Instability of cylindrical shells with inclined stiffeners

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Title:
Instability of cylindrical shells with inclined stiffeners
Physical Description:
xii, 93 leaves. : ill. ; 28 cm.
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English
Creator:
Lee, Ru-Lin, 1934-
Publication Date:

Subjects

Subjects / Keywords:
Cylinders   ( lcsh )
Strains and stresses   ( lcsh )
Buckling (Mechanics)   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 88-92.
Statement of Responsibility:
By Ru-Lin Lee.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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oclc - 13907649
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Full Text








INSTABILITY OF CYLINDRICAL SHELLS
WITH INCLINED STIFFENERS



















By
RU -LIN LEE











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
August, 1965











ACKNOWLEDGMENTS


The author wishes to express his sincere gratitude to

all the members of his supervisory committee. In particular,

he wishes to thank Dr. S. Y. Lu, Chairman of his supervisory

committee, for constant advice and encouragement throughout

the entire period of this research. He would also like to

thank Dr. William A. Nash, Chairman, Department of Engineer-

ing Science and Mechanics, for his valuable suggestions and

financial support throughout the author's entire graduate-

study program.

He is also indebted to Dr. I. K. Ebcioglu, Department

of Engineering Science and Mechanics, Dr. J. Siekmann, De-

partment of Engineering Science and Mechanics, and

Dr. R. G. Blake, Department of Mathematics, for serving on

his supervisory committee and for the various stimulating

discussions he has held with them over the past few years.

Final thanks go to the National Aeronautics and Space

Administration for their sponsorship of this study.













TABLE OF CONTENTS


ACKNOWLEDGMENTS .

LIST OF TABLES .

LIST OF FIGURES .

LIST OF SYMBOLS .

ABSTRACT .

CHAPTER


* C *

* C C C



* C C C C C

* C C


Page

S. ii

. V

. vi

S. viii

* xi


I. INTRODUCTION . .

II. BASIC RELATIONS. . .

2.1 Compatibility Equations .

2.2 Energy Expressions ..

III. INSTABILITY OF A STIFFENED CYLINDRICAL SHELL
UNDER AXIAL COMPRESSION AND INTERNAL
PRESSURE . .. .

3.1 Deflection Pattern and Stress Function.

3.2 Expressions of Total Potential Energy .

3.3 Minimization of Total Potential Energy.

IV. INSTABILITY OF A STIFFENED CYLINDRICAL SHELL
UNDER BENDING AND INTERNAL PRESSURE. .

4.1 Deflection Pattern and Approximate
Stress Function . .


iii









TABLE OF CONTENTS (Continued)


4.2 Expressions of Total Potential Energy .

4.3 Minimization of Total Potential Energy.

V. INSTABILITY OF A STIFFENED CYLINDRICAL SHELL
UNDER TRANSVERSE LOAD AND INTERNAL PRESSURE.

5.1 Deflection Pattern and Approximate
Stress Function . .

5.2 Expressions of Total Potential Energy .

5.3 Minimization of Total Potential Energy.

VI. NUMERICAL EXAMPLES . .

6.1 Cylinder under Bending and Internal
Pressure or Transverse Shear and
Internal Pressure . .

6.2 Cylinder under Axial Compression and
Internal Pressure . .

VII. EXPERIMENTAL INVESTIGATION .

7.1 Models . .

7.2 Test Result .. . *

VIII. DISCUSSIONS AND CONCLUSIONS.. .


APPENDIX A . .

APPENDIX B . .

BIBLIOGRAPHY. .. .

BIOGRAPHICAL SKETCH ..


. .. 74

. 81

. .. 88

. 93


Page

31

35


40


40

41

43

44



44


46

48

48

50

51


\












LIST OF TABLES


Table Page

1. Tests of Stiffened Thin Mylar Cylinders under
Axial Compression or Bending without Internal
Pressure... .. ..... 54












LIST OF FIGURES


Page


Figure


1. Coordinates and Displacement Components of a
Point on the Middle-Surface of the Shell .

2. The Coordinate System of the Stiffened Shells


and Stiffeners . .

3. The Dimensions of the Stiffened Shells under
Different Loadings . .


* 56


* 57


4. Stress Parameter, 12 as
Deflection, n .

5. Stress Parameter, F2 as
Deflection, t .

6. Stress Parameter, (2 as
Deflection, .

7. Stress Parameter, 2 as
Deflection, 7 .

8. Stress Parameter, 92 as
Deflection, .

9. Stress Parameter, 52 as
Deflection, n .

10. Stress Parameter, 2, T
Wave-Length Ratio, C .


a Function of
a Fu n o.

a Function of
*a Fn of

a Function of



a Function of



a Function of



a Function of



as a Function
* & 0 0 0 0


* 58


* 59



. 60



S. .. 61



.. 0 62


0


10


11. Minimum Stress Parameter, 2, as a
Function of Deflection .

12. Minimum Stress Parameter, ~2, 3 as a
Function of Deflection n, at Different
Inclined Angles .. .

vi


. 63



* 64



. 65


* *







LIST OF FIGURES (Continued)


Figure Page

13. Relation Between Minimum Stress Parameter,
12, l/ and Inclined Angles .. 67
14. Relation Between Minimum Stress Parameter,
2,,y aand Rigidity of the Stiffeners. 68
15. Relation Between Minimum Stress Parameter,
32,r ,R and Internal Pressure p 69

16. Comparison of Effect of the Imperfection Ratio
on the Stiffened Shells Under Axial Compression 70

17. Typical Buckling Patterns for Unpressurized-
Stiffened Shell Under Axial Compression 71

18. Comparison of Theoretical and Experimental
Results on the Stiffened Shells Under Pure
Bending ........ ..... 72


vii









LIST OF SYMBOLS


Et3
D Flexural rigidity of the shell =12
12(1- 92)
E Young's modulus for shell

Ej, Ek Young's modulus for jth and kth stiffeners,
k respectively

Ej, E Dimensionless Young's modulus for jth and
kth stiffeners, respectively

G, Gk Shear modulus for jth and kth stiffeners

Gj, G Dimensionless shear modulus for j and
k stiffeners, respectively

Ij, Ik Moment of inertia of jth and kth stiffeners
about center of gravity of jth and kth
stiffeners respectively

I I Dimensionless moment of inertia of jth and
Sk kth stiffeners about center of gravity of
jth and kth stiffeners respectively

J J Polar moment of inertial of jth and kth
j k stiffeners

Jj, Jk Dimensionless polar moment of inertial of
jth and kth stiffeners

K Torsional rigidity of the stiffeners

L Length of the shell

I Total length of stiffeners

m, n Numbers of waves in the axial and circum-
ferential directions

Nk Number of stiffeners in ~1-direction


viii









LIST OF SYMBOLS (Continued)


PJ

p


PC






R

t

tl

U, V, W


w
o

x, y, z







yI' T'2

Th'~Izl



ALe


Number of stiffeners in y2-direction

Transverse load

Internal pressure

Axial load

Dimensionless transverse load

Dimensionless internal pressure

Radius of the shell

Thickness of shell

Thickness of stiffeners

Components of displacements in x, y, z
directions, respectively. Fig. 1

Initial deflection

Orthogonal coordinates on median surface of
shell

Dimensionless parameter defined in Eq. (22)

w
Imperfection ratio = -o

The angles between the stiffeners and the
generator of the cylinder, respectively

Ratio of arbitrary parameters defined in
Eq. (48)

Ratio of numbers of waves in the circum-
ferential and axial direction

Poisson's ratio for shell








Ec' 5ib'



6c b, U


Average axial compression stress, bending
stress, and eccentric compression stress,
respectively

Dimensionless axial compression stress,
bending stress, and eccentric compression
stress, respectively

Laplace operator

Biharmonic operator = (V2)2











Abstract of Dissertation Presented to the Graduate
Council in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy


INSTABILITY OF CYLINDRICAL SHELLS
WITH INCLINED STIFFENERS


By

Ru-Lin Lee

August, 1965

Chairman: Dr. S. Y. Lu

Major Department: Engineering Science and Mechanics


An analytic study of the general stability of thin

circular cylindrical shells stiffened by a system of

inclined stiffeners has been carried out. The cylinder is

internally pressurized and is under three types of load-

ings, namely, axial compression, bending, and transverse

shear load. The stiffeners are not very close but are

discretely located and the eccentricity of stiffeners is

disregarded. The method of the solution is carried out

by the use of nonlinear large deflection theory and the

effects of initial imperfections in the strain-displace-

ment equations are considered. The Ritz-method is used

xi








to find the governing equations of instability of stiffened

shells subject to the three types of loading.

Numerical examples for cylinders with inclined

stiffeners at different angles are worked out. The minimum

strength of shell versus different rigidities of stiffener

are also calculated. The relation between the minimum

strength of pressurized shell and internal pressure is

plotted. The computation was carried out on the IBM 709.

