Citation
Orthotropic cylindrical shells under dynamic loading

Material Information

Title:
Orthotropic cylindrical shells under dynamic loading
Creator:
Mangrum, Elmer, 1936-
Place of Publication:
Gainesville FL
Publisher:
[s.n.]
Publication Date:
Copyright Date:
1969
Language:
English
Physical Description:
xvi, 152 leaves. : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Computer programs ( jstor )
Cylindrical shells ( jstor )
Damping ( jstor )
Eggshells ( jstor )
Equation roots ( jstor )
Inertia ( jstor )
Method of characteristics ( jstor )
Static loads ( jstor )
Structural deflection ( jstor )
Velocity ( jstor )
Buckling (Mechanics) ( lcsh )
Cylinders ( lcsh )
Strains and stresses ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 149-152.
General Note:
Manuscript copy.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
025366980 ( AlephBibNum )
AFP3837 ( NOTIS )
20143414 ( OCLC )

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Full Text








ORTHOTROPIC CYLINDRICAL SHELLS

UNDER DYNAMIC LOADING















By
ELMER MANGRUM, JR.













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












UNIVERSITY OF FLORIDA
1970





































UNIVERSITY OF FLORIDA
3 122 08552 3057
3 1262 08552 3057
































This dissertation is dedicated to

my wife Rita

and my daughter Gaila.











ACKNOWLEDGMENT


I would like to acknowledge the support and encouragement

of General William M. Thames, K. A. Campbell, and

N. E. Munch of the General Electric Company who made

this research possible. I wish also to express my sincere

gratitude to Dr. J. J. Burns for this guidance and sugges-

tions during the course of this research.















TABLE OF CONTENTS


Page


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


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


KEY TO SYMBOLS .............


ABSTRACT . . . . . . . . .


Chapter


I INTRODUCTION ..........


Statement of the Problem . .


Specific Goals of This Research .


Review of Previous Work . .


Contributions of This Work . .


II GOVERNING EQUATIONS OF MOTION


General Equations .......


Axisymmetric Loading . . .


Pressure Loading Form ....


Nondimensional Equations . .


III TRANSFORMATION OF EQUATIONS .


IV INVESTIGATION OF THE CRITICAL VELOCITIES .


. . . . . . vii


. . . . . . viii


. . . . . . xi


xv
........... XV








. . . . . .1


. . . . . . 2


. . . . . . 3


. . . . . . 3


. . . . . . 5


. . . . . . 5


. . . . . . 15


. . . . . . 15


. . . . . . 16


. . . . . . 20
20


. . . 23






TABLE OF CONTENTS (Continued)


Chapter


V SOLUTION FOR DISPLACEMENTS . . . . . .

General Solution . . . . . . . . .

Transformed Displacements . . . . .

Inverse Transformation of the Rotation .. ...

Inverse Transformation of the Radial Deflection

Inverse Transformation of the Axial Deflection

Summary of Deflection Expressions . . .

Solution for No External Damping . . . . ..

Form of the Radial Deflection in Region IV . .

Form of the Radial Deflection in Region VII . .

Form of the Radial Deflection in Region V . .

Form of the Radial Deflection in Region VI . .

Comparison of Solution with Other Results ..

Numerical Results . . . . . . . .

Comparison of Results with Other Solutions for
a Static Load . . . . . . . .

Summary of Deflection Response for Shells under
Various Load Velocities . . . . . .

Region II Response . . . . .. . .

Region IV Response . . . . . . .

Damping Effect on Regions V and VII Response

Region VI Response . . . . . . .
2
Deflection Behavior in the Vicinity of X2CR
2
Effect of Prestress on 1 CR .........

Superposition of Step Loads . . . . .

Study of Material Properties Variations . .

v


Page


S. 42

42

S. 42

S. 43

46

51

S. 55

S. 58

S. 59

S. 62

S. 63

S. 63

. 64

S 65


S 65


S. 69

71

S 71


71


78
S. 78

78

. 82

82






TABLE OF CONTENTS (Continued)


Chapter Page


VI STRESSES . . . . . . . . ... .. ... . 92

Development of Stress Equations . . . . ... 92

Numerical Results . . . . . . . . . 94

VII CONCLUDING REMARKS ................. 98

Conclusions . . . . . . . . ... .. . 98

Suggestions for Future Work. . . . . . ... 99

Appendix

A FOURIER TRANSFORM OF THE FORCING FUNCTION . . 100

B SOLUTION OF EQUATIONS FOR THE TRANSFORMED
DEFLECTIONS . . . . . . . . . . . 102

C EVALUATION OF THE DISCRIMINANT OF A FOURTH ORDER
POLYNOMIAL .. . .. .. .. .. . .. . . 106

D DETERMINATION OF THE CRITICAL VELOCITY
EQUATIONS . . . . . . . . . . . 111

E COMPUTER PROGRAM FOR SOLUTION OF LOAD VELOCITIES
WHICH CAUSE REPEATED ROOTS IN THE UNDAMPED
CHARACTERISTIC EQUATION. . . . . . . . 116

F PARTIAL FRACTION EXPANSION OF A FOURTH ORDER
POLYNOMIAL . .. . .. .. .. ... .. . 123

G CHECK TO SEE THAT SOLUTIONS SATISFY THE GOVERNING
DIFFERENTIAL EQUATIONS . . . . . . ... 125

H COMPUTER PROGRAM FOR DEFLECTION AND STRESS
CALCULATIONS ................... .. 130

I RELATIONSHIPS BETWEEN ANALYSIS PARAMETERS . . 147

BIBLIOGRAPHY . . . . . . . . . . . . . 149

ADDITIONAL REFERENCES ................... 151












LIST OF TABLES


Page


Correlation of Root Type with Region Numbers for
Figures 4.2 through 4.8 . . . . . . . . . 27


H-1 Options Available for Program DEFSTR . . . . .. 131


Table











LIST OF FIGURES


Figure Page


2.1 Cylindrical Coordinate System. . . . . . . 5

2.2 Pressure Loading . . . . . . . . . 16

4.1 Flow Diagram of Computer Program VCRIT which Determines
Load Velocities at which Repeated Roots Occur ...... 28

4.2 Classification of Roots of the Undamped Characteristic Equation
for Variations in the Thickness-to-Radius Ratio Including
Prestress . . . . . . . . . . 29

4.3 Classification of Roots of the Undamped Characteristic Equation
for Variations in the Thickness-to-Radius Ratio with No
Prestress . . . . . ... . . . ... . 30

4.4 Classification of Roots of the Undamped Characteristic Equation
for Variations in E og/E o . . . . . . . . 31

4.5 Classification of Roots of the Undamped Characteristic Equation
for Variations in Gxzo/Ex . . . . . . . .. 32

4.6 Classification of Roots of the Undamped Characteristic Equation
for Variations in Evo/Exo . . . . . . . . 33

4.7 Classification of Roots of the Undamped Characteristic Equation
for Variations in the Circumferential Prestress ...... 34

4.8 Classification of Roots of the Undamped Characteristic Equation
for Variations in the Axial Prestress. . . . . . 35

4.9 Path of the Roots of the Undamped Characteristic Equation
for Increasing Load Speed (Beginning in Region I) . . .. 37

4.10 Path of the Roots of the Undamped Characteristic Equation
for Increasing Load Speed (Beginning in Region V) . . .. 39

5.1 Contour Integration Path for Evaluating Rotation Integral . 44

5.2 Integration Contour for Radial Deflection for ( > 0, bk > 0 47

5.3 Integration Contour for Evaluating Radial Deflection
for p < 0, bk > 0 . . . . . . . . . 49





LIST OF FIGURES (Continued)


Figure Page


5.4 Loci of the Roots of the Characteristic Equation
as the Damping Approaches Zero . . . . . . .. 60

5.5 Static Load Problem . . . . . . . . . 64

5.6 Flow Diagram for Computer Program for Deflection and
Stress Calculations . . . . . . . . . . 66

5.7 Radial Deflection for a Static Load on an Isotropic Shell
(p = 0.3) . . . . . . . . . . . .. 67

5.8 Displacements for a Static Load on an Isotropic Shell
(p = 0.3, h/R = 0.01) . . . . . . . . . .68

5.9 Radial Deflection Shape for Various Load Velocities . .. 70

5.10 Radial Deflection Response for Variations in Radial
Damping (A2 = 2.0) . . . . . . . . . .72

5.11 Radial Deflection Pattern Immediately Above the First
Critical Load Speed . . . . . . . . . . 73

5.12 Change in the Radial Deflection Pattern with Increasing
Damping (?2 = 2.7) . . . . . . . . . .74

5.13 Deflection Wave Form at 2 = 5 and 10 . . . . .. 75

5.14 Effect of Damping for X2 = 500. . . . . . . .. 76

5.15 Effect of Damping for X2 = 2000 . . . . . ... 77

5.16 Deflection Response for Variations in Circumferential
and Axial Prestress . . . .. . . .. .. .79

5.17 Maximum Radial Deflection in the Vicinity of the First
Critical Load Speed . . . . . . . . . 80

5.18 Effect of the Axial Prestress on the First Critical Load Speed. 81

5.19 Variation of Pressure Pulse Length, d, at X = 30 . . .. 83

5.20 Response from a Smooth Sine Wave Type Pressure Pulse
Using Superposition . . . . . . . . . . 89

5.21 Response from a Sharp Pressure Front Using Superposition 90

5.22 Radial Deflection Response for Variations in the Circum-
ferential Modulus . . . . . . . . . . . 91





LIST OF FIGURES (Continued)


Figure


Bending Stress in an Isotropic Shell Under a Static Load
(p = 0.3, d = 1) . . . . . . . . . . .

Surface Stresses in an Isotropic Shell Under a Static Load
(p = 0.3, h/R = 0.1) . . . . . . . . . .


Page












KEY TO SYMBOLS


x, 0, z

Ka ( = x, 0, z)

R

h

U, V, W

0, 77

t

No, Nap (ra,3 =x, z)

Ma' Map (aCe =x, 8, z)

Q( (a = x, 0)

T

N

I

h

Pi

PO

C (a = x, 0)

VY a (", = x, 0, z)

CT ( = x, 0)

Tr (&, p = x, 0, z)

Exo

E0o

E10


coordinate axes

unit vectors in coordinate directions

radius of cylinder (to the middle surface)

thickness of cylinder

displacement in directions of coordinate axes

rotations

time

stress resultants

moment resultants

shear force

axial prestress stress resultant

circumferential prestress stress resultant

moment of inertia

defined in Equation (2-7)

initial lateral pressure

mass density

strain in a direction

shear strain

stress in a direction

shear stress

modulus in x direction

modulus in 0 direction

modulus in normal direction






GOpo (a,p = x, 0, z)

D (o = x, 0, i)

Dx0

Ex (c = x, 0, v)

Gx (c=x, 0, v)

Gx0

I
2
Ka (a = x, 0)

q'(x, 0, t)



C..
1j
H(y)

q

V

t



U, W




F

P

E
1

G

E
o
r




0
I
o
E


shear moduli

defined in Equation (2-12)

defined in Equation (2-12)

defined in Equation (2-12)

defined in Equation (2-12)

defined in Equation (2-12)

defined in Equation (2-12)

correction factors

time varying lateral pressure load

damping coefficient

coefficients of a matrix

Heaviside step function

magnitude of lateral pressure load

constant velocity

constant

defined in Equation (2-29)

dimensionless displacements in axial and normal
directions

dimensionless axial distance

dimensionless axial stress resultant

dimensionless circumferential stress resultant

dimensionless normal modulus

dimensionless shear modulus

dimensionless tangential modulus

dimensionless inertia term

dimensionless load velocity

dimensionless rotatory inertia term

dimensionless damping term





r
o
D1

f(s)

s

c, c. (i = integer)

C.

D(s)



e. (i = integer)

f. (i = integer)
1
\i (i= 1, .... 8)


Xi

a

b

s.
i, j

AiCR

VCR

E



qo

c! (i = 5, 9)

d

Fk

ak, bk

R

Res(a)

a k' Ok


thickness ratio depending upon pressure direction

dimensionless constant

transform of f(p), F[f(P)] = f(s)

complex variable in transform space

constants

constants

characteristic equation

discriminant

coefficients

coefficients

load speeds giving repeated roots in undamped
characteristic equation

approximate load speed roots

constant, real part of complex root

imaginary part of complex root

nomenclature used to trace root loci

the ith critical load speed

dimensional critical load speed

Young's modulus

Poisson ratio

dimensionless pressure load magnitude

coefficients

dimensionless load length

functions defined in analysis

real and imaginary part of complex root

large radius defined in analysis

residue of a function at point a

coefficients in partial fraction expansions


xiii





, a

K K, K'
1 2 1
/3

(0 )o (o = x, 0)

(0 )i (ao= x, 0)

Sk


lengths defined in static problem analysis

coefficients

coefficient, defined in Equation (6-24)

outer surface stress

inner surface stress

elements of determinant

determinant of a matrix of coefficients

integer












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


ORTHOTROPIC CYLINDRICAL SHELLS
UNDER DYNAMIC LOADING

By

Elmer Mangrum, Jr.

August 1970



Chairman: Dr. J. J. Burns
Major Department: Engineering Science and Mechanics


An orthotropic right cylindrical shell is analyzed when subjected to a dis-

continuous, finite length pressure load moving in the axial direction at constant

velocity. The analysis utilizes linear, small deflection shell theory which in-

cludes the effect of axial and circumferential prestress, transverse shear de-

formation, and external radial damping.

The problem is solved using Fourier transforms, and the inverse Fourier

integrals are evaluated for the radial deflection, axial deflection and rotation

by expanding the Characteristic Equation in partial fractions and using complex

contour integration. By studying the discriminant of the undamped characteris-

tic equation the load velocities which give repeated roots are determined. The

loci of these load velocities separate regions in which the form of the displace-

ment solutions differ. The behavior of these load velocity loci is studied for

variations in the three nondimensionalized material moduli, the thickness-to-

radius ratio, the axial prestress, and the circumferential prestress.







By tracing the root loci of the undamped characteristic equation and by in-

spection of the displacement expressions, it is determined that there are five

critical load velocities (velocities at which the displacement becomes unbounded)

for the specific example of an isotropic shell. An increase of the load velocity

above the bar wave speed produces a deflection mode which is predominantly

axial.

The deflection response is investigated for numerous combinations of load

speed, material properties, length of pressure load, axial and circumferential

prestress, and radial damping. The axial prestress has a significant effect on

the first critical velocity of the cylinder; initial compression tends to lower

the velocity. Circumferential prestress has no pronounced effect on the critical

load speeds but does influence the response at higher velocities. Variation of

material properties was found to cause a rapid change in deflection response.

Through superposition, the variation of pressure load length can be utilized

to approximate the response to any desired pressure load. Examples of this

application are demonstrated. A comparison of stresses and deflections against

those predicted by the Timoshenko thin shell theory is shown for a static load.

All of the above numerical work was done using dimensionless parameters

which can be applied to thin shells in general. The calculations were done utiliz-

ing a computer program developed from this research for the calculation of de-

flections and stresses in the shells. The program is written in Fortran and is

operable on the General Electric Company Mark II time sharing service.











CHAPTER I

INTRODUCTION


Statement of the Problem


One of the most commonly used geometries for structural application is the

right circular cylindrical shell. This is particularly true in the aerospace field

and in undersea exploration vehicles. In many aerospace applications the cylin-

drical shell serves as the primary load carrying member for the rocket system

and performs simultaneously as a portion of the pressurized fuel tank. In under-

sea applications the quest for greater depth range has brought about many re-

finements in structural optimization techniques. A result of the many stringent

requirements being placed upon structural systems has resulted in two areas of

rapid advancement: new material technologies and more sophisticated analysis

techniques.

The material technologies for advanced design applications have in many

cases moved away from the isotropic materials and are utilizing orthotropic and

anisotropic materials to satisfy the demanding requirements for more efficient,

lighter weight vehicles. Studies such as that reported in Reference (1)* have

shown that there is indeed an incentive for the application of these advanced

technologies.

Until recent years the mathematical complexity encountered when approach-

ing the dynamic analysis of shells has been so formidable that few results were

available for design applications.


*Denotes entries in the Bibliography.






The new technology demands mentioned previously have brought a response

from the analysts in the past five to ten years and some of the more idealized

dynamic shell problems have been investigated. The problem of particular in-

terest in this work is that of a thin orthotropic cylindrical shell subjected to an

axisymmetric pressure load moving in the axial direction. It is necessary to

consider refinements to the theory such as the transverse shear deformation,

axial inertia, and rotatory inertia effect so that the higher load velocities may

be investigated.

It is known that axial prestress has an influence on results in dynamic

analyses. In this work the effect of axial as well as circumferential prestress

is investigated. The specific loading considered will be a constant pressure

pulse finite in both magnitude and distance which moves along the cylinder

at velocity V. The shell theory utilized is linear, assuming small deflections,

and by superposition it is possible to investigate the effect of various pressure

pulse shapes. External radial damping is also included.


Specific Goals of This Research


The major goal of this research was to obtain a solution for the deflections

and stresses associated with the problem outlined above. The secondary goal,

although perhaps not secondary in importance to those interested in utilizing the

results, was that of developing a computer program for the calculation of deflec-

tions and stresses in the cylinder. Finally, the calculation and presentation of

the effect of the many parameters included in the analysis conclude the goals to

be reached in the study.







Review of Previous Work


A review of the early work on the response of a cylindrical shell to a moving

load is given by Jones and Bhuta (2). Until the work by Nachbar (3), who con-

sidered the dynamic response of an infinitely long cylindrical shell to a semi-

infinite step pressure load, the axial inertia was not considered. Nachbar

included the axial inertia effect and also assumed an external damping effect.

However, due to the damping included, the first resonance condition was missed.

Jones and Bhuta solved the problem with a ring load moving on an infinitely long

cylinder but did not include the transverse shear effect.

Other contributions were made by Reismann (4) who included the effect of

axial prestress, which was significant as had been found in his work on plate

strips (5). Hegemier (6) studied the stability problem for a large class of con-

stant velocity moving loads but limited the velocity range to that lower than the

first critical.

All of the work previewed above was done for an isotropic material. More

recently Herrmann and Baker (7) solved the problem of a moving ring load on a

cylindrical sandwich shell of infinite length. Numerical results were presented

for a core material which is assumed to have material damping. Also, the

problem of a ring load moving on a viscoelastic cylinder was solved by Tang (8).


Contributions of This Work


The following contributions are believed to be original with this work.

1. Analysis of orthotropic monocoque cylindrical shells including trans-

verse shear deformation, axial and rotatory inertia, radial damping,

circumferential and axial prestress under a finite length step load.

2. Presentation of the forms of the solution with no damping for the seven-

dimensional space whose coordinates are the thickness-to-radius ratio,




4

the three material property ratios, axial prestress, circumferential

prestress, and the load velocity parameter.

3. Results indicating the effect of a finite length pressure pulse, and the

capability to approximate any load shape through superposition.

4. Indication of the effect of prestress on the critical velocities of an

orthotropic monocoque shell.

5. Results which show the effects of external damping throughout the load

velocity range.












CHAPTER I

GOVERNING EQUATIONS OF MOTION


General Equations


A cylindrical shell of thickness h and mean radius R is referred to the co-

ordinate system shown in Figure 2.1.


Figure 2.1. Cylindrical Coordinate System


Coordinate x is measured along the shell axis, 0 along the circumference

and z is perpendicular to the middle surface. The unit vectors tangent to the

coordinate lines at a point (x, 0, z) are designated by Kx, Kp, K The dis-

placements in these three directions are ux, u and uz respectively. It is






assumed that the displacements can be represented by the linear relationship in

terms of z

u (x, 0, z, t) = u(x, 0, t) + z4(x, 0, t)


uo(X 0, , t) = v(x, 0, t) + Z7(x, 0, t)


u (x, 0, z, t) = w(x, 0, t) + z z(x, 0, t) (2-1)

where u, v, w are the displacements on the middle surface (z = 0); t denotes

time; 0 and 7r are the rotations of a line perpendicular to the normal surface in

the x-z and 0-z planes respectively. 0z is referred to as the thickness stretch.

Equations (2-1) require that all straight lines normal to the middle surface

of the shell before deformation remain straight after deformation. This is a

good approximation if the shell is thin.

Herrmann and Armenakas (9) derived a linearized theory for the motion of

isotropic cylindrical shells subjected to a general state of initial stress by as-

suming the final state of stress is reached by passing through an intermediate

state, the state of initial stress. Subtracting the initial equilibrium equations

from the non-linear equations of motion and then linearizing by disregarding all

non-linear terms involving the additional stresses, the linearized equations of

motion for a shell under initial stress are obtainable.

Following this procedure, Baker and Herrmann (10) derived a linearized

set of equations for the motion of orthotropic shells. Assuming an orthotropic

cylindrical monocoque shell is under initial lateral pressure p., an axial tension

T, and is subjected to a time dependent radial load, the five equations of motion

have the form:


Nx 1 O x u N 8u N 8w I
+ + T -+ hpu 0 (2-2)
ax R ao aX2 2 2 Rax hp R
ax R ao







1N aN Q xw aV
1 8 xe + 8v N 8w 8av "
i + +-+T +2a = hpov
R 0a ax R aX2 R 2 00a
ax R\80


SaQ aw N 8v
- -+-- -8 + h w+


N Q 2 + 2
R ax R aP 2 a2 0 a2 )
axx R 8


1 o)(w+ v)
(h) h
+2 +
R R R a8 R


+ q'(x, 0, t) = (w + p hw
___ 1 __


Mx 1 Mx
-X+- -x Q x
ax R 8x x


RM aM
1 80 x8M
+ -ax 0
R 80 8x '0


2
2 ) + au
R2 a8 ax


h
o ( w
R ax


I .
= I + u
R


8w R)
+ (h -R)o7
30 o J


I;R"
R


piR [1 +
N

+ pR [i -]


for external pressure


(2-7)


for internal pressure


3Po
I h (2-8)
These equations include the effect of external radial damping, axial and

These equations include the effect of external radial damping, axial and


rotatory inertia, and transverse shear deformation.


(2-3)


+h 1
o 8x


(2-4)


(2-5)


Nh
+ 0
R2
R


)h
+ o


where


(2-6)


h =
0


IV- (1





8

The elongations, shears, and rotations have been assumed small in com-

parison with unity. The strain-displacement relations are therefore taken in

the form of Hooke's law.


au
Dx ax


C i1 av ,
e R z o w)

1 8u + v
Yx0 R + z aO ax


Su aw
xz 8z ax

Sav 1 (w v)
z az R + z \80

The stress strain relationships are assumed in the following form

ax = Exo Cx + EV C
x ox, VQ 6o


(2-9)


0 = Evo x + E0o



*xO = Gx0o YxO


Oz = GOo YOz


Txz = Gxo Yxz (2-10)


Integrating the stresses through the shell thickness the z variable is eliminated

and the stress and moment resultants are obtained.







D
= au + x
x 8x R 8x


= G +
xo ax


D a
R 8x


v Dx0
ex x0 ax R2
R


D
x au
R 8x


D 0
R 80


DX


R

D Gx
R


= G


D
+ x0 8u
R3 a- /


D
+ D + -
x 8x R


w
R


1
R


+D "
v ax


8e



8e


+ D
x0 ax


1
R


au


0 8x
x6 ax


G 0
- R + a
R ag


(2-11)


D

R2


+5
8
80 I


Nx
x0


0. + E au
80 v ax


G
R x


S(G
R


+- v


(Eo
o R


8aw
ax-
ax


+ Do
R3) (W






where


D = Ex I, D
X x0 2' 0


= GxOo 2, Ex


= E, h,
o


G = K2Gh
G = K Goz h


= E 0 12 D = El
o 2' v o 2


= Ex h,


E0 = E0o h


Gx h, G = K Gxz h
o x X o


+ h+ 2 2
12R2


S3
'2 12


The coefficients K and K0 are constants for adjustment and can be taken as
x 0
7r/V2 as discussed by Mirsky and Herrmann (11).

