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
H1 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 ThicknesstoRadius Ratio Including
Prestress . . . . . . . . . . 29
4.3 Classification of Roots of the Undamped Characteristic Equation
for Variations in the ThicknesstoRadius 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 (27)
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 (212)
defined in Equation (212)
defined in Equation (212)
defined in Equation (212)
defined in Equation (212)
defined in Equation (212)
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 (229)
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 (624)
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 thicknessto
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 thicknesstoradius 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) (21)
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 xz and 0z planes respectively. 0z is referred to as the thickness stretch.
Equations (21) 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 nonlinear equations of motion and then linearizing by disregarding all
nonlinear 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 (22)
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
(27)
for internal pressure
3Po
I h (28)
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.
(23)
+h 1
o 8x
(24)
(25)
Nh
+ 0
R2
R
)h
+ o
where
(26)
h =
0
IV (1
8
The elongations, shears, and rotations have been assumed small in com
parison with unity. The straindisplacement 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
(29)
0 = Evo x + E0o
*xO = Gx0o YxO
Oz = GOo YOz
Txz = Gxo Yxz (210)
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
(211)
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 (211) into the Equations of Motion, (22) through (26),
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
(212)
2
+ T 8_U
2
ax
(213)
D
+
R
2
axv
ax
a2v
T 8 x
8x2
S8w
8a
2
+ v
2
8a
+ )]
(214)
+ 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
(2151
802
1
R
Nh
o
R
I "
SI +R u
(216)
D aV
R 8x80
+ xe0 /ax
R \axa o
ax2 )
Nh [
SR2
" I
77+ 
R
E x2
E x2
(217)
82
+ T a2
2
ax
N 2
+ o h
2 2 0o
R a0
32]u
2
at
(218)
+ 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
(219)
h
+
R
2
ao
+ N)
80
(220)
+
Nh
R
(
N) L
ax
(221)
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 (222)
Equations (216), (217), and (218) express the principle of linear momen
tum in the x, 0, and z directions, respectively, while Equations (221) and
(222) 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.. (223)
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) (224)
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
(225)
+ R E
Dx0
+
R2
Axisymmetric Loading
Assuming the loading on the cylinder is axisymmetric, the set of Equations
(224) 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]} (227)
where H(y) is the Heaviside step function, defined as
0, y < 0
H(y) = (228)
1, y>0
(226)
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 (229)
the partial derivatives may be written in terms of a.
ax a8x aa
a a a
=V
at aa at aa
S= 2 (230)
at a82
Using these relationships in Equations (226), letting
u
UU
R
w
W (
R
a R (231)
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
(232)
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 = (233)
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 (233), 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 (234a)
(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
(234b)
(234c)
CHAPTER III
TRANSFORMATION OF EQUATIONS
Equations (234) will be transformed using the Fourier transform
oo
F[f(cP)] = f(s) =f f() eis" dcp (31)
The inverse transform is
00oo
F [f(s)] = f(f) = f f(s) e's ds (32)
where i = vT.
Assuming all the derivatives of f(o) through order (r1) vanish as  +o
the transforms of derivatives of f(o) are given by
k k 
f (s) = (is) f(s) (33)
so
SO
F d is f(s) (34)
F = s f(s) (35)
Applying this to Equations (234) 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
(36)
Collecting coefficients on like displacements allows this set of equations to
be written in the following matrix notation.
[(1+Fr l2 )s2] [i(E P)s]
1 1
[i(E1 P)s]
[(D1 I X)s 2]
[(G+Fr 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
(37)
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 )= (1e [(D I 2)cs2 +(G+Pr )( + Fr l2)]
o isD(s)
isd
= ( ) [(E P)(D I 2) + (G+Pr )(l+Fr A)]
q 1 1 0 0 1
(38)
where
S4 3 2
D(s) = c +i c s +c s +i c s +c (39)
4 3 2 1 O
and
c = (G+Pro) [(1 + Fr1 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)(FPr r 2)
c = E X(D1 I X2)c
3 10
c4 = (G+Fr A)(D 1 X)c
2 2
c = 1+FD rA +I_ A
S 1 + 1 r1 X +1o X (310)
The displacements are found by inverting Equations (38) using transforma
tion (32). 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 (32), where
f(s) represents the deflection expressions (38)] 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 (41)
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 ) (42)
04 2 04
Therefore, the three conditions which will make the discriminant zero are
c 4c c = 0 (43)
2 0 4
c = 0 (44)
c = 0 (45)
Substituting the required coefficients from (310) into (43) gives*
C + CA6 + C4 + C2 + C = 0 (46)
*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
(47)
(48)
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 (49)
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
(410)
The solution of Equation (49) 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
(44). 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
(412)
= 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
(413)
C' = f f D
0 13 1
)n
(414)
If rotatory and axial inertia are neglected Equation (413) reduces to give
one root
C' f
(X)2=  = r (G + F) (415)
CtC r r
2
The third condition (c = 0) from Equation (45) is now investigated. Substituting
for co gives
o2
(1+F r 2) (E+ Pro) (E P)2 = 0 (416)
or
2 2
(f r ) e = e (417)
0 1 O 1
thus
e f e
2 00 1
"4 e r
o 1
X4 = er (418)
4 e r
o 1
Therefore, Equations (46), (413) and (418) can be solved to give eight
roots in X2 which satisfy the conditions for repeated roots. Equation (46) 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 (41) directly gives
c c2 4c
S= 2 2 4 (419)
2c4
If the radical in (419) is zero, the roots are
c c
2 2
s c2 s (420)
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 (413) 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 (418) 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 (41) it is observable that the characteristic equation becomes
s2 (4 2 + c ) = 0 (421)
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 41.