Some tests have been made on the shell with inclined

stiffeners at angles of Y = 300, 450, and 600 under axial

compression and bending loads. Some of the experimental

data are plotted to compare with the analytical results.
$


xii












CHAPTER I


INTRODUCTION


Interest in shell buckling phenomena dates back to

1858 when Fairbairn (1)* performed his experiments on the

buckling of cylinders under uniform external pressure. In

1888, the first theoretical treatment of the problem was

published by Bryan (2) and Love (3), and since that time,

many investigators have been attracted to the various prob-

lems of thin shell instability. The equations for thin

shells were discussed by Timoshenko (4) in his well-known

book. The study of stability of isotropic cylindrical

shells was advanced by Southwell (5). In 1933, a set of

equilibrium equations for cylindrical shells subject to

torsion was derived by Donnell (6) based on the theory of

small deflections. A simplified method of elastic-stabili-

ty analysis for thin cylindrical shells was also discussed

by Batdrof (2).



Underlined numbers in parentheses refer to entries
in the references.








Then Suer and Harris (g) applied Donnell's eighth-

order linear equilibrium equation to the problem of the

stability of cylinders under combined torsion and hydro-

static pressure. They obtained a solution with Galerkin's

method based on an assumed radial deflection function in

the form of an infinite trigonometric series.

In 1961, Seide (9) had presented an analysis of

the buckling of cylindrical shells subject to pure bend-

ing. However, these classical analyses are valid only

for infinitesimal deflection. When the deflection ip not

very small and has a magnitude of wall-thickness, large

deflection theory must be used, and the high order terms


of deflection are to be considered. In general, small

deflection theory predicts buckling stresses approximate-

ly three times as great as those found by experiments.

This evidence is particularly obvious for cylindrical

shells under axial compression. The discrepancy between

the small deflection and experimental results can be

explained by use of large deformation theory which was

advanced by Donnell (10) and von Karman and Tsien (11).

In Ref. (10), Donnell also considered the initial imper-

fection in the shell to cause lower buckling load. A

diamond-shaped buckling pattern represented by a function








of arbitrary parameters was first proposed by von Karman

and Tsien (11) for cylindrical shell under axial

compression. Their results agreed with the experimental

results. The same problem was also solved by Kempner (12)

who used the Ritz-method, but improved the von Karman and

Tsien's solution by considering five free parameters. An

investigation of the buckling of cylindrical shells by use

of Galerkin's method was also studied by Ekstrom (13) and

Lu and Nash (14). The general case of the axially compress-

ed orthotropic and isotropic cylinder with internal pressure

has been discussed by Thielemann (1). The general equa-

tions in tensor notation for isotropic shells were derived

by Dill (16) who also included the effect of initial imper-

fections. The effect of initial imperfection on the buck-

ling of cylinders under different loadings was studied

in Ref. (1i), (18), (19), (20), (21) and (22).

The general instability of ring-stiffened or

stringer-stiffened cylindrical shell by the use of small-

deflection relations have been investigated by many authors,

Nash (23), Alfutov (24), McKenzie (25) and Kan and

Lipovskiy (26). Further study in general instability of

ring-stiffened cylindrical shells subject to external hy-

drostatic pressure with a comparison of theory and experi-

ment was discussed by Galletly, Slankard and Wenk (27).








A detailed treatment of the buckling of stiffened

shells has been discussed by Hedgepeth and Hall (28).

The problem of stability of orthotropic shells was studied

by Becker and Gerard (29). Further study in general

instability of orthogonally stiffened plates or cylindrical

shells has been discussed by Huffington, Jr. (30), Neut

(31) and Becker (32).

In Ref. (32), an analysis of orthotropic cylinders

was made by Taylor whose derivations were based upon the

use of a form of Donnell's equation. Recently, an analy-

tical investigation on the instability modes of ortho-

tropic cylinders subject to compressive or bending load was

made by Block (33). He used the Dirac delta function for

rings in the linear equilibrium equation and considered

the pressure load.

The large deflection relations applied to the prob-

lem of the instability of ring-stiffened cylindrical shells

derived energy method was investigated by Lee (34).

'Experimental work on the ring-stiffened cylinders

or unstiffened cylinders under different loadings has


been performed by MaCoy (35), Peterson and Dow (3),

Lundquist (3.), Harris (38), Fung (39), and Goree (40).








The theory of anisotropic elasticity has been dis-

cussed by Hearmon (41) and Lekhnitsky (42). A general

discussion on the theory of anisotropic shells has been

made by Ambartsumyan (43). A linear solution of aniso-

tropic cylinders was investigated by Cheng and Ho (4).

The purpose of this study is to investigate the

general instability of cylindrical shells with a system

of stiffeners which is mounted at various angles. The

cylinder is internally pressurized and under one of the

following types of loading:

a. Uniform axial compression

b. Uniform bending

c. Uniform transverse load.

The stiffeners are not very close but discretely located

and thus the anisotropic relation does not apply. The

Kirchnoff hypothesis is assumed; the eccentricity of

stiffeners disregarded. The method of the solution is

first to assume the deflection function, then the stress

function is found from compatibility equation which is

carried out by the use of nonlinear large deflection theory.

The expressions for the energy stored in the shell and

stiffeners as well as the work due to the external loads

are obtained. The large deflection terms and the effect

of initial imperfections in the strain displacement








equations are considered. The Ritz-method is used to find

the governing equations of instability of stiffened shells

subject to these three types of loading used. Hence, the

minimum loads are found from these equations.

Numerical examples are worked out for the cylinders

with the inclined stiffeners at different angles. The mini-

mum strength of shell versus different rigidities of stiffen-

er is also calculated. The relation between the minimum

strength of pressurized shell and internal pressure is

plotted in Fig. 15. The computation was carried out on the

IBM 709. The general computer programs written in

Fortran II to find these minimum stresses for all three

types of loading are listed in Appendix B. Some tests have

been made on the shell with inclined stiffeners at angles

of Y = 30, 45, and 600 under axial compression and bend-

ing load. Some of the experimental data are plotted in

Fig. 18 to compare with the theoretical results.












CHAPTER II


BASIC RELATIONS


2.1 Compatibility Equations

In the present investigation, the following

assumptions are made:

a. Elements normal to the unstrained middle sur-

face of the shell remain normal to the strained middle

surface.

b. The material follows Hooke's Law.

Let x and y be measured in the axial and the circum-

ferential direction in the median surface of the underformed

cylindrical shell (Fig. 1), w equal the total radial deflec-

tion, and wo represent the initial deflection in the

radial direction. The strain-displacement relations are



e 1- +_ _.___
-x x 2 2 (




((1)
(: i ( f- 0
a 2y 2 2 B R
ex bJ Lk Lr bur r D J. bura
Z ~ ^ h rx ax b xc ar







In the above relations, the higher order terms of deriva-
tives involving u and v are neglected, since these dis-
placements are small compared to the radial displacement w.
The stresses and the strains in the median surface

of the shell in the case of plane stress are related to
each other by the following equations:
E
5-xET^ (Et6x)
6Y 1- (2 +,) )
E


_rE2' ( ) (2)
-x4 2(1+))

By substituting Eqs. (1) into Eqs. (2), the median surface
stress-displacement relations are obtained


E= C 2 CAX 22F I r 22, :
2^ -)- + ,- +

aLk, -1 ,iU r

I 7W+ ) (xW



+jr ((AT.
ZX$= ax 2ar x 3
(3)

The conditions of equilibrium can be satisfied by using the
well-known Airy stress function F which is defined as










S__F_ 7'" F
X=E -E I 1 6*r F

(4)

Eliminating the variables u and v in Eqs. (3) and (4), the

following relation between stress function F and the radial

component of the displacement w is obtained.


( + F E "-co-
x+2 -- y 2 -ax

(5)

This equation expresses the condition of compatibility be-

tween stress and strain. It was first obtained in the pre-

sent form by Donnell (10).

