A FINITE ELEMENT ANALYSIS OF THE STATIC AND DYNAMIC ...
150
A FINITE ELEMENT ANALYSIS OF THE STATIC AND DYNAMIC BEHAVIOR OF THE AUTOMOBILE TIRE by Prashant D. Parikh, B. of Eng., M.S. in M.E, A DISSERTATION IN MECR^NICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Accepted ^ hiy, 1977
Transcript of A FINITE ELEMENT ANALYSIS OF THE STATIC AND DYNAMIC ...
A FINITE ELEMENT ANALYSIS OF THE STATIC
AND DYNAMIC BEHAVIOR OF THE AUTOMOBILE TIRE
by
Prashant D. Parikh, B. of Eng., M.S. in M.E,
A DISSERTATION
IN
MECR^NICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Accepted
^ hiy, 1977
• / ^
,,' ,/ ACKNOWLEDGEMENTS
t?* z ^ -
I am very grateful and deeply indebted to Dr. Clarence A.
Bell and Dr. C. V. Girija Vallabhan for their guidance and sug
gestions and for the many hours they spent in discussing the
project, directing its course and reviewing its progress. I
would also like to thank the other committee members. Dr. Donald
J. Helmers, Dr. James Strickland, Dr. Elbert B. Reynolds, and Dr.
William P. Vann, for their helpful criticism and advice. The
encouragement shown by Dr. James H. La\\rrence, Chairman of the
Department of Mechanical Engineering, is also appreciated.
I would especially like to thank Dr. C. V. Girija Vallabhan,
whose deep insight and critical knowledge in the field of finite
element method helped me make this research a reality.
I would like to extend my thanks to Mr. Daulatram Lund,
Mr. Young-Pil Park and Mr. Vijay P. Vasani for their help.
Tlianks are extended to Ms. Sue Haynes for the careful typing
of the manuscript.
11
TABLE OF CONTENTS
Page
ACKNOWLEDGENENTS ii
ABSTRACT vi
LIST OF TABLES vii
LIST OF FIGURES viii
NONENCLATURE x
I. INTRODUCTION 1
1.1 Purpose of Research 1
1.2 Description of the Tire and Its Components 2
1.3 Previous Work 4
1.4 Method of Approach 6
1.5 Contents of the Thesis 9
II. THE FINITE ELEMENT METHOD 11
2.1 General 11
2.2 Descretization of the Continuum 13
2.3 Selection of the Displacement Function is
2.4 Derivation of the System Equation 17
2.5 Assemblage of the Stiffness and Mass
Matrices of the Individual Elements 23
2.6 Boundary Conditions 24
2.7 Solution to the Overall Problem 25
2.8 Review of the Methods for Dynamic Analysis 26
Inverse Iteration Using Successive
Over-Relaxation Technique 26 Householdcr-QR A3 gorithm 27
111
Page
Subpolynomial Iteration Method SPI 27
n-step Iteration Method 28
III. DEVELOPMENT OF THE TIRE MATHEMATICAL MODEL 30
3.1 General 30
3.2 Strain-Displacement Relations 30
3.3 Composite Elastic Properties 39
3.4 Displacement Functions 48
3.5 Development of the Displacement
Transformation Matrix 50
3.6 Strain Energy 56
3.7 Kinetic Energy 59
3.8 Transformation into Global Coordinates 63
3.9 External Work Potential 66
3.10 Equation of Motion 68
3.11 Load Vectors for Specific Types of Loading 69
A. Internal Pressure Load 69
B. Centrifugal Force 70
C. Load Vector Due to Ground Contact Loading 71
3.12 Boundary Conditions 72
IV. THE DYNAMIC . NALYSIS 73
4.1 n-step Iteration Method 73
4.2 Determination of the Matrix R and Vectors V 77
4.3 Size Criteria 81
V. NUMERICAL RESULTS 83
5.1 General 83 iv
Page
5.2 Asymmetric Static Analysis 92
5.3 Axisymmetric Static Analysis 102
5.4 Dynamic Analysis 109
VI. CONCLUSIONS AND RECOMMENDATIONS 115
6.1 General 115
6.2 Conclusions 115
6.3 Recommendations 117
LIST OF REFERENCES 118
APPENDIX A 123
APPENDIX B 126
APPENDIX C 128
APPENDIX D 135
ABSTRACT
A mathematical model to represent a radial ply passenger car
tire has been developed for axisymmetric and asymmetric static and
d>Tiamic eigenvalue analysis by the use of a direct stiffness finite
element method. Linear analysis is performed. The tire is consi
dered as a thin shell of revolution. The finite element chosen has a
shape of a conical frustrum with five degrees of freedom at each
node in the local coordinate system of the element. The tire pro
perties have been derived by assuming the tire to be composed of
thin layers of composite materials, linearly orthotropic in nature.
Hamilton's principle has been applied to derive the equation of
motion of the element.
In the case of asymmetric static analysis, fifteen Fourier har
monic terms have been used to represent the as)TTimetric loading and
deformation. The equation for the static case has been solved by
employing the Gauss elimination method. Three different types of
pressure distributions have been assumed to simulate the actual pres
sure distribution in the tire footprint area. The natural frequencies
and the associated set of mode shapes have been evaluated by cmpoly-
ing a method based on a modified version of Lanczos' n-step iteration
procedure.
The analysis predicts experimentally verifiable deformed shapes
under static loading, and natural frequencies of vibration and
associated mode shapes with good accuracy.
vi
LIST OF TABLES
Page
5.1.1 Constuctional Parameters of the Tire 85
5.2.1 Crown Deflection for Dodge's Tire No. 6 97
5.2.2 Crown Deflection for Dodge's Tire No. 7 97
5.2.3 Deflections for Patel's Tire No. 2 97
5.2.4 Static Ground Contact Loading Data for the Tires •. 104
5.3.1 Axisymmetric Deflections at the Crown and Sidewall of the Two Radial Tires Under the Effects of Internal Pressure 107
5.4.1 Natural Frequencies of HR78-15 Radial Tire 110
C.l Meridian Coordinates of the Tires 128
C.2 Meridian Coordinates of HR78-15 Tire for (|) = 0", 45" and 180** for Constant Pressure Distribution in Footprint 129
C.3 Meridian Coordinates of HR78-15 Tire for (f) = 0", 45" and 180" for Cosine Pressure Distribution in Footprint 130
C.4 Meridian Coordinates of HR78-15 Tire for <j) =0", 45" and 180" for Trapezoidal Pressure Distribution in Footprint 131
C.5 Modal Data for HR78-15 Tire for n = 1, m = 1, 2 and 3 132
Vll
LIST OF FIGURES
Figure Page
1.2.1 Cut Section of a Radial Tire Meridian 3
1.2.2 Coordinate System of the Tire 5
1.4.1 Coordinate Geometry of a Conical Shell Element 7
1.4.2 Finite Element Idealization of the Tire Meridian ... g
3.2.1 Typical Shell Elements 31
3.2.2 Nomenclature of a Typical Shell Element 33
3.2.3 Nomenclature of the Conical Shel 1 35
3.2.4 Rotations that Determine Twisting Strains 38
3.2.5 Determination of a 38
3.3.1 Cord and Rubber Orientation in a Single Ply of the Belt 41
3.3.2 Cross Section of a Single Ply Viewed
in the Plane Along the Cords 41
3.5.3 Schematic of the Plies of the Tire 46
3.8.1 Coordinate System and Local and Global Displacements of the Conical Shell Element 64
5.1.1 Meridian Profile of Dodge's Tire No. 6 (Radial) .• 87
5.1.2 Meridian Profile of Dodge's Tire No. 7 (Radial) 88
5.1.3 Meridian Profile of Patel's Tire No. 2 (Bias) 89
5.1.4 Meridian Profile of the Firestone HR78-15 Tire
(Radial) 90
5.1.5 The Axisymmetric and Asymmetric Loading 91
5.2.1 Crown Deflection for Dodge's Tire No. 6 Under Static Ground Contact Loads 93
5.2.2 Crown Deflection for Dodge's Tire No. 7 Under Static Ground Contact Loads 94
viii
c- ^ Page Figure — s _ 5.2.3 Meridian Profile for Patel's Tire No. 2 Under
the Effect of Static Ground Contact Load 96
5.2.4 Different Types of Pressure Distributions in the Footprint Area 98
5.2.5 Meridian Profiles for HR78-15 Radial Tire for Constant Pressure Distribution in Footprint 99
5.2.6 Meridian Profiles for HR78-15 Radial Tire for Trapezoidal Pressure Distribution in Footprint 100
5.2.7 Meridian Profiles for HR78-15 Radial Tire for Cosine Pressure Distribution in Footprint 101
5.2.8 Circumferential Profile of the Crown Under the Static Ground Contact Load, HR78-15 Radial Tire 103
5.3.1 Axisymmetric Deflection of Two Radial Tires Under the Effect of Inflation Pressure 106
5.3.2 Axisymmetric Deflection at the Crown for HR78-15, Under the Effect of Centrifugal Loads 108
5.4.1 Radial Modes of the Radial Tire Ill
5.4.2 Mode Shape of HR78-15 Radial Tire for the Frequency of 181 Hz 112
5.4.3 Mode Shape of HR78-15 Radial Tire for the Frequency of 181 Hz 1^^
5.4.4 Mode Shape of HR78-15 Radial Tire for the Frequency of 297 Hz 114
IX
A j ,
1
^^
h'
F-n
S'
h
\k
I
L
MM
n
n s
nn
N
P
^B'
R s
r.
R j ,
*2 f
, B . . ,
d s
^R-
%-
-1
^ C
S , (J)
•^2
1
C. .
E s
G s
^T
NO^ENCLATURE
Lame parameters
composite elastic parameters
effective diameters of the polyester and steel cords
Young's modulii for polyester cords, rubber matrix and steel cords
assumed ground pressure acting normal to the surface of the tire, in psi
shear modulii for rubber matrix, polyester and steel cords
thickness of the layer
distance from the geometrical central axis of the element to the k layer
length of the element
Lagraingian function of the element
number of elements in the tread region
Fourier harmonic number
number of cords per inch in a layer
number of composite layers in an element
total number of Fourier harmonics
internal pressure, in psig
radii at bead, crown arid tread region
radius of the steel cords
coordinate axes of the element
principle radii of curvatures of the shell element
t total thickness of the element
T kinetic energy of the element
Tol some small tolerance
u, V, w displacements in circumferential, tangential and transverse direction of the element
U strain energy of deformation of the element
V volume fraction of the steel cords in rubber s ^ . matrix
W potential energy of the body forces and
surface traction
z, r, (|) global coordinate axes system
a, 3 rotations in <}> and s directions f
3 cord orientation in a composite layer with
respect to s direction
6 variation of the quantity following the s.ymbol
6 rotations of the shell element 3
p density of the material, in Ibm/in
X eigenvalue
X Raleigh quotient
0) natural frequency, in rad/sec
ip cone angle
V , V , V Poissons' ratios for polyester cords, rubber matrix and steel cords
G, X» Y membrane, bending and transverse shear strains
Matrices or vectors
[ ] denotes a square matrix
XI
{ } denotes a column vector
_ denotes a square matrix or a column vector
A , A displacement transformation matrix
B matrix of the strain-displacement relations
E elastic stiffness matrix for an element
f , F load vectors for an element and entire structure
G elastic stiffness matrix for transverse shear strains
k_, IC stiffness matrices for an element and entire
structure
L lower triangular matrix including diagonal terms
m, M mass matrices for an element and entire structu
re
N , M membrane stress resultants' and'bending moments
CL generalized displacements
R_ reduced tridiagonal, symmetric matric of order
m X m
TT transformation matrix
T„ surface traction vector, in Ibf/unit area —R u displacement vector of an element V displacement transformation matrix of order
n X m V. columns of V matrix X, X nodal displacement vector for an element and
entire structure
Xj, body force vector, in Ibf/unit volume —6 Y eigenvectors
a stress vector
Xll
I
e_, Q_ matrices of trignometric functions
•>(_ shear strain vector
£_ strain vector for membrane nad bending strains
Subscripts
Unless otherwise specified the subscripts mean the following:
a, b nodal points or circles of the conical shell
element
L longitudinal direction
m, n Fourier harmonic number
q generalized coordinates, or displacements
s tangential to the element
T, w transverse direction
Y transverse shear strains
(f), z, r circumferential direction, z direction and radial direction
E membrane and bending strains
Superscripts
Unless otherwise specified the superscripts mean the following:
* antisymmetric motion
g global quantities
n Fourier harmonic number
T transpose of a matrix
q generalized coordinates or displacements
Y transverse shear strains
e membrane bending strains
xiii
CHAPTER I
INTRODUCTION
1.1 Purpose of Research
The modern pneumatic tire is a complex load carrying structure.
Due to the means by which a tire obtains its support, namely air pres
sure, a tire is a very flexible structure. The external loads may
cause the tire to undergo very large deflections, even though the
resulting strains may be small. The recent advances in modern tire
technology owe their origin to yesteryear's trial and error solutions,
empirical relations, and today's knowledge of basic deformation mechan
ics. In the past the use of the automobile tires, by most consumers,
was taken for granted. Motorists were happy so long as their tires
held air pressure and had a recognizable tread pattern. But due to
the increased construction of high speed highways and the growing pop
ularity of automobile travel, motorists have come to expect more from
their tires. Also,the competitive market led the automobile manu
facturers to produce increased numbers of luxurious and comfortable
vehicles suitable for high speeds. One of the majoi* factors contribu
ting to the discomfort in an automobile ride is caused by the vibra
tion transmitted through the tires. The vibrations are induced by the
road roughness and unevenness, and also by the vibrations of the tires
themselve?. The specific purpose of this investigation is to develop
a mathematical model which can satisfactorily predict the deformed
shape of a statically loaded tire, as well as the natural frequencies
and the mode shapes of the tire. The accuracy of this method is
verified by comparing some of the results of the analysis with experi
mentally obtained data. The analysis in this work is conducted for
both radial and bias plied tires with most of the emphasis on radial
tire analysis.
1.2 Description of the Tire and Its Components
Because of its shape, an inflated tire can be considered to be a
toroidal shell. The size, type and load carrying capacity of a tire
is designated by s>Tnbols like A70-12, H78-15, HR78-15, etc. The first
letter identifies the tire load capacity. The second letter R, if pre
sent, means that the tire is a radial tire. The first two numerals
represent the percentage fraction of the tire depth to its maximimi
v\7idth. The last two digits represent the diameter of the wheel rim.
Thus, KR73-15 would mean the tire is an H series radial tire, with a
wheel rim diameter of 15 inches, and the ratio of the maximum depth
to maximum width is 78 percent. In Figure 1.2.1 the cross section of
a typical radial tire is shown. The tire consists of the following
components.
a. A fabric base usually composed of polyester cords embeded
in a rubber matrix usually consisting of two layers with cords ar
ranged radially. This is also known as the carcass.
b. Two layers of cords, usually made of steel wires embeded in
a rubber matrix, with the cords making an angle of about 78 degrees
with the radial direction at the crown. This is also known as the
belt or the intermediate structure.
c. Tread which protects the belts from external physical and
chemical conditions and which ensures traction, braking and cornering
o •H
+->
+-> O o: iw o to
•H X
<
phenomena. This region carries little or no load.
d. The bead of the tire, made of steel wires and enclosed by
rubber, is responsible for the proper connection between the tire and
the rim. Figure 1.2.2 defines the coordinate system of the tire.
1.3 Previous Work
The birth of the mathematical analysis of pneumatic tires
came in the late 1920's. In 1928 Purdy (1)* provided a rigorous solu
tion to the problem of determining the inflated shape and stresses
of the tire. However, his analysis did not include the effects of
loading on the tire. Many researchers, (2, 3, 4, 5, 6) to mention a
few, have tried to analyze the basic deformation characteristics of
an inflated tire under static and dynamic conditions. These research
ers have tried to predict shapes and stresses by making many simplify
ing assumptions. ' In these approaches, the tire was modeled in terms
of its actual constitutive components and the geometry of its conf.truc-
tion. The application of these types of methods is limited due to the
complexities of the mathematics involved. Recent works of Dunn and
Zorovvosky (7), .and Patel and Zorowosky (8) deserve attention. These
researchers have developed a mathematical model for determining the
carcass stresses using the finite element method. However, their
solutions were limited to the static analysis of the tire.
