/ d

O O

Ol,x2/ d

COo~O

o~C! ~

C!

o~0

89

O C4 OO

C!C4

C!O

OLx 3/ d

OLx 3/ d

B CQ

H g

CR CIQ 0

MM C4

HC9 P

D 4

H Q4 CJU f4

O

90

a PCO

QC4

+F4C4

0 UH H

O C4 H C4O

CDO

O O Q O O 0 0 0 Q O QQ 0 do N 4 vj rg Pl ~ Q

lA!g !dd

o O 91

O

0

O

00

C!O

O

OW

QH

H

M

OH

C4

92

OO

Q cQOPH gB AOO

4 f4Q ~

0 g'QQUH

CQ g

ii

O OI

4

93OO

O

O

O

CD

OO

0 co 4 cr W O0 0 0 O

0 0 0 0 0 0 0 0 0 0

g

i Q

oa

W ~

p"'ÃW fxlQH MU

O O

0 C0

VR0

~ s

OC4

C4

C!OW

yLU

-9 4-

C!o

~ a0

R

oUI UH

I

C 0

-95-

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-

A

O

O

II II

CQZs S4H

g 0H 0

00

o~ CO ~

II II II II

W O Q R

M I4l4

X 0 A O V R Cx1O

-99-

4 O Z U O OPl

U H

-100-

5

-10 1-

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-

O A I4

0 O O a0

O

0nl

~ g! e

�

O O

-104-

K K KH H H

II II II II II II

t2 ' Q & i4

OA

tQ

H

KH

D

ELE21ENT NUifBERING OF FINITE ELEMZNT STRESS MODEL

SEE FIGURE 4 0 !

FIGURE 42.

-105-

NODE NUiilBERING FOR FINITE ELEMENT STRESS MODEL

SEE FIGURE 40!

F I GURE 4 l.

NECSRAitE 2;OOP STRESS RATIONS a /ae s ALONG LlNE A-C-ee'

FIGURE 03 ~

-10 7-

MKIBRANE HOOP STRESS RATIO g g /g ~ ALONC LIRE B D ~em ee'

FIGURE .44..

-108-

COMPRESSION

MEMBRANE AXIAL STRESS RATIO, o /a, ALONG LINE B-D.

FIGURE 06.

-110»

BENDINC HOOP STRESS RATIO, a fa, ALONG LINE A-C ~8b be'

FIGURE 47.

BLINDING HOOP STRESS RATIO, ~eh ~b ' ALONG LINE B-D ~PIGURE 48.

-112-

BENDING AXIAL ST",:""S P,;"TIO, a b/ab, ALONG LINE A-C.ab be'

O'IG"P.E 49 .

-113-

BENDING AXIAL STRESS RATIO, a b/ab ALONG LINE B-D.ab be'

FIGURE $0 .

-114-

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

TABLE 4 .

BUCKLING WAVE NUMBERS FOR THE PC SHELL

N=10! AND EQUIVALENT CIRCULAR CYLINDER

-118-

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.

CD

Pl

CDCDPl

IICD g

P4 g40 ~Wl W

Icn gklQ Ll

CD V U

4

CDCD +

NI! H HO ISd! d

-12 4-

NI! 8 HO IS d!

-125-

4. ~»

C 4 V~ eg

'4

4 N

ll

R 0 ~g P!

Ia4

P Up HI4 W

CQ

O

4ll4

iR0

cg a5

o8>III

vs $

A

RNID~acti<

4st

LO QII GR U

C4Q ~fv M

IAC

AB4 H

CQ

-127-

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-

PC SHELL MOLD FOR N=10

FIGURE A-4. CONT.!

-129-

TABLE A-l.

TENSILE TEST DATA

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.

AXIAL SHORTENING TEST DATA{SEE TABLE 1 FOR SHELL GEOMETRY!

MODEL 41: M=6~ M=6

-132-

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 1 FOR SHELL GEONETRY!

NODEL 022: N=l0, N=S

TABLE B-l. CONT.}

AXIAL SHORTENING TEST DATA

SEE TABLE l FOR SHELL GEOMETRY!

MODEL 423: N=l0, M 2

-147-

TABLE B- 1. CONT. !

AXIAL SHORTENING TEST DATA

SEE TABLE 1 FOR SHELL GEONETRY!

NODEL 424: N~10, N=5

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

154

& m

0

I MO

I

4

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

C n.4

4r44

4ICI

4 It.4144

C~ II. V.

v4 ~44444

4

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t4444

4 r4

i

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r 4

4 4 4 4 4IC tr r

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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.

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