In general, wo is unknown. For simplicity, w is

assumed to be proportional to wo Ref. (10), (17), (20),

and (22) ., Thus one can define

OX- = imperfection ratio (6)

where r is independent of x and y. With the relation from

Eqs. (5), (6), the compatibility equation is expressed ass









41 b 2 eLa- r ? l
--T -fz -


2.2 Enerqy Expressions


The strain energy in the shell and stiffeners and
the potentials due to external load are found below.
a. The extensional strain energy in the shell can
be written as:


x1 "^ F JJ F
-2 -F] dxdy


b. The bending strain energy has the following


expression:


U DL27rR Z247+ 2 a
b 2Jo [ o xX X2 aX 2 2
?(rx. aJ),u( a) __ ]9
(9)


where


E t3
D= E(3
12C11- )


c. The potential due to edge bending of the shell


+2((1+-2








rM L ZIM Cos ]1.u ]&,Ct
f o b COSc (10)

where Ub is applied peak bending stress and

-ax~E \ b'F b+F\ Llb1 (,.rx,

d. The work of the external force applied at the
ends of the shell can be calculated as the product of the
applied force and the change in length of the shell as
follow:


TT = (Uf~fl F ( -3AF\jU (11)
tJoo 1 ET Z, 2 W F 7


where Cc is constant stress over the shell thickness.
e. The potential due to the internal pressure p is

S, 1< 1 (.,-- a) dxda
Uo ( (12)

The change in volume due to the displacements u, v are
neglected (11).
f. The potential due to the edge bending on fixed
end by transverse shear load P which applied at the free
end (Fig. 3) is







-- L 1o rP CCS ) d


=--l x COS )2
ITdfol X2R Ex 2 'x



(13)

g. The potential due to transverse shear load P

applying on free end is





STT oJjo 1 E -3xa x ,,



where P is average transverse shear load.

h. The bending strain energy of stiffeners. The

stiffeners are assumed in parallel with the yl, y2 coordi-

nates lines and the principal directions of the cylindrical

shell coincide with x, y lines (Fig. 2). Let the subscript

k be used for kth stiffener which is inclined at an angle

1I with the generator of the cylinders, and is parallel
with yl line and normal to y'2 line. Thus the bending

strain energy in the kth stiffener is



(15)








where Nk denotes the number of the stiffeners in (1

- direction, and Ek Ik, represent Young's modulus and the

moment of inertia of the kth stiffener, respectively. The

eccentricity due to stiffeners is neglected. The limit

Ik is the length of the stiffener in Yl direction.
Similarly, the bending strain energy in the jth stiffener

which is parallel with y2 line and normal to y'1 line

(Fig. 2) is



j=1 Jo =
(16)


The subscript j is used for jth stiffener which is inclined

at an angle of '2 with the generator of the cylinders.

Where Nj denotes the number of the stiffeners in I 2

- direction, Ej, IP, represent Young's modulus and the

moment of inertia of the jth stiffener, respectively. The

limit Ij is the length of the stiffener in 2 direction.

i. The torsion strain energy of the kth and jth

stiffeners are


UT, = 2 J,, \y 2=0 (17)










_= 2 {J II Ur I;Id I-
(18)

where G and J represent the shear modulus and the polar

moment of inertia of stiffeners, respectively. The sub-

scripts j is for stiffeners in 7 2 direction and k in

Y1 direction. In the present analysis, the inclined
angles, Y1 and 72' are considered in axial symmetry

(i.e. = = )












CHAPTER III


INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER
AXIAL COMPRESSION AND INTERNAL PRESSURE


3.1 Deflection Pattern and Stress Function

Let a deflection shape assume the form (12)s


W = b,+ b2 COS C COS' +b+COs 2M t+ COS y (19)
T + + R

where m and n are the numbers of waves in the axial and

circumferential directions, respectively. In Eq. (19),

bl is not an independent parameter but is used to satisfy

the condition of periodicity of circumferential displace-

ment (11). It is found later to be a function of b2, b3

and b4. The boundary conditions of the displacements at

the ends of the stiffened cylinder are disregarded. This

simplification is justified by the experimental findings

of N. Nojima and S. Kanemitsu as discussed in Refs. (11)

and (12). It was found that there is no appreciable length

effect when the length of the cylindrical shell is greater

than 1.5 times the radius of the shell. Therefore, the

long stiffened cylindrical shell is considered hereon.







The corresponding stress function is


_2 pR
j~ = --Be 2 1


X Cos- 3-2 cos + oCnocos

St ,3c- Cos M +x o
Co rtl- 3rC C0--- O5


The coefficients all, a22, a20' a02, a31 and a13
are found in term of b2, b3 and b4 from the compatibility
equation (22). These coefficients are found to be:


S- (- r) ( -
aon-E-l 2--2
ctoz= E--*I- -F')32,u2


2) b 2


a =. =- (i ----2)2)(b+ )
._ E 2 (ci+^( +,'UZ) b2 b4


-- 31 2,, .z-- b-


rF2 2,4 --
(1+4,2 b3b


R
mZ -


(21)


A ==2,,3, 4


(22)


a cos cos os
a, 1os cos + a22.COS cos


(20)


a,- =- -


where


b, b,
~t


'It








Physical considerations lead to the conclusion that

the circumferential displacement, v, must be a periodic

function of y. This is expressed in the condition that

v/ by contains no constant terms or functions of x alone.

Using the second of Eq. (1) together with Hooke's law, the

following expression for ?v/ 3y can be obtained:


__ 1 i6 )f6 1(pA, /^1 /,\ (U(j- S- E (6-^ -i\ M Y ) CL-
(23)


By substituting Eqs. (4), (19), and (20) into Eq. (23) it

is found that


E 18 R 4 -R I o-R
(24)
+ terms involving periodic functions

Since v must be a periodic function of y, the constant

term on the right side of Eq. (24) must be equal to zero.

Thus,


b, -(25)


where


R -- 6 (26)







3.2 Expressions of Total Potential Energy
After substituting the expressions for w (19) and
F (20) into the relations given in Chapter II, section
2.2, the various energy terms are obtained after integra-
tion. They are

UeR 2 1+/Z -z 4 (1/)
S + f+ 2,d -U + aI, + Ctas
TrEPL O ) -

g+ (ba C + )+- (9+4M C+(1+5) 4
(27)

U-bR + IA + 16B ; +-;Z
fTE a-L 24(1-) 2 3 4


(28)




(29)

Pz^( ^) + ?-(( T +6e)

(30)

=Eix i2 r b((ce C, + R -6, C)
+32-
32 (;; ; + 1 +R,


(31)















S(1- r
SP


_ (-F


j'Z


+ (Ca-,uA C)2 (C4 f + 4C3c3 ( 6,4f b2)


(34)


where


E
^- E~Gk



r ^-G=i


E


E
E = E



Ei JE


- IJT



- Ij


1 jT*



- T1=-7^
ST F
, -J ~ fR


tr E ..L
:TEt3L


--2
bz (ct + c + 6 4cic)

S)] +R2


(i-j r
r ,=1


JT E*3L


(32)


Jr EPL


(33)


3t2 4--2
+ 32 (-,,4i: +- b


G6 q[(+C+ACi+ cc(C-,c,)z


+(c,- Ac .1(c(+a c + 66cci(4+ fC)+ R,


16 Lc' c


E~~LI[


}+R,










l =C =7 Ci=Cos Ca = SiN


C=.3i = C3=Cosa5, C4= SIN ,
S (35)

The expressions R1, R2, R3, and R4 are very lengthy and can
be neglected if the stiffeners are located uniformly and
not far apart. They are listed and discussed in Appendix A.
The total potential of the system is the sum of the
strain energies and the potential of the applied loads, as
follows:


U, = Ue+T+ T+ +Tjt ,*+-t.iT, ,,,


In nondimensional form:

---r
+R -2- -u(- ++Z)
JT E'L 4 8

-Z 4(if~+' )2 -, a 4

+ 4 + )+ +

'(1+q, eY(1 a is] 9 (l++ 16 ,U4;' +b!')]









+R, 2 c?+ 6 Uz2C' CZ + U1 4( + + + c C


4+ n 4y (c+ c) (c c ) 44(c f C44+ c,)
j=1

+ G4 J c(,+(C^ t ')2 C i+ (C, c(c2 C(C ] z














3.3 Minimization of Total Potential Energy


to each of the arbitrary parameters must vanish for equi-
librium, this yields
4 o =- AO 0
44 z (2 &bjb16} (C. +A^(4- a


+(C3- C4CI(C4 +,u c3 + 4 C A++ 624


+R,+ ? Rzt R3+FR4

(37)

3.3 Minimization of Total Potential Energy
The variation of the total potential with respect
to each of the arbitrary parameters must vanish for equi-
librium, this yields


(38)







After differentiation, the following three equations are
established:


+ i Ef(Cf+ 6COCc +,4c


4E
+ 7 l ( e + 6_Ae C31 C*- + C.-le)



I--t 4
T -





+ C [2 (24r)(ZT-3+b4) + 4al b + i
(i+1 6L (1 +A)2? 2 a a(i+X2
b( (i+ ) (+r) 4 4 (f+r) b+ _
+ + p ( L a+ 4 +A1+ y
(


39)


-z -
(i-F) 4(1-1 ) b3



+ lU4 + c4 + A


A4 (Iff r)
2 (i1+A?)T
2 (i +x :


+1--


i+ W


-2
4 2.A ( 1I-+)
S(I gsy


^ ~Wk c

j -i *


(40)


_6 l 4(+ bA.
1-r- b. 4(1+A'7 g


A4(1+r)
2(1 +A9Z


-2
6


I-r
1-F)


i *tM" 2


2Al~itr)
-f
(i $U2)2


1,~,








+ 14r) ,
4 4


S29+3(
3 + z 2(9+.Y -


4N Nj
+ t EI c), +
j=- 1 1 3
+ 4 cT rc2r
c*"i i^c


ETjja c,


-kk4 c, C


For brevity, Eqs. (39), (40) and (41) may be
rewritten as:

, =Ar t(A,+A,[+A4,)+, A5


Tr- =b ++ ( +)


' -=H-,+ (H2+ H4)


where


= 6o-,,2"


A,

Az


12(1 'e)Z

(1I+ )2


A3 ([ (2+)(2 1' )
(1 +*)2


(41)


(42)


(43)


(44)


N
+ >
lk=


+2 J


+ 2 (Hs + )







+ i 1a 1
zi^4 ^


+ j E i3 (C, + 6uc:c +,. cl)

jt 2
-I E I*,, [ (C,+Ca,) (Cz,-c,Jt.