The dynamic analysis of a tire has been attempted by Tielking
(9), Fiala (10), Bohm (11), and Dodge (12). They derived the equation
* The numbers in parenthesis indicate the reference numbers as
listed in the List of References.
M
u -^ o
•H
O
o B o
+->
X
o rj
•H
o o
CN
f-. - ^
o
•H
of the motion of the tire by considering it as a prestressed circular
ring or beam on an elastic foundation. They tried to simulate the
effective deformation characteristics of the tire in terms of dis
crete mechanical components, such as springs and dashpots to represent
the sidewall and the inflation support, and a circular beam or ring
to serve in the role of the tread segment. Practical use of this
approach requires that the mechanical properties of the model compon
ents are to be determined experimentally from the existing tires.
These methods are useful in understanding the dynamic behavior of an
already built tire, but it is more difficult to use these methods for
predicting the dynamic behavior of an arbitrary tire which has not
been fabricated. A finite element model developed herein will
eliminate some of the limitations of the previous models.
1.4 Method of Approach
Figure 1.4.1 illustrates the finite element model employed. In
the method of analysis the tire is represented by a number of trucated
conical shell elements as shown in Figure 1.4.2. The direct stiff
ness finite element method is used in this research. The displacements
of the conical element are expressed in terms of a polynomial in the
meridian direction, while in the circumferential direction Fourier
expansion is used (see Figure 1.2.1 for a description of these direc
tions) . Each element has ten degrees of freedom, for each Fourier
harmonic, in the local elemental coordinate system. These degrees of
freedom are u, v, w, a and 3 at the nodal circle of the conical ele
ment. For the description of the kinematics of deformation of the
Reference Azimuth
a
Figure 1.4.1 Coordinate Geometry of a Conical Shell Element
8
Tread Region
Element i
Axis of Rotation
Figure 1.4.2 Finite Element Idealization of the Tire Meridian
middle surface of the conical shell element, Novozhilov's linearized
strain-displacement equations are chosen. The equivalent stiffness
of the tire element is derived by considering the tire to behave as
a multilayered shell of composite linearly orthotropic materials.
First the static analysis of the tire is performed. In the
axisymmetric static analysis, loads due to inflation pressure and
inertial loading are considered. In the asymmetric analysis, static
ground contact loading is considered. Three different types of pres
sure distribution in the footprint area are assumed to simulate the
actual footprint pressure distribution. The natural frequencies and
the associated mode shapes are also evaluated. Computer programs for
both types of analysis are written in the Fortran IV language and
executed on the Texas Tech IBM 570 computer.
1.5 Contents of the Thesis
The contents of this thesis are presented briefly in the follow
ing paragraphs.
Chapter II is devoted to an exposure of the direct stiffness
finite element method. A brief description of the essential features
of this method is presented. Based upon variational principles the
equation of motion for a single element is derived. The methods
available to solve the governing equation in the static and dynamic
cases are also discussed objectively.
Chapter III deals with the state of strain, and with the consti
tutive equations of thin shells. Novozhilov's strain-displacement
relations are derived in this chapter. The stiffness and mass matrices
for the conical shell element are derived in this chapter. A dis
cussion of the boundary conditions is also presented.
10
Chapter IV is devoted to the solution of the equation of motion
to evaluate the natural frequencies and the associated mode shapes.
A tridiagonal reduction method based on a modified version of Lanczos'
n-step iteration procedure, also kno\m as the fast eigenvalue extraction
routine, is employed.
In Chapter V actual results of the numerical computations are
given.
Conclusions and recommendations are presented in Chapter VI.
The advantages and limitations of the analysis employed are shov\rn.
In Appendix A the matrix of the coefficients of the strain-
displacement vector relations for transverse shear strain is pre
sented. A similar matrix for membrane and bending strains is given
in Appendix B. The numerical data for the results presented in
Chapter V are given in Appendix C. The steps involved for incor^
porating zero transverse shear strains are given in Appendix D.
41
CHAPTER II
THE FINITE ELEMENT METHOD
2.1 General
Due to the complexities of the mathematics involved, the exact or
the closed form solution for most real engineering problems exists only
after making simplifying assumptions. For problems involving complex
material properties, shapes, boundary conditions, and loads, some sort
of numerical technique is usually applied. The finite element method
is one of the many numerical methods available. The concept of this
method was developed in the mid 1950's and has been attributed to
Turner, Clough, Martin and Topp (13). Since the advent of high speed
computers, this method has flourished and has been applied to solve a
wide variety of problems. One can successfully apply this method to
any kind of problem where formulation of the field equation based on
variational principles is possible. Unlike many numerical techniques,
the finite element method discretizes the continuum into imaginary
smaller bodies of simple shapes. The assemblage of such small bodies
then represents the entire structure.
Energy and variational principles are used to formulate the basic
equations of the problem, and use of such principles can lead to three
general types of finite element methods. These methods are the dis
placement, force and hybrid methods. The displacement method has
gained wide popularity among engineers because of its simple nature.
Many researchers have tried to give proof of its convergence to the
actual solution (14, 15, 16).
11
12
In the displacement approach, the displacements are assumed to be •
the primary unknown quantities. Generally, displacements at certain
points, called the nodal points, are selected as unknowns. The actual •
loading on the structure is replaced by a set of equivalent loads at
the nodal points. For the static analysis a minimum potential energy
theorem is employed (17). The total potential energy function of the
system is expressed in terms of the nodal point displacements, and
the system equations are derived by extremizing this potential energy
function. In the case of dynamic analysis, the system equation
can be derived by one of two methods. The first method employs mini
mization of the Lagrangian function based on an extended Hamilton's
principle. The second method uses Hamilton's principle and seeks the
extremum of the line integral of the Lagrangian function between two
specified time intervals. In the force method, stresses are taken as
primary unkno\>m quantities, and the system equations are derived by
minimizing the complementary potential energy function of the system
(17).
The displacement method is used in this investigation and the
steps involved in deriving the system equation are as follows (13,17
18, 19):
1. Discretization of the continuum.
2. Selection of the displacement function.
3. Derivation of the stiffness and mass matrices for an element.
4. Formulation of the matrix equation.
5. Application of the appropriate boundary conditions.
13
6. Solution of the nodal displacements.
7. Computation of the internal stresses.
2.2 Discretization of the Continuum
In the discretization process, a decision has to be made regarding
the number, the size, and the shape of the elements in such a way that
the geometry of the structure is simulated or approximated as closely
as possible. The selection is based on two criteria.
a. Degree of accuracy desired.
b. Computational time.
As a rule of thumb, it could be said that the greater the number of
elements, the greater are the number of unkno\vms to be evaluated and,
hence, the greater is the computational time.
At present there are three types of finite elements available for
the analysis of shell structures by the displacement method (20).
a. Triangular or quadrilateral shell element.
b. Conical shell element.
c. Solid elements.
In the formulation based on triangular or quadrilateral elements,
the shell is represented as the assemblage of plates of triangular
or quadrilateral shaped finite elements. These elements can be either
flat or curved elements. Gallagher (21) and Iyer (22) have compiled
a list of such types of elements which were developed by several re
searchers. The triangular shaped shell element has very general appli-
tion, since it can be employed to represent shells of any arbitrary
14
shape. These types of elements can handle nonlinear shell problems
and problems involving geometrical discontinuities. Even though these
elements have the versatility of handling any kind of shell geometry, in
some instances it is economical to use conical or ring type elements.
The analysis based on conical shell elements is restricted to use
on axisymmetrical shells of revolutions only. With the conical shell
element the shell structure is represented by a series of conical frusta
joined together at the nodal circles instead of nodal points. Grafton
and Strome (23), and Popov, Penzien and Lu (24) used flat conical shelf
elements for problems involving axisymmetric deformations and loads.
