• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Nomenclature
 Historical background
 Part 1 -- Experimental Investi...
 Testing program
 Discussion of test results
 Part 2 -- Theoretical investig...
 Analysis
 Application of theoretical...
 Part 3 -- Conclusions
 Reference
 Biographical note
 Copyright














Title: Theoretical and experimental investigation of the buckling of thin cylindrical shells subject to combined torsion and uniform external pressure.
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Title: Theoretical and experimental investigation of the buckling of thin cylindrical shells subject to combined torsion and uniform external pressure.
Series Title: Theoretical and experimental investigation of the buckling of thin cylindrical shells subject to combined torsion and uniform external pressure.
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Creator: Ekstrom, Ralph Edwin,
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
    List of Tables
        Page iv
    List of Figures
        Page v
    Nomenclature
        Page vi
        Page vii
    Historical background
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
    Part 1 -- Experimental Investigation
        Page 7
    Testing program
        Page 8
        Page 9
        Page 10
        Page 11
    Discussion of test results
        Page 12
        Page 13
    Part 2 -- Theoretical investigation
        Page 14
        Page 15
    Analysis
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
    Application of theoretical analysis
        Page 32
        Page 33
    Part 3 -- Conclusions
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
    Reference
        Page 55
        Page 56
    Biographical note
        Page 57
        Page 58
    Copyright
        Copyright
Full Text












- I .- .- T -


DULLIIII5I UI 111111 %,YIIUILu ill Oii~Ibi OUUJCLL

to Combined Torsion and Uniform External Pressure












By

RALPH EDWIN EKSTROM












A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY












UNIVERSITY OF FLORIDA
June, 1960
























ACKNOWLEDGMENTS


The author wishes to express his appreciation to all the

members of his Supervisory Committee. In particular, he wishes to

thank Professor William A. Nash, Chairman, for suggesting the prob-

lem and for guidance and encouragement throughout the course of this

investigation.

He also wishes to thank Professor Herbert A. Meyer, Director

of the Statistical Laboratory, for assistance with the numerical ex-

amples.

Finally he wishes to acknowledge the assistance of Mr. William

S. Goree, Graduate Assistant, Department of Engineering Mechanics, dur-

ing the experimental phase of the investigation.
















ii





TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS .............................................. ii

LIST OF TABLES .............................................. iv

LIST OF FIGURES ................................ .............. v

NOMENCLATURE ................................................. vi

HISTORICAL BACKGROUND ..................................... ... 1

PART I -- EXPERIMENTAL INVESTIGATION................. 7

Chapter

I. TESTING PROGRAM ...................................... 8

1.1 Models
1.2 Testing Machine
1.3 Testing Procedure

II. DISCUSSION OF TEST RESULTS .......... ................. 12

PART II -- THEORETICAL INVESTIGATION ................. 14

III. ANALYSIS .................................... ........ 16

3.1 Basic Relations
3.2 Deflection Pattern and Stress Function
3.3 Periodicity of Circumferential Displacements
3.4 Determination of Deflection Function Parameters

IV. APPLICATION OF THEORETICAL ANALYSIS................... 32

4.1 Method of Solution
4.2 Numerical Example

PART III -- CONCLUSIONS............................... 34

REFERENCES.. .............................................. 55

BIOGRAPHICAL NOTE ............................................ 57
















LIST OF TABLES


TABLE Page

I. DIMENSIONS AND MECHANICAL PROPERTIES
OF MATERIALS USED FOR CYLINDERS..................... 36

II. TEST RESULTS ........................................ 37

III. COMPARISON OF EXPERIMENTAL AND
THEORETICAL BUCKLING LOADS.......................... 42


- iv -














LIST OF FIGURES


Figure Page

1. Coordinates of a point on the middle-surface of
the shell............................................ 43

2. Schematic diagram of testing machine................. 44

3. Testing machine with cylinder in place ............... 45

4. Typical buckle pattern for stainless steel cylinder
under torsional loading.............................. 46

5. Typical buckle pattern for stainless steel
cylinder under hydrostatic loading................... 47

6. Typical buckle pattern for stainless steel cylinder
under combined hydrostatic and torsional loading..... 48

7. Buckling loads for cylinders under torsion alone..... 49

8. Buckling loads for cylinders under hydrostatic
pressure alone................................. ..... 50

9. Buckling loads for cylinders under combined
torsion and hydrostatic pressure for L2/Rt = 4,000... 51

10. Buckling loads for cylinders under combined torsion
and hydrostatic pressure for L2/Rt 12,000.......... 52

11. Buckling loads for cylinders under combined
torsion and hydrostatic pressure for L2/Rt = 17,500.. 53

12. Comparison of experimental load-deflection
relation with classical solution for cylinder
under torsion alone .................................. 54














NOMENCLATURE

3
) Et (flexural rigidity)
12(1 2) 2)

Modulus of elasticity

F Airy stress function

L Shell length

P
--r (critical pressure ratio)
Pcr

R Shell radius


r (--xs)cr (critical shear stress ratio)
Xs cr

al, a2, a3, a4 Deflection function parameters

bl, b2, b3, b4, b5, b6 Stress function parameters

m Number of axial half-waves (m # 1)

n Number of circumferential waves

p Uniform hydrostatic pressure

Pcr Critical value of p for combined loading

P* Critical value of p for hydrostatic loading
alone

t Shell thickness

u, v, w Displacement components of a point in the
middle-surface in the axial, circumferential
and radial directions, respectively

x, s, z Coordinates of a point in the middle-surface
in the axial, circumferential and radial
directions, respectively





1i, L2, Y3 Constants in equilibrium equation

Q Tangent of angle between torsional buckle
and generator of cylinder

xE, Cs, Exs Middle-surface axial, circumferential
and shear strains, respectively

Exs Mean value of shear strain

^ Poisson's ratio

0 6x (7, Us .Middle-surface axial, circumferential
and shear stresses, respectively

Xs ( Mean values of axial, circumferential
and shear stresses, respectively















HISTORICAL BACKGROUND


Interest in shell buckling phenomena dates back to 1858 when

Fairbairn (1)* performed his experiments on the'buckling of cylinders

under uniform external pressure. The first major theoretical treat-

ment of the problem was published by Bryan (2) in 1888 and, since

that time, many investigators have been attracted to the various prob-

lems under the general grouping of thin shell instability. This in-

terest in shell theory in general and shell buckling in particular has

continued and, in fact, increased. In 1959 an international symposium

on the theory of thin shells was held in Delft under the sponsorship

of the International Union of Theoretical and Applied Mechanics and

was attended by 65 invited participants from 14 countries. An exten-

sive survey of shell buckling was given by Fung and Sechler (3) in

1958 and Nash (4) has recently presented a concise summary of current

work. Both papers have comprehensive bibliographies.

Problems in shell buckling are easily formulated, the geometry

involved is relatively simple, since most analyses deal with circular

cylinders or spheres, and experimental data are not difficult to ob-

tain. However, there is frequently a large difference between the

experimental observations and the theoretical predictions. This is

particularly true in the case of axially compressed cylinders where the

buckling loads predicted by the classical theory are roughly three times


*Numbers in parentheses refer to references at the end of the
dissertation.








-2 -


the experimentally determined values. In the case of torsional and

hydrostatic loading these differences, while still significant, are

not as great. Such results indicate that the buckling phenomena are

not fully understood and much of the current work is directed toward

determining more realistic buckling criteria.

Buckling occurs in structural elements having one dimension

much smaller than the others; in shells, for example, where the thick-

ness is small compared with the length and width. Such structures

can buckle under loading such that the internal forces resisting the

external load act along the large dimension; i.e., they are in-plane

forces. The initial deflection is thus in the direction of the large

dimension with no noticeable displacement in the direction of the

small dimension. At the buckling point, two states of strain are pos-

sible, the undeflected shape and a deformed shape. The classical

shell buckling theories leave the magnitude of this deflection unde-

termined, just as in the Euler treatment of column buckling. How-

ever, the more recent theories carry the analysis into the post-

buckling region and show a distinct variation of the post-buckling

deflection with the load. This is an important observation, because

it indicates a decrease in load carrying capacity immediately beyond

the buckling point.

The two most popular approaches to the problem of determining

elastic buckling loads are (a) the equilibrium method and (b) the

energy or variational method. The equilibrium method has been used

in the classical, linearized analyses. It consists of setting up the

equilibrium equations for the shell in terms of the load and the







- 3-


artial derivatives of the displacements, assuming a reasonable deflec

ion pattern which satisfies these equations and, finally, determining

he values of the load for which the shell is in equilibrium with non-

ero deflections. This formulation leads to a system of linear homo-

eneous equations and is recognized as a linear eigenvalue problem.

numerous examples showing the application of this method are found in

imoshenko's book (5). The energy method is ordinarily based on the

principle of minimum potential energy and is the common one for the

arge-deflection treatments. It requires setting up the expression

or the total potential energy---membrane energy, bending energy and



















nonlinear strain-displacement relations and linear stress-strain re-

lations are used. These nonlinearities lead, of course, to non-

linear equilibrium and compatibility equations. It is usually not

possible to solve these partial differential equations exactly and,

in many cases, the difficulties of even a numerical solution are

insurmountable. Thus, although the classical approach may give in-

accurate results, it is frequently the only means for getting any

kind of solution to a particular problem.

Both theories predict the same linear load-deflection re-

lation up to the buckling point. From that point on, they differ;

the classical theory predicting an indeterminate value of deflec-

tion at constant load and the modern predicting a decrease and then

an increase in load with further deflection. Experimental observa-

tions agree in general with the modern theories except that the shell

buckles before the predicted buckling point is reached. Two expla-

ntn hi- h.ni nff.~ar fnr thia. Thp first, introduced bv Donnell


same


___ ______



















proach requires a knowledge of the entire pre-and post-buckling be-

havior of the shell in order to explain the jump phenomenon. An-

other observation by several investigators is that the deformed

shape tends to be a developable surface. This indicates that most

of the strain energy in the undeformed state, which is entirely

membrane energy, appears in the form of bending energy in the de-

formed state. This is reasonable, since the thickness of the shell

is very small and the resistance of the shell to extensional forces

is proportional to the thickness while the bending resistance is

proportional to the cube of the thickness.