Substituting Equations (2-11) into the Equations of Motion, (2-2) through (2-6),

yields

2 / E2
8 u x 8 v 8w 8 v
E + + +
x 2 R 2 R \ax ax8
8x ax


1
R


[G


a2v
axa 0


Dx0
2
R


82O
2
80


(x+ + ) u1
R R ao
( GR 88


2
N u N 8w
2 2 R ax
R a8


1 E R3D aw
T(R R 3 a)


2
ax2
ax


Gx
R


+ 2
802 )


2
8 u
8x80


IV
- hpou +


D 2o
R3 82
R ae8


G 0


E 2
R ax8
R axa8


1 aw v +
R 8 R


I .
= p hv + .
0 R


(2-12)


2
+ T 8_U
2
ax


(2-13)


D
+
R


2
axv
ax


a2v
T 8 x
8x2


S8w
8a


2
+ v
2
8a


+ )]


(2-14)


+ h 1w
0 RiT a









R2E


+ ) v+
R,) +'0


1 2w
18w


N ho) (
R2 R


+ q'(x, 0, t)


Dr a2
R 2
ax


D
R


= w + p hw


2
+D C
x 2x
8x


2
1
0x00


R2 (\2 R


D 2
+D a2,1
R 8x80


Gx 8x -


1 a8v
R 062]


2 Go
D x 2 R 7
xOax


+w )
ao V


o aw
+ h + (h R)r =
R 10 0 (ho


Collecting coefficients on deflections gives


R


+ [ (E +

S[Dx 2

R ax 2
ax


2
8G 1 +
G ) v + (E N) w
x0 8x8 I R V ax


DxO a2
R3 002
B 80


21
I 8t
I a2 = 0
T at2


+ -Go
R 80


E
Sa8u
R ax


+ D

R3 ao


8v
ao)


+ +
ao )


h (1


2
8w
+ T
ax2


hR
R 8
80 Q


+ G I-
ax


+Rh
o ax


+R u
ax


(2-151


802


1
R


Nh
o
R


I "
SI +-R u


(2-16)


D aV
R 8x80


+ xe0 /ax
R \axa o


ax2 )


Nh [

SR2


" I
77+ -
R


E x2
E x2


(2-17)


82
+ T a2
2
ax


N 2
+ o h
2 2 0o
R a0


32]u
2
at


(2-18)


+ 2
ax2)


8 '0
8 0


N av
R2 o


+ aw
ax


+ x/ 802
R ) 0 a







+ + N + (Gx + T)
2 2 xx 2
R 8 a ax


G 2 ]
- -p h- v
R2 o at2

+ 1+ w


Go
+
2
R


D 2 G hN
x8 a8 o
+ ax2+ R+
R 8x R


I 2
I a2
Rat2


(E +- D
0 R


Go
R2


(E +
0 (


a+
at


Do h \
R2 R+


p0a 2
atI


+ + RG0



D a2 Dx a2
R ax2 R3 a02


G) + D
+ Gx+ D a
/ -ax2


2
a 1
(G + T) 2 (G
x 2 20
ax R
SNh a ]
x R ax


= q'(x, 0, t)


I a2
R t2
at


Dx0 2 aa
2 2 2-
R ao at


+ [ .(D +DxO) a ] = 0
a[ 0


+


D 2
Ra 8
3 2
R ao


1
L (Ev


+ [ 1
R [ R


1
[R2


(2-19)


h
+
R


2

ao
+ N)
80


(2-20)


+


Nh
R
(


-N) -L
ax


(2-21)


2
1 8 1
(E + G ) u + ( E
Ril v x0 axaoI 2 0


1 (E
R 2 0


a0
R2 T
Rl


+ Nho 1 +


S+ Nh ) G
x R ax




13

[D 2 D a2 Nh \ a2
x0 8a 0 8 1 o I R
R ax2 R3 a2 R R at


+ [Do Nh ( +
+ -+ G + +- w


1 2
+ [ (D + DxO)

D 2
S 2 D 2 Nh a
ax R 2 0 at
S + Go ( -R + 1 -I = 0 (2-22)


Equations (2-16), (2-17), and (2-18) express the principle of linear momen-

tum in the x, 0, and z directions, respectively, while Equations (2-21) and

(2-22) express the principle of angular momentum about an axis through the

middle surface in the direction of K0 and Kx, respectively.

These equations can be written in the form

C.. u. = q, C.. = C.. (2-23)
1ij J i 1J Ji
In matrix notation the set has the form


C11 C C13 C14 0 u 0

C C C 0 C v 0
12 22 23 0 C25 v
C13 C3 C33 C34 C35 w = q'(x, 0, t) (2-24)

C14 0 C34 C44 C5 4 5

0 C25 C35 C45 C55






where the coefficients are defined as follows:


a2
= (E + T)
ax


+ (G
R


Sa2 a2
+ N a -Poh
/ ao at


12


C
13


D 2
C x a
14 R x 2
ax


D 2

R3 ao2
R ao


= (Gx +T)
xO


2
2
ax


D
R2
R 2


1 E
23 R2


D
2
R


Sh 0
+G +N 1+_
0 R


a2 Do 22
a2 2
ax R a0


= -(G + T)
x


2
a
ax2


G Nh
0 o
+-+ R2


1
R2 (G0
R2 0


2
+N) a
802


I a2
R 2
R at2

1 (E
R


D h
+-Do + N-
R2 1R
R


a a2
+ t p h 2
aat2


( h\
G + N h

1 6
2 D0
HR RLe


a2
C = D
44 X x 2


1
C45 R (D


S a2
= D ax


+D ) a
x0 axa8


1 2
I (E + Gx ax
R v x0 axao


1 a
R (Ev -ax


Dx
2
R


2
I a
R at2


C22


2
2
ao


G a2

R -oth
R 0 at2


Dx0
R


C
25


C
33


C
34


C


a
x


hR


h
-N-
R


+ Nh (1
0


-G
x


a2
a02


a
J Q

a2
at-
at'


C
55


+ D82
R2 a2
R ao


h (
-2 1
R


-G, -N
0


h)
R


-4
at
at2


(2-25)


+ -R E


Dx0
+
R2






Axisymmetric Loading


Assuming the loading on the cylinder is axisymmetric, the set of Equations

(2-24) reduces to the following.


(E + T) p h
XR2 2 0









1a I 2u
-(E -N) -oh
ax x


+ (E + D- + N


Dx a2 u I 82 u
R ax2 R at2


Gx+ + N =
S hR


S+ (E N) -- + I = 0
at2 R ( ax R ax2 R at2

2 2
w aw 8 w
+T) a2 + ph
ax2 at o at2


w- + N ) = q'(x, t)
R x R x


+ N o aw +D a2 i 2
Sx R x ax 2 at2
/ Ox at


0


Pressure Loading Form


A step input in external pressure which is finite in both magnitude and time

and which travels down the length of the cylinder at constant velocity V can be

represented in the form

q'(x, t) = q{H[Vt x] H[V(t t ) x]} (2-27)

where H(y) is the Heaviside step function, defined as

0, y < 0
H(y) = (2-28)
1, y>0


(2-26)




16

This pressure loading is represented schematically in Figure 2.2.


q --- V




R 0


0 x

Figure 2.2. Pressure Loading


Nondimensional Equations


The steady state solution will be investigated. Making the transformation

a = x Vt (2-29)

the partial derivatives may be written in terms of a.


ax a8x aa

a a a
=-V-
at aa at aa


S= 2 (2-30)
at a82

Using these relationships in Equations (2-26), letting

u
U-U
R

w
W -(
R

a R (2-31)






multiplying by R/Ex in the first two equations and 1/Ex in the third gives


T dU
+ d2U
x d/ 2


IV2
E R2
x


Oh V
E
x


dU
d02


E
+
E
S


D
+ x
2
RE
x


2
dp_
d62


d 2 = 0
do2


E V N dU
EEx dx d



+ + --
E R2
x


+ E
x /


d2W
d 02


_VR dW
E d
x


oh V2
E
x


d W
dp2


+ h )
N
ER
x


[H(-R)) H(-Rp Vt )]
0


V2I d2U
R2 E d 2
x


N h\o dW
E R d


IV2 d2
E R2 dq2
x


Gx
E
x


N ho =0
+ Ex
ER
x


/G
x
E
Sx


N h
+E -
E R
x


= q
E
x


dU
d (2


D
x
R2 E
x


D
+ x
R2 E
x


d22
d02


(2-32)


N ) dW
SE x d







Now the

T
E
x

N
x

E
E
x

E
E
x


h
0
R


18

following dimensionless ratios are defined.


= F



-P



= o


G E D
-x E 0 -
E G, E -
1' Ex o Ex ER
Ex x R
(1 +


= r
o


vYp R
2 O R
E
x

Po h
P R
o


I
I R
p R3
o


h3
12R3 P
12R3 p
o,


h3

12R3


2 = (2-33)
E p
x o

The dimensionless velocity parameter is denoted by X and r refers to

the inertia. A distinction will be made in the axial and radial inertia terms for

later investigation, r denoting axial and r denoting the radial inertia terms.


D
x
= D
R2 E
x






Using the dimensionless ratios given in Equations (2-33), the equations of motion

now take the form


(1 + F r ) + (E P) + (D I 2)
1 d$ (P2 1 d 1 0o


dU
(E ) -
1 d


2
d2
d (P 2


= 0 (2-34a)


(G+r2) d2 V dWV
(G+ F -rA ) d X + (E +Pr )W
d2 d4 o o0


- (G + P ro d -qo[H(-R) H(-R Vto)]


(D -I x2) d (G+P
1 0 dP2


.ddw


- (G+P r0) = 0


(2-34b)


(2-34c)











CHAPTER III

TRANSFORMATION OF EQUATIONS

Equations (2-34) will be transformed using the Fourier transform



oo
F[f(cP)] = f(s) =f f() e-is" dcp (3-1)


The inverse transform is

00oo
F [f(s)] = f(f) = f f(s) e's ds (3-2)


where i = v-T.

Assuming all the derivatives of f(o) through order (r-1) vanish as - +o

the transforms of derivatives of f(o) are given by

-k k -
f (s) = (is) f(s) (3-3)

so
SO

F d is f(s) (3-4)



F -= -s f(s) (3-5)







Applying this to Equations (2-34) gives
2 2- 2 2-
(1 + F r I2) s2 (s) i(E P)s W (s) + (D I \2) s2 v(s)


i(E P) s U(s) + (G + F r X2) s W(s) iE s (s)


o q isd
+ (E + P r) W (s) i(G + P r) s (s) [ e 1]



(D I X2) 2 U(s) + i(G + P r ) sW (s) + (D I X2) s2 (s)
1 0 0 1 0


+ (G + Pr ) I(s) = 0


= 0


(3-6)


Collecting coefficients on like displacements allows this set of equations to

be written in the following matrix notation.


[(1+F-r l2 )s2] [-i(E -P)s]
1 1


[i(E1 -P)s]


[(D1 -I X)s 2]


[(G+F-r X)s
-i EXs
+(E +Pr )]



[i(G+Pr )s]
o


[(D Io )s2]
1 0


[-i(G+Pro)s]



[(D1 I 2)s2
+(G+Pr r)]
0


*See Appendix A for the derivation of the transform of the forcing function.


U(s)



W(s)




(s)


0



o isd


- -

0


(3-7)





22

This set of equations can now be solved for U, W, and b. This work is carried

out in Appendix B and the results are

isd
(s) = (1- e [(D I 2) (G+Pr +E P)s+ (G+Pr )(E -P)]
q 2- 1 0 0 1 0 1
o s D(s)


S1 -isd
W )= (1--e [(D I 2)cs2 +(G+Pr )( + F-r l2)]
o isD(s)


isd
= (- ) [(E -P)(D -I 2) + (G+Pr )(l+F-r A)]
q 1 1 0 0 1
(3-8)

where
S4 3 2
D(s) = c +i c s +c s +i c s +c (3-9)
4 3 2 1 O

and


c = (G+Pro) [(1 + F-r1 X)(E +Pr) (E1 P)]


c = E X(G+P ro)(1 + F- r 12)
1 0 1

c = (D I1 X2) {(E0 + r)c (E1 P)[(E1 P) + 2(G + P r)1 }

+(G+Pro)(1+F- r 2)(F-Pr r 2)


c = E X(D1 I X2)c
3 10


c4 = (G+F-r A)(D 1 X)c

2 2
c = 1+F-D -rA +I_ A
S 1 + 1 r1 X +1o X (3-10)

The displacements are found by inverting Equations (3-8) using transforma-

tion (3-2). In order to evaluate these integrals the roots of the characteristic

equation D(s) = 0 must be determined.












CHAPTER IV

INVESTIGATION OF THE CRITICAL VELOCITIES


It is known that the inverse of the deflections [given by integral (3-2), where

f(s) represents the deflection expressions (3-8)] does not exist when there are

repeated roots of the characteristic equation

D(s) = 0

on the real axis. This can occur when there is no external damping and corres-

ponds to a resonant condition as discussed by Jones and Bhuta (2). There are

specific load velocities corresponding to these points and they will be referred to as

critical velocities. The condition which must be satisfied in order to have re-

peated roots is that the discriminant of the undamped characteristic equation

c s + c 2 + c = 0 (4-1)
4 2 0

must be zero. The discriminant of this equation is determined in Appendix C as

2 2
A = 16 c 4(c2 4c ) (4-2)
04 2 04

Therefore, the three conditions which will make the discriminant zero are

c 4c c = 0 (4-3)
2 0 4

c = 0 (4-4)

c = 0 (4-5)

Substituting the required coefficients from (3-10) into (4-3) gives*

C + CA6 + C4 + C2 + C = 0 (4-6)
*See Appendix D for the detailed calculations.

*See Appendix D for the detailed calculations.







where


The


2
C = e -e
8 9 14

C = 2e e e
6 89

C = e2 + 2e e e
4 8 7 9 12

C = 2e e e

2
C = e e
0 7 10

e. are defined as


e = D e + e f f
7 1 5 302

e = D e e I e ef
8 1 0 4 0 5 35

e = e rr e e
9 3 1 O 0 4

e = 4e e f D f
10 3 63 1 1

e11 = 4e3[e6(f3f9 -rDf1) eo rlafDf


e 2= -4e[e (r f +f I e) + eo r (f f rD f)]


e3 = 4e [e6rI oe + e r (rf + fIe )]


e4 = -4e eO r rl e4
14 3 0 1 0 4


(4-7)


(4-8)


and the f. coefficients are defined in Appendix D.
1

If the axial and rotatory inertia are neglected the equation for determining

the location of the repeated roots reduces to

X + C X2 + C = 0 (4-9)
4 2 0

where

C = e2 r2 2
4 3 0


C = 2re 2eD f f (D e +e f f)]
2 3 6 11 o 1 5 30 2


X 2
C = (D e +e ff ) 4e f D f e
0O 15 3 02 33 11 6


(4-10)







The solution of Equation (4-9) gives two roots (X' )2 and (X' )2 which are

1 2
velocities at which repeated roots occur.

The second condition which gives repeated roots is that given by Equatic

(4-4). Substituting for the c4 coefficient gives

(G+ F r?2) (D I A2) [1 +F- D + (I r) ] = 0 (4
1 0 1 0 1


-11)


This can be written


(f3 r2) (D1 I X2) (f + e4 2) = 0

Expanding this and collecting coefficients on 2 yields


[f D -(rD +f I) 2 + I r4] (f + e4 2)
3 1 1 30 0 1


f f3 D + [ e f3D f (rD1 + f3 2


+ [f rI e (rD +fI )]4 + eI rX6
1 0 4 1 30 40


(4-12)


= 0


= 0


Finally


C' X + C 4 + C X2 + C = 0
6 4 2 0
where

C = e I r
6 40

C = f rI e (rD +f I
4 10 4 1 3


C' = e f D f (r D + f3 I1
2 4 3


(4-13)


C' = f f D
0 13 1


)n


(4-14)







If rotatory and axial inertia are neglected Equation (4-13) reduces to give

one root

C' f
(X)2= -- = r (G + F) (4-15)
CtC r r
2

The third condition (c = 0) from Equation (4-5) is now investigated. Substituting

for co gives

o2
(1+F- r 2) (E+ Pro) (E- P)2 = 0 (4-16)


or
2 2
(f r ) e = e (4-17)
0 1 O 1

thus

e f e
2 00 1
"4 e r
o 1


X4 = er (4-18)
4 e r
o 1

Therefore, Equations (4-6), (4-13) and (4-18) can be solved to give eight

roots in X2 which satisfy the conditions for repeated roots. Equation (4-6) gives

four values of X2 and these values are labeled X, X2, X2, and X. The condi-

tion which lead to these roots was that

c 4c c = 0
2 0 4

Solving Equation (4-1) directly gives


c c2 4c
S= 2 2 4 (4-19)
2c4


If the radical in (4-19) is zero, the roots are


c c
2 2
s c2 s (4-20)
1,2 2c 3,4 e4
4 4






The repeated roots are either real or imaginary depending upon the sign of

the coefficients.
2 2
Equation (4-13) gives three more roots which will be labeled X XA, and

X These roots come from the statement that c = 0 and if c2 0 the repeated

roots in this case will occur at an infinite value.

Equation (4-18) adds one more root, making a total of eight. This root is
2
designated A3. The condition leading to this root was that c = 0. From Equa-

tion (4-1) it is observable that the characteristic equation becomes

s2 (4 2 + c ) = 0 (4-21)

which shows a repeated root at the origin.

A computer program was written for the solution of these equations. A

simplified flow diagram of the program is shown in Figure 4. 1 and the details

of the program, named VCRIT, are presented in Appendix E.

The results of a parametric study using the computer program VCRIT are

presented in Figures 4. 2 through 4. 8. The locus of each of the roots Ax (i = 1, .. .8)

is shown on these figures. These curves are the boundaries which separate

these plots into distinct regions which are labeled as Regions I through VIII. In

each of the regions the roots of the undamped characteristic equation have a par-

ticular form as noted on the figures and as listed in Table 4-1.


Table 4-1

Correlation of Root Type with Region Numbers for Figures 4.2 through 4.8


Region Form of Root

I, III, VI ibl, ib2
I a + ib, -a + ib
IV, VII + a, t a2
V, VIII a, ib
a, b are real








































































Figure 4. 1. Flow Diagram of Computer Program VCRIT which Determines
Load Velocities at which Repeated Roots Occur


Insert Data in Separate File

h EOo Gxzo E0o
P, F
R' Exo Exo 'Exo







Pure


10.000


1.000







100







10




2


1







0.1


0.01


0.001 --
0.00001


0.0001


0.001


= 0


Gxzo
0.35
Eo


F = 0.002


2 Real,
2 Imaginary
Roots


0.01


Figure 4.2. Classification of Roots of the Undamped Characteristic Equation for
Variations in the Thickness to Radius Ratio Including Prestress











10,000






1000


t=0


Gxzo
- = 0.35
Exo


Pure
IV Imaginary
10 Roots
2 Real,
2 2 Imaginary
Roots




X Loci, i= 2





















0..00001 .0001 .001 ..01 0.1

h/R




Figure 4. 3. Classification of Roots of the Undamped Characteristic Equation for
Variations in the Thickness to Radius Ratio with No Prestress
.. ...... ....... ...... ........ ............ r e s t r e s











Pure
Imaginary
Roots


2 Real,
2 Imaginary
Roots


0.40


0.80 1.2 1.6


E0 /Exo



Figure 4.4. Classification of Roots of the Undamped Characteristic Equation for
Variations in Eog/Exo









10,000







1000






100



A2










Roots
10 I







0.1







0.01







.001
.00001






Figure 4.5.


0001 001 .01 0.1


Gxz/Ex
xzo x~o


Classification of Roots of the Undamped Characteristic Equation for
Variations in Gxzo/Exo









VII Real Roots


1000







100






10






1


0.001


0.01


Evo/Exo



Figure 4.6. Classification of Roots of the Undamped Characteristic Equation for
Variations in the Evo/Ex


Complex V lotsoo VI

P, Pure


























: . .. . . : *2 R .
S... . .: Imaginary
Roots



Real Roots

IV




2

Complex Roots U


\i Loci, i =








Re 1.
.'e 2 n"imaginary
SRegion Root s



..-- .,,,.. ., .. hR- = -. (01 :- " -

.mag inar' F n.nnl v' X
R oots
-E Po E = 0 n




',,: -' n

". ,. ' I.,,,. : ' , :v , .,.


0.10






0.01






0.001


0 0001'


0.0001











1000














100


Region I
i 2




.....Imnaginary
P .. Roots......


|: : a::: { :: :: :::::::::: ::::: == ==========:::, :.,1 :: .......:::::::::::::: ..
I ?, k.:: :: ::: :i':<.:: ::: 4 :~i::. ..: .:: :::: ::::::::::::::::: :::::. :::::: :::::: :::::: ::::::::!
f: : ::-: : ::: :::::::::::::::::::::::::::: ::: : :: ::::: :::::::::::::: ::::::::: ::::: : ::::::::::
-;} |:::! ..:: .: ..: ....:.... :: i `.. ':* <.:!| ...... ....:" ;.


0.002


0.004


Figure 4.7. Classification of Roots of the Undamped Characteristic Equation for
Variations in the Circumferential Prestress


VII (Real Roots)




.11. ... .. .. ...... .. . ..
.. |. ..|.. .. v .
9 [LYmagiinar -


0

E






O
M -N





ReI R t c1 o=t S L 0
F 0
F C



















Loci, i =1
:::ffil m :~::::-,


.0



.35



1.3


1.001

1.002
3


21

4ii~


-0.002








Real Roots VII


1000












100












10



5


A'

2



1.0












0.1


-0.002 -0.001


0.001


0.002


0.003


0.004


F

Figure 4.8. Classification of Roots of the Undamped Characteristic Equation for
Variations in the Axial Prestress


2 Real,
2 Raginary Imaginary
Roots Roots >




h
= 0.001

- E 0
= 1.0
E
Xo





= 00.35

P =0 Real Roots
= 0







100








Complex .'"...
Soo .o














Roots
; %%, ,
"'. ". 7 ... ": ,. . ,: .
".;. '. .. :. ., .. ,, ,".] :: .? ;.,, #
ix ,, % ..%~ 2 :' ..'.', ,,', d', ''"'''
_."_'; ':: ... :-. .. ..' .,' ... .. "'". l


0.01







The lower three values of X. are approximated by Expressions (4-9) and
1
(4-15) by neglecting the axial and rotatory inertia. The approximations are

quite good over the range of parameters studied.

The critical load speeds are denoted as those speeds at which the displace-

ments become unbounded for an undamped system. This corresponds to the

load speeds which produce a double root on the real axis as will be shown later.

It is instructive to follow the path of the roots of the undamped characteristic

equation in the complex plane as the load speed increases. As an example,

the roots will be traced for material properties corresponding to an isotropic

shell with h/R = 0. 001 and positive axial and circumferential prestress.

Following the vertical line for h/R = 0.001 in Figure 4.2 for increasing X2

will give a path crossing all of the boundaries separating different types of

roots. Starting at the low load speed, the roots are all on the imaginary axis.

(This would not be the case if prestress were not included, as shown by Fig-

ure 4.3.) As a means of tracing the location of the roots, Figure 4.9 is util-

ized which shows the complex plane. The roots in Region I appear on the plane

on the imaginary axis and these particular roots are designated s ,1 s2,1'

s ,, and S ,. The nomenclature s. denotes the ith root location and j indi-
3,l 4,l 1,j
cates the relative position of the roots. For instance s,4 denotes the position

of the first root and at that time (load speed) the other roots are located at

points s 2,4 s and s The arrows indicate the direction in which the

root is moving for an increasing load speed. At the first speed two pairs of roots are

moving toward one another on the positive and negative parts of the imaginary

axis. They meet, and the first repeated root location is established which cor-

responds to ,1 in Figure 4.2. As the load speed increases Region II is entered.

The roots are complex as can be seen in Figure 4.9. Next, the complex roots

approach one another in pairs on the negative and positive real axis. This gives




37

the first repeated roots on the real axis and this load speed is designated as the
first critical load speed, hXCR.


/

/
/ 3,2


[ .3,
v1,3 43,r 3,- 3,


S=A CR
A1CR


1,5
Complex s Plane

4,i1


I Re(s)

I
/
/
/
/
/
' 54,4


8i'j = Root I at load speed j


Figure 4. 9. Path of the Roots of the Undamped Characteristic Equation
for Increasing Load Speed (Beginning in Region I)


//








Now the roots separate and go in opposite directions along the real axis, the

larger roots eventually becoming unbounded. This speed corresponds to the

boundary line between Regions IV and V. The condition causing this occurrence

is that c4 -) 0. Repeated roots occur at infinity and the second critical load

speed has been determined. As the load speed increases further, the large roots

come in from ico along the imaginary axis. At location 5 the two large roots

are imaginary and the other two are still on the real axis approaching the origin.

This corresponds to Region V in Figure 4.2.

The locations of the roots are.now transferred to Figure 4.10 to avoid undue

complication of the picture. Two roots meet at the origin while the other two

are yet large and imaginary. This double root corresponds to A4 in Figure 4.2

and is the third critical load speed. An increase in load speed now causes the

roots to become all imaginary which is in Region VI. The last four values of
2
X. (i = 5, 6, 7, 8) are very close together. In fact 2 is approximately 1000, and
1 5
2 2 2
X6, A7, and A are almost nondistinguishable at 1002. Increasing the load speed
2
past A5 causes the large imaginary roots to proceed to an unbounded imaginary

value and reappear on the real axis so that the roots are now two real and two

imaginary. Also As must be a critical speed because Condition (4-4) is satisfied

at load speed As. This region is not distinguishable on Figures 4. 2 through 4.8

and is not given a number.

Further increase in load speed moves the imaginary roots out and eventually

they reappear on the real axis with the other pair, there to remain. This posi-

tion is indicated by s (i = 1, 2, 3, 4). It is interesting to note that A7,8
1,10 6,7,8
also must correspond to a critical load speed, XsCR.

The lowest critical load speed for an isotropic shell is therefore XCR = X2.

Figure 4.5 shows that for a material with a very low shear modulus, the first

critical speed becomes 3 = A2CR which corresponds to the shear wave speed.











Im ( s)
I cu





s 40
1.5

A. C4C11


S
3.-


S S S S S S
4,. 4_ 1031 2 '.10 1,.


/

/
/


I
I
-LT-pt~-S


S
J
s <
4.-

-i


A .

4.A
4,

-I


Complex s Plane






\







S S
SHeks)


I


/
/
/


CR -\






Root i at load speed j


Figure 4.10. Path of the Roots of the Undamped Characteristic Equation for
Increasing Load Speed (Beginning in Region V)


/.


A


S

s
2.

S
2.

s
2.


S
3.1 )


Y_ Y_ __ _____ _Y _Y


A, -.


s 4





40

When this load speed goes to zero, the critical small deflection buckling load

has been reached. From Equation (4-15) this load is seen to be approximately

F = -G (4-22)

This corresponds to the second critical load for engineering materials with

a more realistic value for shear modulus.