Later, Percy, Plan, Klein and Navratna (25) applied these elements to
problems involving non-symmetrical loads and deformations by expanding
them in a Fourier series. However,.their analysis included geometric
nonlinearity only for axisymmetrical dformations and loads; that is
when the loading is axisymmetric. Later, Jones and Strome (26) devel
oped a doubly curved shell element. In some cases, the doubly curved
shell element is superior to the flat conical shell element since it
has the advantage that coordinates, slopes and radii of curvature are
everywhere continuous functions and are identical to those of the ac
tual shell structure.
The third type of finite element is a solid shell element. There
are different kinds of solid shell elements. This element is well
suited for thick shell analysis or in the regions of the shell where
shell walls have been made considerably thicker, for example, at the
junction of a shell structure with an external pipe fitting.
In this investigation the tire is modeled by the use of flat coni-
15
cal shell elements as developed by NASA (27). This type of element has
the capability of including both the coupled and decoupled longitudinal
strains and strains due to changes in curvatures, in addition to the
inclusion of transverse shear strains.
2.3 Selection of the Displacement Function
In order that the finite element method converge to the exact solu
tion, it is required that the displacement behavior of each element be
represented very closely with the actual displacement pattern of the
element, and that inter-element compatibility should be satisfied. If the
chosen displacement function is not compatible and does not properly
represent the actual deformation within the element, the finite element
solution may not converge to the exact displacement solution of the
structure even when a finer mesh size is employed. On the other hand,
a good accuracy can be expected even with coarse mesh size if the
displacement function is compatible and properly represents the defor
mation field of the element.
There are two methods to represent the deformation field: a)
by selecting a polynomial in the coordinate system with the required
number of unknov>m constants, and b) by selecting a set of proper
interpolation functions. In this investigation the deformation pat
tern is approximated by a simple polynomial function. Polynomials are
used very widely because mathematically they are easy to handle. Trig
nometric functions can also be used to describe the displacement pattern.
For one dim.ensional system, the exact displacement within the element at
any time t can be represented by a n degree polynomial as.
16
u(x,t) = q^(t) + q2(t)x + q^(t)x^ + + q^(t)x"'^^
or
u(x,t) = Jcx)a(t)
where
i^x) = (l ,x,x^ , x"") and
q (t) = (qi(t),q2(t), > % ^ ^ ) - (2.3.1)
Based on the concept of vector space the coefficients of this poly
nomial are called generalized displacements. In order that the finite
element m.ethod converge to the exact solution, the displacement func
tion must satisfy the following three conditions (28, 29).
a. The displacement model must be continuous within the element
and must be compatible at the inter-element boundaries.
b. It must include rigid body displacement of the element.
c. It must include a state of constant strain within the element.
Relaxation of one of these conditions might give an unrealistically
flexible or stiff matrix (17) . However, a model which does not satisfy
the first two conditions, but which does satisfy the third condition,
can give acceptable convergence (21). Also, in selecting a polynomial,
care should be taken that the order of the polynomial is at least equal
to the order of the highest derivative of the displacement appearing in
the strain-displacement relationship (17).
17
2.4 Derivation of the System Equation
In this section the general procedure for deriving the system
equation for dynamic analysis is discussed. In deriving the equation
it is assumed that the damping in the system is negligible. Hamilton's
principle is used to obtain the equation of motion for a single element.
Hamilton's principle can be stated as follows (17). "Among all possible
time histories of displacement configurations which' satisfy compatibil
ity and the constraints of kinematic boundary conditions at time t and
t_, the history which is the actual solution makes the Lagrangian function
a minimum." As an element moves and deforms under the influence of the
loads, its Lagrangian function L is defined as
L = T - U - W (2.4.1) )<
QI
where T is the kinetic energy given by
T = i 2
pii u d(vol) (2.4.2)
Here p is the density of the material and the "dot" denotes the deriva
tive with respect to time. u_ is the element displacement vector, and
the integration is performed over the volume of the element. In dy
namic analysis the displacements, velocities, strains, stresses, and
loading, are time dependent but, for the sake of convenience, the sub
script t to denote the time is omitted. The strain energy of deform.a-
tion in a linear elastic continuum is given by
ri
r 6 y
vol .K
18
" - J T e a d(vol)
vol
Or in indicial notation
U = i { o. .z.. d(vol) (2.4.3) 2 j ij 13 ^ ^
vol
Here e. . and a.. are the strain and stress tensors respectively. For
an elastic material the stresses and strains are related by the gen
eralized Hooke's law as
a. . = E. .,,£,, (2.4.4) 13 ijkl kl
-i r T £ F £ d(vol) (2.4.4a)
Here, E is the elastic stiffness matrix. In indicial notations,
U = - E. ., ^e. .£. H(vol) (2.4.5) 2 J i3kl 13 kl ^ ' vol
The displacements at the nodal points are assumed as primary
unknown quantities, so the kinetic energy expression and the strain
energy expression have to be transformed into nodal point quantities
before Hamilton's principle can be applied. This can be achieved
Here E. ., , is the fourth order elasticity tensor. Substitution of C i3kl >
equation (2.4.4) into equation (2.4.3) yields, ^
ri
vol ^ >
19
from strain displacement equations. From continuum mechanics the
strain-displacement realtion in indicial notation is.
Ir ^ e. , = -x-yu. . + u. .) . ij 2^ 1,3 3,1"
(2.4.6)
Here, a coma denotes differentiation with respect to the coordinate
axes. For a cartesian system, the strain-displacement relations in
matrix notation can be written as.
Where
e = B q - -q^
B = -q
0
0
0
0
0
a.
0
8,
0
0
8.
0
3.
(2.4.7)
Here, 3's denote — . and £ is the vector of generalized coordinates 9X^
The displacement in matrix notation is expressed by,
or in indicial notation as.
u = 0 q
u. = . . q . I 13 '3
(2.4.8)
20
Using equation (2.4.8), the nodal displacement vector x can be ob
tained at each node, and can be written as
X = A £ (2.4.9)
From equation (2.4.9) we get
a = A"^ X ' (2.4.10)
and
u = $ A ^ X (2.4.11)
Here the matrix A is known as the displacement transformation matrix
Substituting equations (2.4.7) and (2.4.11) into equation (2.4.5) the
strain energy of the element is given by
1 r '^A'-^^B ^E B A"-^X d(vol) (2.4.12) - -q - -<i- -
vol
Let us define
B = B A'-^
Here, B is the matrix of the coefficients containing the strain-
displacement vector relations., Now the strain energy function U can
be expressed as
U = y l ^ ^ ^ i2i d(vol) (2.4.13)
;S
vol
21
In a similar way using equation (2.4.11), in equation (2.4.2) the
kinetic energy function can be written as ,
T 1 r T -1 T -1
^"^jj^Kt: i 1 X d(vol) . (2.4.14)
vol
The potential energy of the body force X^, in Ibf/unit volume, and
surface traction T , in Ibf/unit area, is given as,
W = -J \ B ^^ol) - [u^Tj^ d(area) . (2.4.15)
vol area
Using equations (2.4.9) and (2.4.11), it follows.
W =
r T r 1 • \ ^ ^ ^ ^ ^ d(vol) - I x^A"^ ' T d(area) . (2.4.16)
Substitution of equations (2.4.13) and (2.4.14) and (2.4.16)
into equation (2.4.1) for the Lagrangian function gives.