Table 41
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 (49) and
1
(415) 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 (44) 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
LTpt~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 (415) this load is seen to be approximately
F = G (422)
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_#
(423)
(424)
(425)
for an isotropic material. Since, in the nomenclature used in the present work,
V oR(1 p2)
= Eh (426)
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 )
(427)
(428)
(429)
Now taking p = 3 it is found that
2 h
XCR = 0.55075 0.15 R (430)
hCR h (431)
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 (38) 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) (51)
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 (52)
Inverse Transformation of the Rotation
Using the inverse transformation given by Equation (32) 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*
(53)
(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
(1e ) i( ds
s sk
(55)
(56)
S= Fk(E )
k=l
*See Appendix F for the derivation of a sample partial fraction expansion
expression.
(57)
c
9
~ 27r
CO
where
(54)
then
qo
(sk sm)
Letting
isd
1e sd
f (s) _e eis (58)
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)
CF
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 (59)
(59
The integral around the closed path shown in Figure 5.1 is
Sfk(s) ds
fk(s) ds +
R
R
(510)
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 (510) 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 kbk P+i ak [
bk(d+)+i ak(d+k )
H(0) e
H(d+ (P) (511)
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 (512)
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 (57).
4
q = Fk()
qo
k=l
Inverse Transformation of the Radial Deflection
The radial deflection is obtained by inverting W to give
isd
se
s
c' s2 + c'
7 8 eis ds
D(s)
(515)
The term (c' s + c' )/D(s) can be expanded in partial fractions to give
7 8
isd
1e
s
2
c sk + c
k7 e
4
S(sk sm
m=l
"k)
 k
eis ds
(516)
k m
c'
7
c 
7 c
4
c'
8
c 
8 c
4
*See Appendix E for this expansion.
(513)
(514)
00
27 i f
Wqo
q%
where
(517)
00
_ 1f
27r i f
Defining
w Fk(0)
 k
2r i
isd
s(s sk e
(518)
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 (518) can be evaluated. Breaking the integral into two parts gives
19)
(520)
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)
Roo 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
Ro 
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 (521)
s(s s ) s(s s ) k'
0 C
P
The second integral in Equation (521) can be evaluated as
lim ds = ai Res(O)
p s(s Sk)
p0 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)
pO Ck
Integral (521) 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
(523)
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
(522)
The integrals have the values
S ds + lim ds =0, <0
s( k) p0 (k)
p
or
e ds = i Res(O) = i < 0 (524)
Ss(s sk) sk
m bk > 0
Similarly
Ses ds = (p +d < 0 (525)
Ss(s sk) sk
o bk > 0
Equations (522), (523), (524), and (525) 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
7r + 2sir e H(+d) + H(Pd)
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(Pd) (526)
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)(^)]
(527)
Inverse Transformation of the Axial Deflection
The axial deflection in the splane can be written as
isd
= c' 1 ei)
5 (s)
isd
(1 e )
6 2 (s
s D(s)
From Equation (51) it can be seen that the first term in Equation (528) can be
written in terms of . Thus
isd
(1 e ) eis d
s !( e ds
2
s D(s)
(529)
 7
CO
0
q
(528)
c
5
c'
e
w Fk()
1 o
S27
0
The integral in Equation (529) can be written as
4
Sisd ak d k
(1 e ) is s k
2 e sk ds
s k=
k1
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 (530) 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 (532) 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 (sk)
S> 0
bk >0
(533)
00
 00
F (( )
uk
then
(530)
Fk ()
k
"6 k
27r
(531)
o
(532)
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
(537)
S i +d
sk
00
00
where
O+d > 0
bk >0
(534)
(535)
(536)
Res[ufk(sk)]

Sk
(538)
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 (530) 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(Od) ,
2 s k
bk >0
(539)
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
(540)
1 1k
2 sk
The solutions given by Equations (57), (519), and (531) with the correspond
ing Fk functions (513), (527), and (540) 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
(541a)
(541b)
4
Su k(P)
k=l
(541c)
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)]
(541d)
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)]
(541e)
( )
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)] (541f)
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+FD
e = E P+2(G+ Pr) f = FPr
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
(542)
The derivatives of the deflections are
4
__ = E Fk,
k=l
4
q = jl w y Fi(0)
k=l
4
S= uFk() +) (543)
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, () (544)
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 (541)
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 (541) 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 (541b),
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 Fia ia (0+d) H
 e H(O) e 1 H(0d)
a
4 ~ i a 20 ia (+d) d) (545)
3P ia ia ( +d)
+ e H( ) e H( 4)
 e H(0) e H(04d) (545)
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
S72
7 2 8
2a (a a2)
2 1 2
Substituting expressions (546) into (545) 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(1d)
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(pd)]
C a +
7 2 e c (a 0) H() cos [a (0+d)] H(0+d)
21 2
W
qo
(546)
W
qo
(547)
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)] (548)
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)] (549)
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)] (550)
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(pd)
c a +c
2 \
+ 7 2 2e 'cos (a2 )H( ) cos [a (0+d)]H(=Pd) (551)
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(Pd)]
qo a2 b2
2 2e cos(ao)H(p)cos [a(+d)] H( d)
a (a + b )
c b2 +c
+ [eb H(P) ebO H(P) eb(I+d) H(P+d)
2b2 (a + b )
+ eb(d) H(d)] (552)
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(Pd)]
c c b2 b b2 b ((+d)
+ 8 7 2 [e2 H(O) e 2H() e 2 H(Od)
2b(b b)
+ e d)H(4+d) (553)
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
(554)
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 (555)
The second term appearing in Equation (554) is missing from Equation
(555). 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 (541) 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 thicknesstoradius
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 (553) 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 (547). Crossing 3 into Region V, the
roots are real and imaginary. Equation (552) 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 (551).
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   11
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