If prestress is not included, Region I disappears as shown by comparing

Figures 4.2 and 4.3. Decreasing the tangential modulus drastically or increas-

ing the normal modulus results in the same effect as shown by Figures 4.4 and

4. 6. Circumferential prestress has essentially no effect on the critical load

speeds but axial prestress has a pronounced effect as shown by Figures 4. 7 and 4. 8.

It is instructive to make a comparison of these results with those obtained

in Reference (2). Three critical velocities were derived there, which are given as


-2 Eh Eh 22
VCR2 2 2
p R[3(1- )]2 6pR (1- )


2 E
VCR, =
Po(l - )


2 E
VCR3 -
0ol_#


(4-23)




(4-24)




(4-25)


for an isotropic material. Since, in the nomenclature used in the present work,

V oR(1 p2)
= Eh (4-26)

The corresponding expressions in terms of 2 are


/2 P h
XCR = 6 R

2 R
ACR2 h

R2 R (I- 2)
XCR3 hW(1- )


(4-27)


(4-28)


(4-29)







Now taking p = 3 it is found that

2 h
XCR = 0.55075 0.15 R (4-30)


hCR h (4-31)

2 R
XCR3 = 0.91-

h2
2 h
From Figure 4. 3 (which is a plot of Xi versus R for an isotropic material

with p = 3) for the case of zero axial and circumferential initial stress, XCR1,
2 2 2 2
XCR2, and 2CR3 from Reference 2 agree extremely well with 2A, X and
3 2 5B, 6,7, '

XA respectively. The two velocities, X and X, arise in the present results be-

cause of initial prestress considerations and shear deformation, respectively,

which were not included in the referenced results. X2 corresponds to the dili-

tational wave speed, X corresponds to the shear wave speed, X4 corresponds

to the bar wave speed, and 6,7, corresponds to the plate wave speed.
5,b,7 ,8












CHAPTER V

SOLUTION FOR DISPLACEMENTS


General Solution


Transformed Displacements

From Equations (3-8) the transformed displacements can be written as

isd c' s + c'
U(s) 1- e 5 6
Ss2 D(s)


isd c s2 + c'
W(s) 1 -e 7 8
q is D(s)


cI
=- (1 eis) (5-1)
qo D(s)

where

c' = f (e + e)
5 73 1


cl = e e
6 3 1


c = f(f +e X2)
7 7f1 4

c' = e (f r x2)
8 3 0 1

c' = e f +e (f -r X2)
9 1 7 3 0 1

f = D -I
7 1 o (5-2)






Inverse Transformation of the Rotation


Using the inverse transformation given by Equation (3-2) the rotation is

defined as


c'
9
27r


isd
J (1 eisd) ei(s ds
-0e d(s)
__ Dfs)


By partial fraction expansion this can be put in the form*


(5-3)


(1 eisd) eis


4

s ds
k= 1


1

S[Ds(s)]
d s S : k


1
4


m k


m=
m=1


9
c =
9 c
4

Defining


c k
27r


00


isd
(1-e ) i( ds
s sk


(5-5)


(5-6)


S= Fk(E )
k=l











*See Appendix F for the derivation of a sample partial fraction expansion
expression.


(5-7)


c
9
~ 27r


-CO


where


(5-4)


then


qo


(sk sm)







Letting

isd
1-e sd
f (s) _-e eis (5-8)
s s k

the integral to be evaluated is

00

f fk(s) ds
o00

where fk(s) is an analytic function except at the simple pole s = sk. Defining the

complex root in general to be

sk = ak + i bk '

the Cauchy integral theorem is used to evaluate this integral.



Im(s)


C-F





-a Re(s)



Figure 5.1. Contour Integration Path for Evaluating Rotation Integral


Assuming first that bk > 0 the Cauchy integral theorem gives



Sfk(s) ds = 27T i Z Residues (5-9)
(5-9






The integral around the closed path shown in Figure 5.1 is


Sfk(s) ds


fk(s) ds +


-R
-R


(5-10)


fk(s) ds


From Reference (12) it is shown that if f(s) 0 uniformly as R -* then


lim f f(s) eis ds = 0,
R -o C-
R


(4 > 0)


Therefore the first integral on the right in Equation (5-10) goes to zero as

R c and


c fk(s) ds


fk(s) ds


= 27 i Res(sk)


i (ak+i bk)


00

= /



= 27 i l


+ 27- i -e d(ak+i bk)] i (ak+ibk)
+ 27Ti -e e


-ic 9 k-bk P+i ak [


-bk(d+)+i ak(d+k )
H(0) e


H(d+ (P) (5-11)


where

bk > 0


H = Heaviside step function.

Similarly, when bl < 0 the integration path is in the lower half plane and the

result is


F i -bbk (+i ak H(-) -bk(d+p)+i ak(d+P)
F = ic 9 "ke H(-p) e


H[-(d+0)] ,


bk < 0 (5-12)


Therefore


(+d>0




46

The general expression for Fk can be written in the form


Fk = -i sgn(bk) c9 ak e ak H[sgn(b


-bk(d+P)+i ak(d+P)
-e


where H is the Heaviside step function and


sgn(bk)


k)


H[sgn(bk)(d+P)]l


bk >0


bk<0


The rotation is given by Equation (5-7).

4

q = Fk()
qo
k=l


Inverse Transformation of the Radial Deflection

The radial deflection is obtained by inverting W to give


isd
s-e
s


c' s2 + c'
7 8 eis ds
D(s)


(5-15)


The term (c' s + c' )/D(s) can be expanded in partial fractions to give
7 8


isd
1-e
s


2
c sk + c
k7 e
4

S(sk sm
m=l


"-k)
- k


eis ds


(5-16)


k m


c'
7
c -
7 c
4


c'
8
c -
8 c
4


*See Appendix E for this expansion.


(5-13)


(5-14)


00

27 i f


Wqo
q%


where


(5-17)


00
_ 1f
27r i f






Defining



w Fk(0)


- k
2r i


isd
s(s sk e


(5-18)


The deflection is then given by the sum


= VFk) (5-
qo
k=l

By contour integration in the upper and lower half planes, the integral in

Equation (5-18) can be evaluated. Breaking the integral into two parts gives


19)


(5-20)


w k e i- ds ei( +d)s
F ( ds- ds
wk 2 [ i s(s sk) s(s sk


The contour shown in Figure 5.2 is used to evaluate these integrals for 4 > 0,

bk >0.



Im(s)

C
R

C s

-R p R Re(s)


Figure 5.2. Integration Contour for Radial Deflection for 4 > 0, bk > 0


From Cauchy's integral theorem


Sw k(s) ds = /wfk(s)ds +
C-
R


+ fk(S) ds
P


-p

-R


wfk(s)ds + /wfk(s) ds
C
p


= 2n i Res




48

From Jordan's Lemma (12) it can be seen that the integral on contour

C- -*0 as -~ i.e.,
e


lim i ds 0, > 0
S s(s sk)
R-oo C~

and

e i(p+d)s
lim J id)s ds 0, O+d > 0
s(s Sk)
R-- C-

Also


lim f fk(s) ds + f(s) ds P fk(s) ds
0
R- -T P -00
R-o -

which is the Cauchy principal value of the improper integral. If the integral

exists, then this is the correct value for the integral and the symbol P can be

dropped. Therefore

00
-0J es P ds + lim e ( -s ds = 27r i Res(sk), ( > 0 (5-21)
s(s s ) s(s s ) k'
-0 C
P
The second integral in Equation (5-21) can be evaluated as


lim ds = ai Res(O)
p- s(s Sk)
p-0 C k
P
Since the integration is clockwise

Cf = -T7r

and

( eips \ =
Res(O) = ks= -1
SSk)s=0 k










lim ds
O s(s sk)
p-O Ck


Integral (5-21) now may be written


a0


- 00


is
e
s ds
s(s sk)


where

0 >0

bk > 0

Similarly


f ei(~I+d)s
- s(s Sk)


i(+d)
Tri 2r i ei(Od)
ds +
sk sk


(5-23)


where

p+d > 0

bk > 0


Now the case is investigated where bk > 0, 0 < 0. The contour shown below

is used for this case.


Im(s)

,_T__ sk

; Re(s)


-~a
R k C-






Figure 5.3. Integration Contour for Evaluating Radial Deflection for p < 0,
bk >0


7Ti
Sk


7T
Sk


i sk
+ 27ri e
sk


(5-22)






The integrals have the values


S ds + lim ds =0, <0
s( k) p-0 (-k)
p
or


e ds = i Res(O) = i < 0 (5-24)
Ss(s sk) sk
-m bk > 0

Similarly


Ses ds = (p +d < 0 (5-25)
Ss(s sk) sk
o bk > 0

Equations (5-22), (5-23), (5-24), and (5-25) can be combined to give the solution
for wFk(() when bk > 0 as


wFk() = k i + e i H(P) + -i H(-)
w 27ri sk s sk

/ o (~,,+d)S \ ]1
7-r + 2sir e H(+d) + H(-P-d)
k sk k
bk > 0

or finally

Ik 1 isk 1
w k + e H((P) + H(- P)
w Fk(P) s -k


S[( + e ( )sk H(+d) + H(-P-d) (5-26)

where bk > 0.





51

The case where bk < 0 is now investigated. Integrating in the lower half

plane it is found that


ds
s(s sk)





ei s
i ds
s(s sk)


eiS eSk
ds = 27i -
s(s sk) sk


bk < 0
bk < 0


7T1
+ i
Sk


ifsk
= -27ri -
sk


bk <0


It is evident, therefore, that the general expression for w Fk(() can be written

in the form


sgn(bk) Ok
s k


i(fsk ) H[-sgn(bk) 0]
+ e k H[sgn(bk) ] + 2
/2


[ 1 i(0+d)S k]
1 +e M )skH[sgn(bk)(+d)]


+ H[-sgn(bk)(-^)]


(5-27)


Inverse Transformation of the Axial Deflection


The axial deflection in the s-plane can be written as


isd
= c' 1 ei)
5 (s)


isd
(1 e )
6 2 (s
s D(s)


From Equation (5-1) it can be seen that the first term in Equation (5-28) can be

written in terms of . Thus


isd
(1- e ) eis d
s !( e ds
2
s D(s)


(5-29)


- 7


CO


--0


q


(5-28)


c
5
c'
e


w Fk()


1 o
S27
0







The integral in Equation (5-29) can be written as


4
Sisd ak d k
(1 e ) is s k
2 e --sk ds
s k=
k-1


where
c,
6
C = -
6 C
4

Now defining


C6 k
27


00


-00


c
5 9 (0) +
9 9O


isd i(s
(1 -e ) e ds
s'(s sk)
s2(s s )


4

uFk(Z)
k=l


The integrand in Equation (5-30) can be written in two parts as


eis
e -ds -
s2(s s k)


0 ei(o+d)s
J/ e ds
- s 2(s -s)


The contours shown in Figures 5.2 and 5.3 are used to evaluate the integrals

in Equation (5-32) for the case bk > 0. Define


ufk(s)
uk


S ei s
s2(s sk)


ei(+d)s

s (s sk)


By a procedure completely analogous to that just described for the radial deflec-

tion, the results can be written


j e ds = 7riRes(O) + 27riRes[ fk(sk)],
-0 s (s-k)


S> 0
bk >0


(5-33)


00
- 00


F (( )
uk


then


(5-30)


Fk ()
k


"6 k
27r


(5-31)


-o


(5-32)


uf (s)








ei(P+d)s
ds
s (s sk)


= 7riRes'(0) + 27riRes[ f(sk)] ,


-bk ( +i ak
e
(ak + i bk)2


Res[ uf(sk)
u k k


-bk(P+d) +i ak( -+d)
e

(ak + i bk)2


To find the residue of the functions at s = 0 they are expanded in a Laurent series

about the point s = 0.


eics
e
2
s



1
s sk


I + +l
2 s
s


1
sk


2 3
2! 3!


+-
sk


2
+ s
+5
2
sk


1
i (P+s
k
+ k
s


es2s
s (s sk)


1
+ 1
2
sk


+i
s k


2
2!


The residues can be evaluated as


Res(0)


1
2
s k


1
2
sk
k


(5-37)


S i +d
sk


00

-00


where


O+d > 0
bk >0


(5-34)


(5-35)


(5-36)


Res[ufk(sk)]


-


Sk


(5-38)


Res'(O) =









eis
e ds = -
s (s s ) k
k


_00


00

/
-CO


77i
Sk


Sk


eisk
S k
2/ k


- + i(P+d)] +
sLk J


Integrating in the lower half plane for ^ < 0, Equation (5-30) can now be written

in the form


F (P)
k


ck
sk


+ i )


H( )


+ i( +d))


H(O+d)


1 +1 +i sk H(
+ ( s H(-P)
I + s k


- k ) H(-O-d) ,
2 s k


bk >0


(5-39)


Performing the integration for bk < 0, the final result can be written as


sgn(bk) ic ak
2
sk


1
2 (1


+i sk) +e k


SH[sgn(b) + ( + ) H[-sgn(b
x H[sgn(bk)p + *(l+i 0 sk) H[-sgn(bk)P]


i(+d)k H[sgn(bk)(P+d)]


+ [1 +i(+d)sk] H[-sgn(bk)(0+d)]
2


so that


ei(S+d)s
s S ds
s2(s s )
k


bk >
S>


i(4+d)sk
e
2
sk


bk > 0
d +0 >0


Fk( )


-- [1+i(P+d)sk] + e


(5-40)


1 1k
2 sk






The solutions given by Equations (5-7), (5-19), and (5-31) with the correspond-

ing Fk functions (5-13), (5-27), and (5-40) are substituted into the governing
equations of motion in Appendix G to show that the solutions satisfy the differen-

tial equations.

Summary of Deflection Expressions

The following expressions summarize the solution for the deflections.
4


= F k(F )
k=l

4

= wFk( )
k=l


k-- 1
c 5 (qS) +
c q
9 0


(5-41a)


(5-41b)


4

Su k(P)
k=l


(5-41c)


where


I isk P is k(0+d)
Fk( ) = -sgn(bk) ic9 ak e k H[sgn(bk)P] e sk H[sgn(bk)(P+d)]
(5-41d)


sgn(bk) f3k
wFk( s
Sk


- + es H[sgn(bkl + H[-sgn(bk)p]


[- +e sk H[sgn(bk)(P+d)]


+ H[-sgn(bk)(P+d)]


(5-41e)


( )
qo







sgn(bk) i c6 ak
u Fk( 2
sk


sgn(bk)





H(y)


r 1 i0sk
-(1 + isk) + e [sgb H[sgn(b

+ (1 + i sk ) H[-sgn(bk)

1 i Sk(d) )
- -2 [1 + i sk(+d)] + e H[sgn(bk)(P+d)]

+1 [1 +isk(l+d) H[-sgn(bk)(P+d)] (5-41f)


bk >

bk <


y<

y>


1 C7 sk +
k 4 k 4 k k m
(sk- sm) R (sk- sm)
m=l m=l

Sk= ak + i bk (k = 1, ..., 4) are the roots of the characteristic equation

D(s) = Cs4 + i c s3 +C 2+ic s+c = 0
4 3 2 1 0







and the coefficients are defined as

c fl e4 2


c = e (fe )
O 380 1

c = -EXe f
1 38

c = f (e c- e ) +e f (f rX2)
2 70 1 2 38 2

c = -Eff c
3 7

c = (f rX2)f c
4 3 7

c = f (e + e )
5 73 1

c = ee
6 1 3

c = f7(f + eX2)
7 71

c' = e f
8 38

c' = ef +c'
9 17 8

e = E +Pr f = 1+F
o o o o

e = E -P f = 1+F-D


e = E -P+2(G+ Pr) f = F-Pr
1 0 2 0

e = G+Pr f = G+F
3 O 3

e = I -r f = Dr +If
4 0 1 4 11 00

e = e f -e e f = r f +rf
5 01 12 5 12 0

2
e ef -e f = eD -If
6 00 1 6 4 1 01

f = D I2
7 1 0

2
f = f -rX2
8 0 1


(5-42)







The derivatives of the deflections are

4

__ = E Fk,
k-=l

4
q = jl w y Fi(0)
k=l

4
S= uFk() +) (5-43)
c 9 q 0 u
k=1

where

F'k() = i sk Fk()=
lk esk +c
k( k kc Fk()
F9 k 9

C6 ak C
F'() wFk(0) F, () (5-44)
uk c sk +c
7 8

0, O+d / 0


Solution for No External Damping


For the case of no external damping, the solutions as given in Equation (5-41)

are not directly applicable. It will be noticed that the forms of the solutions are

dependent upon the signs of the imaginary parts of the complex roots. Figures

4.2 through 4.8 show that there are regions in which there are only real roots

of the Characteristic Equation with no damping, thus causing a problem of non-

uniqueness of the solutions.

Following the method of Achenbach and Sun (13) the undamped solution will

be obtained uniquely by assuming the undamped solution is the limit of the damped

solution as the damping approaches zero. In this manner the sgn functions in the







Fk functions for Equation (5-41) can be determined. Figure 5.4 provides an

example of the behavior of a set of roots as the damping, E, approaches

zero. This figure shows the type of the four roots (sij = root i with damp-

ing e ) for heavy damping to be two complex and two imaginary, root 1

being very near the origin. As the damping is decreased, the imaginary roots

approach one another and finally meet and separate which gives four complex

roots. Meanwhile the other complex roots are also approaching the real axis.

This establishes the correct sign for the imaginary part of each root in the limit

as e 0 and the roots all approach the real axis.


Form of the Radial Deflection in Region IV


The roots are all real, having the form a a From Equation (5-41b),
1 2
the radial deflection expression (when the proper signs are established) in

Region IV becomes

W = 2 + [H(+d) H()]
qo a a a a



aP1 [a le- H(-@-d)
+0 F-ia -ia (0+d) H
- e H(-O) e 1 H(-0-d)
a




4 ~ -i a 20 -ia (+d) d) (5-45)
3P ia ia ( +d)
+ e H( ) -e H( 4)



- e H(0) e H(04d) (5-45)



























































s.. = Root i for
External Damping
External Damping c.


e -
0
1
E =
2
E
3


0

0.1

1.0

10.0


Figure 5.4. Loci of the Roots of the Characteristic Equation
as the Damping Approaches Zero






Expanding the i3. coefficients gives


P/ = -/P

1 = -~


P3 -f4


Sa + c
7 1 e
2a (a2 a)


c a2 +c
S-72
7 2 8
2a (a a2)
2 1 2


Substituting expressions (5-46) into (5-45) gives


2
c a + c
7 1 8
2a (a a2)
11 2


e +e 1


) H(-0)


i a,(j+d) + e -i al( +d) H(-1-d)
Sia (+d) -i a (0 )

2
c a +c ia 6 -ia \
72 8 2 +e2 HU
2a (a -a a /
21 2




e + e H(C+d)
c
S 2
a a
1 2


c
- 2[H(0+d) H(O)]
a a
1 2


2
C a +c
7 1 e
S(a a
1 1 2


X [cos (a ) H(-() cos [a (o+d)] H(-p-d)]


C a +
7 2 -e c (a 0) H() cos [a (0+d)] H(0+d)
21 2


W
qo


(5-46)


W
qo


(5-47)





62

Therefore, for the three distinct regions of the cylinder, the solutions are:

Solution behind the load (P < -d)

2 +c
- 1 cos (a ) )- cos [a (P+d)] (5-48)
o a (a1 a
11 2


Solution under load (-d < 0 < 0)

2 2
c c a +c c a +c
W 8 71 8 7 2 8
2 2 cos (a ) + 2 2 cos [a (P+d)] (5-49)
Sa a2 a (a a2) a (a a2 2
12 11 2 21 2


Solution ahead of load (0 > 0)


cW 2 + c
7 72 2 cos (a4) cos [a ( }+d)] (5-50)
a2 (a2 a2 2 2
o a(a a2)
21 2


Form of the Radial Deflection in Region VII


As in Region IV, the roots are a a However, there can be no deflection
1 2
ahead of the load in this region because the load speed is greater than any of the

wave speeds in the material. This zero displacement comes about mathematically

because the roots all approach the real axis from the negative imaginary direction.

The solution for this region has the form



S2 2 2 2 2
o a a a(a- )
12 11 2


x {cos (a4 )H(-p) cos [a (O+d) ]H(-p-d)


c a +c
2 \
+ 7 2 2e 'cos (a2 )H(- ) cos [a (0+d)]H(=P-d) (5-51)
2 2 2 2
a (a a )
21 2






Form of the Radial Deflection in Region V


The roots in this region have the form a, ib. The radial deflection in this

region is given by the expression

W c
q [H(-Ip) H(-P-d)]
qo a2 b2


2 2e cos(ao)H(-p)-cos [a(+d)] H(- -d)
a (a + b )

c b2 +c
+ [e-b H(P) ebO H(-P) e-b(I+d) H(P+d)
2b2 (a + b )

+ eb(d) H(--d)] (5-52)



Form of the Radial Deflection in Region VI


In this region the roots are all imaginary, of the form +i b i b This
1 2
gives an exponentially decaying solution as in Regions I and III. The radial de-

flection expression is given below.

W c
S[H(M+d) H(4)]
qo b b
1 2

c c b -b b -b ((+d)
+ 271 e H( ) e H(- ) e H( +d)
2b (b b )
1 1 2 b (+d)
+ eb H(-P-d)]

c c b2 b -b2 b ((+d)
+ 8 7 2 [e2 H(-O) e 2H() e 2 H(-O-d)
2b(b b)

+ e d)H(4+d) (5-53)







Comparison of Solution with Other Results


As a comparison of the results of this analysis with another theory, the

static problem of a distributed pressure load on an isotropic shell was considered

as shown in Figure 5.5.


Figure 5.5 Static Load Problem


For the static problem shown above the roots of the characteristic equation,

excluding prestress, are complex. The solution for the region under the load

as given by the present theory can be reduced to


a L
K 2 eaL 2
1 K. 2 <


-aL
1
cos aL e
2


cos a L)


+ -a L
+ K 2 e sin a L + e
2


-a L


sin aL )


L -
2 R'


where


(5-54)


L -
1 R





65

This problem is solved by Timoshenko (14) and the deflection given by his

theory, when put in a compatible form, becomes


eW -2 -/ 1
= K' 2 -e cos P3 e cos L (5-55)


The second term appearing in Equation (5-54) is missing from Equation

(5-55). This additional term arises because of the inclusion of shear deflection

which was not present in the Timoshenko theory. A comparison of the results

of these deflection expressions is made in Figure 5.7.


Numerical Results


A computer program was developed for the calculation of the displacements

and stresses determined in this research. The general expressions given by

Equations (5-41) were programmed for the displacement solutions and the stress

calculations are discussed in Chapter VI. A flow diagram of the computer pro-

gram is shown in Figure 5.6. The details of the program can be found in Ap-

pendix H. It is written in Fortran for time share computer application.


Comparison of Results with Other Solutions for a Static Load


The radial deflection for a static distributed load on a cylindrical shell is

given in Reference (14). As a check on the solution this static problem was

solved using the present results and the comparison is shown in Figure 5.7.

The results agree very well. The effect of variation in the thickness-to-radius

is also illustrated in Figure 5.7. The rotation and axial deflection are shown

for this static problem in Figure 5.8. In addition to showing the form of the

displacements for the static load, Figures 5.7 and 5.8 serve as a basis against

which the dynamic displacements can be compared. The deflections are sym-

metric about 4 = -0.5 for the static load.










Initiate Program


Figure 5.6. Flow Diagram for Computer Program for Deflection
and Stress Calculations







S .I


iA1L1 111 ilhAIT


. I .I I I I, I


4 4 4 4 4 4 4 4 44 1 Jr


L


Figure 5.7. Radial Deflection for Static Load on an Isotropic Shell (p = 0.3)

















-1.0


Figure 5.8.


-I---


Displacements for a Static Load on an Isotropic Shell
(p= 0.3, h/R= 0.01)


0.15


0.10


0.05


U/q
0


-0.05





69

Summary of Deflection Response for Shells under Various Load Velocities


A summary of some of the types of deflection patterns assumed by a shell

for increasing load speed is shown in Figure 5.9. For the particular properties

used for this example, the various regions (root types) associated with each

waveform can be found by inspection of Figures 4.2 through 4.8. For example,

for no damping, positive prestress corresponding to internal pressure,

h/R = 0.001 and the material properties given in Figure 5.9 (properties are

those corresponding to an isotropic shell as shown in Appendix I), Figure 4.2

can be used to associate load speed with root type.

Following the vertical line of h/R = 0.001, it is evident that X2 = 1 lies in

Region I where the roots are all imaginary. This gives a critically damped

exponentially decaying solution as shown by Equation (5-53) and is shown in

Figure 5.9(a). As the load speed increases Region II is entered where the roots

are complex. This is the form of the static load problem roots, and if no pre-

stress existed Region II would extend from zero load speed up to the first criti-

cal, which is at X The solution for 2 = 2 is shown in Figure 5.9(b), and is

exponentially decaying.

The response becomes sinusoidal after crossing X = X At a load speed

just greater than 2 the deflection response has a very short period. A small

amplitude wave train precedes the load and a large amplitude wave follows it.

As the load speed increases the sine wave period increases as shown in Fig-
2
ure 5.9(d) for X = 30. These sinusoidal deflection patterns are in Region IV

where the roots of the Characteristic Equation are all real. The mathematical

expression for W/qo is given by Equation (5-47). Crossing 3 into Region V, the

roots are real and imaginary. Equation (5-52) gives the radial deflection, and
2
Figure 5.9(e) shows the response to be a long period sine function for A = 500.











































h
R


E0




E,,o


Ex
Ex0


0.001



1.0



= 0.35



= 0.30


= 0.004

= 0.002

= 1.0

= 0


Figure 5.9. Radial Deflection Shape for Various Load Velocities







Jumping to Region VII brings a longer period sinusoidal oscillation as illus-

trated by Figure 5.9(f) and the response in this region, where the roots are again

all real, was discussed previously. The radial deflection is given by Equation (5-51).