-h '•i vol I J
X^A"^ i Iij d(area) . (2.4.17)
rea
According to Hamilton's principle,
t 2
6 J L dt = 0 . (2.4.18)
Taking the variation under the integral sign (Leibnitz' rule)
m
n
r Q]
r / T T \ 7\ T - l T - l T T T - 1 T \ V
px A $_ $. A X - 2i B E B X + 2x^k $_ XLg d ( v o l ) -• K
22
J « f [ PA-^Vi A-ld(vol)i - 6J ( B' E B d(vol)x
1 T I -1 T - T r -1^ T + 6x I A ^'x^d(vol) + 6x A ^ rTj^d(area) (2.4.19)
Integration of the first term by parts with respect to tim.e, gives,
t^
J t
1^ I P^ ^ 1 . ci(vol) 2i dt =
-•'-'
•T I -1' T -1 62C_ pA ^ £ A. d(vol) x
T I -1^ T -1 6x pA <l> «J> A d(vol) X dt (2.4.20)
J t
According to specified boundary conditions at times t and t^
6x(t ) = 6x(t ) =0J SO the first term on the right hand side of
equation (2.4.20) vanishes. Substituting the remaining term into equa
tion (2.4.19) we get
J t
6x
1
- I T -1 pA $ $ A d(vol) X -
J B E B d(vol) X
-l' T 1-1 A r X d(vol) + I A
$ T^d(area) — —K
= 0. (2.4.21)
Since the variations of the nodal displacements 6x are arbitrary, and
the integrand equals zero, the expression in parenthesis must vanish
Thus, the equation of motion for a finite element is.
r<
m X + k X = f (2.4.22)
23
Here m is the element mass matrix given by.
m = r -l' T -1 J A i $ A d(vol). (2.4.23)
k is the element stiffness matrix, given by.
k = JB'^E B d(vol) (2.4.24)
and £ is the force vector, given by
f -l^T A ^ X i = I A ^ X^d(vol) +
r ^
A~^ j ' T_d(area). (2.4.25) - - - ^
In a static case, when 21 = 0, the equation (2.4.22) reduces to.
]i X = i (2.4.26)
For the dynamic case if f = 0, the equation (2.4.22) takes the
form of the free vibration equation of the element as,
m X + l£ 21 = 0- (2.4.25)
When £ = £(t) the equation (2.4.22) is the equation of motion for
the forced response of the system
m X + k X = f(t). (2.4.28)
2.5 Assemblage of the Stiffness and Mass Matrices of the
Individual Elements
The element mass and stiffness matrices, and the force vector
derived in the previous section tiave to be transformed from the local
<
24
coordinate system of the element, if used, to the global coordinate
system of the entire structure. Also, they have to be assembled to
get the field equation for the entire structure. The equilibrium of
the nodes allows the individual elements of the stiffness and mass
matrices and load vector to be added directly into the appropriate
locations of the matrices. Since the total potential energy of the
structure is the sum of the potential energy of the individual ele
ments, the direct addition of the stiffness and mass matrices and
load vector does not violate the principle of minimum potential
energy. So the equations (2.4.26) and (2.4.27) are modified as
L^= L and m + KX_=0^ (2.5.1)
Here, j(, M, and F are the stiffness and mass matrices and load vector
for the entire structure.
2.6 • Boundary Conditions
The finite element formulation is not complete unless the boundary
conditions at the element nodes are specified. Without the boundary
conditions the global stiffness matrix, in most instances, will be
singular. The physical explanation for this can be given as follows:
imposing some sort of kinematic restraints, the system equation can
not prevent rigid body displacements; hence, the resulting displace
ment solution has no practical meaning. There are two types of boundary
conditions, geometric and natural boundary conditions. In the finite
element formulation, the natural boundary conditions are implicitly sat
isfied (17); so, only the geometric boundary conditions have to be con-
25
sidered. The geometric boundary conditions are further classified as
homogeneous and nonhomogeneous boundary conditions. Homogeneous
boundary conditions are the ones in which some of the nodal points
of the structure are completely restrained against any kind of motion;
that is, the specified displacements at these points are zero. This
can be easily accounted for by striking out the rows and columns of
the stiffness and mass matrices associated with these degrees of freedom,
along with the coefficients of the force vector. The diagonal terms
should be made unity. If a nonhomogeneous or non-zero bounday con
dition is specified, a similar procedure should be applied. The stiff
ness matrix should be post multiplied by a vector consisting of the
specified value, and with all othe values zero. Then the resulting 'q
vector should be subtracted from the force vector. Then, the procedure J
for the zero displacement case should be applied to both the matrices -(
and the corresponding force vector components should be replaced by
the values of the specified displacements.
2.7 Solution to tlie Overall Problem ;
The assembled stiffness and mass matrices will have the following
properties: They v\fill be
a. S>'Tiimetric and positive definite
b. Banded.
There are many direct and indirect methods available (30, 31, 32,
33) for the solution of the governing equation in the static case. In
this investigation Gauss' elimination method was selected after careful
review of all the methods available.
26
Once the solution for the displacements is obtained, strain and
stress in each element can be calculated using equations (2.4.4) and
(2.4.6). It is possible that the compatibility of stresses and strains
along the inter-element boundaries are violated. This is an inherent
characteristic of any finite element formulation based upon the dis
placement approach. One way to overcome this problem is to use a re
fined mesh size and compute the strains and stresses at the centroid
of the element. An alternative approach might be to take an average
of the stresses at the nodal points instead of at the centroidal point.
2.8 Review of the Methods for Dynamic Analysis
In dynamic analysis the characteristic equation for the free vi- •q
y bration of the system can be derived by substituting X = Y_ sin(cijt) into >
the equation (2.5.2); it is ^
K Y = u^M Y (2.9.1)
Here there are n real and distinct natural frequencies and corres-
ponding set of orthogonal eigenvectors or mode shapes. If the size "
of the matrices are large, a large amount of computer time will be re
quired to extract all the eigenvalues and eigenvectors. However, in
most structures most of the dynamic loading is contributed by the first
few modes and, hence, the effect of higher modes can be neglected (35)
and it is only necessary to evaluate the first few modes and the first
few natural frequencies. Based upon literature surveys, the following
methods and their relative merits were considered.
Inverse Iteration Using Successive Over-Relaxation Techniques
(35) The advantage of this method is that it can take into account the
27
handedness of both the stiffness and mass matrices, and also the decom
position of the stiffness matrix has to be done only once. Both the
eigenvalues and eigenvectors are iterated simultaneously. The conver
gence of this method can be accelerated by a proper selection of the
optimum over-relaxation parameter. Unfortunately, there is no simple
method by which this parameter can be calculated with the least ex
penditure of computer time. Without the over-relaxation parameter,
the solution requires a large number of iterations to converge. Also,
when the problem has repeated eigenvalues, or the eigenvalues are close
to each other, this technique has difficulty in evaluating the corres
ponding eigenvectors.
Householder-QR Algorithm (56). This method involves the tridiago-
nalization of the stiffness matrix by the Household algorithm, and
evaluation of the eigenvalues by QR sequence. Eigenvalues and eigen
vectors are not computed simultaneously. This method requires the full
storage of both the matrices. However, this method has the capability
that only a limited number of eigenvectors can be calculated, even
though all the eigenvalues are determined. This method is excellent
when the sizes of the matrices are relatively small.
Subpolynomial Iteration Method SPI (57). This method was devel
oped at Texas Tech University in June 1973, and is a combination of many
separate techniques. This method takes into account the handedness of
both the stiffness and mass matrices. In SPI, lower and upper bounds
of the first m roots are estimated by using the Sturm sequence. After
determining these bounds, a Lagrangian polynomial, called a subpoly
nomial, is passed through these points. The first root of this poly-
<
28
nomial is calculated by using the method of bisection. This root is used
as a shift point for the first eigenvalue which is calculated by the
method of inverse iteration. After this eigenvalue is determined, an
other polynomial is passed through this first eigenvalue and the rest
of the points. Its second lowest root is calculated and used as a shift
point for the second eigenvalue. This process is repeated until the
desired number of eigenvalues are extracted. The main disadvantage of
this method is the large ajnount of computer time used. The stiffness
matrix has to be decomposed each time a shift point is calculated. Also,
the accuracy of this method depends upon the upper and lower bounds
selected.
n-Step Iteration Method (38). As pointed out earlier, in most
engineering problems the higher frequencies are not of primary signi- X
ficance and, hence, can be neglected. Recently, several methods have
been developed for the evaluation of the eigenvalues and eigenvectors of
a large system which deals with a reduced set of generalized coordinates,
rather than dealing with the nodal coordinates or the actual degrees of ; J
freedom. In methods dealing with this concept, reference should be made ^
to the works by Hestness and Kraush (39), Jenning and Orr (40), Dong,
Wolf, and Peterson (41), Guyan (42), Bathe and Wilson (34), and many
others. The essence of the eigen reduction method is the transformation
from the nodal coordinates of a finite element formulation to a smaller
number of generalized coordinates via a transformation matrix, In some
variations of this method, static condensations is also used to obtain
the reduced system matrices. The method selected in this investigation
is based upon a modified version of the Lanczos' n-step iteration
29
procedure (30). The main advantages of this m.ethod are the computa
tional efficiency, storage, and the fact that this method is applica
ble to both positive definite and positive semi-definite stiffness
matrices.