E l4 +x (C4+ ) c4-(c 3-)c)-f


+iAcf]


(cs-,ac4f(cC++,c,)] (4


o 9 /(42


A4

=B, (L=
= (i +i +


S3 2-
4 4(i+e)2


(1 + X)
(i~f 72


2 Ce ai;E,+4cA
C c,2 cU + "U4 1Tj0 cy*
-, c1j C3 C 4iz
jz=l


(46)


A4=4(i + "f) I (+,)
4 + AJ)

As= (1+ r)l
16


(C,-A cC (c


+

k=l












H, 3 (1-z) 1


Ha 4
H4=(1+r) +



6- la+ 8
H3 +



+ G %C C, c +


(47)


Ejjl 1 c C4


E7,I c^


%=^3


(48)


Eliminating b2 and


from Eqs. (42),


(43), (44),


the following equation is obtained:


M,I T+ M$ + M3 =0


where


Az
4X,


c4
03


(49)


Ej I)i


31~
(3


(1+ U











- A5)

A s)


F2 D3
/RD


M, T f 2 a fD
- (-r F
-+ 2F l(5+B -A5)
F3 t


-F(


x (H! +5)


A2 -As (B,+


+ H )


- P1 (i__ pl pz


F32
IFD3
+ Dj


(H+ +


+- A


_ ZPaFD 12
+ z+D


(50)


As


34A5?


+
F3
:B,4


H,
+
F.-


(55+ "?
B61

As)


/
*)


2 (B + t) X


2 = A, ( Bs+


B3c)


(15+


(51)


SM,- (H,+ (,


86\_ B1
7t/


Ps3= (Az + A-3 + \A4V


Ht










F2 = A (Hs ) H, A

F, =(A2 + A.3 + A41) (H, ) -(H + H4Lz) A5
(52)

Equation (49) is the governing equation for the '
stationary load of a stiffened shell under axial compression.

The numerical calculations have been carried out on

the IBM 709 and the computer program used is attached as

Appendix B.













CHAPTER IV


INSTABILITY OF A STIFFENED CYLINDRICAL SHELL
UNDER BENDING AND INTERNAL PRESSURE


4.1 Deflection Pattern and Approximate Stress Function

When a cylindrical shell is subjected to either

pure bending or eccentrically applied compression, an

approximate form of the deflection pattern is assumed



Sn x 2M rr2 714
L=b,= os 6i-0 (bcos Cos ,Cos b cos R


(53)


This approximate deflection pattern is obtained by

multiplying the pattern of the axial compression by

cos (y/2R) which signifies the localized buckling on one

side of the stiffened shells. The effects of the end

restraints are also neglected.

The corresponding stress function F is proposed





S29


F _6 Y- + 6b Iecos + E k -t+ cost cos

+ a COS 2 X COS2 4- zoCS2 CO52rS +a, COD
+ R z Cos Cos Cos
+a3,coS 3 co5 I + ,13 coS5- cos3--
R R
(54)

The stresses 6 and 6b are the average axial
compression and peak bending stress, respectively, and are
positive for compression.
The coefficients a20, a02, all' a22, a31 and a13 in
Eq. (54) can be expressed in terms of b2, b3 and b4 by the
Galerkin method. This method establishes the following
set of equations:


IL 2JR
oo O

JL ZJTR Q
Jo
L 12~R
0 0,
Loo
27Rl


COs MCOS^ 0 dx o

COS 2 x COS2n9_0_ 9 ax = 0
R R
COS mx dC9 x= 0
R
COS2^- ?j dx = 0
'mx 3__
COS Cos o L~cx = 0
R R
CoS c 1x CO5 r d Oc = 0
R F


(55)








where


S i-- o- E [ (

:xa S


'b--2- -Yur r' JL + ?W
*^2x-a -LJ -LJ _b


R x2 R


Z~Ob It


After the expressions for Lr in Eq. (53) and F in Eq. (54)

are substituted in Eq. (55) and the integration carried

out, the following relations are founds


0- E-20
E E"


o2 -E, = a26 (I _

- =..( ( + --)
z -' E( f -+ AO- )


" --k --

a' ==Ef
1 E*

where
E Z





where r,/ ,


-(3+) b4 (1.- r2)
4 (7+ a)
3 A4 +T4+
- (14- F) b b.3
4 (9qt + i


(57)


and bi are same as given in Eqs.


(6) and


(22), respectively.
In order to find the bI as a function of b2, b3 and


b4, we derive as in section 3.1 the relation:


3LJ0
..b Yf


(56)


T ;+ F2r)]









b, r -f [ +(2 -2
i- = -~+ -) b- r 1
40 + 16
,(i-F)= -(+1 )+ p I 4 ( .+;)

S(2d+ 1 + 4

(58)


4.2 Expressions of Total Potential Energy
The total potential of the system is the sum of the

strain energies and the potential of the applied loads. By

adding the Eqs. (8), (9), (10), (11), (12), (15), (16), (17)
and (18), the total potential may be expressed as follows:


Z= e+ Ub+Tc+U,+U+ ,+UT+
(59)

If w, F and b1 as given by Eqs. (53), (54) and (58)
respectively are substituted in Eq. (59) and the integration
carried out, the total potential is obtained in the follow-
ing form:

JZ -2 -- (-F
Tz R -2 -2 b- 3(1--
2 )TELk 2 L 32
r)- -
-z 3(1-F2) 6 b AI
4 4 0 16 2 1












tb r +
(z .L1


c4
+ 2


+2


(


[48


- 2


-2
16


= P 16 b2
4=, 1 6 -


+n )
6,4,4.Zm
t- --


Cn2 +
"n I


(,+)+ +


4


3 (c1, 6,2 C IC + A C4)
2


b2
37!


C, + 24 z
c ~ZL


48$4+24 ,z


* Ej {16 p 2Z


mZ t 7n4) .3 C31 C4r


+ 24 McC' ci


+


--N


+2


4
(C
\?*4


+ -2.c ++4t
7?yy


Sc
-- _


[19


c4j


T[c, ac) (c(- XAcf + (c, cf (c+ Ar


[(c +'cif C 4(i-Me)C2c + (ci+c)Cj+7j


32 m4)


c ci


+ 32


2o8


- 2 a1 i-
T MZ


'1 4 0-2 4 a


+2 ,(~++ a+- +-- + -+] 4 +6 ++-


(2 +


4 m2


92 2-mz


- + P 2


+ 224 (I
+fl"-^


+,a+ c:) +


4 ad+


--216
+ 16,b-g


+]
wt+


+cz1
+ In4.


lb :' L


C c z+ C +4 +







+b (48,e+ 24+) +~ 5324) c4.c


X(C4-*c.#)+ (C3-xC+)#(C4+,aC.3)Y ]+- [ (-Cz+.C4L
















(60)


where R, R2' R3' R4 are also the portion of the energy of
C4 4











the stiffeners which have the similar expressions as RI,

R2, R3 and R4. Since they are neglected in the numerical

calculation, their detailed expressions will not be

elaborated here. From the discussion in Ref. (14) it is

indicated that m has a greater magnitude when R/t is

greater. In the present analysis, we are concerned only

with extremely thin shells; hence this ratio is large.