Because Region VI covers such a limited range in velocity the response was

not included in the summary but is discussed later.


Region II Response


A study of the response of an isotropic shell at a load speed below the first

critical was made to determine the effect of external damping. These results

are shown in Figure 5.10 where the damping ranges from very light to very heavy.

Of course, when damping is introduced the root form is no longer the same as

that of Region II.


Region IV Response

2
The short period sinusoidal response of the radial deflection at X = 2.7 is

shown in Figure 5.11. As the radial damping is increased this response is

changed drastically as shown in Figure 5.12. The response for a damped sys-

tem, which was in Region IV with e = 0, approaches closely that of the Region II
2
behavior. Figure 5.13 shows the radial response at X = 5 and 10. The maxi-

mum amplitude remains constant as the period of the wave increases for greater

load velocities.


Damping Effect on Regions V and VII Response

2
The effect of damping on the wave forms for X = 500 and 2000 is shown in

Figures 5.14 and 5.15. The amplitudes of the sinusoidal deflection response are

initially decreased, and, as the damping becomes greater, the response becomes

critically damped and the deflection approaches zero with an increase in distance

from the load.










































































= 0.001
= 0.01
= 0.11
= 1.0

= 0


Figure 5.10. Radial Deflection Response for Variations in
Radial Damping (" = 2.0)


0







-0.2


-0.4


-0.6






-0.8






-1.0






-1.2


q0o





























G2
S- 0.35
E
xo

= 0.30
I Eo

I P = 0.004
1 I
F = 0.002

d = 1.0

= 0








-11
-2 0






I

-1 - -___________



















-3
-2 -1 0 1

Figure 5. 11. Radial Deflection Pattern Immediately Above the First Critical
Load Speed









- d


R





-1.0


0


X- =2.7
I h
Il- = 0.001
\ I
I / Efl
IE0 = 1.0

-1
Gxz
= 0.35
Exo

EXo
0.30
xo

-2 P = 0.004
F = 0.002
d = 1.0
e = 0.01
--- = 0.10


Figure 5.12. Change in the Radial Deflection Pattern with Increasing
Damping (X2 = 2.7)










--- d



-1.0


2



(a) X2 =10
1


-1.5


-0.5


S = 0
d = 1.0
F = 0.002
P = 0.004
h
S= 0.001
R
E00
S= 1.0

XZO

S= 0.35
w Ex
qo

E" = 0.3
Ex


0


-1
\ T^ /2


Figure 5.13. Deflection Wave Form at X2 = 5 and 10


-1.0












d



-1 0



w

3.0 qo



2.0



1.0
1



0 2


1 0 2 500
h
0.001
R

-2.0 E00
= 1.0
EX

Gxzo
-3.0 = 0.35
EXO
Eo
-p = 0.30
xo
P = 0.004
F = 0.002
d = 1.0


Figure 5.14. Effect of Damping for X = 500










d


L R
L CL---^ --


G
XZO
Exo

EV

-o
Exo

P
F
d


0 -
0


= 0.35


= 0.30

= 0 004
= 0.002
= 1.0


Figure 5.15. Effect of Damping for X2 = 2000







Region VI Response


This region has a deflection pattern which is almost totally axial. Three

critical velocities have been crossed to get into this velocity range which corre-

spond to the longitudinal, shear, and bar wave speeds. Therefore, there can

be no bending effect transmitted. The behavior is like an axial compression on

a membrane which expands radially, as shown by Figure 5.16. The effect of

prestress is observable in this figure. The maximum deflection is increased

by about 20 percent when going from external hydrostatic pressure to internal

hydrostatic pressure.

This axial mode of deflection also appears in other velocity ranges. For

instance, the broken lines in Figure 5.16 show the behavior at X = 1001. This

is the range between X = 1000 and A2 = 1002 where the roots are real and

imaginary. Another example is shown in Figure 5.22 where E0o/Exo is less

than 0.08 and this occurs in Region VIII as shown in Figure 4.4.

2
Deflection Behavior in the Vicinity of CR


The first (lowest) critical load speed occurs at iCR = = 2.552. Figure

5.17 illustrates the unbounded response of the deflection as that speed is ap-

proached. The effect of damping on the maximum deflection is also illustrated.

There are four other critical velocities as discussed in Chapter IV.

2
Effect of Prestress on 2
'CR

The effect of the axial prestress on the location of the first critical load

speed is shown in Figure 5.18. This effect is also observable in Figure 4.8,

since the first critical load speed is at A2. The circumferential prestress does

not have a significant effect on load speed, as shown in Figure 4.7.
















-1.0 0 0






0.8

h 0.001

E0
= 1.0


-.. .\ Gxz
Ex- 0.35
E



.qo Eo .\\O.30
Ex

U --xo


o -_.. _







-6.0 -4.0 -2.00 2.0 4.0



\2 = 940
P = 0.004, F =0.002 _
2











S P=0,F=0
-22


P=-0.004, F=-0.002

2 = 1001
-4 --*
P =0.004, F = 0.002







Figure 5.16. Deflection Response for Variations in Circumferential
and Axial Prestress























2.5 2.552 2.6


Figure 5.17. Maximum Radial Deflection in the Vicinity of the First Critical
Load Speed


w
max
qo






81
























C
oI

C4
-e









S ________
cM ----- ------------- 1-1









0 0
oC













I C










- - _______ 00
IIn n c,

















I 0 C






















a
on eooo
























co , a -
^







Superposition of Step Loads


The effect on the radial deflection can be observed in Figure 5.19(a)-(f)

where the load length was varied from 0.1 to 5.0. By superposing various com-

binations of step loads it is possible to approximate any shape of load desired.

As an example of this type of application, the radial deflection response from

a symmetric sine wave type load and a sharp edged pressure front was deter-

mined. The results of these calculations are presented in Figures 5.20 and

5.21, respectively.


Study of Material Properties Variations


A look at the effect of decreasing the E0o/Exo ratio is summarized in Fig-

ure 5.22. Starting in Region IV, as can be seen in Figure 4.4, the ratio is de-

creased from 1.0 (as for an isotropic material) to 0.04. As the ratio is lowered,

the maximum deflection gets large rapidly, and becomes unbounded as 2= 2CR

is approached. After crossing A3CR into Region VIII, the strength in the cir-

cumferential direction is of course very low and the material is of little interest

for engineering applications.

The same type of response as in Figure 5.22 will be obtained by increasing

Evo/Eo significantly. This can be observed by inspecting Figure 4.6. For

other types of material property variations, the general response can be pin-

pointed by observing the type of roots at the particular location through the use

of Figures 4.2 through 4.8 and using the numerical results presented here show-

ing similar calculations of deflections.














qjOR


(a) d = 0. 1


Figure 5.19(a). Variation of Pressure Pulse Length, d, at A2 = 30


A
h
R
E0o


EL



Exo
E0
10

P
F
c
d


0

= 30

-0.001


1.0


= 0.35


-0.30

0.004
- 0.002
=0
- 0.1




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PAGE 1

ORTHOTROPIC CYLINDRICAL SHELLS UNDER DYNAMIC LOADING By ELMER MANGRUM, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTLU. FULFILLMENT OF THE REQUIREMENTS FOR THB DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970

PAGE 2

UNIVERSITY OF FLORIDA 3 1262 08552 3057

PAGE 3

This dissertation is dedicated to my wife Rita and my daughter Gaila.

PAGE 4

ACKNOWLEDGMENT I would like to acknowledge the support and encouragement of General William M. Thames, K. A. Campbell, and N. E. Munch of the General Electric Company who made this research possible. I wish also to express my sincere gratitude to Dr. J. J. Burns for this guidance and suggestions during the course of this research. ill

PAGE 5

TABLE OF CONTENTS Page LIST OF TABLES vii LIST OF FIGURES viii KEY TO SYMBOLS xi ABSTRACT xv Chapter I INTRODUCTION 1 Statement of the Problem 1 Specific Goals of This Research 2 Review of Previous Work 3 Contributions of This Work 3 n GOVERNING EQUATIONS OF MOTION 5 General Equations 5 Axisymmetric Loading 15 Pressure Loading Form 15 Nondimensional Equations 16 m TRANSFORMATION OF EQUATIONS 20 IV INVESTIGATION OF THE CRITICAL VELOCITIES 23 IV

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TABLE OF CONTENTS (Continued) Chapter Page V SOLUTION FOR DISPLACEMENTS 42 General Solution 42 Transformed Displacements 42 Inverse Transformation of the Rotation 43 Inverse Transformation of the Radial Deflection ... 46 Inverse Transformation of the Axial Deflection ... 51 Summary of Deflection Expressions 55 Solution for No External Damping 58 Form of the Radial Deflection in Region IV 59 Form of the Radial Deflection in Region Vn 62 Form of the Radial Deflection in Region V 63 Form of the Radial Deflection in Region VI 63 Comparison of Solution with Other Results 64 Numerical Results 65 Comparison of Results with Other Solutions for a Static Load 65 Summary of Deflection Response for Shells under Various Load Velocities 69 Region 11 Response '^1 Region IV Response 71 Damping Effect on Regions V and VII Response ... 71 Region VI Response 78 Deflection Behavior in the Vicinity of ^.^qj^ 78 Effect of Prestress on Xi^p 78 Superposition of Step Loads 82 Study of Material Properties Variations 82 v

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TABLE OF CONTENTS (Continued) Chapter Page VI STRESSES 92 Development of Stress Equations 92 Numerical Results 94 Vn CONCLUDING REMARKS 98 Conclusions 98 Suggestions for Future Work 99 Appendix A FOURIER TRANSFORM OF THE FORCING FUNCTION .... 100 B SOLUTION OF EQUATIONS FOR THE TRANSFORMED DEFLECTIONS 102 C EVALUATION OF THE DISCRIMINANT OF A FOURTH ORDER POLYNOML\L 106 D DETERMINATION OF THE CRITICAL VELOCITY EQUATIONS Ill E COMPUTER PROGRAM FOR SOLUTION OF LOAD VELOCITIES WHICH CAUSE REPEATED ROOTS IN THE UNDAMPED CHARACTERISTIC EQUATION 116 F PARTL\L FRACTION EXPANSION OF A FOURTH ORDER POLYNOMIAL 123 G CHECK TO SEE THAT SOLUTIONS SATISFY THE GOVERNING DIFFERENTIAL EQUATIONS 125 H COMPUTER PROGRAM FOR DEFLECTION AND STRESS CALCULATIONS 130 I RELATIONSHIPS BETWEEN ANALYSIS PARAMETERS .... 147 BIBLIOGRAPHY 149 ADDITIONAL REFERENCES 151 VI

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LIST OF TABLES Table Page 4-1 Correlation of Root Type with Region Numbers for Figures 4 . 2 through 4 . 8 27 H-1 Options Available for Program DEFSTR 131 Vll

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LIST OF FIGURES Figure Page 2 . 1 Cylindrical Coordinate System 5 2.2 Pressure Loading 16 4.1 Flow Diagram of Computer Program VCRIT which Determines Load Velocities at which Repeated Roots Occur 28 4.2 Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness-to-Radius Ratio Including Prestress 29 4 . 3 Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness-to-Radius Ratio with No Prestress 30 4.4 Classification of Roots of the Undamped Characteristic Equation for Variations in Eq /E^ 31 4.5 Classification of Roots of the Undamped Characteristic Equation for Variations in Gxz /^x 32 4.6 Classification of Roots of the Undamped Characteristic Equation for Variations in E,, /E„ 33 J^o •'^o 4.7 Classification of Roots of the Undamped Characteristic Equation for Variations in the Circumferential Prestress 34 4.8 Classification of Roots of the Undamped Characteristic Equation for Variations in the Axial Prestress 35 4 . 9 Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region I) 37 4.10 Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region V) 39 5.1 Contour Integration Path for Evaluating Rotation Integral ... 44 5.2 Integration Contour for Radial Deflection for > 0, b, > . . 47 5.3 Integration Contour for Evaluating Radial Deflection for 49 Vlll

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LIST OF FIGURES (Continued) FJRure Page 5.4 Loci of the Roots of the Characteristic Equation as the Damping Approaches Zero 60 5.5 Static Load Problem 64 5 . 6 Flow Diagram for Computer Program for Deflection and Stress Calculations 66 5.7 Radial Deflection for a Static Load on an Isotropic Shell (M0.3) 67 5.8 Displacements for a Static Load on an Isotropic Shell (M 0.3, h/R = 0.01) 68 5.9 Radial Deflection Shape for Various Load Velocities 70 5.10 Radial Deflection Response for Variations in Radial Damping (A^ 2 . 0) 72 5.11 Radial Deflection Pattern Immediately Above the First Critical Load Speed 73 5 . 12 Change in the Radial Deflection Pattern with Increasing Damping (> 2 = 2.7) 74 5.13 Deflection Wave Form at A^ = 5 and 10 75 5 . 14 Effect of Damping for A^ = 500 76 5.15 Effect of Damping for A^ = 2000 77 5.16 Deflection Response for Variations in Circumferential and Axial Prestress 79 5.17 Maximum Radial Deflection in the Vicinity of the First Critical Load Speed 80 5.18 Effect of the Axial Prestress on the First Critical Load Speed . 81 5.19 Variation of Pressure Pulse Length, d, at A =30 83 5.20 Response from a Smooth Sine Wave Type Pressure Pulse Using Superposition 89 5.21 Response from a Sharp Pressure Front Using Superposition . . 90 5.22 Radial Deflection Response for Variations in the Circumferential Modulus 91 IX

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LIST OF FIGURES (Continued) Figure Page 6 . 1 Bending Stress in an Isotropic Shell Under a Static Load ()Li = 0.3, d = 1) 96 6.2 Surface Stresses in an Isotropic Shell Under a Static Load (M = 0.3, h/R= 0.1) 97

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KEY TO SYMBOLS X, e, z coordinate axes K^ (fv = X, 6, z) unit vectors in coordinate directions R radius of cylinder (to the middle surface) h thickness of cylinder u, V, w displacement in directions of coordinate axes 4>, f) rotations t time N N (a , /3 = X, 0, z) stress resultants M^, M (a, /3 = X, 6, z) moment resultants shear force axial prestress stress resultant circumferential prestress stress resultant moment of inertia defined in Equation (2-7) initial lateral pressure mass density strain in a direction shear strain stress in a direction shear stress modulus in x direction modulus in e direction modulus in normal direction XI Q^(a = X, 0)

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Gq,o (a,(i ^ X, e, z) shear moduli D {a = X, 0, p) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) correction factors time varying lateral pressure load damping coefficient coefficients of a matrix Heaviside step function magnitude of lateral pressure load constant velocity constant defined in Equation (2-29) U, W dimensionless displacements in axial and normal directions


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1

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£ , H , £ lengths defined in static problem analysis 1 2 3 K , K , K' coefficients 12 1 /3 coefficient, defined in Equation (6-24) (a ) {a ~ X, 6) outer surface stress a o (a ). {a = X, e) inner surface stress a 1 S, elements of determinant A determinant of a matrix of coefficients n integer XIV

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Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ORTHOTROPIC CYLINDRICAL SHELLS UNDER DYNAMIC LOADING By Elmer Mangrum , Jr . August 1970 Chairman: Dr. J. J. Burns Major Department: Engineering Science and Mechanics An orthotropic right cylindrical shell is analyzed when subjected to a discontinuous, finite length pressure load moving in the axial direction at constant velocity. The analysis utilizes linear, small deflection shell theory which includes the effect of axial and circumferential prestress, transverse shear deformation, and external radial damping. The problem is solved using Fourier transforms, and the inverse Fourier integrals are evaluated for the radial deflection, axial deflection and rotation by expanding the Characteristic Equation in partial fractions and using complex contour integration. By studying the discriminant of the undamped characteristic equation the load velocities which give repeated roots are determined. The loci of these load velocities separate regions in which the form of the displacement solutions differ. The behavior of these load velocity loci is studied for variations in the three nondimensionalized material moduli , the thickness-toradius ratio, the axial prestress, and the circumferential prestress. XV

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By tracing the root loci of the undamped characteristic equation and by inspection of the displacement expressions, it is determined that there are five critical load velocities (velocities at which the displacement becomes unbounded) for the specific example of an isotropic shell. An increase of the load velocity above the bar wave speed produces a deflection mode which is predominantly axial. The deflection response is investigated for numerous combinations of load speed, material properties, length of pressure load, axial and circumferential prestress, and radial damping. The axial prestress has a significant effect on the first critical velocity of the cylinder; initial compression tends to lower the velocity . Circumferential prestress has no pronounced effect on the critical load speeds but does influence the response at higher velocities. Variation of material properties was found to cause a rapid change in deflection response. Through superposition, the variation of pressure load length can be utilized to approximate the response to any desired pressure load. Examples of this application are demonstrated. A comparison of stresses and deflections against those predicted by the Timoshenko thin shell theory is shown for a static load . All of the above numerical work was done using dimensionless parameters which can be applied to thin shells in general. The calculations were done utilizing a computer program developed from this research for the calculation of deflections and stresses in the shells. The program is written in Fortran and is operable on the General Electric Company Mark II time sharing service . xvi

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CHAPTER I INTRODUCTION Statement of the Problem One of the most commonly used geometries for structural application is the right circular cylindrical shell. This is particularly true in the aerospace field and in undersea exploration vehicles. In many aerospace applications the cylindrical shell serves as the primary load carrying member for the rocket system and performs simultaneously as a portion of the pressurized fuel tank. In undersea applications the quest for greater depth range has brought about many refinements in structural optimization techniques. A result of the many stringent requirements being placed upon structural systems has resulted in two areas of rapid advancement: new material technologies and more sophisticated analysis techniques . The material technologies for advanced design applications have in many cases moved away from the isotropic materials and are utilizing orthotropic and anisotropic materials to satisfy the demanding requirements for more efficient, lighter weight vehicles. Studies such as that reported in Reference (1)* have shown that there is indeed an incentive for the application of these advanced technologies. Until recent years the mathematical complexity encountered when approaching the dynamic analysis of shells has been so formidable that few results were available for design applications . Denotes entries in the Bibliography,

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The new technology demands mentioned previously have brought a response from the analysts in the past five to ten years and some of the more idealized dynamic shell problems have been investigated. The problem of particular interest in this work is that of a thin orthotropic cylindrical shell subjected to an axisymmetric pressure load moving in the axial direction. It is necessary to consider refinements to the theory such as the transverse shear deformation, axial inertia, and rotatory inertia effect so that the higher load velocities may be investigated. It is known that axial prestress has an influence on results in dynamic analyses. In this work the effect of axial as well as circumferential prestress is investigated. The specific loading considered will be a constant pressure pulse finite in both magnitude and distance which moves along the cylinder at velocity V. The shell theory utilized is linear, assuming small deflections, and by superposition it is possible to investigate the effect of various pressure pulse shapes. External radial damping is also included. Specific Goals of This Research The major goal of this research was to obtain a solution for the deflections and stresses associated with the problem outlined above. The secondary goal, although perhaps not secondary in importance to those interested in utilizing the results, was that of developing a computer program for the calculation of deflections and stresses in the cylinder. Finally, the calculation and presentation of the effect of the many parameters included in the analysis conclude the goals to be reached in the study .

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Review of Previous Work A review of the early work on the response of a cylindrical shell to a moving load is given by Jones and Bhuta (2). Until the work by Nachbar (3), who considered the dynamic response of an infinitely long cylindrical shell to a semiinfinite step pressure load, the axial inertia was not considered. Nachbar included the axial inertia effect and also assumed an external damping effect. However, due to the damping included, the first resonance condition was missed, Jones and Bhuta solved the problem with a ring load moving on an infinitely long cylinder but did not include the transverse shear effect. Other contributions were made by Reismann (4) who included the effect of axial prestress, which was significant as had been found in his work on plate strips (5). Hegemier (6) studied the stability problem for a large class of constant velocity moving loads but limited the velocity range to that lower than the first critical. All of the work previewed above was done for an isotropic material. More recently Herrmann and Baker (7) solved the problem of a moving ring load on a cylindrical sandwich shell of infinite length. Numerical results were presented for a core material which is assumed to have material damping. Also, the problem of a ring load moving on a viscoelastic cylinder was solved by Tang (8) . Contributions of This Work The following contributions are believed to be original with this work. 1 . Analysis of orthotropic monocoque cylindrical shells including transverse shear deformation, axial and rotatory inertia, radial damping, circumferential and axial prestress under a finite length step load. 2. Presentation of the forms of the solution with no damping for the sevendimensional space whose coordinates are the thickness-to-radius ratio,

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the three material property ratios, axial prestress, circumferential prestress, and the load velocity parameter. Results indicating the effect of a finite length pressure pulse, and the capability to approximate any load shape through superposition. Indication of the effect of prestress on the critical velocities of an orthotropic monocoque shell. Results which show the effects of external damping throughout the load velocity range.

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CHAPTER n GOVERNING EQUATIONS OF MOTION General Equations A cylindrical shell of thickness h and mean radius R is referred to the coordinate system shown in Figure 2.1. Figure 2.1. Cylindrical Coordinate System Coordinate x is measured along the shell axis, e along the circumference and z is perpendicular to the middle surface. The unit vectors tangent to the coordinate lines at a point (x, e, z) are designated by K , K K . The displacements in these three directions are u , u and u respectively. It is

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assumed that the displacements can be represented by the linear relationship in terms of z u^(x, 9, z, t) = u(x, e, t) + z>p{x, e, t) u^(x, e, z, t) = v(x, 9, t) + ZT7(x, e, t) u^(x, 0, z, t) = w(x, 9,t) + z^^(x, 9, t) (2-1) where u, v, w are the displacements on the middle surface (z = 0); t denotes time ; ip and tj are the rotations of a line perpendicular to the normal surface in the x-z and 9-z planes respectively, ip is referred to as the thickness stretch. Equations (2-1) require that all straight lines normal to the middle surface of the shell before deformation remain straight after deformation. This is a good approximation if the shell is thin. Herrmann and Armenakas (9) derived a linearized theory for the motion of isotropic cylindrical shells subjected to a general state of initial stress by assuming the final state of stress is reached by passing through an intermediate state, the state of initial stress. Subtracting the initial equilibrium equations from the non-linear equations of motion and then linearizing by disregarding all non-linear terms involving the additional stresses, the linearized equations of motion for a shell under initial stress are obtainable. Following this procedure, Baker and Herrmann (10) derived a linearized set of equations for the motion of orthotropic shells. Assuming an orthotropic cylindrical monocoque shell is under initial lateral pressure p., an axial tension T, and is subjected to a time dependent radial load, the five equations of motion have the form: X , 1 9x , „ 9 u , N 9 u N 9w u . I , /o o\ IF^R-gr^T^-*-;^^ -R gT = ^ V ^ R ^ (^-2)

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1^ R dd " Ox xp « 8x^ R^^^ a/ hn V I •• N r^o aw ^, (2-3) N^^aO^^^aO, 93^ _N/ ^av afw ^ av + q'(x, 0, t) = ^w + p hw (2-4) aM . aM„ X ^ _1 Ox ax R ax -Qx-Nf (.^If) I'^' + ^u (2-5) R ae ax ^e 5[-('4)if^*o-^ (2-6) where h 2 h 2 I = 1,3 12 p.R 1 + r^ for external pressure N = \^['-£ (2-7) for internal pressure (2-8) These equations include the effect of external radial damping, axial and rotatory inertia, and transverse shear deformation.

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8 The elongations, shears, and rotations have been assumed small in comparison with unity. The strain-displacement relations are therefore taken in the form of Hooke's law. ^ 8u ^x ax ^e R +z \9e ^J "^xe R + z ae ax au ^ aw "^xz az ax 9v ^ 1 /aw \ -o ON ^0z ^ ai "-RTiy-derV <2-^) The stress strain relationships are assumed in the following form a = Ex C + Ey C. X ^o x ^o 9 = E„ C + Eq C. '^o X "o 6 ''xz *^xzo \z (2-10) Integrating the stresses through the shell thickness the z variable is eliminated and the stress and moment resultants are obtained.

