Title: Orthotropic cylindrical shells under dynamic loading
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00098431/00001
 Material Information
Title: Orthotropic cylindrical shells under dynamic loading
Physical Description: xvi, 152 leaves. : ill. ; 28 cm.
Language: English
Creator: Mangrum, Elmer, 1936-
Publisher: s.n.
Place of Publication: Gainesville FL
Publication Date: 1969
Copyright Date: 1969
 Subjects
Subject: Cylinders   ( lcsh )
Buckling (Mechanics)   ( 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
Bibliographic ID: UF00098431
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 001144326
notis - AFP3837
oclc - 20143414

Downloads

This item has the following downloads:

orthotropiccylin00mangrich ( PDF )


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




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs