 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 middlesurface 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 loaddeflection
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 halfwaves (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
middlesurface in the axial, circumferential
and radial directions, respectively
x, s, z Coordinates of a point in the middlesurface
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 Middlesurface axial, circumferential
and shear strains, respectively
Exs Mean value of shear strain
^ Poisson's ratio
0 6x (7, Us .Middlesurface 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 inplane
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 postbuckling
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
argedeflection treatments. It requires setting up the expression
or the total potential energymembrane energy, bending energy and
nonlinear straindisplacement relations and linear stressstrain 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 loaddeflection 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 preand postbuckling 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 loadingaxial, 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
eighthorder 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 fullhard, 0.006" and 0.0075" thick. The aluminum
was Type 3003H19, 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 thickwalled 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 thickwalled 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) Onefourth of critical load from (a) applied
and then hydrostatic pressure applied until buckling occurred.
(c) Step (b) repeated for onehalf and threefourths
of critical load from (a).
(d) Hydrostatic pressure applied until cylinder
buckled.
In many cases, including all those for hydrostatic pressure
alone, the "snapthru" 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 postbuckling 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 nondimensional
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 eighthorder 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
secondorder straindisplacement 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 straindis
placement relations introduced by Donnell (7) and the usual form of
Hooke's law.
The straindisplacement relations, derived on the basis of
Donnell's simplifying assumptions, in the middlesurface 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 middlesurface 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 )2jx,),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 mrx 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 halfwaves 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 03 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 
R3/6
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 sZi
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:
S100 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 ^17^2k 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 RI 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 I2  S  1 7 2? 
L2 +
rl4 L Yr"1
4 4L (t L L 2 
,a r0 [
.tS 2v\Y~ b 1
 'l~F FLK Tp. 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^
4S 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>+ 4t L4 .4
QLr +4 1kk
 .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 L1b 23 5
 4. . t1
  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, EK 'i,'",
+ b
i23 1
E;'K n
 4
,i 
1'
F Z nKrL
j ^V
,EZC~t WBn'Clt
I^bL" 1 ^
a2. L,4
e Ek2 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~'LIZ%4h
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 L4 F
P IL 2 1Z1
k3 2 {2 '23c]
;2 2 4 z1_
4LA4
___ _s icZ~C~Wr k4T 3LcIEt~~ ~ 7*~Cr 
oz12Z t k46
+cz 2iL L2 35
If /,? I k4 y
422
+ 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 ua1a TI tl wa nnrfCf4hP n datP1rminp tIho InnAAi
LLLiij.LLL=CXiL CluCiLuio.
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 IIContinued
1.4 0
'S 41 0k *
4 J M 
4 I . U^
ou u
,4 C: 41 co
A
Sr C.1 .M^ 1JC
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 IIContinued
"4
r44
c0
*
44
"4a)
0. 0 0U
0J U u
4 .. 0. 4E4P
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 IIContinued
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{ rl 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
rJck 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. 60Typical 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\
050
\\
\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 loaddeflection 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. 389413.
. Bryan, G. H., "On the Stability of Elastic Systems," Proceedings
of the Cambridge Philosophical Society, vol. 6, Part 4,
1888, pp. 199211.
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 1114,
1958. New York: Pergamon Press, 1960, pp. 115168.
i. Nash, W. A., "Recent Advances in the Buckling of Thin Shells,"
Applied Mechanics Reviews, vol. 13, no. 3, 1960, pp. 161164
i. Timoshenko, S. P., Theory of Elastic Stability. New York:
McGrawHill 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. 302312.
. 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. 795806.
Ebel, H., "Das Beulen eines Kreiszylinders unter axialen Druck
nach der nichtlinearen Stabilitatstheorie," Der Stahlbau,
vol. 27, no. 2, 1958, pp. 4553.
. Donnell, L. Ho, "Stability of ThinWalled Tubes under Torsion,"
National Advisory Committee for Aeronautics, Technical
Report 479, 1933.
i. Suer, H. S., and Harris, L. A., "The Stability of ThinWalled
Cylinders under Combined Torsion and External Lateral or
Hydrostatic Pressure," Journal of Applied Mechanics, vol. 81,
Series E, no. 1, 1959, pp. 138139.
C C
iurobLul. ULLpuviJLieutU rlL.IU. ULOOCLL.L.A.JLL, uILV.VJ.y
California, Berkeley, 1935.
* Lipp, J. E., "Strength of ThinWalled 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 IllApplication 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
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