Ebel (8) has traced the development of the various treat-

ments for the buckling of a cylindrical shell under axial compression.

The foundations of each method are examined and particular attention

is paid to the types of approximate solutions for the resulting equa-

tions.

The majority of the investigations of shell buckling have

considered only one type of loading--axial, torsional, hydrostatic

or bending. Numerous small deflection analyses consider the effect

of two or more types of loadings and most are based either on the

set of equilibrium equations given by Timoshenko (5) or on Donnell's

eighth-order linear equilibrium equation (9). Since shell buckling

theory is widely applied in missile and aircraft design, several in-

vestigators have considered the combined effect of internal pressure







- 6 -


and some other type of loading. Internal pressurization has marked

stiffening effect on a shell and this is an important factor in de-

signing for minimum weight. With the increasing importance of sub-

marines and the interest in optimum design, the problems of uniform

external radial loading or of hydrostatic loading become important.

The higher speeds and greater maneuverability of modern submarines

subject them to more severe loadings than formerly. The present

investigation is a contribution to this area and is believed to be

the first large deflection analysis of the buckling of a cylindrical

shell under any type of combined loading.















PART I


EXPERIMENTAL INVESTIGATION



Fung and Sechler (3) have indicated the areas for which no

work on combined loading of shells was found up to 1958. Cases which

are of practical interest and which include uniform external pressure

are (a) hydrostatic pressure plus bending, (b) hydrostatic pressure

plus torsion and (c) hydrostatic pressure plus bending and torsion.

Since that time the results of tests conducted at North American

Aviation Company on circular cylinders under various combinations of

loads have been published. One group of tests (10) covers the case

of hydrostatic pressure plus torsion, but results are given for only

one geometrical configuration. Therefore, as background for the

theoretical investigation of Part II, it was necessary to conduct a

series of tests in order to have experimental data for comparison

purposes and also to have a basis for formulating a reasonable de-

flection pattern.















CHAPTER I


TESTING PROGRAM



1.1 Models

The models used in this series of tests were unstiffened cy-

lindrical shells constructed from stainless steel and aluminum strip.

Three lots of Type 301 stainless steel strip were available; -hard,

0.008" thick and full-hard, 0.006" and 0.0075" thick. The aluminum

was Type 3003-H19, 0.008" thick. Measurements taken prior to forming

the cylinders showed that there was no significant variation in the

thickness of the material. The material properties are shown in Ta-

ble I.

All of the models were made by rolling the metal strip a-

round a thick-walled steel tube and joining the ends with a 3/4" lap





constructedd from structural steel. The machine will accommodate cyl-

inders of 4", 6", 8" and 10" inside diameter of lengths up to 48". A

schematic drawing of the machine is shown in Fig. 2 and Fig. 3 shows

the machine with a cylinder in place ready for testing.

The cylinder is held in the machine by two end flanges made

of 2" lengths of the same thick-walled tubing used to form the cyl-

inders. This tubing was welded to 3/8" circular steel plates and

each welded joint was carefully machined so that the end of the cyl-

inder would bear uniformly on the plate during the application of

axial loads. Steel bands, as seen in Fig. 3, clamp the cylinder to

:hese end flanges. For tests involving hydrostatic pressure, a

commercial joint sealing compound, Gasoila, was found to be satisfac-

tory for the moderate pressures involved. This method of holding

the cylinders approximates the condition of clamped ends.

Axial loading is applied by means of a worm gear jack at the

base of the machine. This jack has a capacity of 5 tons and a gear

ratio giving one inch of travel for 64 turns of the handwheel.

Strain gage readings from the compression dynamometer just above

the jack indicate the axial load. The thrust bearing just above the

dynamometer allows the cylinder to twist while under combined axial

and torsional loading. Torsional loading is applied by means of a

lead screw which pulls the cable wrapped around the movable base plate

to which the lower end flange is fastened. The magnitude of this

FnrreP in mPasurPd hv a tpnion dvnamnmPtpr hoetwon tho lnad neraw snd








- 10 -


the cable. A dial indicator mounted on the machine frame measures

the total angle of twist of the cylinder. Since the required hydro-

static pressures are less than one atmosphere, uniform external pres-

sure is applied by partially evacuating the air from the cylinder.

A manifold mounted on the frame of the machine is used to control the

amount of vacuum and the difference between the internal and the ex-

ternal pressures is shown by a mercury manometer. The weight of the

lower end flange and base plate is balanced by an axial load applied

with the jack.


1.3 Testing Procedure

To insure a uniform state of loading on the cylinders, a

level and a dial indicator were used to align the end flanges after


















(a) Torsional load applied until buckling occurred.

(b) One-fourth of critical load from (a) applied
and then hydrostatic pressure applied until buckling occurred.

(c) Step (b) repeated for one-half and three-fourths
of critical load from (a).

(d) Hydrostatic pressure applied until cylinder
buckled.

In many cases, including all those for hydrostatic pressure

alone, the "snap-thru" type of buckling occurred and, hence, the

buckling point was easily determined. In all the tests where tor-

sion was involved a sudden, definite drop in the torsional load was

observed at the buckling point, even for those cylinders which

buckled gradually. The buckle pattern formed across the seam with

no distortion in shape and none of the seams failed during buckling

or in the early stages of the post-buckling region.