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N ^x ax ^ ^ 37 ^ ^ (^^ ^ 90 ; N 'e . ''e R w + R^ 8v do _i 92. + E ^ j^2 de V 9x N xe D P G ^ + _2ii M + _Ji£ 9u xe ax R ax R dd N, ex G 9v 5ii 9i ^ /5s£ + "x0 \ 9u xe ax r2 ap \ R ^3 j 39 M D D _xau^Q 9i+_ii92L R ax X ax R ae M, R \ap R R a^y "^ ax D M ^ = xi a^ +^ + D ^ x0 R \de dx x6 ax D M ex M /M _ 1 9ii\ + n 9ZL R lae R ae i xe ax «x = Ox* -If G, aw % = ^ Rr, .^ R 86 (2-11)

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10 where D > D xe E Ex I ,

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11 ^ ' \ do J ^3 do R ax xyax ^^2 .R^ R-* R N r" ot? ^ 1 / a w av ^ ^ afw _ _N, L+2^ ^ ax^ r" V ^^ a/, -f l^^^ l\ + h f 1 + !!^W + R 9H + Rh ^1 ft y oy Rya0 ax " o ax J + q'(x, 0, t) = ^w + p hw (2-15) ^ ax^ ^ ax^ ^ 9^9^ R^ W ~ ^ do^ D 2 + xe a T? R axa( °x^^lf Nh o R aw ax = Iip + ^ u (2-16) ^ ( 9liL 1 9w _ j_ afv\ ^ ^ _af£ ^ ^\o_l_£± afv ' R^ae^ Rae -R ^^^^ r axae r ^axao ^^2 + D ^ X0 ^ 2 ax [^ , aw r r^ ^ ^ ^ Nh R 2 ^-^^^ If^^--)^ It, +^v (2-17) Collecting coefficients on deflections gives E ^^ + ^ ( G + ^ ) -9l + T -^ + ^ -^ ^ ax" r" \ ^^ rV a/ ax" r" ae" 9^ h 9M ^ p h — u ae at J (E + G ) -^ — ^ V xo' dxdO V + ff<^.-N)^] w \o a^ 2 3 P ax R do 1 £. « at^ 4) = (2-18)

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12 1 a u + — lE^ + — + N ) -^ + (G ^ + T) -^ R' R do xe R' z(^^^M^^^i^(-^)up h -^ V D^ ^2 D ^ 2 G, h N e a ^ xe a + _£ + _o ^3 2 R 2 R _2 L R ae ax r I 8=1 ax (2-19) ^(^.-^>ir u + M^. -'i^) -i (•'*) t 86 4 k + ^ + N ^) (G^ + T) ^ 4 (G^ N) ^ R^ \ ^ R^ R / X ax^ R^ ^ ae^ ^^ ^ ^P o'iViX R i ax r4^ + RG, + Nh 1 + „ Q o\ R o\l al rjj a^ h q'(x, 0, t) (2-20) Kjl ^i lill A L ^ ax^ " R= ae^ " ^ at^J "" L Nh \ X R o \ _a_ ax w J. (D + D -^ — R ^ V xe) dxde TJ = (2-21)

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13 D 2 xO d n Dx D. C.2 , / Nh _£ 1_ + 1 G + -=^ 1 iC _i R -^ + G + r" ' Nh o R 1 + R _a_ w ^ u^ xe' dxde ] 2 2 R 80 " Nh o R ,-Tf ^^ I at (2-22) Equations (2-16), (2-17), and (2-18) express the principle of linear momentum in the x, O, and z directions, respectively, while Equations (2-21) and (2-22) express the principle of angular momentum about an axis through the middle surface in the direction of K and K , respectively. a X These equations can be written in the form C. u. ^i' ij C. In matrix notation the set has the form (2-23) ^11 *^12 ^13 ^14 ^12 ^22 ^23 25 ^13 ^23 ^33 ^34 ^35 ; c c c 14 34 44 45 c c c c 25 35 45 55 u

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14 where the coefficients are defined as follows : C = (E + T) -^ + — ( G + -^ + N I -^ ° at 12 1 a R ^ [^ xe' dxde C = :^(E N) 7^ 13 R ^ V ' dx C = -rf D 2 D 2 2 X a xe a i a 14 ^ ax= r" do^ " ^ at^ 22 ""^ ax^ R^ V ^ R^ de~ R~ " af 2 — Po^ -.2 c = -^ 23 25 R D =(. " ^ * °e * ^ 1,1 * f _a_ ae ^ ax^ R= ae^ ^ R^ ^ at^ 33 -(G^ + T) ^ -^ (G + N) ^ + ^ E + ^ + N ^ "^ ax^ R^ ^ ae R^ \ ^ R^ ^ / J. >• 9 _i u a 34 -K^'^f^ 35 R D / h ^.RG^.Nh (l.f d_ dd 44 D -9l +^ ^ ^ ax^ R^ 80^ G N ^ x R at= 45 ^ (D + D ) — — ^ p xe dxde R 55 xe -^)-S (2-25)

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15 Axisymmetric Loading Assuming the loading on the cylinder is axisymmetric, the set of Equations (2-24) reduces to the following. (E + T) ^ p h ^ + i(E N) ^ + ^ ^ 1 9!i = ox at 9x 9t -L/F \r\ ^ /noT-\ 9 w , . aw , , a w R <^. ^) ^ (^x "" T) 77 ^ ^ aT ^ Po h -J ox at ^afuiafu/^ ^N^\|w^D 9!i.i9!i ^ ax^ ^ at= \ ^ ^ / ^^ ^ ax^ at^ (g^ + N ^J^ (2-26) Pressure Loading Form A step input in external pressure which is finite in both magnitude and time and which travels down the length of the cylinder at constant velocity V can be represented in the form q'(x, t) = q{H[Vt x] H[V(t t^) x] } (2-27) where H(y) is the Heavi side step function, defined as 0, y < H(y) { (2-28) 1, y >

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16 This pressure loading is represented schematically in Figure 2.2 TTTiTtTTTtTTTTTYT V R -. Vt Figure 2.2. Pressure Loading Nondimensional Equations The steady state solution will be investigated. Making the transformation a = X Vt (2-29) the partial derivatives may be written in terms of a. d 9x

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17 multiplying by R/E in the first two equations and 1/E in the third gives 1 + jr \ d"u X / d0 2tt P h V j2 din .p 2 I E d<^' N \ dW E Jd0 D ,2, R^E d0^ X ^ E R^ d0^ X Ejd<^ 2x ^ X 1 dfw E E / , ,2 X X / dcp tVR dW ^ PQ ^ dfw E dd) E ,,2 X ^ X d0 /E, D, \ X E R ^ ^x ^ / 2x , ji ^\ d£ E E R / d0 ^ [H(-R0) H(-R<^Vt^)] D d^U V^I d^U R^E d(f x R^E d(^' fx + Ji l^o \ dW E E R I d(^ X X / D R^E df£ d(^^ E R^ d0^ X ^ X ^ N o \ , _ rt r^ E" R-j'^ ° X X / (2-32)

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18 Now the following dimensionless ratios are defined. T X E X

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19 Using the dimensionless ratios given in Equations (2-33), the equations of motion now take the form (1 + F r x^) ^ + (E p) 1^ + (D I A^) ^ = (2-34a) (E _p)^ _(G+F-rA")^^ -.A^ ME 4-Pr)W (G + P r^) ^ = -q^[H(-R0) H(-R(^ Vt^)] (2-34b) (D I x'") ^ (G + P r ) ^ MD I A^) ^ ^ ° d0^ o' d0 ^ 1 o ' ^^2 (G + P r )i^ = (2-34C)

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CHAPTER ni TRANSFORMATION OF EQUATIONS Equations (2-34) will be transformed using the Fourier transform 00 F[f(0)] f (s) = J f(cp) e"'^'^ d0 (3-1) The inverse transform is 00 " ds (3-2) F V'(s)] = i{ the transforms of derivatives of f((/)) are given by {\s) = (is)^f(s) (3-3) so I dcf.^ J is f (s) (3-4) F ^^ -s^f(s) (3-5) 20

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21 Applying this to Equations (2-34) gives (1 + F r \^) s^ U (s) i(E P) s W (s) + (D I X^) s^ 4(s) i(E P) s U (s) + (G + F r X^) s^ W(s) i e X s W(s) + (E^ + P r^) W (s) i(G + P r^) s ip (s) ^o , isd ,, ^ IS ' ' (D^ Iq ^^) s^ U(s) + i(G + P r^) s W (s) + (D^ I^ X^) s^ 4(s) + (G + P r^) i^ (s) = (3-6) Collecting coefficients on like displacements allows this set of equations to be written in the following matrix notation. [(l + F-r^X^)s^] [-i(E^-P)s] [(D^-I^X^)s^] [(G + F-rX^)s^ [i(E^-P)s] -i€Xs + (E +Pr )] ^ o o 2, 2 [-i(G + Pr^)s) [(D^-I^X^)s^ [(D^-I^X)sl [i(G.Pr^)s] ,(G.Pr)] U(s) W(s) i^(s) o , isd , , •:— e -1 1 s ^ ' (3-7) *See Appendix A for the derivation of the transform of the forcing function.

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22 This set of equations can now be solved for U, W, and ijj. This work is carried out in Appendix B and the results are ^^-^ = ^^ ~ ^ — ^ [(D I A^) (G + P r + E P)s^ + (G + P r ) (E P)] q 2 TT, , ^ 1 o ^ o 1 ^ o' ' 1 ' ^o s D(s) ^^^ = ^^ " — ^ [(D I X^)c s^ + (G + P r ) (1 + F r A^)] q . — , , ^^ 1 o ' ^ o' ^ 1 '^ ^o isD(s) ^^ = ^ — — ^ [(E P)(D I A^) + (G + P r )(1 + F r A^)] ^o D (s) (3-8) where — 4 3 2 D(s) = c s +ic s +c s +ic s + c (3-9) and c = (G + P r ) [(1 + F r A^) (E + P r ) (E P)^] o 1^0 01 c = e A(G + P r )(1 + F r A^) 1 ^ o^ 1 ' c = (D 1 A^) {(E + P r )c (E P) [(E P) + 2(G + P r )] } 2 1 o "^ O O 1 1 o ^ + (G + P r ) (1 + F r, A^) (F P r r A^) c = e A(D I A^)c 3 ^10 c = (G + F r A^)(D^ I A^)c 4 ^ '^10 c = 1 + F-D^-r^A^ + I^A^ ^^_^^^ The displacements are found by inverting Equations (3-8) using transformation (3-2). In order to evaluate these integrals the roots of the characteristic equation D(s) = must be determined.

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CHAPTER rV INVESTIGATION OF THE CRITICAL VELOCITIES It is known that the inverse of the deflections [given by integral (3-2) , where f (s) represents the deflection expressions (3-8)] does not exist when there are repeated roots of the characteristic equation D(s) = on the real axis. This can occur when there is no external damping and corresponds to a resonant condition as discussed by Jones and Bhuta (2). There are specific load velocities corresponding to these points and they will be referred to as critical velocities. The condition which must be satisfied in order to have repeated roots is that the discriminant of the undamped characteristic equation cs'^+cs^+c=0 (4-1) 4 2 must be zero. The discriminant of this equation is determined in Appendix C as A = 16c c (c^ 4c c )^ (4-2) O 4 ^ 2 O 4' Therefore, the three conditions which will make the discriminant zero are c^ 4c c =0 (4-3) 2 O 4 c^ = (4-4) 4 C = (4-5) o Substituting the required coefficients from (3-10) into (4-3) gives* C X^ + C\^ + C X^ + CX^ + C =0 (4-6) 8 6 4 2 O *See Appendix D for the detailed calculations. 23

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24 where C = e^ e 8 3 14 C = 2e e e 6 8 9 13 c 4

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25 The solution of Equation (4-9) gives two roots (X' )^ and (X' )^ which are 1 2 approximations for X^ and X . These are the square of the two lowest load velocities at which repeated roots occur. The second condition which gives repeated roots is that given by Equation (4-4). Substituting for the c^ coefficient gives (G+F-rX^) (D^ -I^X^) [l+F-D^ + (I^-r^)X^] = (4-11) This can be written (f^ rX^) (D^ I^X^) (f^ + e^X^) = (4-12) 2 Expanding this and collecting coefficients on X yields [f3D^-(rD^-.f3yx^ + I^rX^] (f^-.e^X^) = + [f rl e (rD +f I )]X^ + e I rX^ = lO 4^1 3 0" 40 Finally C' X^ + C' X* + C' X^ + C' =0 (4-13) 6 4 2 O ^ ' w^here C' = e I r 6 4 O C' = f rl e (rD +f IJ 4 10 4^130' C; = f^f3D^ (4-14)

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26 If rotatory and axial inertia are neglected Equation (4-13) reduces to give one root C' f p The third condition (c = 0) from Equation (4-5) is now investigated. Substituting for c gives (1 + Fr^x"^) (E^ + Pr^) (E^ P)^ = (4-16) or (f r A.^) e = e (4-17) ^ o 1 ' o 1 ^ ' thus A

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27 The repeated roots are either real or imaginary depending upon the sign of the coefficients. Equation (4-13) gives three more roots which will be labeled A. 3, X^, and 2 A.^. These roots come from the statement that c^ = and if c / the repeated roots in this case will occur at an infinite value. Equation (4-18) adds one more root, making a total of eight. This root is 2 designated Xg. The condition leading to this root was that c =0. From Equation (4-1) it is observable that the characteristic equation becomes s^ic^s^ + c^) =0 (4-21) which shows a repeated root at the origin. A computer program was written for the solution of these equations. A simplified flow diagram of the program is shown in Figure 4. 1 and the details of the program, named VCRIT, are presented in Appendix E. The results of a parametric study using the computer program VCRIT are presented in Figures 4. 2 through 4. 8. The locus of each of the roots A ^(1 = 1, ...8) is shown on these figures. These curves are the boundaries which separate these plots into distinct regions which are labeled as Regions I through Vin. In each of the regions the roots of the undamped characteristic equation have a particular form as noted on the figures and as listed in Table 4-1. Table 4-1 Correlation of Root Type with Region Numbers for Figures 4. 2 through 4. 8 Region

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28 Insert Data in Separate File z E,, h ^Oq Gxz„ E^,^ R ' Ev ' Ey ' Ey Aq Xq Aq P, F Start VCRIT Yes Read and Print Data Calculate Coefficients Solve Fourth Order Equation in A^ Print Roots Solve Third Order Equation in X^ Print Roots Solve First Order Equation for Last Root Set Axial and Rotatory Inertia to Zero Repeat Solutions Using Simplified Equations More Data? No Stop Figure 4. 1. Flow Diagram of Computer Program VCRIT which Determines Load Velocities at which Repeated Roots Occur

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29 10.000 1.000 100 0.1 0.01 ^ 0.001 ^ "'••'•' ' •' 0.00001 Pure Imaginary Roots 1.0. t = TT-ii = 0.3. -^ = 0.35 P = 0.004. F = 0.002 2 Real, 2 Imaginary Roots 0.0001 0.001 0.01 0.1 h/R Figure 4.2. Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness to Radius Ratio Including Prestress

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30 10,000 1000 100 0.1 = 0.35 0.00001 .0001 .001 .01 0.1 h/R Pure Imaginary Roots 2 Real, 2 Imaginary Roots Figure 4.3. Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness to Radius Ratio with No Prestress

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31 1000 Pure Imaginary Roots 2 Real, 2 Imaginary Roots o.iot^ 0.40 0.80 1 .: l.G 2.0 Ee^Ex^ Figure 4.4. Classification of Roots of the Undamped Characteristic Equation for Variations in Eg /Ex„

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32 10,000 1000 0.01 vir Real Roots Pure Imaginary. Roots .00001 1.0 Figure 4.5. Classification of Roots of the Undamped Characteristic Equation for Variations in G^zq/^Xq

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33 1000 100 VII Real Roots O.K 0.01 0.001 0.0001 0.0001 0.001 0.01 0.1 1.0 10 Figure 4.6. Classification of Roots of the Undamped Characteristic Equation for Variations in the E^; /Ejj^

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34 1000 100 VII meal Roots)" 2.0 1.0 .Complex. Roots > A Loci .1-1' 0.1 liegion I j Pure " Imaginary Roots »^...^.^^^t^aj»QA.iau.^ji^tfnilfi r^ r -^ — -^--^ ^.....a...-..^^>.wwa '^^ -^^^^^-' -0.002 0.002 P 0.004 Figure 4,7. Classification of Roots of the Undamped Characteristic Equation for Variations in the Circumferential Prestress

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35 1000 100 — 1.0 0.1 -0.002 -0.001 0.003 0.004 Figure 4.8. Classification of Roots of the Undamped Characteristic Equation for Variations in the Axial Prestress

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36 The lower three values of X. are approximated by Expressions (4-9) and (4-15) by neglecting the axial and rotatory inertia. The approximations are quite good over the range of parameters studied. The critical load speeds are denoted as those speeds at which the displacements become unbounded for an undamped system. This corresponds to the load speeds which produce a double root on the real axis as will be shown later. It is instructive to follow the path of the roots of the undamped characteristic equation in the complex plane as the load speed increases. As an example, the roots will be traced for material properties corresponding to an isotropic shell with h/R = 0. 001 and positive axial and circumferential prestress. Following the vertical line for h/R = 0.001 in Figure 4.2 for increasing X^ will give a path crossing all of the boundaries separating different types of roots. Starting at the low load speed, the roots are all on the imaginary axis. (This would not be the case if prestress were not included, as shown by Figure 4.3.) As a means of tracing the location of the roots. Figure 4.9 is utilized which shows the complex plane. The roots in Region I appear on the plane on the imaginary axis and these particular roots are designated s , s , s„ , , and S^ , . The nomenclature s. . denotes the i^^ root location and j indi3,1' 4,1 l,j J cates the relative position of the roots. For instance s ^ denotes the position of the first root and at that time (load speed) the other roots are located at points s^ ^ , s„ , and s, , . The arrows indicate the direction in which the ^ 2,4' 3,4 4,4 root is moving for an increasing load speed . At the first speed two pairs of roots are moving toward one another on the positive and negative parts of the imaginary axis. They meet, and the first repeated root location is established which corresponds to X in Figure 4.2. As the load speed increases Region II is entered. The roots are complex as can be seen in Figure 4.9. Next, the complex roots approach one another in pairs on the negative and positive real axis. This gives

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37 the first repeated roots on the real axis and this load speed is designated as the first critical load speed, ^ipr>. Complex 8 Plane 8. . = Root 1 at load speed j Figure 4.9. Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region I)

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38 Now the roots separate and go in opposite directions along the real axis , the larger roots eventually becoming unbounded. This speed corresponds to the boundary line between Regions IV and V. The condition causing this occurrence is that c^ -*• 0. Repeated roots occur at infinity and the second critical load speed has been determined. As the load speed increases further, the large roots come in from +i
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39 Im(s) X 3. J 1 ,'. I A, ^ CH / / () s s s -s$^-^s s s \ \ \ \ \. \ =8.1 s o 2.V 2.P^ <1 3 l.t> Complex s Plarn'CK 3.t 3,6 S S S S s.^ 1.1 i;i ij.. . << «o — «o »o 'CR Ab.y.e A r CR 4 .5 x^ ^ \ \ \ \ I "J~T{f\s) / / / / / 2.9 s. . 'Root i at load speed j Figure 4. 10. Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region V)

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40 When this load speed goes to zero, the critical small deflection buckling load has been reached. From Equation (4-15) this load is seen to be approximately F = -G (4-22) This corresponds to the second critical load for engineering materials with a more realistic value for shear modulus. If prestress is not included, Region I disappears as shown by comparing Figures 4.2 and 4.3. Decreasing the tangential modulus drastically or increasing the normal modulus results in the same effect as shown by Figures 4 . 4 and 4. 6. Circumferential prestress has essentially no effect on the critical load speeds but axial prestress has a pronounced effect as shown by Figures 4 . 7 and 4.8. It is instructive to make a comparison of these results with those obtained in Reference (2). Threecritical velocities were derived there, which are given as p^R[3(l-M)]2 6p^R^(l-M) VcRs = ^ (4-24) VCR3 = f (4-25) '^o for an isotropic material. Since, in the nomenclature used in the present work, V^pR(l-M^) ^ = -^Eh (4-26) 2 The corresponding expressions in terms of A. are ^CRs = f (4-28) ^CR. = #(1-M^) (4-29)

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41 Now taking /i = . 3 it is found that ^CR = 0.55075 0.15^ (4-30) ^CR^ = f (4-31) ^CR3 = 0.9lf 2 h From Figure 4. 3 (which is a plot of \. versus -^ for an isotropic material with ^ = . 3) for the case of zero axial and circumferential initial stress, A(^j^ , 2 2 2 2 '^CRc» ^"^ ^CR-^ ^^^^ Reference 2 agree extremely well with \ , X , and A^ respectively. The two velocities, A and A , arise in the present results because of initial prestress considerations and shear deformation, respectively, which were not included in the referenced results. A corresponds to the dilitational wave speed, A^ corresponds to the shear wave speed, A^ corresponds to the bar wave speed, and A , ^ corresponds to the plate wave speed. 5 , o , y ,8

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CHAPTER V SOLUTION FOR DISPLACEMENTS General Solution Transformed Displacements From Equations (3-8) the transformed displacements can be written as 2 TT/ \ 1 isd c' s + c' U(s) ^ 1 e _5 6 % s^ D(s) m7/<,\ 1 „isd c' s + c' W(s) ^ 1 e _7 8 ^o D(s) q (1 e^-) -^ (5-1) ^o D(s) where c' = f (e + e ) 5 7^ 3 l' c' = e„ e 6 3 1 c' = f (f + e x^) 7 7^ 1 4 ' C = e (f^ r X^) 8 3 O 1 ' c' = e f + e (f r A^) 9 17 3^ O 1 ' f = D I ;^^ 7 1 o (5-2) 42

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43 Inverse Transformation of the Rotation Using the inverse transformation given by Equation (3-2) the rotation is defined as ^ — I / M^ e'*^ d. ,5-3, % ^"^ ^ D(s) By partial fraction expansion this can be put in the form* where Q^k " T"; ; " ~4 ^^^ n (\ ^j |At^(«)i s=s, --km k m=l c\ (5-4) % = r (5-5) 4 Defining CO!, /« /I isd, . , — CO then 4 ^ ^ y; F,(c^) (5-7) *See Appendix F for the derivation of a sample partial fraction expansion expression.

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44 Letting 1 isd f, (s = e ^ (5-8) k the integral to be evaluated is 00 f ys) ds _ 00 where fi^(s) is an analytic function except at the simple pole s = s. . Defining the complex root in general to be \ = ^k ^ ^ \ • the Cauchy integral theorem is used to evaluate this integral .

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45 The integral around the closed path shown in Figure 5.1 is R 0) R^co c^ Therefore the first integral on the right in Equation (5-10) goes to zero as R 00 and (|) fj^(s) ds = / f (s) ds = 2ir i Res(s. ) — 00 i0(a^+ib^) = 27r i e , (j) > lirilid(a.+ib,)-l i(^(a +ib ) + 27ri|-e " ''e ^ ^ , + d>O Therefore -bj^ <^+i aj^ cp -h^(d+(t>)+i a^(d+(P) r -b. H = Heaviside step function. Similarly, when b, < the integration path is in the lower half plane and the result is j -h.(t>+ia,(t> -b (d+0)+ia,(d+0) ^k " ^''s^kr H(-<^) e "" "" H[-(d+0)l(, bj^ < (5-12)

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46 The general expression for F, can be written in the form Fj^ = i sgn(b^) c^a^\e -h^(d+ct>)+isi^(d+(P) H[sgn(bj^)0] H[sgn(bj^)(d+(^)] where H is the Heaviside step function and 1. sgn(bj^) = b, > k -1, b, < k (5-13) (5-14) The rotation is given by Equation (5-7). 4 k-l Inverse Transformation of the Radial Deflection The radial deflection is obtained by inverting W to give q^ — f Ini J 00 2 isd c' s + c' e 7 e i
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47 Defining w k^^' 27r i 7 s(s s, ) ^ d (5-18) The deflection is then given by the sum k=l (5-19) By contour integration in the upper and lower half planes , the integral Equation (5-18) can be evaluated. Breaking the integral into two parts gi in ves F, (0) k 2-n i I ,i0s s(s Sj^) ds / gi(0+d)s s(s Sj^) ds (5-20) The contour shown in Figure 5.2 is used to evaluate these integrals for > 0, b, > 0, k

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48 From Jordan's Lemma (12) it can be seen that the integral on contour Co ^OasR-*«>, i.e., lim / i(ps R->°o C s{s s^) ds 0, > R and lim / s(s s. ) ds -> 0, 0+d > R->oo c^ Also lim £-0 ZP R P / f, (s) ds which is the Cauchy principal value of the improper integral . If the integral exists, then this is the correct value for the integral and the symbol P can be dropped. Therefore / i0s ds + lim / i0s P ds = 27r i Res(s, ), (p > (5-21) The second integral in Equation (5-21) can be evaluated as lim / p->o ;; i0s s(s-sj^) d^ = cviRes(O) Since the integration is clockwise a = -IT and Res(O) = icps s s k / s=0

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49 so /i0s -^ s(s s, 7r 1 lim / -; rds = r, J s(s s, ) s, p-*0 ^ ^ k' k Integ

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50 The integrals have the values / i(ps ds + lim / i(ps _e s(s s. ) "" ""' J s(s s, ) ds = 0, (p < or / ^^f^ds = -.iRes(O) = ^, c/xO b, > (5-24) Similarly / b,>0 (5-25) Equations (5-22), (5-23), (5-24), and (5-25) can be combined to give the solution for FyXcp) when b, > as wV*) = Irr 1 (1;; ^ ^ »'*'']"<*) ^ f;; «<-*> ; T * ^ ;<*^>M H(^,d) + |1 H(-*-d)] b^>0 or finally F, (0) = — w k^^' s, / 1 i<^s \ I + e '^ H{+ 0. k

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51 The case where b, < is now investigated. Integrating in the lower half plane it is found that i<^s. / id)S /• id)s ds + / TTZ ^T ds = 2-ni — s(s s, ) J s(s s, ) or / "» . . i0s, 10s ^ k ^ ds = -27r i ^+ T^ , < s(s Sj^) s, s, ' k k bj^<0 It is evident, therefore, that the general expression for ^\S4>) can be written in the form sgn(bj^) /3j^ w k^^' s 1 , ''^\ 2^' H[-sgn(b )0] H[sgn(b^)0] + ^ r i(0^)s -1 l^-i + e ''J H[sgn(bj^)(0+d)] + H[-sgn(b^)((^4d)] Inverse Transformation of the Axial Deflection The axial deflection in the s-plane can be written as (5-27) ^ s^D(s) (5-28) ^o = D(s) From Equation (5-1) it can be seen that the first term in Equation (5-28) can be written in terms of ip . Thus q^ c' q^ 2Tr J ^2 j^ isd. . , e ) 10s J ' e ^ ds (5-29) 9 'O s D(s)