In this chapter the main structure of the finite element method
was presented. The next two chapters specifically deal with the
application of these principles to the tire problem using conical
shell elements.
X >>
J;
CHAPTER III
DEVELOPMENT OF THE TIRE MATHEMATICAL MODEL
5.1 General
In this analysis the tire is approximated by thin shells of revo
lutions. The tire is subdivided into imaginary smaller subregions of
elements which are conical shell elements. The properties of these
shells are assumed to be constant with respect to the axis of the
shell. However, the loads and deformation need not be axisymmetric.
The deformations are expressed in different polynomials in the meri
dian direction, while in circumferential direction they are expanded
in a Fourier series to represent the antisymmetric phenomenon. The
tire geometry is a complex geometry, neither its thickness nor its
material properties remain constant over the meridian. However, in
the development of the model it is assumed that all the geometric pro
perties, and material properties, remain constant for one element.
3.2 Strain-Displacement Relations
In Figure 3.2.1 the motion of a typical shell element is speci-
1 2 3 fied by a set of orthogonal curvilinear coordinates X , X , X .
X 3 X (X = 1, 2), and X , are the curvilinear coordinate lines embeded
3 in the middle surface of the shell. The axis X is normal to the
undeformed middle surface of the shell element.
The position vector of a point on the middle surface of the
undeformed shell is denoted by R, and that of any arbitrary point is
denoted by R. u and 0 are the displacement and rotation of tlie middle
surface respectively. For the shell element shown the tensor component
30
n X >
^
31
Deformed
>>
Figure 3.2.1 Motion of a typical shell element
of the middle surface membrane strain, bending strain and the trans-
verse shear strain are given by the following relations (43).
•''~7^;c^"^;X-2u3b,^)> (3.2.1)
X^. =
2 An ;c en ;A^ ' (3.2.2)
^Ir =- — C U„ . - b, + /^ AC ,n
3,A - ^ n ' '^ ^ c ^ - > C = 1> 2 (3.2.3)
32
Here, the semicolon and the coma denote the covariant and ordinary
differentiations, respectively, ti 's and 6 's are the covariant and
contravariant components of the displacements and rotations, respectively,
"a" is the determinant of the first fundamental form of the metric ten
sor and b are the components of the second fundamental form of the
metric tensor and e is the permutation symbol. Nov\;, if it is assumed AC
that the shell is thin, then the physical components of the strains
can be derived from equations (3,2,1, 3,2.2 and 3.2.3) (44),
The membrane strains are.
(3.2.4) ^11
22
-- 1 ^ +
^lax^
_ 1 8u '^
^28X^
8A, u 1 w
V ^ ' ^ 2 ^ w
^1 *23X^ " 2
(3.2.5) I 01
-f In
, _ ^ 2 i _ . u _ > , ^ ^ a L _ r Y _ ^ (3.2.6)
The bending strains are.
X
1 n2 3A, L-^ +_1__JL , (3.2.7.)
11 ^ 1 3X1 A^ A^ 3 2 '
_ l _ i § ^ Q^ ^2 (5.2.8) ^22 A^ ,^2 ' A^ A^ ^1 '
, _^2 3_. ei ^ ^ ( ! i ) . (5.2.9)
' 1 2 - ^ 3 x 1 ^ ^ 2 ^ ^2 3 X 2 ^ ^ '
33
< - - ^
R, R.
Figure 3.2.2 Nomenclature of a Typical Shell Element
>
-I .n
s
34
And the transverse shear strains are^
.. - i 9^ _ y_ - a (3.2.10)
^ = 1 9lL-_ H_ + 3 (3.2.11) ''? A 2 R ^ ^2 3X 2
1 2 Here, u, v, w are the displacement components in the direction X , X ,
3 1 2
and X , and 3 and a are the rotations in the X and X directions as
shown in Figure 3.2.2. A and A^ are the Lame parameters, and R, and
R- are the principle radii of curvatures.
The above relations give the strain displacement relations for
any shell of revolution. For a conical shell of revolution, the n
following relations are available ( 4). (refer to Figure 3.2.3) X i> SI
A = r; R_= r/sin<& ,
xl = <J> and x2 = (|) . (3.2.12)
<
The Gauss condition for a conical shell is also given by reference (44)
as.
dr = r.cos^ ,
and J
d^ $
Lim r J ^ J (r. d, ) = ds, T* -v 00 Q q>
^ = ^ - ^ , (3.2.13)
35
X >>
J'
Figure 3.2.3 Nomenclature of the Conical Shell,
36
In the following paragraphs derivations for one strain-displacement / •
relation for each type is explained in detail. Tie other eouatlons
can be derived by using a similar approch.
The membrane strain component, equation (3.2.5) is,
22 " 2 3X2 A^ A^ ^,1 ' R^ *
Using equations (3.2.12) and (3.2.13), the above equation can be written
as
1 3u V 2 w . (}) r 3({) A,A 3$ r
m
1 3u V ^ w . q = — -r-r + — C O S $ + — Sm"?) , n?
r 3(t) r r ™ 1 , 3u . , , . J]
= 7 ^ 3^ - ^ ^^^^ •*• "" ^ ' - (3.2.14) .,
3v (3.2.15) and e = r—
s 3s .
e_, = T— - — ( u sinif - '^T ) • (3.2.16) 's^ OS r 3c{)
The transverse shear strain equation (3.2.10) is,
Y ^ Y ^ ^ 1 9w _ v ^ ^ ^
2 ^2 3x2 R2
Lising equat ions (5 .2 .12) and (3.2.13) we ob ta in ,
1 9w ^ • ^ , D ^* = F 3 i - r " " * ' ^ '
= - I T - -<=°s"l' ^ f • (3 .2 .17) r 3(f) r
n
37
and
3w ^s - 37 - ^ • (3.2.18)
The bending strain, equation (3.2.8), is
X = l_i§_ . _ a _ ^ ^ A 2 A A 1
2 3X^ ^1^2 3X1
Using equations (3.2.12) and (3.2.13) we get.
X _ 1 _33 ^ _a_ 3r
<() ~ r 3(}) rr^ 3$ '
= F ^ 3^ •" " ^ ° ^ ^ ^ '
1 . 33 . , ^ = 7 C ^ + ct sin4; ). (3.2.19)
3a. '^ ^S 3 s n
^s^ = - If ^ 7 ^ ff - ^ " ^ ^ ^°^^ ^ (3.2.21)
Here 6^ is the rotation about a normal to the shell surface (Figure
3.2.5) and is given by reference (27) as,
Q 1 ^ 3u u . , 1 3v . r^ ^ ^^^ % = - 2 ^ 3 ^ * ?^^"* - TH^ • ( •2-22)
The above relations are based upon Novozilov's linear strain-
displacement relations, and these relations are used in this investi
gation to compute the strains of the middle surface of the tire element
Mang (43) in his dissertation points out that the analysis based on
Sander's strain-displacement equations will yield the same results.
38
(Reference 27)
Figure 3.2.4 Rotations that Determine Twisting Strains
Normal Surface
sini|j
Generator of Cone
eference 27)
Axis of Cone
Figure 3.2.5 Determination of a
in X
J]
The strain matrix can be written as.