Therefore, for practical purposes 1/m2 and 1/m4 are

negligible compared to unity. Thus, from Eq. (60), the

total potential can be simplified as follows:







= -2
2 -2 f -2 b
= -' 6 6b- 236 (1-r)
U? 2


36
32


--2r
(4


- ( I-r1)


-Az-


--2 (1+ )2
+ai, 4- 4


az o
12zo0 .a


+t aoz


S ( q +- 2
%+ 0-3


+ C,4 +


+ C,'cc) +24-+p t+ct+ 244 +c]


+A*4ct+ 6 c4)


j I6C~


xt c [( ,+uc,fc(c,-a,f +c(C, -, C.t2 (C c]+24

W. --1 2 '
+ c {-- (c., J, C4 (C4-A c)
+ U b 41 +1 6= iP2 L B


+ (c, -A c4 (C4 +,U C3] +


24 ,3 AM cfC4 +24,C. c +


aa -71- ) 3
+~~;)I ib (t.+,uzJ +4 (,-"A )
t iz "l ,)


where


2- 1
a^=0- lb


3 A2Z
a


16


[f;


+2z4; +*c 24 ;]


I6w*^x
^l^


(61)


I 6
zg r2}


a 4 (1C+2
a22 p +


Z -(- 16


-i( +cr








-2
3
a0- -(I r) 3(b-

a,- ( i-+ 3.4 T-
8 2(1+Aa )2
r 3 3 AA 2 b3z b


3 AU
a13= (- (fff)) b b-
4(1+ J)A)2

(62)

4.3 Minimization of Total Potential Energy
The total potential energy must be a minimum when
the structure is in equilibrium and this condition leads to
the following equations



o -0o o
S63 b 2) b4
(63)

Substitution of Eq. (61) into Eq. (63) leads to
three equilibrium conditions as follows:



1-r A, + (A + A3 +AA32) + b As (64)









S--2z
o"= +, + 8B41 + b6z
=-T -





where *

=I + -23+


- (I (1+^ )2 -
A -

A-2-- f+ )

~I (1 +)2


A4=(+r) -I+q(
{-4 ( L.+U zl)l +


`- 256
S = 2 El [( 4 CN a^ccr -I-cI4C)
N+ EI [13 (C4+6 6'azc;c +eC4)]

t.^Z GfsCC, +AAC^jGC.-AtCcfjc,-tcc,-+tcj+^ +
-=r 8
=6i~ t8[c~~c(-c)t~,p~G~C)


(67)


(65)




(66)


7)


t~Jl
it I+)


+ i T-


4


+ a
r/ +


+ -2H+
4z+ b, (H+


+ 3
8









3 = -' 2 -



= (-F+) A+ s.

5)2(1321W ?2


S16(1+ A ( )2


+ =^ i^


'4 +

Zcecj~


)+
32 (1 +AL' ) a


N-
44

3G j ,OC32C4


T = + 33

2(1-

H,- n-
36 A-
H4 32(1+ +-


H1= -(1+i r)
6 128 ),


+
)2

a1+)


3AA2
16 (ltAjl


(68)


(q+,te)a1


32








S = 3 E6JI.&

+ 3 TA


N'

C2cf + 3EIi c
cc l + 3


Eliminating


Sand 12 from Eqs. (64),


(65) and


(66), the following equation is obtained:


MI3 =0O


where


-As)


Z D D, H


M2' --a2 )P


+-(-
+ 2(15+


Fz D3 D -z2
C,35 + $- 5As
+ ) F1t-a AsBs + E
*+y ^Z H,t


( RS


(69)


(70)


P, D3
F3
]6 _
7;1


i


M 2


Di(B,+
-t)]
F -
8,+B +i~










54 H )(H + ( + AS)2

---2
= = (- + -,H +; + ,+ -A 52
+ -5-3- -2 Az

(71)


=B



zt )-H,A
F3 7 ( 2
F3= (A+A31+M A )(S+ ()-(H+H4 A,

(72)

and

-= 6 (73)

Eq. (70) is the equation for the stationary load of
a stiffened shell under bending load. Numerical calcula-
tions of the minimum stress parameter <2 are given in
Chapter VI.












CHAPTER V


INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER
TRANSVERSE LOAD AND INTERNAL PRESSURE


5.1 Deflection Pattern and Approximate Stress Function

When a long cylindrical shell is subjected to trans-

verse shear load at one end, an approximate form of the

deflection pattern can be assumed as in the bending case

in the previous chapter in Eq. (53). The deflection

shape is


= b, + cos2 ()(b)bacos-cos- +C coS +b45c


Corresponding to this deflection pattern, the stress

function F is proposed


=o + + a,, COs 7L

+ 6t2z COS 27n COS 29 + 2 COS 2 .a o5n
a2z Cos2xCoss + +a0 Ccos

-- (3 Co5 3M COS^ + ,3C05 COS

(74)








The relations between the coefficients all, a22, a02,

a20, al3, a31, and b2, b3, b4 are given in Eq. (57),

and BE is the same as given in Eq. (58).

5.2 Expressions of Total Potential Energy

The total potential energy of this system is the

sum of the strain energy and the potential of the applied

loads.


-= UtUIb+UT +Ui+UT,3t U J (75)


If w, F, and I as given by Eqs. (53), (58), and

(74), respectively, are substituted in Eq. (75), and the

integration carried out, the total potential is obtained

in the following forms


T, R 2 2- ( 2 2-+
T3 ETAT L +- 5 '4 2(1i 3 I- (+ P
-2 2-2
b6 P P 41- 62, b
16 4 6 4 4
2 -2 -2


-2 (1+-a z 4( 1+ f 2 g -ZZa
+ a,, 42 + a22 + a20 + ioz +
*+ -l4f +2 ^ ^ ^ 6" B^










a2 +U 2 (I+ 2 + (I -r )2 )
S 4 p2 13 4 g 192 (I-)



+ I2
+ b(44 24 + -)+(76)
(76)


where .J LU .'U ,lTUTTj

obtained in the bending case Eq.


where and l are given in
and
PL
E JT Rf


For the

chapter, 1/m2

in Eq. (76).
trained in the


are the same as those

(60).


Eqs. (22) and (6),


(77)


same reason as mentioned in the previous

and 1/m4 can be neglected compared to unity
The total potential energy is finally ob-
following forms


--i5 2 2 2 + 1) 22
,3 -P p (1i+9) P P + T+ ( P2
2 -P2
402 K?^ --^^. ^)-
^cr 4(t^") -+/ S^4C^+^)+ it


4 2











192 (1- -.0 24] (78)
+ U.+ +T, + U
where all, a a02, a20, a31, and a13 are given in

Eq. (62).
5.3 Minimization of Total Potential Energy

By variation of the total potential energy in

Eq. (78) with respect to each of the arbitrary parameters,
three simultaneous equations are obtained. From these

three equations, two arbitrary parameters (b2 and ) can

be eliminated. Finally, a governing equation for.the

stationary load of a stiffened shell under transverse shear

load is obtained:

2
M1 M, 23+ M3 = 0 (79)

where

P (8
D3 ?- (80)


MI, M2, and M3 are given in Eq. (71).













CHAPTER VI


NUMERICAL EXAMPLES


The minimum stresses are to be found from Eqs. (49),

(70), or (79) for the various types of loading. In each

case, the stress parameters ( f 2 and 3) are func-

tions of three free parameters 1~, 1, and U.. While A.

is the wave-length ratio, the expression i and 1,' can

be considered as deflection parameters ratio. 'The minimum

values are found numerically at various inclined angles,

stiffeners rigidity and the internal pressure. Numerical

calculations were made on the IBM 709. Program I is for

finding minimum stress ac for axial compression case.

Program II is for finding minimum stress parameters, 42

and 13. The minimum stress can be found with any given

numerical data. In the following, some examples are given.

6.1 Cylinder under Bending and Internal Pressure or
Transverse Load and Internal Pressure

In this part of calculation, the cylinder is assumed

perfect, i.e., J = 0 [Eq. (6)], and the angles of the

inclined stiffeners with the generator are equal, i.e.,








(l = Y2 = Poisson's ratio is 1/3. The general
characteristics are assumed:


1,= Ij I r=j T. Tj


K ---E I N




j (81)


By substituting Eq. (81) into Eq. (70), the value of 12

as a function of n ~1 and ,4. is found. First, 2

versus n is plotted for various values of UA. at a fixed

value of l1, in Figs. 4, 5, 6, 7, 8, and 9. Minimum 12

found from each of these curves is called 12' n It

should be equivalent to the value found from the relation

S2/ b = 0. Then (2, t versus / is plotted for
various Y 1 in Fig. 10. Minimum 2, found from each of

these curves is called 2, ,A, plotted in Fig. 11. It

should be equivalent to the value found from the relation

S2, /)a =7 0. This minimum value is the dimensionless
critical stress ( b + 3/2 6 )cr for the shell with in-

clined stiffeners at an angle of f = 450 with the

generator. By the same procedure as in the previous case,








the curves of 2,I versus t 1 for various inclined

stiffeners at an angle of Y = 00 (stringers), 300, 600,

and 90 (rings) are found and shown in Fig. 12. The

minimum value of each of these curves is plotted in

Fig. 13. The relation between minimum stress parameter

2,. and stiffeners rigidity, E I A is shown in
Fig. 14. The relation between minimum stress parameter

2, and internal pressure p is also plotted in

Fig. 15.