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52 The integral in Equation (5-29) can be written as 4 2tt J isd, . , k=l a, s s. ds where c ^6 = C Now defining F, (0) c a o l^k r ii_,^i!ii^,. S (S Sj^) then (5-30) ]M1 _!^ M^ +yF,(0) q 9 ^o k=l The integrand in Equation (5-30) can be written in two parts as F, (<^) 6 k 27r / i0s ds s (s Sj^) / i(0+d)s ds (5-31) (5-32) s (s Sj^) The contours shown in Figures 5.2 and 5.3 are used to evaluate the integrals in Equation (5-32) for the case b, > 0. Define i0s i{(p+d)s f, (s) u k^ ' s (s Sj^) f; (s) = u k^ ' 2, S (S Sj^) By a procedure completely analogous to that just described for the radial deflection, the results can be written i(ps f — ds ^ 7riRes(0) +27riRes[ f, (s, )] , 0>O ^ (^ \) b^ > k (5-33)

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53 /ei(0+d)s J ~ -ds = 7riRes'(0) + 27riRes[^f|^(s^)l, (p^ > (5-34) S (S Sj^) b, > k where -\(l> + ^\4> ^^«fufk(\)] = r (5-35) -bj^(0+d)+iaj^(0-+d) ^^«Iu^k(\)J ^ i (5-36) i\ + i bj^) To find the residue of the functions at s = they are expanded in a Laurent series about the point s = . e = J+ ii _ ^ _ i*^ s ^ s2 g2 s 2! 3! -1= J1 . ^ . ^ , e""~ _ 1 / 1 , \ ^ -• . .2 i0s / i<^ ^ s^(s-sj^) \\s^ s s/ \ 2 The residues can be evaluated as Res(O) = -1_ i A ^3_3^j \ k Res'(O) = J_ i ±±A \' '^ (5-38)

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54 so that / i(j)S s (s Sj^) as. -f^(t-*)--5 b, > k > / i((/)+d)s ds s (s Sj^) -'tk'^H i((/)^)S, + i((p+d) + 27ri . e b, > k k d +(p > Integrating in the lower half plane for <^ < 0, Equation (5-30) can now be written in the form i<^s. F, (0) = i -^-^ u k^^' s, -nt-*)^^ -i(t^'<*^^')^^ H(0) i(0-hd)s,^ H(0+d) + — ,/l+i0s^ 2 \ s, H(-0) ^ / 1 + i(cp-Hi) s^ H(-0-d) 1 , b, > (5-39) Performing the integrations for b, < 0, the final result can be written as F, (0) sgn(bj^)ic^aj^ [-2(l+i<^sj^)-fe ^J X H[sgn(bj^)0] + |(1 +i Sj^) H[-sgn(bj^)0] , i((/)-Kl)s 1 ^[l+i{(p-Hi)s^]+e '^ H[sgn(bj^)(0-kl)] + i[l+i(0+d)Sj^] H[-sgn(bj^)(0+d)] (5-40)

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55 The solutions given by Equations (5-7), (5-19), and (5-31) with the corresponding F. functions (5-13), (5-27), and (5-40) are substituted into the governing equations of motion in Appendix G to show that the solutions satisfy the differential equations. Summary of Deflection Expressions The following expressions summarize the solution for the deflections. 4 2 Fj^(0) (5-41a) k=l 4 m^ k=l (5-41b) }M1 = .^^ ±i^ +Y F,(0) c q Z^ u k^^' (5-410) 9 'O k=l where F^ict» F, (0) iSj^(0+d) ( IS

-2 +e H[sgn(bj^)0] +2 H[-sgn(bj^)0] ^ is,((^+d)' H[sgn(bj^)((^+d)] + 1 H[-sgn(bj^)(<^+tl)] (5-41e)

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56 F, (0) u k^^ sgn(bj^)ic^aj^ [-1 (1 +is^0) +e i ct> Sj^-i H[sgn(bj^)0] + i-(l +iSj^ -1, bj^<0 H(y) y < y >i a Px c s, + c 7 k 8 k 4 ' ^k 4 n (\-v n <^k-v m=l m=l kT^m s, = a. + i b, (k = 1, , . . , 4) are the roots of the characteristic equation D(s) = c s +ic s^+c s^+ic s+c = 4 3 2 1 O

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57 and the coefficients are defined as c^ = -exe f 1 3 8 C 2 = f (e c e e ) +e f (f rx^) 7 O 12 3 6 2 C -exf C 3 7 c^ = (f^ rx^)f c 4 3 7 c' = f (e + e ) 5 7^ 3 l' c^ = e e 6 13 < = fA^V^) c' = e f e 3 e c' = e f +c' 9 17 8 e=E+Pr f=l + F o o o o e E P f, = 1 + F D L -L 1 1 e = E P + 2(G + Pr^) f = F Pr 2 1 2 o e=G + Pr f=G + F J O 3 e. = I r, f = D r + I f ^Ol 4 1100 ®5 " ^n^ ®.^^ f = r f +rf 5 0112 5 120 (5-42)

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58 The derivatives of the deflections are

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59 F^ functions for Equation (5-41) can be determined. Figure 5.4 provides an example of the behavior of a set of roots as the damping, e, approaches zero. This figure shows the type of the four roots (s.. = root i with damping €.) for heavy damping to be two complex and two imaginary, root 1 being very near the origin. As the damping is decreased, the imaginary roots approach one another and finally meet and separate which gives four complex roots. Meanwhile the other complex roots are also approaching the real axis. This establishes the correct sign for the imaginary part of each root in the limit as c -* and the roots all approach the real axis. Form of the Radial Deflection in Region IV The roots are all real, having the form ±a , ±a . From Equation (5-41b), the radial deflection expression (when the proper signs are established) in Region IV becomes O L 1 1 2 2 J /3 r ia la {
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60 600 400 200 ^^ -6 0O -400 -200 -400 -600 (>s 23 s. . = Root i for External Damping

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61 Expanding the /?. coefficients gives /?. -/?, c a + c 7 1 e 2a (a^ a^) 1 1 2 c a + c '^^2 e 2a (a^ a") 2' 1 2' Substituting expressions (5-46) into (5-45) gives (5-46) W q o ca+c l/ia(;& -ia^ 2a^(a^ aT) 11 2 [ia (0+d) -ia (0+d)' e ^ +e ^ H(-0-d) c a +c l/iad) -iad)\ 7 2 e 1/^ 2^^^ ^ H(0) 22 ? 2a (a a ) 2' 1 2' •[• ia^(0+d) -ia^(0+d) + e ] H(0+d) o2 „2 a a 1 2 [H(0^) H(0)] or 2 2 a a 1 2 [H(0-Hi) H(0)] ca+c 7 1 e a^(a^ a^) 1^ 1 2' ^ ["cos (a^0) H(-0) cos [a (0+d)] H(-0-d)l c a^ + c 7 2 af (a^ a") 2^ 1 2' cos (a 0) H(0) cos [a (0+d)] H(0+d)| (5-47) a^\ L 2 2 J

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62 Therefore, for the three distinct regions of the cylinder, the solutions are: Solution behind the load (0 < -d) f" = ^"7 f {cos (ad)) -cos [a (0+d)]} (5-48) 1^ 1 2' Solution under load (-d < < 0) XT, c ca+c ca+c -= — ^^— i cos (a 0) + ^ ^ 2. cos [a ((/)+d)] (5-49) q 22 2, 2 2, ^ 1^' 2, 2 2, ' 2^'^ " ^ ' o a a a (a a ) a (a a ) 12 i'' 1 2' 2^ 1 2' Solution ahead of load (0 > 0) [cos (a^0) cos [a^(0+d) ]| (5-50) 2 ^ Wc a + c _ 7 2 8 % a^(a^ a^) 2^ 1 2' Form of the Radial Deflection in Region VII As in Region IV, the roots are ±a , ±a . However, there can be no deflection ahead of the load in this region because the load speed is greater than any of the wave speeds in the material. This zero displacement comes about mathematically because the roots all approach the real axis from the negative imaginary direction. The solution for this region has the form TTT ca+c W e r^^, J, TX, J 1X-, T 1 8 [H(-0) H(-0-d)] q 22^ \ T /J 2, 2 2, o a a a (a a ) 12 1^ 1 2' X /cos (a 0)H(-0) cos [a {(p+d) ]H(-0-d)l 2 , ca+c + 7 2 8 2 2 2, a (a a ) 2^ 1 2 icos (a^0)H(-0) cos [a^(0+d)]H(=0-d)l (5-51)

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63 Form of the Radial Deflection in Region V The roots in this region have the form ±a, ±ib. The radial deflection in this region is given by the expression W ^e % a^ b^ [H(-0) H(-0-d)] c a + c 7 2 2 a (a 1fcos(a0)H(-0)-cos [a(^+d)] H(-(^-d) 1 + b ) L -J u2 . c b + c ^ ; a I fe"''*^ H(c^) e^^ H(-<^) e'^^^^^) H(0^) 2b (a + b ) •+ e^('^"^)H(-0-d)] (5-52) Form of the Radial Deflection in Region VI In this region the roots are all imaginary, of the form ±ib , ±ib . This gives an exponentially decaying solution as in Regions I and III. The radial deflection expression is given below. e ^ = -T^ [H((/)-Hl) H(0)] ^o b b 1 2 c c b^ r -b b -b {(p+d) _!_ e ^ ^ ^-^ \e ^ H(0) e ^ H(-0) e ^ H{(p+d) + 6 ^ H(-0-d)J r b„ -b <^ u vv"; e ^ H(-0) e H(0) e H{-
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64 Comparison of Solution with Other Results As a comparison of the results of this analysis with another theory, the static problem of a distributed pressure load on an isotropic shell was considered as shown in Figure 5.5, -• i' T R K^ ^2 ^ Figure 5 . 5 Static Load Problem For the static problem shown above the roots of the characteristic equation, excluding prestress, are complex. The solution for the region under the load as given by the present theory can be reduced to — = K 2 e cos aL e cos aL -a L -a L + K I e sin a L + e sin a L 2 \ 2 1 (5-54) where ^a = -f\ t

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65 This problem is solved by Timoshenko (14) and the deflection given by his theory, when put in a compatible form, becomes W ( -^^2 -^^1 — = KM2-e cos /3i^ e cos (3£^j (5-55) The second term appearing in Equation (5-54) is missing from Equation (5-55). This additional term arises because of the inclusion of shear deflection which was not present in the Timoshenko theory, A comparison of the results of these deflection expressions is made in Figure 5.7. Numerical Results A computer program was developed for the calculation of the displacements and stresses determined in this research. The general expressions given by Equations (5-41) were programmed for the displacement solutions and the stress calculations are discussed in Chapter VI. A flow diagram of the computer program is shown in Figure 5.6. The details of the program can be found in Appendix H. It is written in Fortran for time share computer application. Comparison of Results with Other Solutions for a Static Load The radial deflection for a static distributed load on a cylindrical shell is given in Reference (14), As a check on the solution this static problem was solved using the present results and the comparison is shown in Figure 5.7, The results agree very well. The effect of variation in the thickness-toradius is also illustrated in Figure 5.7. The rotation and axial deflection are shown for this static problem in Figure 5.8. In addition to showing the form of the displacements for the static load, Figures 5.7 and 5.8 serve as a basis against which the dynamic displacements can be compared. The deflections are symmetric about (p = -0.5 for the static load.

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66 Figure 5.6. Flow Diagram for Computer Program for Deflection and Stress Calculations

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T h 67 L -1.0 -0.8 -0.6 I -0.4 -0.2 W/q Present Theory From Reference (14) -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 Figure 5.7. Radial Deflection for Static Load on an Isotropic Shell (fi = 0.3)

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68 U/q -0.15 Figure 5.8. Displacements for a Static Load on an Isotropic Shell (/i= 0.3, h/R = 0.01)

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69 Summary of Deflection Response for Shells under Various Load Velocities A summary of some of the types of deflection patterns assumed by a shell for increasing load speed is shown in Figure 5.9. For the particular properties used for this example, the various regions (root types) associated with each waveform can be found by inspection of Figures 4.2 through 4.8. For example, for no damping, positive prestress corresponding to internal pressure, h/R = 0,001 and the material properties given in Figure 5.9 (properties are those corresponding to an isotropic shell as shown in Appendix I), Figure 4.2 can be used to associate load speed with root type. Following the vertical line of h/R = 0.001, it is evident that A = 1 lies in Region I where the roots are all imaginary. This gives a critically damped exponentially decaying solution as shown by Equation (5-53) and is shown in Figure 5.9(a). As the load speed increases Region II is entered where the roots are complex. This is the form of the static load problem roots, and if no prestress existed Region n would extend from zero load speed up to the first critical, which is at A . The solution for X = 2 is shown in Figure 5.9(b), and is exponentially decaying. The response becomes sinusoidal after crossing \ ~ X . At a load speed just greater than A the deflection response has a very short period. A small amplitude wave train precedes the load and a large amplitude wave follows it. As the load speed increases the sine wave period increases as shown in Fig2 ure 5.9(d) for A =30. These sinusoidal deflection patterns are in Region IV where the roots of the Characteristic Equation are all real . The mathematical expression forW/q is given by Equation (5-47). Crossing A into Region V, the roots are real and imaginary. Equation (5-52) gives the radial deflection, and 2 Figure 5.9(e) shows the response to be a long period sine function for A = 500.

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70

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71 Jumping to Region VII brings a longer period sinusoidal oscillation as illustrated by Figure 5.9(f) and the response in this region, where the roots are again all real, was discussed previously. The radial deflection is given by Equation (5-51). Because Region VI covers such a limited range in velocity the response was not included in the summary but is discussed later. Region II Hcsi)onse A study of the response of an isotropic shell at a load speed below the first critical was made to determine the effect of external damping. These results are shown in Figure 5. 10 where the damping ranges from very light to very heavy. Of course, when damping is introduced the root form is no longer the same as that of Region II. Region IV Response 2 The short period sinusoidal response of the radial deflection at A = 2 . 7 is shown in Figure 5.11. As the radial damping is increased this response is changed drastically as shown in Figure 5.12. The response for a damped system, which was in Region IV with e = 0, approaches closely that of the Region II 2 behavior. Figure 5.13 shows the radial response at X =5 and 10. The maximum amplitude remains constant as the period of the wave increases for greater load velocities. Damping Effect on Regions V and VU Response 2 The effect of damping on the wave forms for X = 500 and 2000 is shown in Figures 5.14 and 5.15. The amplitudes of the sinusoidal deflection response are initially decreased, and, as the damping becomes greater, the response becomes critically damped and the deflection approaches zero with an increase in distance from the load.

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72 w Figure 5. 10. Radial Deflection Response for Variations in Radial Damping (x2 = 2,0)

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73 — • w -1 -2 -3 R 'Xo 2.7 0.001 1.0 . 35 . 30 p

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R k74 J__ — q. -1.0 4> Figure 5. 12. Change in the Radial Deflection Pattern with Increasing Damping (A^ = 2 . 7)

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75 u

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76 U=: |

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77 L q. T" R nnr -1 L


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78 Region VI Response This region has a deflection pattern which is almost totally axial. Three critical velocities have been crossed to get into this velocity range which correspond to the longitudinal , shear , and bar wave speeds . Therefore , there can be no bending effect transmitted. The behavior is like an axial compression on a membrane which expands radially, as shown by Figure 5 . 16 . The effect of prestress is observable in this figure. The maximum deflection is increased by about 20 percent when going from external hydrostatic pressure to internal hydrostatic pressure. This axial mode of deflection also appears in other velocity ranges. For instance, the broken lines in Figure 5.16 show the behavior at \ = 1001. This is the range between A = 1000 and X = 1002 where the roots are real and imaginary. Another example is shown in Figure 5.22 where En /E^ is less than 0.08 and this occurs in Region VUI as shown in Figure 4.4, 2 Deflection Behavior in the Vicinity of An CR 2 _ ,2 The first (lowest) critical load speed occurs at Aip,„ = A = 2.552. Figure 5.17 illustrates the unbounded response of the deflection as that speed is approached . The effect of damping on the maximum deflection is also illustrated . There are four other critical velocities as discussed in Chapter IV. 2 Effect of Prestress on A . ^r^ ^^CK The effect of the axial prestress on the location of the first critical load speed is shown in Figure 5.18. This effect is also observable in Figure 4.8, since the first critical load speed is at A^ . The circumferential prestress does not have a significant effect on load speed, as shown in Figure 4.7.

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79 _u Figure 5. 16. Deflection Response for Variations in Circumferential and Axial Prestress

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80 2.4 2.r) V2 2.6 2.7 W -4 -10 -12

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81 o o

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82 Superposition of Step Loads The effect on the radial deflection can be observed in Figure 5 . 19(a)-(f) where the load length was varied from 0.1 to 5.0. By superposing various combinations of step loads it is possible to approximate any shape of load desired. As an example of this type of application, the radial deflection response from a symmetric sine wave type load and a sharp edged pressure front was determined. The results of these calculations are presented in Figures 5.20 and 5.21, respectively. Study of Material Properties Variations A look at the effect of decreasing the Eq /E^ ratio is summarized in Figure 5.22. Starting in Region IV, as can be seen in Figure 4.4, the ratio is decreased from 1.0 (as for an isotropic material) to 0.04. As the ratio is lowered, the maximum deflection gets large rapidly, and becomes unbounded as ?^^ =^ ^IpR is approached. After crossing ^^cn ^"^° Region VIII, the strength in the circumferential direction is of course very low and the material is of little interest for engineering applications. The same type of response as in Figure 5.22 will be obtained by increasing E,, /Ej. significantly. This can be observed by inspecting Figure 4.6. For other types of material property variations, the general response can be pinpointed by observing the type of roots at the particular location through the use of Figures 4 . 2 through 4 . 8 and using the numerical results presented here showing similar calculations of deflections .

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83 R — (L Hd -o; 1 (a) d = 0.1

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84 l*cl • ""'' — q.-I-0.2 0.35 0.3 p

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85 ^ -0.5 0.35 p

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86 R — q.p

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87 -q.-f6 = 0.001 p

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88 t t -5 q.p

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89 :
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90 T 3q„ -1.0 -0.5 4t = 0.001 0.35 0.30 p

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91 '9o = 0.5 122 E0„ = 0.2 0.04 Figure 5.22. Radial Deflection Response for Variations in the Circumferential Modulus

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CHAPTER VI STRESSES Development of Stress Equations For the axisymmetric loading the stress and moment resultants become N =E^+^M + E^ (6-1) X X ax R ax I' R ^ ' N=(^E+^^^ + e|^ (6-2) ^x = ^ i ^ °x i <^-=' «e'X I ^ '^^ If ('^-^) Assuming the bending stress is linear across the thickness of the shell, the stresses can be put in the nondimensional form (6-6) E d<^ 1 d4> ^1^ E^ X ^ ^ ^o N^ JTT O'fl ^ = E W + E ^ = ^ (6-7) E o 1 dd) Ev 92

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93 "^x _ du ^ di^ o R !^ X ^ ^ Xq GQ X G ^V^ d(p The stresses are therefore „ / , ^ dW \ ^xz HL. = ^ ym ^^ Him ^uim (6_n) ^Xo% ^ % ' % ^ o ^xo % ° % ^ % ^ ^ -!^^ = i Ji/UM ^i!!^ Ey q 2 R I q q Ev q ^o ^o

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94 (a ). X inner q Ex

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95 The moment, M , in Reference (14) is assumed to be ,2 M = D ^-f (6-22) Combining Equations (6-21) and (6-22) and assuming a linear stress distribution through the thickness, the bending stress is determined. <^Xb 3 / -^^2 -^^1 \ 'e sin(S£ +e sin /? £ I (6-23) 2^3(1 M^) where 4 ^ 3(1 M^) 2 2 R h (6-24) The stress given by Equation (6-23) is shown in Figure 6.1 in comparison with the results derived in this work. The results given by (6-23) are good for thin shells and get worse as the shell becomes thicker. The axial and tangential surface stresses are shown in Figure 6.2 for the static load problem. The computer program developed can be used to calculate the individual stresses and/or the surface stresses as desired.

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96 " -1.0 (p x/R R Present Theory Equation (6-23) x/R Figure 6.1. Bending Stress in an Isotropic Shell Under a Static Load (/i = 0.3, d = 1)

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97 x/H Figure 6.2. Surface Stresses in an Isotropic Shell Under a Static Load (;i = 0.3. h/R = 0.1)

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CHAPTER VII CONCLUDING REMARKS Conclusions An analysis has been made of an orthotropic infinite circular cylinder subjected to an axisymmetric discontinuous pressure loading of finite length moving in the axial direction with a constant velocity. Fourier transforms were used to solve for the steady state solution and proved to be very useful for this type of application. The problem of nonuniqueness-of-solution arose for the case of an undamped cylinder . This was handled by taking the undamped solution as the limit of the damped solution when the damping approached zero . Investigation of the higher load speeds showed that there are five critical load speeds, the two highest being very close together and corresponding approximately to the plate wave speed. Lowering of the tangential modulus ratio was found to cause a rapid increase in deflections . An increase of speed above the bar wave speed was found to cause a mode of deflection which is mainly axial in nature as opposed to the low speed bending type deflection pattern. The method of superposition was used to demonstrate the adaptability of the results to other forms of loads. Although the shell theory used was linear , small deflection theory , the results are useful for loads in the range for which most structures are designed. The general response picture can be obtained from the results presented here and the location of the critical load speeds observed . The computer program developed here can be used to investigate in detail particular problems of interest. Because of the increasing frequency of applications of cylindrical shells 98

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99 in dynamic loading conditions in military, government, and industrial programs, the critical load speed environment should be considered in the design requirements. Programs utilizing thin walled launching tubes for launching missiles are a specific example of such applications. Suggestions for Future Work For large deflections the theory used in this analysis becomes invalid. A natural extension of this work would be the investigation of the stability through use of large deflection theory. Also the increased use of composite materials for structural applications makes the inclusion of anisotropic material a worthwhile endeavor . In this analysis the axial and circumferential prestress were considered. Another parameter which may prove to have a significant effect is that of torsional prestress. The computer program developed for the calculation of stresses and deflections in the shell is by no means efficiently programmed. Extensive use would warrant an investigation to reduce run time.