39
z =
. ^
i 's<t>
<I>
^S<})
(3.2.23)
and the transverse shear strain as
Y = Y
<!>
(3.2.24) 3
Textbooks by Saada (46) and Sokolinkoff (47), also proved to be of
valuable aid in deriving these equations.
3.3 Composite Elastic Properties
For a tire the elastic constants are formulated by considering
it to behave as a multilayered shell composed of orthotropic thin
layers. Based on the elastic stiffness of the rubber and cord layers,
the effect of the rubber matrix is considered to be insignificant so
far as the load carrying capacity vvhen the resulting deflections are
primarily due to membrane strain is concerned. Based on its construc
tion, a typical radial tire may be divided into three different regions
1. The region from the crown to the shoulder, or the tread
40
region, where a tire element consists of two layers of steel cords
and two layers of polyester cords.
2. The sidewall region, which consists of two polyester cord
layers.
3. The bead region, where the tire walls have been made stiffer
by addition of extra layers of either steel or polyester cords.
Figure 3.3.1 shows the typical orientation of the cords in each
of the steel cord layers, and Figure 3.3.2 shows a cross-section of the
layers. The steel cords are seen to lie at an angle with respect to
the meridian direction. This angle varies from the crown to the
shoulder due to the manner in which the tire is fabricated. The cord
population, or the number of cord ends per inch, also varies with the
location. At any radius r, if a unit length of ply thickness is con-
rn
sidered, then there will be n (r) steel cords embeded in the rubber
matrix. The relation to compute the ends per inch is given in refer
ence (8) as.
n (r) s^
n (r )r cos(3'(r )) ^ ^ ^ C
r cos(B'(r)) r < r < r^ T - - C (3.3.1)
Here, n fr^) is the uncured, or green steel cord density or ends per s C
inch at the crown. The cord orientation with respect to the s axis at
any radius r can also be approximated (8).
sin(3'(r)) r sin(B'(r^))
r < r < r^ T - - C
(3.3.2)
41
Transverse Direction T . 17
<!> L Longitudinal Direction
Cords
>• s
Figure 3.3.1 Cord and Rubber Orientation in a Single Ply of the Belt.
J]
Rubber
Figure 3.3.2 Cross Section of a Single Ply Viewed in the Plane Along the Cords.
42
To satisfy physical continuty, the law of mixture for a composite
material states that the strain level in the direction of the cords
will be same for both the cords and the rubber matrix, even though
they might be under different loads. Thus, the effective modulus in
the longitudinal direction, L, can be given as (49),
E, = E V + (1-V ) E„ L s s s^ R • (3.3.3)
Where V is the volume fraction of the cords given by reference (48)
as
V^ = n (r) IT R2 / h . s s^ ' s ' ' (3.3.4)
The effective modulus in the transverse direction T, normal to
the cords is also given as (49)
^T = -R C1^2Vp / (1-Vp . 3 3_ ^
And the effective poisson's ratios for the composite are given as (49)
\T = W * R(I- S)
v~. = "T T E™ / E. (5.3.6)
The effective shear modulus is (50)
G^ (1+V^) + G^ (1-V^) S;r " S G (l-V ) + G„ (1+V ) ' (-.•>.3.7)
S S K S
From reference (51), the relationship between inplane stresses
and strains in the L and T directions (assumed to be principle direc
tions) can be written as.
Where,
a^ > =
•LT
'11 12 0
^21 ^22 '
0 0 66
•< ''T
'LT
Qii = i/^ 1 - ^ L I T ^
Q22 = E^/( 1 - v^^v^^)
12 = 21 = \TS = L \
(3.3.8)
43
^66 = ^LT (3.3.9)
These material properties, however, have to be rotated by an
angle 3' to coincide with thj s and <) axis systcn. After trans
formation the tranformed equation relating stresses and strains is as
follows (50).
'11
Q. 16
'12
'26
'16
12 Q22 ^26
66
4 <P
[ 'S(|)
(3.3.10)
44
Here,
Where,
^11 " " l "" 1^2^°^^23 ) + U2Cos(43 )
Q22 = " i - U2COSC23') + U^cos (43 ' )
Q12 = 1 4 - U3C0S(43 ' ) ,
^66 = ^5 - V°^C4B ) ,
and Q^^ = - ^ U 2 s i n ( 2 3 ' ) + U ^ c o s ( 4 3 ' ) . ( 3 . 3 . 1 1 )
l ^ l 4 ^ ^ Q l l ^ ^ Q 2 2 ^ 2 Q ^ 2 ^ 4 Q ^ 6 ^ >
" 2 ^ ( ^ 1 2 - Q 2 2 ^ >
S = i ^ Q l l ^ ^ 2 2 -2Qi2^ ' ^Q66^>
"4 =1-^^11 ^^•22^^Ql2 - ^ ^ 6 6 ^ '
15 = 1-^^11-^^22 - 2 Q i 2 ^ 4 Q ^ 6 ^ ' ^'''''^^
The details of this transforriatJon arc given in Reference (SO)
The material properties for the second steel cord layers can be
evaluated in exactly same manner except that tiie transformation
angle should be -3 .
For the polyester cord layers equations (3.3.1) through (3.5.9)
will give the required results. No transformation of the coordinate
axis is needed as the L and T directions coincide with the principle
directions s and ^ of the structure.
45
Assuming that the material properties for each ply are known, the
total effective constituent relations relating strains to stress resul
tant are given in reference (50) as.
N
N 4>
N Scj)
M
M <!>
M Scl)
r- ' » « » » I -, A,, A, A,, B, B B 11 12 16 11 12 16
» » I I » I
^12 ^22 ^26 ^12 ^22 ^26
16 ^26 ^66 ^6 ^26 ^66
hi \2 he il ^ 2 ^ 6
I I t f » I
^12 - ^22 ^26 1 12 ^22 ^26
^^6 ^26 he ^^6 ^26 ^66
'4>
-scj)
7
^S(f)
Or,
/ —
N
M
t !
A B
t t
B D L J
X
Or,
N = E £ (3.3.13)
Here, A., has units of Ibf/in., B.. has units of Ibf, and D . has 3J ij ij
units of Ibf-in. From reference (50)
nn A. . = 2 Q. . ,, . ( h, - h, ^ ) ij ^^^ il(k)^ k k-1
itn
B:. = i^.^H«cs-'^L^
46
k=nn k=nn-l
k=3
k=2
k=l
7f
nn h ^ nn-1 nn-2
hj hj hp
Figure 3.3.3 Schematic of the Plies of the Tire
47
1 '''' - 3 3 ij 3 ^^^ ^ij(k)' k k-1
i,j = 1 , 2 and 6 (3.2.14)
where the h's are the ply thicknesses as shown in Figure 3.2.3.
The matrix G which relates the transverse shear strains to shear
stresses is expressed as.
G = Si 0
0 G22
where G^^ = A ^^/2( 1 + ^^ ) i = 1, 2 (3.2.15)
andO^^'s are the poisson's ratios of the composite. The detailed
discussion on the derivation of these ratios is not presented in
this thesis, because here it is assumed that the transverse shear
strains are small and can be neglected. The steps involved for incorpo
rating the zero transverse strain condition are given in Appendix D.
The relations developed here neglect the effect of curvature
and slippage between the layers. These properties vary from crown «
to bead as different types of layers are used in fabricating the
radial tire. Hence, it is required that they be evaluated for all
the elements selected for representing the tire meridian geometry.
Also, these properties will change with the changes in the internal
inflation pressure, as the strain levels in the cords will vary.
However, the analysis involved is very complicated; hence for the
48
purpose of this analysis the elastic relations are assumed to remain
constant with respect to the pressure!
5.4 Displacement Functions
In developing the model it is assumed that each node of the
element has five degrees of freedom in the local coordinate system
of the element. As shown in Figure 1.4.1, it has three displacements
and two rotations. The harmonic dependence with respect to the azi
muth position is given by the following set of equations (27).