If ;3 is replaced by 2 in Eq. (79), the equation

is the same as the one obtained in the bending case.

Therefore, the minimum stress for the case of transverse

shear can be found by using 3 instead of 2, ,

in all the figures from Fig. 4 to Fig. 15. Where



3, 2.)cr.




6.2 Cylinder under Axial Compression and Internal
Pressure

By the same procedure as in the previous section we

can find the minimum stress, ( from Eq. (49) by Program

I in Appendix B. In this part of calculation, a numerical





47


example is given for comparison of the effect of the

imperfection in shells at various inclined angles.

This is plotted in Fig. 16.














CHAPTER VII


EXPERIMENTAL INVESTIGATION


A group of tests for the general instability of,

rings and/or stringers stiffened shells have been made by

many authors, but up to the present time, no work has been

done on the experimental investigation of the general

instability of inclined stiffened cylinders subject to

axial compression or bending. Therefore, it was necessary

to conduct a series of tests in order to compare experi-

mental with the theoretical results.

7.1 Models

The models used in these series of tests were

stiffened cylindrical shells constructed from Du Pont

Mylar of 1000 gage (0.01 inches), type A. Tests indicated

that the Mylar sheet has a Young's modulus E varied from

550,000 psi to 780,000 psi and Poisson's ratio of 1/3.

In the numerical calculation to follow, the value of E

is taken to be 700,000 psi.

All of the models were made by rolling the Mylar

sheet around a thick-walled steel tube and joining the








ends with a 3/4 inch wide strip of double-faced Scotch

tape. It appears that these joints might stiffen that

part of the cylinder appreciably. However, reports from

various investigators indicate that this is not the case,

since buckling waves appear across the joint with no

noticeable change in pattern. The present tests have

confirmed this observation. The inside radius'of the

cylinders was 4 inches, while the length was 9 inches.

The stiffeners were made by cutting the Mylar tape to a

band-width of 3/8 inch. The thickness of each layer of

the tape was 0.01 inch. All the specimens had stiffeners

made of two layers of the Mylar tape (i.e., nominal thick-

ness of the stiffeners = 0.02 inch). The double-faced

Scotch tape was used to join the layers of Mylar tape. It

was also used to bond the stiffeners with the cylindrical

shells. The adhensive tape was not effective when more

than two layers of Mylar tape were used as stiffeners.

The spacing between two neighboring stiffeners was

1 1/4 inches. The stiffeners were inclined at angles

of ( = 300, 450, and 600 with the generator of the

cylinders. In order to provide additional information

for the numerical example of the previous chapter, the

dimensions of the stiffened shells used in the experiments

were the same as those considered in that example.









7.2 Test Result

The cylinders were mounted vertically on the test

machine. The upper adaptor has two pivot pins fixed

diametrically at its edge. The pins are supported by

horizontal bearings which are fixed to the frame of the

test stand. When a moment is applied to make the upper

adaptor rotate about its pivot pins, this transmits the

bending moment to the cylinder. The lower adaptor is

connected to a circular plate which can be moved up and

down so that the axial loading can be transmitted to the

cylinder. Bending load is applied by means of a lead

screw which pulls a cable up through a bearing which is

fixed on the frame of the machine.

The bending moment and the axial compressive loading

were found after calibrating the readings from a strain

gage indicator. The test results of the stiffened shells

subject to axial compression or bending without internal

pressure at an angle of dI = 300, 450, or 600 are tabu-

lated in Table I. Some of the results are plotted in

Fig. 18. The deformed pattern after buckling at p = 0

are shown in Fig. 17. The buckle pattern formed across

the seam with no distortion in shape and none of the seams

failed during buckling.













CHAPTER VIII


DISCUSSIONS AND CONCLUSIONS


In the previous chapters, nonlinear analysis on

cylinders with inclined stiffeners have been made by the

energy method. From the numerical examples the following

observations are made:

1. From Fig. 13, it can be observed that the mini-

mum stress parameter 2 increases with increasing

stiffeners' rigidity, and the optimum inclined angle

varies with the rigidity. In the present example, it has

been found that the most effective inclined angle is in the

neighborhood of f = 600.

2. In Fig. 14, strength of cylinder with stiffener

inclined at Y = 450 is compared with that of ring-

stiffened cylinder. At smaller stiffeners' rigidity, the

ring-stiffened cylinder is stronger, but at higher rigidi-

ty, the inclined stiffened cylinders have more strength.

The minimum stress of ring-stiffened shells approaches

to the buckling stress found from small-deflection









solutions as a limit, while the inclined stiffened

cylinder continues to increase with increasing rigidity

(E I ). In the same figure, the consideration of

torsional rigidity (for example K = 1/2) is compared with

the result including the bending rigidity only (i.e.,

K = 0). The difference between these results increases

with rigidity, E I t However, the general relation of the

minimum stress with other parameters will remain the same.

3. Example of the variation of 2 versus

is shown in Fig. 15. For other combinations of Y, K and

E I T, the curve will be similar but different numerically.

4. The minimum stress of cylinder under axial

compression versus the various inclined angles is shown in

Fig. 16. Imperfect cylinder (r = 0.3) has lower strength

as expected. The evaluating of f can be referred to

Ref. (22). The effect of imperfection for stiffened

cylinders is smaller in comparison with the unstiffened

cylinder.

5. The buckling patterns of the unpressurized-

stiffened shells subject to axial compression are shown

in Fig. 17. The deformed pattern after buckling is

diamond-shaped and across the inclined stiffeners as

expected.








6. In Fig. 18, it can be observed that the theory

for the bending case is in reasonable agreement with tests

on the Mylar inclined-stiffened cylindrical shells.

This study has presented the approximate solutions

by energy method. Nevertheless, the results of this

analysis should give some insight into the problems of

the general instability of stiffened shells with inclined

stiffeners.










TABLE I


TESTS OF STIFFENED THIN MYLAR CYLINDERS UNDER AXIAL
COMPRESSION OR BENDING WITHOUT INTERNAL PRESSURE


Cylinder Radius = 4 inches
Mylar Nominal Thickness of Shell, t = 0.0075 inches
Thickness of Stiffeners, t1 = 0.02 inches

Inclined Dimensionless Dimensionless Dimensionless
angles rigidity of axial bending
stiffeners compression stress_
i Ti lc 6b
450 0.059 0.267 0.306
0.274 0.317
0.117 0.282 0.312
0.294 0.329
0.308 0.317
0.302 0.306
0.47 0.287 0.321
0.339 0.382
0.322 0.323
0.94 0.368 0.435
0.328 0.388
0.335 0.412
0.336 0.376
1.47 0.333 0.365
0.335 0.365
___0.339 0.358
300 0.124 0.281 0.306
0.315 0.321
0.992 0.302 0.353
0.322 0.335
0.314 0.353
600 0.134 0.288 0.388
0.322 0.322
1.072 0.302 0.353
0.322 0.370
0.322 0.370













































Fig. 1. Coordinates and Displacement Components of
a Point on the Middle-Surface of the Shell.
































x
yl Y2








_y



Fig. 2. The Coordinate System of the Stiffened
Shells and Stiffeners.















(a) Under Axial Compression P
c


)


(b) Under Pure Bending M


(c) Under Transverse Shear Load P

Fig. 3. The Dimensions of the Stiffened Shells under
Different Loadings.




















0
4J
u
1 II
\ + 0* ^




S\ .IS O 0<
0
14
0 II


41
el'

00
S\o



Co N
41H
S--\ H Ic
W r


O0 4.
0


tp I







c a ok 0
* H

10 ^^ ^^^ ^ ^
?1 "" ^ ^- -^^ ^ ^ 1
9 N --^'-^^'" ^^^l a ^



nl ^--^<-- r9" V d
d d
























0
*-J


4i



44
0

0
*4
4J


0
0f


S
S,

0
b4


II



0
l'


0
II








14


II





to
II













I1
'-1
>0;=


I p "i i ,
N O
a S ;
























0


0
-H






41
c
r-I










0
0

r.



)H




0
14





0
m3
M,


0
II



O





0
L



II







IN


0
I|
'-
u,










v


N
RH


__ __ __












U,
C;
0




F3


0 0

II
04 I4
C14


0 0
r; II



4-4
3j 0

o 0




k

4IJ
CO II





0 0

00


0 o
30 w
*I 0


4J 0
? II

u to








Cel





62



In








eq




44
0
N 4







1f
0 \ 0 II
v.-I 0



4 1. ra- \ i

o 0










L eq


o H

ooo
cc '. in *r H -
B B B cc
0 0 0 0 0





63



in
N










co
0 Ila,

0"
1"

0
S II


0





S1 I




SIto










Cl 4 I I



ri
U.-N 40


m -I


















NN N
S
0 -1


O CO 0 0
0 co
r4 0 0 0


II II U II II II II
i-


co r.
.. q
S
0 0


rD in
*
o o


e1 N
S *
0 0


Sl S

o 0


0

o






-H
D.