CHAPTER II


DISCUSSION OF TEST RESULTS



The experimental values of buckling loads, number of lobes

and angle of the lobes are tabulated in Table II and non-dimensional

plots of these loads are given in Fig. 7 through Fig. 11. The buck-

ling loads under torsion alone are shown in Fig. 7 and, for compar-

ison purposes, the theoretical solution of Batdorf (11) is also plot-


















these cases, the relation was similar in form to that shown in Fig.

Accurate measurements of the angle of the lobes was difficult

but it is reasonable to assume a linear variation of the angle with

the torsional parameter T.

In all the tests under hydrostatic pressure alone, the buck-

ling pattern formed completely around the cylinder. In some of the

other cases, the entire pattern did not appear initially and it was

necessary to estimate the number of lobes.

Figures 4 through 6 are typical buckling patterns for one

particular cylinder.
















DADT TT


THE



As was pointe


ICAL INVESTIGATION



t in the introduction, much of the rec


loads has been directed toward the cases of internal pressure plus

axial compression and internal pressure plus torsion. However, some

types of loadings related to the present investigation have been con-

sidered. In 1935, Imperial (12) considered the case of combined


envalue approach outlined by Timoshenko (5) and I


:ne comDinea errecc or axiai compression anUa Lrslon Dy slmiiar metnoal

Suer and Harris (10) applied Donnell's eighth-order linear equilibrium

equation to the problem of the stability of cylinders under combined

:orsion and hydrostatic pressure. They obtained a solution by Galer-

cin's method based on an assumed radial deflection function in the

form of an infinite trigonometric series.

The present investigation is based on the usual shell theory

second-order strain-displacement relations and includes the effect of

Initial imperfections by using the hypothesis advanced by Donnell (7)

:hat the shape of these imperfections is the same as the buckled form

)f the cylinder. The assumed deflection function is based on experi-


I ADq TT








- 15 -


y Galerkin's method and, therefore, the condition of compatibility

f strains is satisfied in the least squares sense.

One of the deflection coefficients, a4, is determined from

he condition of periodicity of the circumferential displacements and

he remaining three from the equilibrium equation in the radial direc-

ion. Again, Galerkin's method is applied to determine average values

If these three coefficients in terms of the applied load.

Finally, a numerical example is worked out and compared with

he experimental data.





CHAPTER III


ANALYSIS



3.1 Basic Relations

This analysis is based on the finite deflection strain-dis-

placement relations introduced by Donnell (7) and the usual form of

Hooke's law.

The strain-displacement relations, derived on the basis of

Donnell's simplifying assumptions, in the middle-surface of a thin-

walled cylindrical shell of radius R are:


ex 'r z ^
, = 3 --L[^l
tr K(D W\
5 z .5 3) I d

da- ` + < + K( `ewv d
6.^ J x d" 76-

where 6x,
strains, respectively, and u, v, and w are the displacements of a

point in the axial, circumferential and radial directions respec-

tively. The imperfection factor, K, is defined as:

K = + 2wo = constant, [2]
w
where wo represents the initial radial deviation from perfect shape.
This implies that the significant component of initial deflection is

the same shape as the final displacement. The coordinate directions







- 17 -


The middle-surface membrane stresses are derived from Equa-

tion [i] and Hooke's law as:


(TI
-^-(

ys -~l^)f


ax a sx R

+ K +3 3^ K
DX as a x -S


where Ox and 7s are positive in tension, E is the modulus of elastic-

ity and ) is Poisson's ratio.

Chien (14) gives a generalized equilibrium equation in terms

of membrane stress resultants which for the present case reduces to:


DV4w N = aVg-, -&! 4ISf
D )2-jx,),5 06s2


where N,, Ns and Nx are the membrane stress resultants, p is the ex-
Et3 2
ternal radial pressure, D = -)is the flexural rigidity andV

is the Laplace operator,

By defining the Airy stress function as:


X )^- ) ~ J_ [5]


Equation [4] becomes:


t Tx v A25 '9 .5a [6]t


Donnell also obtained the following compatibility relation

in terms of the Airy stress function and the radial deflection:














V = K )s- A
L ),-CD-( 2') Y hj ) R [7]


For any assumed form of w, F may be found from Equation [7]

as the sum of a homogeneous solution of this equation and a particu-

lar solution. For the case of combined hydrostatic pressure and tor-

sion, the homogeneous solution is:

F = b x2+ b xs + b x2 [8]

where:

2bi = the mean circumferential stress,

-b2 = xs the mean shearing stress [91

2b3 = Fx the mean axial stress.