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APPENDIX A FOURIER TRANSFORM OF THE FORCING FUNCTION Let L denote the transform of the pressure function given on the right hand side of Equation (2-34b); the integral to be evaluated is ^f = f % [H(-R0) H(-R0 Vt )] e"^^*^ d0 (A-1) or %/ [H(- < (A-4) H(-(/)-d) c + d > (/) + d < Integral (A-2) therefore can be broken up into two integrals, -d I f f e-'^^d4> f e-' -is* d0 (A-5) (A-6) 100

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101 Furthermore , the second integral can be written -d f e-^^'^dc/) = f e'^'^d


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APPENDIX B SOLUTION OF EQUATIONS FOR THE TRANSFORMED DEFLECTIONS From Equation (3-7) the set of equations to be solved is (l+F-r^X^)s^

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103 Collecting coefficients on powers of s gives the expression for the determinant in the form A = (G + FrA^) (D^ -I^A^) ^(1 + Fr^\^) -(D^ -I^\^) + iekCD^ l^X^) I (D^ I^\^) (1 + F r^A^)l s^ (1 + F r A" ) (D -I X^) 1 ' ^ 1 o ' + i (E + Pr ) (D I A ) ' ^ o o' ^ 1 o ' + (1 + F r^\ ) (G + Pr^) (G + F rA ) (G + Pr ) -(D^-Io^*^) (E^ P)^ + 2(E^ P) (G + Pr^) ieA(l + F r^A ) (G + PrQ){ + (G + Pr^) (1 + F-r^A^) (E + Pr^) (E^ P)^ Solving for U(s) using Cramer's rule gives (B-4) U(s) = ^ % , isd -. -:— (e -1) IS ^ ' -i(E^-P)s (G+F-rA^)s^-ieAs +Eo+Pro i(G+Pr^)s or, after expanding the determinant, U(s) = f , isd ,. r (D -IA^)s^ ^ 1 o ' -i(G+Pr^)s (D -I A^)s^+(G+Pr ) ^ 1 o ' ^ o' (B-5) (G + Pr^) (D^ I^A ) + (E^-P)(D^ -Io^^|s" + (E^ P) (G + Pr^)^ ] (B-6)

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104 Defining the coefficients in A as c = (G + Pr ) o o (1 + F r X^) (E + Pr ) (E P)^ c^ = -e\(G + Pr^) (1 + F r^X^) (E^-P) + 2(G + PrQ) + (G + Pr^) (1 + F r^r ) (F Pr^ rX^) c^ = -€Xm I \ )c 3 ^ 1 O ' c^ = (G + F vX^) (D^ I^A^)c c then A where = 1 + F D^ rX^ + IX^ 11 o s^ D(s) 4.3 2 . Tiis) =cs +1CS +CS +ies + c ^ ' 4 3 2 10 Using this nomenclature Equation (B-6) can be written isd U(s) _ 1e' (D^ l^X^) (G + Pr^ + E^ P)s^ + (G + Pr ) (E P) o 1 (B-7) (B-8) (B-9) (B-10) Now W(s) is determined. Using Cramer's rule again, the transformed radial deflection is given by W(s) = ^ (1 + F-r A^)s^ ^ 1 ' i(E^-P)s (D -I A^)s^ ^1 o ' r^(l-e^^^ IS ^ ' (D -I X^)s^ ^ 1 o ' -i(G + Pr^)s (D -I A^)s^ + (G + Pr ) ^ 1 o ' ^ o' (B-11)

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105 Expanding the determinant gives W(s) = l^(i.e^«d) (1 + F r A^) (D I \^)s^ 1 1 o ' + (1 + F r \^) (G + Pr )s^ 1 o' P.-Io^Ts^ (B-12) or W(s) _ il isd. -^ 1 (D^ Iq^^) cs^ + (1 + F r^X^) (G + Pr^) ^o isD(s) Finally, the rotation expression is determined as (B-13) Hs) = 4(l+F-r^\^)s^

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APPENDIX C EVALUATION OF THE DISCRIMINANT OF A FOURTH ORDER POLYNOMIAL th The discriminant of an n order polynomial ax+ax +...+a x + a = o 1 n-i n (C-1) is defined as (15) A = a' 2n-2 b S S n-i n n-i 2n-2 (C-2) The preceding determinant is called Vandermonde's determinant and the elements of it are defined as Sq = n (C-3) (-4)' 2a 3a ka. \-x a o a a 1 o (k1,2, ...,n) (C-4) For a fourth order equation of the form 4 2 ax+ax+a = O 2 4 (C-5) 106

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107 the S, 's become '^ = (-t)^ — \ a = (C-6) i-i)^ 2a 2 2a a a a? -o (C-7) (-t) 2a, a o a o a_ = (C-8) or 2a, 4a o [4a^ a 2a a^ o 4 o 2 J 2a o^ 4

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or 108 2a 1

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109 The discriminant can now be evaluated. A = a 4

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110 Expanding this further gives A = a< (2a a -a) o J 4 ^ o 4 2' ^ o 2 -I 8/0 2, . ^2 (2a a a ) 4 — 2^04 2 2 ^ a a o o 4a + — ^ (a^ 3a a ) 4^2 04' a o -—(2a a a^) -4 — 2 ^ o 4 2' 2 a a •— o o (C-16) A = -4 (2a a a^ ") [-16a a + 8a^ 4a^] \04 2/ 04 2 2' +4a^(a^ 3a a ) (-16a a + 8a^ 4a^) 2^ 2 04'^ o 4 2 2' = 16(a^ 4a a ) -(2a a a^) + a^(a^ 3a 2 o 4 o ^ '^ '^ '^ 4 2' 2' 2 a ) O 4'J = 16(a 4a 2 . a ) -4a o 4 |_ o 2 2 44 a +4aaa -a +a -3a 4 2 4 2 2 . a a o 2 4J 2 222 = 16(a 4a a ) (-4a a + a a a ) ^2 04'^ 04 024' 2 2 16(a 4a a ) (a a ) (a 4a a ) ^2 o 4' ^ o 4' ^ 2 o 4 Finally — 22 A = 16a a (a 4a a ) o 4^ 2 04 (C-17)

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APPENDIX D DETERMINATION OF THE CRITICAL VELOCITY EQUATIONS Equation (4-3) is the first expression satisfying the conditions for which repeated roots occur . It is repeated here for reference . c 4c c 2 O 4 (D-1) With the definition of terms as given by Equation {D-2), e = E + Pr o o o ^ = E^-P e = E P + 2(G + Pr ) 2 1 ^ O' e^ = G + Pr 3 o e = I r 4 o 1 e e f e e 5 O 1 12 e = e f e 6 o o 1 = 1 + F = 1 + F D 1 1 = F Pr 2 o = G + F = D r + I f 4 1 1 o o r f +rf 12 O e D -If 6 4 10 1 7

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112 The coefficients in Equation (3-10) can be written in the following form. c f + e X^ 1 4 2 c = e (f e e ) 3 e o i' c = -eXe f 1 3 8 c = f(ec-ee)+ef(f rA^) 2 Y O 12 3 e 2 c = -e\f c 3 7 c (f rA^)f c 4 3 7 c = f (e + e ) 5 7 3 1 c, = e e 6 13 c = f (f + e A^) 7 7^ 1 4 ' c = e f e 3 8 c = e f + c {D-3) 9 17 8 The coefficients in Equation (D-3) are substituted into Equation (D-1) and 2 terms are collected which multiply various powers of A . Proceeding to this end, the first coefficient is c . 2 c^ (D I A^) [e (f + e A^) e e ] + e (f^ r A^) (f rA^) 2 ^1 O'O^l 4 12 30 12 = Def +DeeA -Dee lOl 104 112 I e f A^ o O 1 2 4 + IeeA -leeA o 1 2 o O 4 + e f f e (r f + rf )A^ + e rr A^ 302 312 O 31

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113 c = D(ef-ee)+eff 2 lOl 12 3O2 + (Dee + I (e e e f ) e (r f + rf )lx' 104 O^ 1 2 01' 312 O" + (e rr I e e )A ^31 004' c = De +eff +(Dee -le -ef)X +ferr -lepu"* 2 15 302^104 o 5 35'^ ^31 00 4^ c^ = e +eX +e\"* 2 y e 9 where e =De+eff 7 15 3 O 2 8 = Dee -le -ef e 104 05 35 e = err-Iee m-4^ 9 3 1 o o 4 \^ *> Now this coefficient is squared cl = S + 2e^e^A^ + (ej + 2e^eg)x'' + 2e^eg\'' + e^A^^ (D-5) c = e[e(f-r\)-e] o 3^ o^ o 1 ' 1' / r 2, 2 = e^(e f -e)-eerA 3 O O 1 3 1 = ^<\-v/) (D-6) c = (f rx^) (D I A^) (f + e A^) 4 ^3 '^1 0^1 4 = (f rA^) [D f + (D e I f )A^ I e a''] 3 1114 01' 04 = f D f + [f (D e I f ) rD f ]A^ 311 '3^ 14 01' 1 1' + [r(I f D e ) f I e ] A^ + rl e A^ (D-7) Oil 4304' 04 ^ '

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114 Let D e I f = f 1 4 O 1 9 then 4 6 ; 4 c = f D f + (f f rD f )X (rf + f I e )A + rl e A 4 3 1 1 ' 3 9 1 l' ^9 3 4 O 4 and 4c c = 4e {(e e r X )c } O 4 3 6 O 1 ' 4-' = 4e^ {(e^ f D f ) + [e (f f rD f ) e r f D f ] x^ 3 6 3 11' ^ 6^ 3 9 1 l' O 1 3 1 l' [e^(rf + f^I e ) + e r (f f rD f )] a^ 6^ 9 3 4' O 1^ 3 9 1 l'' + [e^rl e + e^r (rf + f I e )]A^ e r rl e A® } 6 4 O 1^ 9 3 4' O 1 O 4 J Finally (D-8) 2 4 6 e 4c c =e +eA+eA+eA+eA (D-9) 4 lO 11 12 13 14 ^ ' where e = 4e e f D f 10 3 6 3 11 e = 4e [e (f f rD f ) e r f D f 1 11 3' 6' 3 9 11' O 1 3 1 1' e = -4e [e (rf + f I e ) + e r (f f rD f )1 12 3 6' 9 3 O 4 O 1^ 3 9 1 1 ' e = 4e [e rl e + e r (rf + f I e )] 13 3^ 6 O 4 O 1^ 9 304'' e = -4e e r rl e m-10) 14 3 o 1 o 4 \^ ^"J) Equation (D-1) now becomes C A® + C A^ +C A^ +C X^ +C = (D-11) 6 6 4 2 ^ '

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115 where C = e^ e e 9 14 C = 2e e e 6 8 9 13 C = e^ + 2e e e 4 8 7 9 12 C = 2e e e 2 7 e 11 C = e e (D-12) O 7 lO ^ ' If the axial and rotatory inertia are neglected r 1

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APPENDIX E COMPUTER PROGRAM FOR SOLUTION OF LOAD VELOCITIES WHICH CAUSE REPEATED ROOTS IN THE UNDAMPED CHARACTERISTIC EQUATION The program shown here is used for calculation of the location of the repeated roots of the undamped characteristic equation using the polynomials defined in Chapter IV. The program is written in Fortran for use on General Electric Company Mark II time sharing system. The only requirement for running the program is that a data file which is named "EMFILE" must be established which includes the data for the cases to be solved. EMFILE is set up such that there are seven numbers in each line of the file, the first being the file line number; the next six are the dimensionless parameters h/R, Eg /Ex^, G^^, /-^Xq' ^u /^Xq' -^' ^^^ ^ ^^ *^^^ order. 2 Starting the program will cause the eight roots for A to be calculated and printed. Also, the program automatically calculates the set of roots in which the axial and rotatory inertia are neglected. The following is a listing of the program . 116

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117 v/C'ErOf:X*G^ZP1F':X,R:i ILP^F 120 2 FO'-^MATCV/) I 130 ^=4 140 IF(P-0)7,8,8 150 7 RO = H/2;G0T-3 9 160 H ^0 = -H/2 170 9 01 = CH)**2/12 IHO XKX= 3. 1 4159/(12. )t. 5 190 n=(XKX)T2*GXZ0EX 200 PRI>JT*" H ETHETA/EX GXZ/EX E^JU/EX P 210 & F G" 220 230 PRIMT lO,M,ET0EX,GXZ0EX,Filj,p,F,G 240 10 F0r
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118 260 P^IMT* 270 EOIJ=ET0EX*C1+HT2/12) 280 R=H 290 XI0=Ht3/12 300 R1=H 310 EO=EOU+P*RO 320 E1=E1U-P 330 E2 = EUJ-P + 2*(G+P*R0) 340 E3=G+P*R0 3b0 15 E4=XI0-R1 360 Fl=l+F-01 370 F2=F-P*R0 380 F0=1+F 390 F3=G+F 400 E5=E0*F1-E1 *E2 410 E6=E0*F0-E1 *E1 420 F4=D1*R1+XI0*F0 430 F5=R1*F2+R*F0 440 F6=E4*D1-XI0*F1 450 F9=D1*E4-XI0*F1 460 E7=D1*E5+E3*F0*F2 470 E-8=D1*E0*E4-XI0*E5-E3*F5 480 E9=E3*R*R1-XI0*E0*E4 490 E10=4*E3*E6*F3*01 *F1 500 El 1=4*E3*(E6*(F3*F9-R*D1*F1 )-E0*Rl*F3*Dl *F1 ) 510 E12 = -4*E3*(E6*(R*F9 + F3*XI0*E4)+E0*R1 * ( F3*F9-R*D 1 *F 1 ) ) 520 E13=4*E3*(E6*R*XI0*E4+E0*R1*(R*F9+F3*XI0*E4))

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119 530 H:1 4 = -4*E3*F:0*R*R1*XI0*E4 bAQ C0=E7+E7-Ein 550 C2=2*E7*E8-E1 1 560 C4=E8*E8+2*E7*E9-E12 570 C6=??*E8*E9-E13; C8=E9*E9-El/i 580 C0P = F1 *F3+n>l 590 C2P=E4*F3*D1-F1 CR*D1 +F3*X 10 ) 600 C4P=F1*R*XI0-E4*(R*D1 +F3*XI0) 610 C6P=E4*XI0*R 630 PKIMT 82 630 82 FOR.^ArC/" CO ' C2 C4 640 4 C6 C8" ) 650 90 PRINT 20.C0#C2,C4, C6,C8 660 20 F3RIMATC5E1 4. 6//) The above steps calculate the coefficients for the polynomials and print those for the fourth order polynomial ,

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120 'S70 Ad ) = nojA(y ) = r: j;A(3)=n4;A( > 1 X, 1 P2E20 . 7 ) 740 PS'InJT 60 750 60 FORMA I C/" IHE ^^EC^'MST. COEFF. OF THE POLYM. A'^Ef) 760 Pf^INl, 770 /i = ,M+l 780 L=M+1 790 IF((-1 )*+M)l 10, 190, 100 800 too Pi^l,\]r 102 810 102 FOi^MAK/" CO C2 C4 B20 & C6 C8") The above steps solve for the roots of the fourth order polynomial and print the reconstructed coefficients as a check on the validity of the roots.

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121 830 IF (-^-:>)67, 73, 73 HAO 67 DO 70 I=L*5 RSO 70 A(I)=0 860 73 P-^INT 20,A( 1 ), A(2),A C3),AC4),A(',>) 870 P.r^IIMT, 880 i^^IXT 104 JtgO li)/i Ffl^MAfC COP^TME C2P'UMF: r./^prs'iv,:-: 895 *C6P'>ITMF") 900 PJUMT 66,C0P,C2P,C4P,C6P 910 66 H"j;>; iAr(4Fl 4. 6// ) 9;£0 A( 1 )=C0F; A(2)=C2PJA(3 ) = C4P;A{4)=C6P 930 7S N=i\)-1 940 IF(2-N)80, 130* 1 40 9b0 140 IF(-,M)80, 1 50, 1 bO 960 110 PF^INJT 112 970 112 F0RMAr(/" COPRIME C2PRIME C4PRIMF 975 &C6PRIME") 980 IF(M-5)72,83,83 990 72 0081 I=L,5 1000 81 ACI)=0 1010 83 PRINT 20,A(1 ),A(2)*A<3),A(4),A(5) 1020 PRINT, 1030 G0T0 75 1040 130 VCR = E6/(p:0*R1 ) 1050 PRINT 120,VCR 1060 PRI^JT 122 Lines 830 to 1060 direct the solution and printing of the other four roots for ^^ .

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122 1070 IP.P. FGRMATC ,\)EGLECT. R0TAT. Af^D AX. I-nJE^TIA, THE 1080 & CTF.FF. ". f^00TS ARF:"/) 1090 PRINT 99 1100 99 F?)KMAT(" ***** + ***>4: + ****:> + .i-* + -+-^ 1105 5+***+*+") 1110 IFf-XI0)30,40,40 1120 30 CONTIiNJUE 1130 XIO=0 1140 R1=0 11 SO GOTO 15 . 1160 1 )^RI.NT> 1170 150 PRINT* 1180 190 FOR.^ArC THE FINAL CRITICAL VELOCITY IS "F15.8//) 1190 PRI^JT 152 1200 152 F'5RMAT(" THE FINAL GRIT. V/EL. APPR. INFINITY"//) 1210 PRI^]T 160 1220 160 F0R.V1ATC" 1230 & ••///) 1240 1250 40 G0T0 5 1260 300 ST0P 1270 FNO ' This section of the program makes the simplification of neglecting axial and rotatory inertia and directs the solution for the corresponding roots .

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APPENDIX F PARTIAL FRACTION EXPANSION OF A FOURTH ORDER POLYNOMIAL The characteristic equation D(s) can be factored as D(S) = c^(s s^) (s s^) (s S3) (s s^) (F-1) so that (F-2) 2 2 c's+c' cs+c, ... 5 6 _ 5 6 _ A(S) ^(s) " (s s^) (s Sg) (s S3) (s s^) " B(s) Let B(s) s s^ ^ " ^2 s ~ S3 s " ^4 The coefficients j3j^ can be found (16) from the expression A(Sk) /3. = (F-4) B'(Si^) which means that A over the derivative of B is to be evaluated at s = Sj^. This derivative is d „, > , . d B (s) = -^ B(s) = (s s^) ^ [ (s s^) (s Sg) (s s^) ] + (s s^) (s S3) (s s^) (F-5) Furthermore ^[(S-S^)(S-S3)(S-SJ] = (s-S^)^[(S-S3)(S-S^)] + (S S3) (S s^) and ^[(S S3) (s s^)] = (s S3) + (s s^) 123

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124 Therefore d ,, ^ [ (s s^) (s S3) (s s J] = (s s^) [ (s S3) + (s s J ] + (s S3) (s s J and B' (s) = (s sj ^(s s^) [ (s S3) + (s s J] + (s S3) (s s^) + (S S^) (S S3) (s s^) Finally the coefficients can be condensed to the expression ^k ~ -^ . P 7^ k (F-6) P=l

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APPENDIX G CHECK TO SEE THAT SOLUTIONS SATISFY THE GOVERNING DIFFERENTIAL EQUATIONS The governing differential equations have the form f dfu^e ^ + f ^ (G-1) dU ,f 2 dW ^, dW ^ „, e -jT (i rX ) ex -rr + e W e 1 d0 ^ 3 ' ^^2 d0 o 3^ = -qjH(-0)-H[-(04-d)] -, d^U dW ^ ^ d^ i_ e„ -r-r + f^ — ^ "dc/,^ d

) K i) = *^— ^ F, (0) w k ^^' CO!, k^^' q k u^k <'^) = r \^^^ 9 (G-4) Substituting into the first 8 C E-sfF, ((^) +E — F, ^+e E— ^F, -f Z) sf F, , k k^^' 1 c„ k J 1 , c„a, k 7 V k k k -ik9-^ kak k ? = 4 E k=l c f c 8 c k c 9 9 TT (^^l + ° J t ^f C_ ' 7 k 8 7 k Fk(0) ? = The bracketed term must be zero if Equation (G-3) is satisfied by the solution. 125

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126 (c e c \ f -^ ^^ f 1 « % ^9 7 f c e c o . 6 6 IB ? s, + = (G-5) Look at the second term to see if it is zero because both terms in Equation (G-5) must be zero. fc -ec = fc, -eef = f(c,-e c„) 86 IB 86 138 8^6 13' = f (e e^ e e ) = 8^13 is' Now the first term is inspected. f„c e c f c 8 5 17 7 9 f [f (e + e ) e (f + e A ) (e f + e,f )] 7^8^3 l' 1^1 4 ' ^17 3 8" = f [e (f -f f e \ )] 7 18 7 1 4 " = ef[f -rX^-D +IX^-1-F+D 1 7 O 1 10 1 (I r )\ ] =0 ^ o i' ' Thus the solution satisfies Equation (G-1). Now Equation (G-3) is checked. 4 E k=l c^ f^c, (c s, + c ) o52,76 ^7k 8' r 2 1^ — s, + + e„ f s, e 7 Cg k Cg 3 Cg 7 k 3 (G-6) or (c e c \ „ /f 7 Cg Cg 7/ k V ^ ^-^6 ^3^8 \ ? , + e 1 = c c 3, 9 9 ,) = Look at the second term. fc +e(c-c) = e(fe+c-c) = e(ef-ef) = 76 38 9 3^71 8 9' 317 17 Now look at the first term f [c + e (f + e \ ) 7 5 3 1 4 ' -ef -ef] =f[fe +e(f+e\-f+f)-fej =0 17 3 8 7 7 1 3^ 1 4 8 7 7 1

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127 So Equation (G-3) is satisfied by the solution. Now Equation (G-2) will be checked. c — k^ k ^ ( CgSj^ ^k "* ^k ) ^ i E s F + E ( '9 k '^ '^ k \ "g'k (f r\ ) ^ 3 ' is^(c s. + c) _ Jk^ 7 '^k ' "a' k 9 ^" 'c" ° ^k €Xl] {-•^'.)-'. i(c s, + c ) ^ 7 k s' F, + A, s, c k k k 9 e E(i\F^ = {H(-0) H[-(0+d)]} (G-7) where sgn (b^ c^aj^ 2 s, {H[sgn(bj^)0] H[-sgn(b^)0] H[sgn(bj^) (0 + d)] + H[-sgn(b^) (0 + d)]} /3 Aj^ = sgn(bj^ 2i^ {-H[sgn(bj^)0] + H [ -sgn (bj^)c^ ] + H [ sgn (b^) (0 + d) ] H[-sgn(bj^) ((/) + d)]} Collecting terms gives k=l I — le. c s , ^ ^, e c, f r\ ) 15k .16 3 ' 1 + c s, 9 k 9 is, (c s, + c ) c k ^ 7 k e ^ le ., + — (c s, + c ) + (c s, + c ) c ^ 7 k e' s, c ^ 7 k ' 9 k 9 ie s, 3 k ^k ^ W ^ %\ {H(-0) H[-((^+d)]} (G-8)

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128 Looking first at the term multiplying F. , which is *9 + e. eXc eXc s, + — c k c (e c e c J 8 . . ^ O 8 16' — + 1 Multiply and divide this by (is, ) and get — I— { c (f rX^) sf + ie\c sf + sf [ e c c (f rA^) CIS, '^ 73 k 7k k'l5 83 e^c + e c 7 3 9 + ieXc s, (e c e c ) } 8 k ^ o a 1 6' • Look at the coefficients on powers of s. e c + e c, O 8 16 e (e f ) + e (e ej O ^ 3 8' 1^13' e(fe -e) = -c 3^83 r o e\c„ = e f eX = -c 8 3 8 1 e c c (f rA. ) 15 8^3 ' e c + e c O 7 3 9 = fe(e+e)-ef(f-rX)e f c 71^3 l' 38^3 ' OY + e (e f + e f ) 3 17 3 8 -f (ec-ee) ef (-e +f rX ) 70 13 383 3 f^ (e 6 e ) 7 ^ 13 l' f (e c 2e 6 6^) e f (-G Pr + G + F r\^ ) 7^0 13 1 38 O f [e c e (6 +2e )] e f (f r\ ) 70 1^1 3' 38^2 ' -f (ec-ee) -ef(f-r\) = -c 70 12 3 8 2 2

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129 Similarly eXc = -c 7 3 c (f rA^) = c SO the term multiplying F, is the characteristic equation which is evaluated at s = s, . Of course this equation goes to zero for all roots s, . The remaining identity which has to be proven is 4 E (e^Bj^ + c^Aj^) = -{H(-0)-H[-(0+d)]} (G-9) k=l Noting the identity sgn (b ) 2"-^ { H ( sgn (bj^) (^ ] H [ -sgn (bk) (^ ] H [ sgn (bk) (0 + d) ] + H[-sgn(bk) ((^ + d)]} = {H(-0) H[-(0 +d))} (G-10) which holds whether b, is less than or greater than zero, the remaining statement which is to be proved is y^ /_i 6 k o*^ k k=l \ s, s, k k = 1 or 4 a Z-« — ec -e(cs, +c) = 1 Substituting for c. (i = 6, 7, 8) gives 4 a, y; -^ [ e f (f + e A ) sf + c ] = c (G-11) .*-' S,07^14'k0 4 ^ k=l k This is an identity and is programmed to be checked with each case calculated using the computer program.

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APPENDIX H COMPUTER PROGRAM FOR DEFLECTION AND STRESS CALCULATIONS The computer program DEFSTR calculates the deflections and stresses in the cylinder. There are two required functions to perform in order to operate the program as set up for operation on the General Electric Company Mark II Fortran time sharing system. The first is the establishment of a data file (called EMDATA in the program) . This file has ten elements on each line of the file. The first element is the line number and following that are the dimensionlessdatah/R, Eg^/E^^, G^z^E^^, E^^/E^, P, F, e , X^, and d. When the data file has been established and the program DEFSTR called and asked to run, a request for data is received which must be in the form: KEYl, KEY2. KEYl and KEY2 are integer values which allow the selection of the type of analysis to be done and the type of output desired. KEYl deals with the inclusion or neglection of inertia terms ; KEY2 allows the amount of analysis and output to be regulated as shown in Table H-1. 130

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131 Table H-1. Options Available for Program DEFSTR

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132 120 IF(AX.EQ.O)G0T0 7000 J IF CB. EQ. n )G'^T'^ 7000 130 AL=AL0G(ABS(AX) ) J gL=AL0G (A8S (B ) ) 140 ALIM=-80.0 150 IF(AL+gL.LT.ALIM)G0T0 7000 160 Pi<0D = AX*9;G0 ra 7010 170 7000 Pf^TDsO 180 7010 MULT=PR3D 190 i^etiir^^;e>jd yOO FtJNCTISN DI\/(AX,B,C*D) PIO REAL MULT P.P.n Yl=v)iJLT(AX>C) J Y2 = .v|ULTCB*D) 230 CL=Y1+Y2 240 Y3 = f>^ULTCC#C); Y4 = Ml)LT CD# D) 250 DL=r'3 + Y4 260 IF(CL.E-3.O)G3T0 7015 270 AL=ALOGCABSCCL)) 280 701 5 AL=0 290 8L=AL3G(ABS(DL>) 300 ALI'>1=-80.0 310 IF(AL-3L.LT.ALIiv|)G0T0 7020 320 QU0T=CL/DLiG0T0 7030 330 7020 aU0T=O 340 7030 DIV=QU0T 350 RETURiMJEND The steps in lines 100-350 define two functions for the multiplication and division of complex numbers.