0


S4J



&I

oil
1 I




4 1









1 0

k

to
0- 0


Vc
14t 'B 41













HQ


I I I I I I I I


I J i I !






65











,N
r --
4J
U


I







O II



4

00
^ IPI


o0 0













^ as -
O O






t N N
qo 1*




+1 -


r-0


-HI
Ak '-















*
C4





N U
44
N 44



0
co
0




0

I4 f-4 -
*- :

C4 i- 4


0 0
o N o



4 o N
0 W





00
Se 1H














S e-1,
I V 4 0







/ 14 4. 41







N *
0 0 .0
M 0
HH g ;m V






67















NI
0)
II jH 0
li .0 4)
In a a

S\ 0


r 04
0



o 0




o0 #
4J r-






i M
o

0
.0





-0
0
\o *4) o#a







So




cn \D r- m
r 8 o a C; o








h(


















0 o





I \I
\ I 0U



In i0




1t
4J





41



a0'
ri


4 I W









o M aO LA m C 4 *
'-1 00 0 0 C0 0 0 0
(\ N 3






















W4






0 $ II
o n4



I l ol |
I 4'4t) II

o
:ci elqa



to II
$4





So
r m0 a 0IH



S00
0 Ht








0 Ln 4o Ln o
C;








0 IoL


C.'
















0.4 T = 0


T'= 0.3




0.3







0.2.







0-.11 I I a i
00 300 60P 90

Fig. 16. Comparison of Effect of the Imperfection Ratio
on the Stiffened Shells Under Axial Compression

(E I = 2 K = 0 p= 0)
























I I


I
a4
N
*H
o



m C




) HE
S0

a4
!= E
0






U
u









*d H
f(





U
-. 4m
1i-, i-


~clI


















So r0
S -I






o0 <








0 <


0


m














I~k
r4










CO






4
9)





(a k


14.C
S400

4 44
S.4

Oa
1 1

0
044

4C1





U 0


tn 0 9 N r0



HM




































APPENDICES




































APPENDIX A













APPENDIX A


ENERGY Ri IN STIFFENERS


To neglect the energy terms R1, R2, R3 and R4 in

Eq. (37), we may refer to Ref. (3). In that study, the

numerical results have indicated that the minimum stress

is only slightly affected by these terms. Also, it is to

be noted that the minimum stress is nearly independent of

the locations of the stiffeners. In the present analysis

and experimental results, it indicates that the shells

buckle in multiple waves pattern. This implies that m2

and n2 are much greater than unity. Moreover, from the

experiment the buckling stress is not affected by small

change in the stiffeners' locations. Thus, the summations

of sine functions of the energy of the stiffeners may be

assumed very small when compared with the constant terms.

The approximate solution can be much simplified by

neglecting these Ri terms (i = 1, 2, 3, 4). However, the

expressions for R1, R2, R3 and R4 are expressed below for

the purpose of reference.







*t (-Flr
1- 2


N {C-
S(c, + Cv )(c, Czt x
-fe~I z *'


L mC
X LZmCi


Sil m7nc, ) +


S sin Atz
LZn C ( a


- z a~ L (C


+ LCjP Ci FL-^(jinC) sin(7e-nc A)


SL(mc, +3n C.) s c 3 ) (


x-f- 2zb,


R sm7nmclnc' 4 z A z (c--a c) C
L ( c ) (L(3 cn,) LmC n'C


--.(mq-3 C. ) n L rnC+nCz)

z Ta cI ( cfac~x LL(smC ) .A3in 1-? si((.A)
+L.R s n( nc,4n (A)ns acc



L ( nc) ?. L Z ,c++n0 2)

X Sin(z m c C2)()+-" ZL(m,-c) st z( c,-n (A)


8[(c C c


RL(mC 4-C 5in Z(mc,+n7CCl)-)-







-Vc-c 2)


R -z 4
5L z (7nZ ct t C + a
L(a ?( nc aci) i z


x 5sizt rtCne) Z'4 C 4nc LnC


j 2


{.j (c. C2(c3-a4 4)2 mCe si 2Cn)3,)
4 L.rt C3R


+ R 3inzn4(-+22
?-r C4 4 4


-(C3+-4) C4 Fi x
L(IC3- ncC4)


X sin ( C3-n c4)( )+ I( C, in (,) in (,+s3 (nc




(mCf c5n (mc3--nc4) ( ) 2 (C-A +.C4) C x


S7csin (mc _) 3C4)( L(mc~ +nc4) 5 (n tn C4)
L C 3 37 C4 L 4C ) f- )l


+ bb (C3 -,C4)2 C


L 5 mc(nc 3 c3-nc4) ()
L(3m~c3-?zc4)


R (lnc) 4 C2
L(m c 3+n c4) 4) ] 4 C3


R ZL M (z3-n C4)
-2
+ (f3+ 4tC4) s2 2(Me+n )


R
xn ce


2 (bZ


RC
2L(Z n C+n e4)







3-C4 R ) 2 44
-Atl, C4-) L( SiX 2(-, C.3-n'?C4F)( 4


2 V


PR Si 471 C4 .
ILC 4 R jI
Ri


R siz(m cnc
ZL(mCz+tC) R


+2L(meC- -C) 57 L z(-t cf C -?)
2 t mcK z


X in Zmc.
\ZL77C,


fni C3
m cDt


*+2 RgL ,l
2?Lnc,


sin 4?C g(C +


SiYlZ ( eZn ( )4) 1 ?C2c:A

$LL4?U 2 ^ 4
LflTC!L^ ^


x t- Cze)(CZ-*c1 bnc si Smc nc,) (L)
(cz -,a ^ -d L )


L(mc,-C,) sin (m c1-n c) a CaTU L (m x+n gy

x5i(m +n c,) (C, -- -ae ( c nC C,- 3 n Ce+7C)


+2 C, CC [(C, -c) (c2-,C, (3m CL+4 -f sin (i3" C ,+ X)


+ sxc) (iAc.(? -7t C2) () ( aC-aC,)(c+ C)(m -n ) x


X5Cn (a3C1,-?Z)( L( C14- C)
^ ~ d~/ i '-^Vft^L c^C,)


xs51n(fWlC4X-N4 2 FC,/
R3


R O_~


R
x
I-L C4


sin (m c,+ )(}J











2L(MC- -nY C)


4
ff -ft-f


SR[ R sn z(
I8 2L (nC 3 + C4) C


itn C4 )
R-


2L(R s in m3g-nc -n'C :t4 ))
.2 L(-m C3-7 Cf) R 4


2mC3
R


t(7n1 C3)

-L
' 3 -niC-


ZL JnC4


Sin. 4n C4 (i)
R.


+ i4


Ms5 nC4( -+C3z C4


mC C_3


-Z~2a 3 c 3c4
tkbZ b3 C- 3 C4


(C, + C4) (C4- C) ')
L (mL zcn c4)


R1S;?t (&L
L ( C 3 7n C4)


+(c, C) ( a c ) nic,+n4) (

4 s3n (m -3n C )) ()J-i+ CCz b X
L. (M C3 3n C4) 3


x c-3 z a c4) (c4-- C,)


. (3S 7Z C + c


+ L (mcc-nc,) n )
L (n mc3 ?t c4) A~


c.- C4) ( c4 + i c)) x
cs c4)(c2,7 eC) x


sin L(mc+Y'C,)(4) ,


stinZmc -nc c(
a^ a


C -flC)()


SAl4 1C2 C 2 R
1 2 L(7~LnCj cZ


S2


4,m e,()


x sin (m C, -- C 4)


(4







R T5A;" (cMC-- nc4) (1) + x
. L(3mc3,-nc) L(,c~- nc4)