3.2 Deflection Pattern and Stress Function

A pattern for the radial deflection, consistent with the ex-

perimental observations presented in Part I, is:


w = aisin mr-x consn + a2 sin n(s + x) sin
L R R L


+ a3 ( cos 2rx ) + a4


where al, a2, a3, a4 and are parameters, m is the number of

axial half-waves and n the number of circumferential waves. The

first term on the right corresponds to the classical axial compres-

sion pattern, the second to the large deflection torsion pattern (15)

and the third accounts for the axially symmetric deflection under

uniform radial loading. The last term, a4, is included to allow






- 19 -


uniform radial deflection prior to buckling. This deflection function

does not satisfy the clamped end condition, since w = a4 at x = o and

x = L and 7 0 at x = 0 and x = L. However, the effect of the
end conditions is rapidly damped out in a narrow region around the end

and the basic deflection pattern l is thus the dominant one. This

problem, the effect of edge conditions, is discussed at length by

Goldenveiser (16).

A stress function satisfying the compatibility relation [7]

and in agreement with the deflection function was found to contain 38

terms. An analysis based on this stress function would be almost im-

possible to carry out and, therefore, the following approximate stresE

function was assumed:

F = blx2 + b2xs + b3s2 + b4 cos 2f x + b5 sin mLx cos ns
1 L L R

[11
+ b sin n(s +(x) sin TX
R L

where the coefficients b4, b5, and b6 are found from the compatibility

relation [7] to be:


b K E2L n- 4 f' (L
b4 =i" 8 4. 1TR 3




2,-
v IT-' 2 MI.TrL ,r ~ r -















pF L R


4 it L Z 2 -1=Z.
fr + r -
^ZeU ^R L PL- (Z_^
L R [(


.3 Periodicity of Circumfer

Physical considerate

umferential displacement, v


ions of


ential Displacements

Lons lead to the conclusion

', must be a periodic fun(
r ,


x alone. From equations L1J and L~J ,I -s


ition of [10


'b -4,-
-


I and [:


:2 0-3 pc o


-... ..-------. -4 --- -- 1 -
:o zero yields the following i

)e a periodic function of s:

2C kb, K
&4= 3 +1 Z

Mhis shows that a4 may be expi
.4^


1] in [13] gi


L /


..- -3 --- L- J ---- L J
:elation which is the condition t



Z z Z/Px
SCLI C95 -


ressed in terms of the other coef
r 1


-EK kLK


[14]


:hat v


_ Ir _


L r. L RI












4- 3 b,
0 4 = F


LUuV.J L fc^JL.l

ress functi,


4 Determine

The tl


i condition could be obtained by c,

Mn.


:ion of Deflection Function Parame

iree remaining deflection paramete

;he equilibrium equation [61 is sa


TA F cx C s n, D i bY. I Z- br '
I + *i^\aL\j

Sin ____Xn)SSI Lx aC + b
/ k x f2 '3 3 + (


L [t j RL R^ 3 I2L -


aftn Z -- In Z5 i
TV\ r -5 2



^~~~- z ^\\^


- ti -


<.L-/^- ;.; t







- 22 -


- Jrn Cos os
L Le


2), (Stdr)
L C v




dos 2L.
fw COS


)1Z 7


8y- ,Z
b63


c b^j


[17]


/2 > ,> /' 9
Selz^


+ ,; co; L c L In Csl'n


*>7 -n(S ,L r')
eo4
R R


-__
t


/2


- 0


where:


r, '' .^



=^4 4-
^ ^ - i-


1- /4-




12.
~7T"


)13 [4 -
R-3/6
ni^


P -
/02 z,-


2k----- b +
nj3 -


.Z z fr .


,z m A2,r,, 1
,eF z /


+ 4-



- n^_a
e t/


,4-


2 h2 d
~" d'


Wm /;Ly -n
C, d s-Z--i


C


3= 4


-Z /













sidual when approximate values of a,, a2 and a3 are substituted ir

S. Galerkin's method will be used to determine weighted average

Lues of these parameters so that H is a minimum and will be appliE

the following form:

S10-0 I


0 0
Ls results in three simultaneous nonlinear algebra


F4m'trLDo b 4h'- L- L~ C Z 4 zt
t L2- LZ fz 2. 3
ItL2 1 s a; y L^-ia? b5- ^ 2L j-*

LL f3 --

-1
Ll- ki _- i I -


, r r.


nL r/6 iJ p


+ Nt -Lt-- I I- l -Z 5 1


LJ


+ +
X [K~rr 2Li






- 24 -


L L' (l I


1 -^17^2-k 1- 1. g+
0- Z 61CC



4 Ci r -Z R
*1Z/ p L L1


5{ThL~) { E4t2



.34 n~ l2 ? t /4 .


I 1RrA4 F~Yji~ hL '4f. -~


O

I R-I- 61


,- J J


' / -C L


S I L J


L


'-N


I,


I






- 25 -


L ~ ___ 14 nLr
3 1~ h3


t4 r^^^ji 1 *Y)
7 u-- I-2 -- -S -- 1 7 2? -


L2- +




rl4- L Yr"1


4- 4-L (t L L 2 --


,a r0- [

.tS 2v\-Y~ b 1


- 'l~F FLK T-p. nx 2"!


A. -1


- r r)-L 2- 0,, Or ,


L > 't \( ELK ly ^rt- r
1,&- c f
t 1^3 -Z ^


r-.^ib 'iX^8;^-t
r n I 'L .