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133 360 '^EAL 'lULF 370 DIMEiMSI0N A ( 5 ) , A I ( 5 ) , R>? ( 4 ) ,CI^ C 4 ) , D^R (3 ) , DC^ (3 ) , RA ( 4 ) , 380 &CBC4),0 C121 ),EP1(?(4),E:P1 I ( 4 ) * E:P2R ( 4 ) , EPf? I (4),SGi\(4), 390 *FP2r^C4)*FP2I(4),FPRC4),FPI ( 4 ) , FLR ( 4 ) , FL I <4),FR(4),FI(4), 400 WFP I (4), WFR(4^> 410 ADFI ( 4), DWFRC 4 ),r)WF I (4), 0?:P1RC4),I)RP1 I ( 4 ) , 0FP2R f 4 ) , 420 &DUFR(4),0UFI < 4 ) , UFPl R ( 4 ) * UFP 1 I ( 4 ) , IJFP2R ( 4 ) , UFP2 I ( 4 ) , 430 ^ IIFPI (4),IJFLR(4),UFLI ( 4 ) ,P'53R ( 1 2 1 ) , PI ) T CI 2 1 ) , WOOR ( 1 2 1 ), 440 &UgOR<121 ),U30I (121 )*SXVJ(121 ),ST'^(i;^l )*SXg(121 )*ST9C121 ), 4b0 ^>S X:^ ( 1 2 1 ) , S r0 ( 1 2 1 ) , S X I ( 1 2 1 ) , S T I ( 1 2 1 ) 460 ^.*'JFR(4)>UFI (4),RR! (4),CR1 ( 4 ) , RR2 ( 4 ) ,CR2 ( 4 ) 46 b '< CA(4),R=^(4),FP1R(4),FP1 I ( 4) , WFPl R (4 ) > WF I ( 4 ) , DFR ( 4 ) , 467 * DFP2I (4)>UFPR(4), WIQI (121 )>SX^(121 ) 470 I\'PiJT,KFYl *KF:Y2 480 5 REAOC'EMDATA"* 1 )OUM\/, H* ET1 EX, GXZ0EX, E 1 ll,P,F,EP,V*0 490 10 F0R'-1AT(V/) 500 PRI>JT 15 510 15 F0RMAT(" H ETHETA/F.X GXZ/FX E^JU/^X P 520 ft F EPSIL0:M LAM'3l.)AtP D" ) 530 PRIMT 2O>H,RTOFX,GXZ0FX,F1U*P>F,EP, V/,D 540 20 F0RMAr(F6. 4, 5F9. 6,F8.6,F8.3,F5.2) 550 PRI'MT 30 560 30 FORMAT ("*+*****************************+*** 57 fiM*^.* ********************** ****"// ) The above steps reserve the required locations for the program variables and request for the input KEYl, KEY2. Then the first set of data is printed and calculations begin.

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134 b80 N=4 590 L3=0 600 IF(P-0)7,B.8 AlO 7 f^0 = H/2iG3T'3 9 620 8 R0 = -H/2 ^ 630 9 D1=(H**2)/12 640 XKX= 3. 1 41 59/(12. )t. 5 650 G=(XKX) t2*GXZ0EX 660 E:0lJ = ETf^EX*Cl+HT2/12) 670 R=H 680 L=0 690 G0r'^(4O,46#44*42»4O#4O*4O,4O*4O)»KEYl 700 40 XI0=CH**3)/12 710 R1=H 720 0010 48 730 42 PRINT 41 740 41 F0RMAT<" NEGLECTING R0TAT0RY INERTIA GIVES;"//) 750 XI0=0 760 R1=H 770 G0T0 48 ISO AA PRINT 43 790 43 F3RMAT(" NEGLECTING AXIAL INERTIA GIVES:"//) 330 XI0=CH**3)/12 810 R1=0 820 G0T0 48 830 46 PRINT 47 ' 840 47 F0RMAT(" NEGLECTING AXIAL AND R0T. INERTIA GIVES:"//)

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135 HbO XI0=O 860 ^\=n H70 G!=n:iij-p+2+(G+p*i^0) 9;?o t-:3=G+p^^o 930 f:^(=xio-ki 940 Fl=l+F-01 9S0 F^=F-P*R0 960 F0=1+F 970 F3=G+F 980 E5=E0*F1 -Fl *H:'-^ 990 E6=F0*F0-E1 *E1 1000 F4=D1 *R1 +XIO+FO 1010 F5 = -^l i
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136 1110 C5=(F7*(E3 + E1 ) )/C4 1120 n6=(El*E3)/C4 1130 G7=(F7*(F1 +E4+V) )/C4 1140 C8=J = 4 1220 CHK=0 1230 Ad )=C0;A(2) = 0iA(3)=C2iAC4)=0;AC5)=C4 1240 Aid )=0JAI(2)=C1 ;AI(3)=0iAI(4)=C3;AI(5)=0 1250 120 CALL CD0 WNH( A. A I ,N, RR,C>^ ) 1260 140 PRINT 150 ' 1270 150 F0RMAT(/" REAL PART IMAG. PART 1280 & REAL PART IMAG. PART") 1290 PRINT 160,RRd ),CRd )*RR(2)*CR(2) 1300 160 FORMATC R00T 1 " 1 P2E1 4. 7,", R00T 2 •'1P2E14.7) 1310 PRINT 170,RR<3)*CRC3),RR(4),CR(4) 1320 170 F0RMAT(" R00T 3 " 1 P2E1 4. 7,", R00T 4 "1P2E14.7) 1330 PRINT 180 1340 180 F0RMATC/" THE REC0N. C0EFF. 0F THE P0LYN. ARE:") 1350 PRINT 190 1360 190 F0RMAT(/" RE/IM--C0 CI C2 1370 & C3 04") 1380 PRINT 1 10,Ad )*A(2),A(3),AC4),A(5)

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137 Lines 890-1380 direct the calculation of the coefficients, the solution of the Characteristic Equation for the roots and the printing of the reconstructed coefficients. 1390 IF(EP)1 100, n00#f?05 1400 205 CONTINUE 1410 PRINT 1 10#AI(1 ),AI(2),AI<3)*AI(4),AI(5) 1420 1100 IFCKEY2-R)200*1000, 200 1430 1000 G0T3(1O1O*1O2O* 1030*1040), L 1440 1010 G3T0(79O,79O,79O,79O,46,44,42,44,44>,KEY1 1450 1020 G0T0(79O#79O, 790, 790, 790, 790, 790,42, 42), KEYl 1460 1030 G3T0(79O, 790, 790, 790, 790, 790, 790*790, 46), KEYl 1470 1040 G0T0 790 1480 200 055 204 K= 1 , 4 1490 RR\ (K)=RR(K) 1500 204 CRl CK)=CRCK) 1510 00 700 .J=l,51 1520 9( J)=-1.02+.02*J 1530 00 500 K=l,4 The logic in lines 1390-1530 uses the keyed input to direct control to the desired location.

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138 154OCCALCULAri0N 0F ALPHA(K) 1550 IF (J.GT.1)G0T0 209 1560 1=1 1570 0^ 206 -^=1,4 1580 IF(K-M)208*206,208 1590 208 ORR( I ) = RRC'-<)-.^R(M) 1600 OC^{CI)=C'^(K)-CK(M) 1610 1=1+1 1620 206 CONTINOE 1630 AR = 0'^-?(l ) JXAI = 0CK<1 ) ;BR = ORR (2 ) ;8 I = DCR C2 ) 1640 XCR = ^ULTBI)+MULT(XAI,-3R) 16 50 AR=XCRiXAI=CIi3R=DRR(3>iBI=DCR(3) 1660 XCR=MULT(AR,BR>-MULTCXAI,BI);CI=MULT(AR*BI)+MULT(XAI,BR) 1670 AR=i .o;xai=o;br=xcr;bi=ci 16 80 XCR=DI\/(AR*XAI,BR#BI);CI = 0IV(XAI,-AR*BR>BI) 1690 RACK) = XCRJCA(K)=C I If the deflections or stresses are requested, the coefficient a, (used in the calculation of the partial fraction expansion of the Characteristic Equation) is determined in lines 1540-1690.

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139 1700n CHECK T.3 SEE IF S0LIJTI0M SATISFIES THE OIFF. EQUATI3NS 1710 Ar{ = RR(K);XAI=CfR(K)JBR = RR(K)if3I=CRCK) 1720 XCR=MULT(AR,BR)-MULT(XAI,BI)JCI=MULT(AR»BI)+MULT(XAI,^R) 173 RMi|'v1=XC^lC^JUM=CIlAR=RA(K) JXAI=CACK) 17 40 XCR=DIV/(A*^,XAI,BR,9I);CI = 0I\/(XAI*-AR,BR,RI) 17 50 AR = E0+C7+C4 + RNU'«1 + C0J XA I = E0*C7*C4*CNUri 1760 Bse=XCf^J9I = CI 1770 XC-l = MlJLT(AR,RR)-M'JLT(XAI,8I>iCI = MiJLT(AR,BI ) +-^iJLT C XA I ^'-S^ ) 1780 CHK=XC^+CHK 17 90 IF(K.NE. 4)G9T0 6001 IROO IJLIM = ABS(CHK+C4); I F ( UL IM-EPS IL0N ) 600 1 * 6003 , 6003 IRIO 6003 PRINT 6004 1820 6004 FP1RMAT(/" THE D-E. IS N3T SATISFIED BY THE SOL.") The expression derived in Appendix G, Equation (G-11) is checked in the above logic to ascertain that the solution satisfies the Differential Equations. 183OCCALCULATI0N 0F BETACO 1840 6001 C0'MTINIJE 1850 AR=RR(K);XAI=CR(K);BR=RR(K)iBI=CR(K) 1860 XCR = MULT(AR,BR)-MIJLTCXAI,BI);CI = MULT(AR»BI)+MULTCXAI,BR) 1870 AR=(C7*XCR)+C8 1880 XAI=C7*CI 1890 BR=RA(K) JBI=CA(K) 1900 XCR=iMULT(AR*BR)-MULT(XAI,BI)JCI = MULT(AR*BI)+MLJLT(XAI>BR) 1910 R>3(K) = XCK 1920 CB(K)=CI 1930 209 CPItNjriNJlJE 1940C CALCULATION 0F E T0 THE IS(K)0 AND IS(K)C0+D) P0WERS 19S0 AR= -0C J)*CR
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140 1960 r)F:LrA = -30 1 1970 IF(Ai^.LT.nF"LrA)G0T'3 511 1980 IFCAR.GF. 1 )G3 rP)21 1 1 9 90 XCrJ = MIJLTCEXP(AK)>CGSCXAI ) ) ;C I = vllJLT (EXP ( AR ) * S I^J ( XA I) ) 2000 GO r-^ 21? 2010 211 Er>lK(K) = OJEPl I (K) = 2020 GO ra 217 2030 212 EP1R(K) = XC^^;EP1 ICK)=CI 2040 217 C0iNiTIiMUE 2050 AR= (OC J)+r))*CRCK)i XA 1= (0 < J) +D ) *:^R (K ) 2060 IFCAR.Lr.DELTA)G0T0 213 2070 IFCAR.GT. 1 )G0T0 213 20 80 XCR=f'1ULTCEXP(AR),C0SCXAI)>;CI = VIULT(EXP(AR)>Sr^JCXAI)) 30 90 G'3T0 214 2100 213 EP2R(K)=0;EP2I
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141 2140C CftLCIILATI0N OF SG.M(BCK)) AND THE HEAVISIOE FlJ,\)CTigMS PA'^0 IFC.J.nr. 1 )G3 TP) 32S 2160 A\y = A3SCCW(K) ) 2170 IFCAV.LT. 1 H:-4)CR350 2450 340 HO=l i HgN=0 2460 G3 rg 3 90 2470 350 H3 = 0i H^iM=l 24 SO GCITP) 3 90 2490 360 IKCT (J) )370,380j3R0 2500 370 H0=O; H0M=1 2510 G0Tg 3 90 2520 380 H0=i; H0iM = O 2530 390 IF (SGN (K) ) 400, 430*430 2540 400 IF(0CJ)+D)41O, 420*420 2550 410 H3D=1;H0DM=O 2560 G0T0 460 2570 420 H0D = O;H0DN=1 2580 6010 460 2590 430 IF(0CJ)+D)44O*45O*45O 2600 440 H0D=OJH0DN=1 2610 G0TO 460 2620 450 H0D=1 ;H0DiN = O 2630 460 D0 465 Ll=l>4 2640 RRCLl )=RR1 (LI ) 2650 465 CRCLl )=CR1 The sgn function and the Heaviside step functions are determined in steps 2140-2650.

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143 2660C CALCIILAIE F(K) AND F(i<)PRIME 2670 FPlr{CK) = M'^*EPlKBR*BniCI = DIV(XAI*-AR,BR*BI) 2910 AR = SGN(K)*XCRJ XA I = SGN (K > *C I ; BR = WFPR (K ) ; B 1 = WFP I (K) 2920 XCR = MULT(AR,BR)-MULT(XAI»BI )iC I=MIILT ( AR, B I ) +MUL F ( XA I *BR ) 2930 WFR(K)=XCRi WFI(K)=CI 2940 AR=R3(K);XAI=CB(K)1BR=FR(K)IBI=FI
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144 29b0 XCK = 'yiULr(AR#BR)-i'^ULT(XAI*Bl) JCI = ^^IJLT(A^,3I)+>^ULr(XAI,i;!'"0 ;?9A0 BK = -C9*RA(K) ; B I = -C 9*CA (K ) i AR= XCR S XAI=CI 3970 XCR = 0IV(AR>XAI,BR*BI);CI = DW(XAI*-AK,B^»8I ) 29B0 DWFR(K)=XCRi DWFI(K)=CI 2990C CALCULATION 0F UF
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145 The F, functions and their derivatives are calculated in lines 2660-3200. k :v->10 500 OONJTINUE 32;?0C CALCULAngN 0F DISPLACEi'^lE.NJTS 3230 PgORC J) = F"RC1 )+FR<2)+FR(3) + F>^(4) 3^40 P09ICJ)=FI(1 )+FIC2>+FI(3)+FI(4) 32S0 W'30>^(.J) = l^JF,R(l )+WFR(2)+WFR(3) + WFR(4) 3260 W0CH ( J) = WFI(1 )+WFI(2)+WFI <3)+WFI<4) 3270 U'ZI^ORC J) = (C5*P0QR( J) )/C9 + UFK( 1 ) +UFR (2 ) + UFR ( 3 ) + UFR ( 4 ) 32 80 UOi)I(.J) = (C5*P0QIC J))/n9 + UFI CI ) +i.)FI (2 ) +IJF I C3)+'JFT (4) 3290r; CALCULATI0N 0F FHF DFRIVATIVES 0F THE 1 SPLACE^E^JT.S 3300 DP = OFr^( 1 )+DFR(2)+DFRC3>+DFR(4) 3310 DW =r)WFU 1 ) + 0WFR(2)+DWFR(3)+DWFR(4) 3320 niJF = DUFR(l ) +DUFR (2 )+DUFR (3 ) +OUFR (4 ) 3330 DU=-(C5*DP)/C9+0UF 3340C CALCULATI0N 3F THE STRESSES 33 50 SXMC J)=ElU*W3gR( J)+D1*DP+0U 3360 STi'^C J) = E0U*W3OR( J)+E1U*DU 3370 SXBC J)=.5*H*C0U+0P) 33 80 STB( J) = (. 5*H)*(E1 U*DP-ET0EX*W0G)R ( J) ) 3390 SXZ(J)=G*CP0QR(J)+DW) 3400 SX0( J)=SXM< J)-SXBC J) JST0 ( J)=STMCJ)-STB( J) 3410 SXI(J)=SXM(J)+SX8(J)iSTI(J)=STMCJ)+ST9(J) 3420 700 C0,MTINUE 3430 G0T0(5O5*5O5, 505, 505*540, 540. 580)*KEY2 3440 505 00 522 J=l,51 3450 IFC J.HT. 1 )G0T0 515 3460 PRINT 510

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146 3470 510 FORMATC/ 6X* 1 H0 > 1 5X, 1 HW,20X, 1 HU, 1 RX, 3HPS I ) 3480 515 Pr^IiNJT 52O*3CJ)#W0QR( J),IW9R( J),P(?)OR( J) 3490 520 F0RMATC1F1O. 5, 1P3E20.6) 3500 522 C!3^JTI^IUE 3510 G9T0(1OOO,54O*54O*58O)*KEY2 3520 540 DO 575 J=W51 3530 IFC.J.GT. 1 )G0T3 560 3540 PRIMT 550 3550 550 Fg;^>lAT(/3X, 1 H0*8X,3HSXM»1OX*3HSXB, 10X*3HSTM, lOX, 3560 &3HST3> 10X>3HSXZ) 3570 560 PRI^JT 570* ( J) * SXM ( J) , SXB ( J) * STMC J) * STR ( J) , SXZ C.J) 3580 570 F0RMATC1F6.2,1P5E13.6) 3590 575 C0NTINUE 3600 G0T0(1OOO* 1000, 5 80* 5 80 #5 80, 1000),KEY2 3610 580 00 610 J=l,51 3620 IFCJ.GT. 1 )G0T0 600 3630 PRIMT 590 3640 590 F0RMAT(/3X, 1H0,8X,3HSX0,1OX,3HST0, 10X,3HSXI, lOX, 3650 &3HSTI, 10X,3HSXZ) 3660 600 PRINT 570,0 < J) , SX0 ( J), ST0 ( J) , SXI ( J) , STI ( J) , SXZ( J) 3670 610 C0INJTINUE 3680 G0T0 1000 3690 790 PRINT 800 3700 800 F0RMATC///////) 3710 810 G0T0 5 3720 820 ST0PJEMD The displacements and stresses are calculated and printed as requested and control is diverted to EMDATA to pick up other cases as directed in lines 3210-3720.

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APPENDIX I RELATIONSHIPS BETWEEN ANALYSIS PARAMETERS Dimensionless and Physical Parameter Relationships Prestress P = r^ = N E N Ev h (I-l) F = T E Ev. h (1-2) Material Properties X K^ Gv7.. h x__220_ Ev h = K 2 ^^O X E-s (1-3) '°^:.^ " '^\' -0-' ^O L 1 + ihZRi: 12 (1-4) E (1-5) Other D D R=E X Ev I R E^ h 12R^ h (h/R)^ 12 (1-6) 147

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148 h o X 1 2 ^ X 1 -M (1-8) For external pressure h = I (1-9) O 2 r = iH^ (I-IO) Isotropic Material Hooke's Law for a homogeneous isotropic material can be written E (e^ + M e J (I-ll) ^ (e + M e J , (1-12) 1 -M assuming the normal stress is zero. Comparing these expressions with the assumed stress-strain relationships given by Equation (2-10) E ^Xo

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BIBLIOGRAPHY 1. N.E. Munch and E . Mangrum, Jr., "Application of Advanced Structural Technologies and Materials to Large Launch Vehicles," Society of Automotive Engineers Preprint 680695 , Presented at the Aeronautic and Space Engineering and Manufacturing Meeting, Los Angeles, California, October 7-11, 1968. 2. J. P. Jones and P.G. Bhuta, "Response of Cylindrical Shells to Moving Loads," Journal of Applied Mechanics , Volume 31, Trans. ASME, Volume 86, Series E, 1964, pp. 105-111. 3. P. Mann-Nachbar , "On the Role of Bending in the Dynamic Response of Thin Shells to Moving Discontinuous Loads," Journal of the Aerospace Sciences . Volume 29, 1962, pp. 648-657. 4. H. Reismann, "Response of a Prestressed Cylindrical Shell to Moving Pressure Load," Developments in Mechanics , Proc. 8th Midwestern Mech. Conf. at Case Listitute of Technology , April 1-3, 1963, Pergamon Press, Elmsford, New York. 5. H. Reismann, "Response of a Prestressed Elastic Plate Strip to a Moving Pressure Load," Journal of the Franklin Institute , Volume 277, No. 7, July 1964. 6. G.A. Hegemier, "Listability of Cylindrical Shells Subjected to Axisymmetric Moving Loads," Journal of Applied Mechanics , ASME, June 1966. 7. G. Herrmann and E.H. Baker, "Response of Cylindrical Sandwich Shells to Moving Loads," Journal of Applied Mechanics , March 1967. 8. Sing-Chih Tang, "Response of Viscoelastic Cylindrical Shells to Moving Loads," Journal of the Acoustical Society of America , Volume 40, No. 4, 1966, pp. 793-800. 9. G. Herrmann and A. E. Armenakas, "Dynamic Behavior of Cylindrical Shells Under Initial Stress," Proceedings of the Fourth U.S. National Congress of Applied Mechanics (American Society of Mechanical Engineers, New York, 1962). 10. E.H. Baker and G. Herrmann, "Vibrations of Orthotropic Cylindrical Sandwich Shells Under Initial Stress," AIAA Journal , Volume 4, No. 6, June 1966, pp. 1063-1070. 11. I. Mirsky and G. Herrmann, "Nonaxially Symmetric Motions of Cylindrical Shells," Journal of the Acoustical Society of America , Volume 29, No. 10, October 1957, pp. 1116-1123. 149

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150 12. F.B. Hildebrand, Advanced Calculus for Engineers , Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1948. 13. J.D. Achenbach and C.T. Sun, "Moving Load on a Flexibly Supported Timoshenko Beam," Int. J. Solids and Structures , 1965, Volume 1, pp. 353-370, Pergamon Press Ltd. , Great Britain. 14. S. Timoshenko and S. Woinowsky-Krieger , Theory of Plates and Shells , Second Edition, McGraw-Hill Book Company, Inc. , 1959. 15. G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers , McGraw-Hill Book Company, Inc., 1961. 16. Yu Chen, Vibrations; Theoretical Methods , Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1966.

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ADDITIONAL REFERENCES W.A. Nash, "Instability of Thin Shells," Applied Mechanics Surveys . Edited by Abramson et al . , Spartan Books, New York, New York, 1966. H.E. Lindberg, "Buckling of a Very Thin Cylindrical Shell Due to an Impulsive Pressure," J. App. Mech . , Volume 31, Trans. ASME , Volume 86, Series E, 1964, pp. 267-272. J.N. Goodier and I.K. Mclvor, "The Elastic Cylindrical Shell Under Nearly Uniform Radial Impulse," J. App. Mech . , Volume 31, Trans. ASME , Volume 86, Series E, 1964, pp. 259-266. W. Stuiver, "On the Buckling of Rings Subject to Impulsive Pressures," Trans . ASME 32 E (J. App. Mech.) 3 , September 1965. M.H. Lock et al. , "Experiments on the Snapping of a Shallow Dome Under a Step Pressure Load," AIAA Journal 6, 7, pp. 1320-1326, July 1968. D.L. Anderson and H.E. Lindberg, "Dynamic Pulse Buckling of Cylindrical Shells Under Transient Lateral Pressures," AIAA Journal 6, 4, pp. 589-598, April 1968. J.C. Yao, "Nonlinear Elastic Buckling and Parametric Excitation of a Cylinder Under Axial Loads," Trans. ASME 32 E (J. App. Mech.) 1 , pp. 109-115, March 1965. N.M. Gregoryants, "Problems of Equilibrium Stability of Cylindrical Shells Under Suddenly Applied Loads," Priklodnaya Mekhanika 2 , 2, pp. 49-56, 1966. R.R. Archer and C.G. Lange , "Nonlinear Dynamic Behavior of Shallow Spherical Shells," AIAA Journal 3, 12, pp. 2313-2317, December 1965. M.P. Bieniek et al. , "Dynamic Stability of Cylindrical Shells," AIAA Journal . Volume 4, No. 3, pp. 495-500, March 1966. V.L. Prisekin, "The Stability of Cylindrical Shell Subjected to a Moving Load" (in Russian), Izvestiyn Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk (Mekhanika i Mashinostroenic) , No. 5, 1961, pp. 133-134. Sing-Chih Tang, "Dynamic Response of a Thin-Walled Cylindrical Tube Under Internal Moving Pressure," PhD Dissertation, The University of Michigan, January 1968. 151

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152 P.G. Bhuta, "Transient Response of a Thin Elastic Cylindrical Shell to a Moving Shock Wave," Journal of the Acoustical Society of America , Volume 35, 1963, pp. 25-30, H. Reismann, "Response of a Cylindrical Shell to an fiiclined. Moving Pressure Discontinuity (Shock Wave)," Journal Sound Vib . , 1968, 8 (2), pp. 240-255. H. Reismann and J. Medige, "Forced Motion of Cylindrical Shells," Journal Engr. Mech. Div . , Proc. Am. Soc . Civil Engr . , EM5, October 1968. Atis A. Liepins, "Asymmetric Nonlinear Dynamic Response and Buckling of Shallow Spherical Shells," NASA CR-1376, June 1969.

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BIOGRAPHICAL SKETCH Elmer Mangrum, Jr. , was born July 6, 1936, at Buffalo Valley, Oklahoma, and in June 1954 was graduated from Buffalo Valley High School, In June 1956 he received his Associate in Science degree from Eastern Oklahoma A&M College at Wilburton, Oklahoma. In January 1959 he received the degree of Bachelor of Science with a major in Mechanical Engineering from Oklahoma State University . He joined the General Electric Company in January 1959, and qualified for the Advanced Engineering Program offered by the General Electric Company. This program is an intense, three-year training program which requires the solution of practical engineering problems using advanced mathematical methods. This was in addition to the training on the job which gave a broad exposure to different disciplines through six job assignments in various departments. Mr. Mangrum specialized in solid mechanics and graduated from the Advanced Engineering Program in June 1962. In 1962 he transferred to Daytona Beach, Florida, and later enrolled in the Graduate School at the University of Florida. Part-time studies were begun, initially through the Florida Institute for Continuing Educational Studies, and later through the Graduate Engineering Education System (GENESYS). He was in the first graduating class of four people from GENESYS in December 1965 when he received the Master of Engineering Degree from the University of Florida with a major in Engineering Science and Mechanics and a minor in Mathematics. Mr. Mangrum continued part-time study and in September 1968 took a year's leave of absence from the General Electric Company to attend the University of

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Florida to pursue the work toward the degree of Doctor of Philosophy. Since September 1969 he has been actively engaged in research to complete the requirements for this degree. Elmer Mangrum, Jr. , is married to the former Rita Anice Chesnut and they have one child. He is a member of Phi Theta Kappa, Pi Tau Sigma, and the ASME,

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee . It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1970 'TV^^^t , College of Engineering Dean, Graduate School Supervisory Committee /; -^'fX^-x Chairman , -.^ i^ L> ^ i^ dUiJiA\/\

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79 Yri'