S (c(]nc) ) ]b( I3 (, f c+- nc4.)
2L (m c,


x.snz( C-zn c4) (i) +


SZL ,-- y S 2,-7 ) Cf A)C
(7n cl-n cW) "K^S


































APPENDIX B








APPENDIX B
COMPUTER PROGRAM I
STIFFENED CYLINDRICAL SHELLS UNDER
AXIAL COMPRESSION
1 READ INPUT TAPE 5, 10,EEG,CDQRtHPIIIlDIFJIJDJ
1F,KI,KDOKF
10 FORMAT (8F5.3,914)
DO 50 K=KIKFKD
Z=.OI*FLOATF(K)
DO 50 I=IIIF,ID
X=.O1*FLGATF(I)
DO 50 J=JIJFIJO
Y=.O01*FLOATF(J)
YS=Y**2
XS=X**2
XQ=XS**2
CS=C**2
CQ=CS**2
US=D**2
DQ=DS**2
QS=Q**2
QQ=QS**2
RS=R**2
RQ=RS**2
HM=1.-H
HP=1.+H
HMS=1./HM**2
SI=E*(CC+QQ+XQ*(DQ+RQ)+6.*XS*(CS*OS+QS*RS)+G*I((C+X*D)
1*(D-X*C))**2+((C-X*D)*(D+X*C))**2+((Q+X*R)*(R-X*Q))**2
2+((Q-X*R)*(R+X*Q))**2)/2.)/4.
S2=E*XQ*(DQ0RQ+G*(CS*DS+QS*RS))
S3=E*(CC+QC+G*(CS*DS+QS*RS))
Gl=1./(l.+XS)**2
G2=1./(9.+XS)**2
G3=1./(1.+9.*XS)**2
Al=3.*(1.+XS)**2/32.+SI
A3=-(2.*(2.+H)*(I.+Z)*GLZ/2.)*XS
A4=HP*4.*XQ*(G1*(1.+Z)**2+G2*Z**2+G3)
A5=HP*(I.+XQ)/16.
Bl=3.*XQ/8.+S2
84=HPe2.*XQ*GI*Z**2
85=HP*XQ*.5*(G3+(l.+Z)*Gl)
86=-XS*.25*G1
C1=3./8.+S3
C4=HP*2.*XQ*GI
C5=HP*XQ*.5*(G2+1t.+1./Z)*G1)
C6=-XS*(Gl+(1.+H)/8.)*.25/Z
H1=C5+C6/Y-A5
H2=B5+B6/Y
D2=AI*H2-B1IA5
D3=IG1+A3*Y+A4*YS)*H2-B4*A5*YS
F2=AI*(C5+C6/Y)-CI*A5
F3=(G1+A3*Y+A4*YS)*(C5+C6/Y)-A5*(.25+C4*YS)










Wl=(D2*D3*Hl*HI/F3**2+F2*H2*H2/F3-H2*H*(F2*D3/F3+D2)/
1F3)*HMS
W2=-HMS*3*XS*P*(D2*03*H1*(HI+A5)/F3*2+F2*H2*(H2-A5)/F
13-(2.H2*(HL+AA5)+A5*A5-H2*A5-(H1+A5)*A5)*.5*(F2*D3/F3+
202)/F3)
W3=HMS*2.25*XQ*P*P*(D2*D3*(H1+A5)**2/F3**2+F2*(H2-A5)*
1*2/F3-(HI+Ai)*(H2-A5)*(F2*D3/F3+D3)/F3)+D2*D2+(D3*F2/F
23)**2-2.*D2*D03F2/F3
DISC=W2**2-4.*WI*W3
IF (DISC) 50, 60, 70
60 XIR=-W2/(2.*WI)
X2R=X1R
GO TO 80
70 S=SQRTF(DISC)
X1R=(-W2+S)/(2.*W1I
X2R=(-W2-S)/(2.*Wl)
GO TO 80
80 WRITE OUTPUT TAPE 6, 20, ZtX,tYXIRX2RPtWIW2,W3,E
20 FORMAT (10F8.3)
50 CONTINUE
GO TO I
END




84


Symbols Used In Computer Program I


E = 'ElJ G = K

C = C1 D = C2

Q = C3 R = C4

H = P = p

Z = X x '

Y = X2R = c










COMPUTER PROGRAM II
STIFFENED CYLINDRICAL SHELLS
UNDER BENDING OR TRANSVERSE SHEAR
1 READ INPUT TAPE 5, 10,E,G,C,D,Q,R,HP,IIIDtIFJI,JD,J
IF,KIKDKF
10 FORMAT (BF5.3,9I4)
DO 50 K=KI,KFKD
Z=.0OIFLOATF(K)
DO 50 I=IIIFID
X=.01OFLOATF(I)
DO 50 J=JIJF,JD
Y=.001FLOATF(J)
ZS=L**2
YS=Y**2
XS=X**2
XQ=XS**2
CS=C**2
CQ=CS**2
DS=D**2
DQ=DS**2
QS=Q**2
&Q=US**2
RS=k**2
RQ=RS**2
HM=1.-H
HP=l.+H
HMS=1./HM**2
S1=3.*E*(CQ+QQ+XQ*(DQ+RQ)+6.*XS*(CS*DS+QS*RS)+G*((IC+X
I*D)*(D-X*C))**2+((C-X*D)*(D+X*C))**2+((Q+X*R)*(R-X*Q))
**2+((Q-X*R)*(R+X*Q))**2)/2.)/8.
S2=1.5*E*XQ*(DQ+RQ+G*(CS*DS+QS*RS))
S3=1.5*E*(CQ+QQ+G*(CS*DS+QS*RS))
Gl=l./tl.+XS)**2
G2=1./(9.+XS)**2
G3=i./(l.+9.*XS)**2
A1=9.*(I.+XS)**2/64.+S1
A3=-(1.5*(2.+H)*(1.+Z)*Gl+3.*Z/8.)*XS
A4=HP*9.*XQ*(Gl*(I.+Z)**2+G2*ZS+G3)/4.
A5=HP*9.*(l.+XQ)/256.
81=9.*XQ/16.+S2
b4=HP*9.*XQ*G1LZS/8.
65=HP,9.*XQ*((l.+Z)*Gl+G3)/32.
L6=-(3.*XS*GL)/16.
CI=9./16.+S3
C4=HP*9.*XQ*G1/8.
L5=HP*9.*XU*((1.+1./Z)*Gl+G2)/32.
C6=-3.*XS*(HP/8.+Gl)/(16.*Z)
HI=C5+C6/Y-A5
H2=B5+B6/Y
02=A1*H2-B1*A5
03=(G1+A3*Y+A4*YS)*H2-B4*A5*YS
F2=AI*(C5+C6/Y)-CI*A5










F3=(G1+A3*Y+A4*YS)*(C5+C6/Yl-A5*(.25+C4*YS)
W1=HMS*.25*(D2*D3*HI*Hl/F3**2+F2*H2*H2/F3-HIlH2*(F2*D3
I/F3+021/F3)
W2=-HMS*1.5*XS*P*(D2*D3*Hl*(HL+A5)/F3**2+F2*H2*(H2-A5)
I/F3-(2.*H2*(HI+A5)+A5*A5-H2*A5-(HI+A5)*A5)*.5*(F2*D3/F
23+D2)/F3)
W3=HMS*2.25*XQ*P*P*(D2*D3*(Hl+A5)**2/F3**2+F2*(H2-A5)*
1*2/F3-(Hl+A5)*(H2-A5)*(F2*D3/F3+D3)/F3)+D2*D2+(D3*F2/F
23)**2-2.*D02D3*F2/F3
DISC=W2**2-4.*W1*W3
IF (DISC) 50, 60, 70
60 XIR=-W2/(2.*WI)
X2R=X1R
GO TO 80
70 S=SQRTF(DISC)
XIR=(-W2+S)/(2.*W1)
X2R=I-W2-S)/(2.*Wl)
GO TO 80
60 WRITE OUTPUT TAPE 6, 20, Z,XYXIR,X2RPtW,W2,W3,E
20 FORMAT (10F8.3)
50 CONTINUE
GO TO 1
END








Symbols Used In Computer Program II


E = El 7

C = C1

Q = C3

H = F



Y = f


G = K

D = C2

R = C4

P = p

x =

X2R = 2 2 = 2(b + 3/2') or

(X2R = 2 3 = for transverse
load)












BIBLIOGRAPHY


1. W. Fairbairn, "On the Resistance of Tubes to
Collapse," Phil. Trans. Roy. Soc., Vol. 148,
pp. 389-413, 1859.

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

3. A. E. H. Love, A Treatise on the Mathematical Theory
of Elasticity, fourth Edition. Reprinted in the
U.S.A., by Dover Publication, New York, N. Y.,
1944.

4. S. Timoshenko, and J. Gore, Theory of Elastic
Stability, McGraw-Hill Book Company, New York,
N. Y., 1961.

5. R. V. Southwell, "On the General Theory of Elastic
Stability," Phil. Trans. Roy. Soc., Series A,
pp. 187-213, 1914.

6. L. H. Donnell, "Stability of Thin-Walled Tubes under
Torsion," NACA Report 479, 1933.

7. S. B. Batdorf, "A Simplified Method of Elastic-
Stability Analysis for Thin Cylindrical Shells,"
NACA Report 874, 1947.

8. H. S. Suer, and L. A. Harris, "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,
pp. 138-139, 1959.

9. P. Side, and V. I. Weingartern, "On the Buckling of
Circular Cylindrical Shells Under Pure Bending,"
Journal of Applied Mechanics, Vol. 28, No. 1,
pp. 112-116, March, 1961.