- L ,


4- k',n4- 6f^
4-S a-


















.Z. k:: % 7t +"{" '" ZL--


iZ7C8,4 ,2- z- LL
+7- /- 4- L2- [, 2_ E '- j,




4-[ J tZL -i 1 d dj^^





] L Z n '-L^ "
p>+ 4--t L4 .4-


Q-Lr +4- 1k-k






- .1 -


u L 3 L - :3


4 O4 L" { L L4-

3 I 24z 3 L4- o-


L 4 L Lt L4- LL- 3


+ \-4-






4- L RC aL t L




2 ar a, L4 R L1--b 23 5
















- 4. -. t-1


-- -- ---s - - - --
2
valid for 200 '-L- = 20,000, has been derived by Donnell (9):
Rt

/7" [20]


,is may be modified to apply to the case of combined loading by assuming

,at the angle varies linearly with T, where T is the ratio of the crit-

al torsional load under combined loading to the critical torsional

,ad under torsion alone:


T )c [21]


mus, since the angle is small, it may be assumed that o also varies

,nearly with T and, therefore, in all cases 0 is given by:

y= -T /. 73Z r?


ite that 0=0 for hydrostatic pressure alone and for this type of

lading equation[19] becomes:



-__ 4_ r, I.'FK -2- 84- br 4






- 29 -


ILK FkR3
-4 R3


i|, E-K 'i,'",
+ b-
i23 1


E;'K n
- 4
,i -


1--'


F Z nKrL
j ^V


,EZC~t W-Bn'Clt
I^bL" 1 ^


a2. L,4-
e E-k2 -p_4
It1 ( 14


ILI


-4

LLL4 I1 4 {L +LL


'4


Cir~L
41


:L L e-
+ + I


4- I ,
4- L2- { I


/0 7 Z.)
^IL^S3 ,LL ~Z


/Z7 z. 8 L1- -+


-t- z 8 + 1 -


to
L -
4" ELT
2 a4 2 + 4-


S*- 3b


5 7~'LI-Z%4-h


1-%O


z/ L 4 7/-
+ Zc6 L
12119 f P


---_
e L


[23a]


+ 84


+ (1_ 5


I [ I ( C "z i rN 4 4 -
, -- ^ -T 5 --- t J
+ 3 J


i F~ir~nZ~LC








* fg ^ft- r ^ l- f ? b, ^A- I n n-

t_ I t 8 -
t ~z^- 4
r,[ .L rZ Lb^'^ II^ 2- 125~v~r CLZ ^






-31-









6Z z-


E AC, 2 2. Tr L L + -Z]h
-L2 L-4 F




P- IL 2 1Z1







k3 2 {2 '23c]
;2 -2 4 z1_








4LA4
___ _s icZ~C~Wr k4T 3L-cIEt~~ ~ 7*~Cr -
oz12Z t k46










+cz 2iL- L2- 35
If /,? I k4- y





4-22-

+ 6 L I 6 t + 4 _
z R~'84 f8 nL I P r'L7I8Z



+o z'i. 6!K~ Q -5














CHAPTER IV


APPLICATION OF THEORETICAL ANALYSIS


4.1 Method of Solution

The terms in equations [19a] [19b] and [19c] involve products

of the three deflection coefficients. However, each equation is quad-

ratic in al, a2 and a3. In order that real solutions exist for each of

these nine quadratic equations, the discriminant of each must be great-

er than or equal to zero and, for any particular pair of values of n











erence is small, it may be reasonably assumed that n


"77t [24]

and, since n must be an integer, the nearest integer to the value

given by [24] will be used.


4.2 Numerical Example

The theoretical solution of Chapter III is, in a sense, a set

of characteristic equations. As a test of the validity of this solu-


tions were computed and compared with the experimental values for a

cylinder having the same dimensions. These experimental values were

taken from the data for Cylinder A as given in Table II. This di-


































































the least values of the loading parameters for which the deflec-

" r"Vmntf-= w I ua-1a TI- tl wa nnrfCf4hP n datP1rminp tIho InnAAi
















LLLiij.LLL=CXiL CluCiL-uio.


ts compared v


same geometry


cling load












TABLE II


TEST RESULTS


CD
--4
m
Go ca
S4. 0J
i00 d
C4 uS.-k

4 44. E- 4r
o^ U U
ca Q, tO ^= CM*
*-^X e Xa4- > m' -
W~~L vX *r l liI <^ e
*Or '^' b ^-/ '^ F 3>l0 'L
0~ C~ -^ U U'^ l, a^v~


A 3 19.85 0.0075 17,500 400 4,800
3,820
2,420
1,210


B 3 17.75 0.006


C 2 16.2


17,500 500 4,400 0
2,260 0.46
1,130 0.66
0 0.86


0.0075 17,500 267 8,300
6,850
6,570
5,840
5,370
4,770
3,520
0


0
0.69
0.98
1.13
1.22


6
5
5
5
5


1,260






1,460




1,455


0.75
0.50
0.25
0

1
0.51
0.26
0

1
0.82
0.79
0.70
0.64
0.57
0.42
0


0
0.56
0.80
0.92
1

0
0.54
0.77
1

0
0.15
0.31
0.46
0.62
0.77
0.94
1


687
484
332
196
0

659
338
186
0


--1,130
--1,040
-- 987
-- 888
-- 777
-- 679
-- 370
148 0


0
1
2
3
4
5
6.1
6.5










TABLE II--Continued


1.4 0
'S 4-1 0k *
4 J M -

-4 I -. U^
ou u
,4 C: 41 co
--A
Sr C.1- .M-^ 1-JC
0 -4 -4 -4 x

z 75 14 4 I
;L oi v a a vo '- ^f.^ pi ~ ;


D 2 14.49 0.006 17,500 333 6,790 0 1 0
0 1.88 0 1

E 3 18.97 0.01 12,000 300 2,820 0 1 0


F 2 13.43 0.0075 12,000 267 8,550
8,360
7,760
7,450
6,240
5,360
4,760
3,580
0


0
0.49
0.98
1.47
1.97
2.45
2.94
3.43
3.65


1
0.98
0.91
0.87
0.74
0.63
0.56
0.42
0


0
0.13
0.27
0.40
0.54
0.67
0.80
0.94
1


G 2 12.00 0.006 12,000 333 6,820 0 1
0 2.74 0


1,490


1,110

1,030










1,030


-- 759 14 4
137 0 0 4

-- 910 10 6

-- 1,040 9 5
996 9 5
937 9 5
848 8 5
788 6 5
625 5 5
551 5 5
402 4 5


117


0 0 5


- 665
112 0










TABLE II--Continued


"4
-r44
c0
*
-4-4
"4a)

0. 0 0U

0J U u

4 .. 0. 4E-4P
i C4 v X 0 4J
'OC '-' m v J1 ~
0 ~ < N ~ U5 S C U K )
g ~ i fl 4J.R 4 < '' ____ < Q________< C


H 4 10.97 0.0075 4,000 534


5,170 0 1 0
4,330 0.24 0.86 0.17
4,170 0.49 0.82 0.33
3,390 0.74 0.67 0.50
2,780 0.98 0.55 0.67
1,850 1.22 0.37 0.83
0 1.47 0 1


I 2 8.00 0.008


4,000 250 10,700 0 1 0
0 6.18 0 1


J 2 7.75 0.0075 4,000 267 10,200 0 1 0
9,550 0.98 0.94 0.16
8,350 1.96 0.82 0.31
7,780 2.45 0.77 0.39
0 6.37 0 1


1,430 11 6
0 0 5


400


409


1,540
1,210
950
825
0


10 6
8 6
8 6
7 6
0 6


1,000
856
803
656
656
347
0


9 9
7 8
7 8
6 8
5 8
4 8
0 8










TABLE II--Continued


C
41 0
00 3
Al $4 94 bp
$4 Li1 U" '-'
.-4 .. Cd >
4J Car
*J '- *.! 4 OJ' O ''- MC
o C'5 -. US,K..^ ,^S f
4--- ) .- ^S ^,
g p{ r-l AJ _] (4st P~i v- (^e Q | < c


K 2 7.75 0.0075 4,000 267 9,850
9,250
8,650
8,240
7,500
7,130
6,650
0


L 2 6.93 0.006
q


4,000 333


M 2 6.93 0.006 4,000 333


7,410
6,950
6,220
0

7,470
5,080
0


338


0
0.98
1.47
1.96
2.45
2.94
3.42
6.12

0
0.42
1.56
8.40

0
2.10
4.01


1
0.94
0.88
0.84
0.77
0.72
0.68
0

1
0.94
0.84
0

1
0.68
0


0
0.16
0.24
0.32
0.40
0.48
0.56
1

0
0.10
0.38
1

0
0.52
1


1,210
929
878
825
723
671
620
0

1,010
677
534
0

1,010
290

























Vacuum i Mercury
lip Manometer





ige -..



Lead \
r --- I Screw
-Cable /


e Dynamometer
,/Torsion
Dynamometer


rings Axial Load
r-Jc---k Dynamometer
rm Gear Jack --* 9 1


I


I































AM


U


l


m
























































L2
R = 3", L 1985"11, t = 00075", = 17,500, 1 400,
Rt t


= 4920 psi
xs




Fig. 4.--Typical buckle pattern for stainless steel cylinder
under torsional loading.








































































R = 3", L = 19.85", t = 0.0071


= 17 ';O R = 400. D


Fig. 5.--Typical buckle pattern for stainless steel cylinder
_--J - L J - 1 JJ~


Sit



















































R = 3", L = 19.85", t = 0.0075",

2 R
L 17,500, = 400, T = 2840 psi, p = 1.4 psi
Rt t xs




Fig. 60--Typical buckle pattern for stainless steel cylinder
under combined hydrostatic and torsional loading.


















10


















10____2
1/









1 10 102 103



Rt
Fig. 7.--Buckling loads for cylinders under torsion alone






















102

*P S
-S

PcrL2
DTrz

10 _

Batdorf (11) -







1 10 102 103 1C


Rt
Fig. 8,--Buckling loads for cylinders under hydrostatic pressure







- 51 -


X P + T 1

\


N



0.50 --,7
00

S\ A


0.25





0
0 0.25 0.50 0.75 1
T

Cylinder R/t
V H 533
A L 333
0 M 333
0 J 267
SK 267


Fig. 9.--Buckling loads for cylinders under combined torsion
and hydrostatic pressure for L2/Rt = 4,000.
































0,50 ___
\





\


\
0.25




0\

0 0.25 0.50 0.75
T


,ll







- 53 -


P + T2 2 1



0,75
\




P\
0-50---


\\

\25

a




0 0.25 0.50 0.75 1
T

Cylinder R/t
A A 400
C 267







Fig. 11.--Buckling loads for cylinders ufder combined torsion
and hydrostatic pressure for L /Rt = 17,500.






- 54 -


6,000


Classical

5,000




4,000


G /

3,000 ..




2,000
/.
//

/-
1,000




0 L ____ ---------
0 2,000 4,000 6,000

Cylinder A: R = 3", L = 19.85", t = 0.0075"


Buckling loads:


8,000

s\tl


Classical 8,100 psi
Experimental 4,920 psi


Fig. 12.--Comparison of experimental load-deflection relation with
classical solution for cylinder under torsion alone.















REFERENCES


I. Fairbairn, W., "On the Resistance of Tubes to Collapse," Philo-
sophical Transactions of the Royal Society, vol. 148, 1859,
pp. 389-413.

. Bryan, G. H., "On the Stability of Elastic Systems," Proceedings
of the Cambridge Philosophical Society, vol. 6, Part 4,
1888, pp. 199-211.

3. Fung, Y. C., and Sechler, E. E., "Instability of Thin Elastic
Shells," Structural Mechanics, Proceedings of the First
Symposium on Naval Structural Mechanics, August 11-14,
1958. New York: Pergamon Press, 1960, pp. 115-168.

i. Nash, W. A., "Recent Advances in the Buckling of Thin Shells,"
Applied Mechanics Reviews, vol. 13, no. 3, 1960, pp. 161-164

i. Timoshenko, S. P., Theory of Elastic Stability. New York:
McGraw-Hill Book Company, 1936.

'. Karman, T. von, and Tsien, H. S., "The Buckling of Thin Cylindri.
cal Shells under Axial Compression," Journal of the Aero-
nautical Sciences, vol. 8, no. 8, 1941, pp. 302-312.

. Donnell, L. H., "A New Theory for the Buckling of Thin Cylinders
under Axial Compression and Bending," Transactions of the
American Society of Mechanical Engineers, vol. 56, no. 11,
1934, pp. 795-806.

Ebel, H., "Das Beulen eines Kreiszylinders unter axialen Druck
nach der nichtlinearen Stabilitats-theorie," Der Stahlbau,
vol. 27, no. 2, 1958, pp. 45-53.

. Donnell, L. Ho, "Stability of Thin-Walled Tubes under Torsion,"
National Advisory Committee for Aeronautics, Technical
Report 479, 1933.

i. Suer, H. S., and Harris, L. A., "The Stability of Thin-Walled
Cylinders under Combined Torsion and External Lateral or
Hydrostatic Pressure," Journal of Applied Mechanics, vol. 81,
Series E, no. 1, 1959, pp. 138-139.


C C




















iurobLul. ULLpuviJLieutU rlL.IU. ULOOCLL.L.A.JLL, uILV.V-J.y
California, Berkeley, 1935.

* Lipp, J. E., "Strength of Thin-Walled Cylinders Subjected to C
bined Compression and Torsion." Unpublished Ph.D. disserta
tion, California Institute of Technology, 1935.

. Chien, W. Z., "The Intrinsic Theory of Thin Shells and Plates,
Part Ill--Application to Thin Shells, Quarterly of Applied
-- . -, -' 1f/ --- 1_ 'IA 1In


__ ~1



















Iowa. His undergraduate work was done at the University of Minnesota

and the University of Missouri (BS, 1948). He has master's degrees

from the University of Missouri (Mathematics, 1949), Purdue Universit

(Engineering Sciences, 1955) and the Massachusetts Institute of Tech-

nology (Armament Engineering, 1956). Since September, 1957, he has

been enrolled in the University of Florida working toward the Doctor

of Philosophy degree in Engineering Mechanics.

While a graduate student at the University of Missouri, the

author was a teaching assistant in the Department of Mathematics. HC

was an Instructor in Mathematics at Westminster College, Fulton,

Missouri, from September, 1949 until August, 1951. He was then em-

ployed as a Supervisory Mathematician at the U. S. Naval Avionics

Facility, Indianapolis, Indiana until August, 1957. He has been em-

ployed by the Department of Engineering Mechanics at the University

of Florida as a Research Associate from September, 1957, until the

present time. He also served as an Instructor in Mathematics at the

Indianapolis Extension Center of Purdue University from September,
















proved by all members of that committee. It has been submitted to th

Dean of the College of Engineering and to the Graduate Council, and

was approved as partial fulfillment of the requirements for the degre

of Doctor of Philosophy.



June 6, 1960



ea, College of Engineeri



Dean, Graduate School



SUPERVISORY COMMITTEE:



Chairman



S/t6y-z&PVL










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AUTHOR: Ekstrom, Ralph
TITLE: Theoretical and experimental investigation of the buckling of thin
cylindrical shells subject to combined torsion and uniform external
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