N N
uCs,({5) = / ^ % ^ ^ ^ sin(ncj)) + u^(s) + \ \^^^ cos(nd))
n=l n=l
N • N \ '1 vCs,(t)) =y^ Cs) cos(n({)) + v^(s) + y v^(s) sin(n(;))
n=l n=l
N wCs,(l)) =y w (s) cos(n({)) + w^Cs) + y %^^^ sin(n(f))
n=l n ^
aCs,(f) = y a (s) cos(n(j)) + a^(s) + ^ a^(s) sin(n(j))
n=l n=l
N N * N * 3(s,(|)) =y 3^Cs) sin(n(})) + 3^(5) + V 3 «.s) cos(n(j))
n=l n=l
Or, in matrix notation these equations can be written as,
u(s,(f) = iH„(s) + i u (s) (3.4.1) -n" n
49
Here, the quantities without star and with star represent the
symmetric and antisymmetric parts respectively. The rotations a and 3
are treated as independent constants because the element has the trans
verse shear flexibility.
The motions corresponding to an individual Fourier harmonic are
elastically uncoupled. Also, the motions corresponding to the starred
and unstarred quantities are uncoupled because the trigonometry func
tions used in expanding the displacements are orthogonal in the range
of 0 - 27T
For each Fourier harmonic, n, ten generalized coordinates are
defined in the local coordinate system. Here, the axial, circumferen
tial, and radial displacements and rotation along the circumferential
line will be time dependent, but for the sake of convenience, the sub
script t, (to denote time) will be omitted. The four displacements,
as expressed in terms of the ten generalized coordinates are as
follows (27):
u = q, + q^ s n ^In ^2n
V = q, + Qy. s n ^3n ^4n
w = q- + q s + q_ s + qo s n ^5n ^6n ^7n ^8n
c n 9n " ^lOn '
(3.4.2)
or in matrix notation
TEXAS TECH UBRAR\3
u n
n
w n
'<})n
I s
1 s
or, more simply, as
1 = 2 3 1 s s s
1
J
f \
"in
'2n
>
50
u = $ q_ -n — -kl (3.4.2)
Other choices of polynomial expansions are possible, but the above
have been selected for the reasons given in reference (27), and are sum
marized below:
1. V/hen !/;->- i.e., when the conical element takes the shape of
a flat circular plate with a hole, the displacements, u and v , become n n
uncoupled, and those two displacements (for a flat circular plate) can
very well be approximated by a linear polynomial.
2. For w , cubic expansion is used which is very similar to the
expansion generally used for plates and beams.
3. Shear strain rather than the rotations a and 3 is used be
cause of simplicity in the analysis.
3.5 Development of the Displacement Transformation Matrix
Before the displacement transformation matrix can be derived,it
is necessary to express the relationship between the strains and the
51
azimuth position ^ for the Fourier harmonic terms. From reference (27)
this relationship is.
N
= ^ cos (nA) e
n=0
N
= y cos(n(j)) e > (^n
n=0
N
•s<|) X^ sin(n (})) E:
scj)n
n=0
<1>
n=0
N
= ^ c o s ( n ( l ) ) X
n=0 ^n
•set) = \ sin(n <t>) X S({)n
n=0
N \ COS(n <{>) Y
sn
n=0
<\> sin(ncj5) Y
<^n n=0
Or, in matrix notation these relations can be expressed as
N
6, ^ -n —n and Y =
52
where,
^ = ^ S ' % ' %r ^s' ^' ^s^ '
Y = •»• Y^, Y. }, — 8 9
T ^ , , ^ ^ ^sri' (f)n' s<l)n* ^sn' (f>n* S({)n ^'
1« " 1 Y^„» Y.„ >, —n sn <pn
T 6 = { cos(n({)), cos(n({)), sin(n({)), cos(n(|)), cos(n({)), sinCncJ)) },
'T and 6 = { cos(n6), sin(n(})) }. (3.5.1)
The set of equations (3.5.2) are used to derive the relationship
between tne Fourier components of strain and displacements. Inserting
equation (3.4.2) into strain-displacement relations (equations 3.2.1A
through 3.2.20), we get,
3v n
'sn 3s '
£ = —( nu + v sinijj + w cos\jj ) , (()n r n n n
3u ^ e = (u s±n\b + nv ) ,
s(i)n 3s r n n '
3w n —
sn 3s n
Y = ( nw + u cosij.' ) + 3 , '(()n r^ n n ^ ' n
3a _ n ^sn " 3s *
53
c n = 7^ - ^ + .^^^^ > '
3u
^s^n = If + 7 ^ - - n - ^^^^ - h^'^^ 3 ^ - F^ ^^-^ + 7 n )>•
(3.5.2)
In equation (3.5.2), the subscript n refers to the n Fourier term, and
this subscript modifies nearly every dependent variable.
Now, the displacement transformation matrix A can be derived —n
by evaluating the generalized coordinates q 's in terms of the nodal
point displacements and rotations. Equation (3.5.2) can be used to
express a and 3 as follows:
3w n
n 3s sn
Substituting for w from equation (3.4.2) it follows,
2 a = q, + 2q_ s + 3q^ s - y . (3.5.3) n '6n 7n 8n sn
Similarly, for 3 ,
^ = 7 ^ ^%n " %n^ ' 7n^^ " 8n^^^ ^ "°" In " 2n "
" V " lOn
(3.5.4)
Evaluation of equations (3.4.2), (3.5.3), and (3.5.4) at the
nodes of the conical shell element, and writing the resulting set
54
of equations in matrix notation gives
X = A q + A Y -n -n -4i —Yn sn (3.5.5)
Here, the bar notation indicates the complete subset of the matrix
A , and is given as follows: -n ^
A -n
1 • :
COSrl) ' '• r • • a : .
1 '. i '
cosi|^ : £cosi|j
b b
1 :
.
: £
;
1 : •
I 1 '.
n : • r-_- : '
a .
: 1 : £
• : ^
• n " n£ ' 1 • r
b b
: ' '
'. 2£
' 2 • n£
• '"b
:
: ' '
'. 3£
' 3 • n£
' ^b
1 '.
: :
: 1 : £
(3.5.6)
55
The matrix A, is a column vector with -1 in the 4 and 9^
columns and zeroes elsewhere, Y in terms of the generalized coordi sn
nates is given as:
• n L Y n \% •
(3.5.7)
Here, B
Substituting equation (3.5.7) into equation (3.5.5) we get.
s the partition of the B matrix given in Appendix (A)
X = A -n -n %. " -yn [ Y n J Si
(3.5.8)
Defining the matrix A such that.
A = A + A —n -n —Yn B [
.n -L ^ Y n -t , (3.5.9)
then,
X = A q , --1 —n -Ti *
or.
-1
% " - n • (3.5.10)
Here the matrix A is known as the displacement transformation matrix n
It is so named because it transforms the nodal displacements of the
elements into displacements within the element.
56
3.6 Strain Energy
The total strain energy of the element is the sum of the energies
for the symmetric and antisymmetric deformations, and can be written as
*
"T = " * " • (3.6.1)
Here U^ is the total strain energy, and U and U* are the strain energies
for the symmetric and antisymm.etric deformations, respectively. If only
the unstarred quantity is considered, the strain energy equation for a
linear orthotropic shell of revolution is given as
N N £ 27T
J L I J
s O E e e + Y e G e Y ^ rd({)ds " ' n ~ n - ~ m - m -^n —n - - m m '
)
"=° •"=" ° ° (3.6.2)
Here, the matrices E and G are given by the equations (3.2.13) and
(3.2.15). Also, in the above equation it is assumed that the m.embrane
strains and bending strains are decoupled from the transverse shear
strains, r is the radius of the shell at any point and is given by.
r = C^ + C^s,
where
S = a' " S = "-i. (3.6.3)
Here, the subscripts a and b denote the two nodal points of the element,
and £ is the length of the element. Use of equation (3.5.1) yields
57
£ 27r
N ^'^ r r
u =
n=0 m=0
sl £ E 8 e + ll e'^ G e' Y 1 rd(})ds -n^Ti m-m -Hi-^ — ^ m m ^
J J 0 0
(3.6.4)
The terms n 51 m represent the coupling between the Fourier terms,
but due to the orthogonality of the trignometric functions in the range
of 0 _;? (j) < 2Tr they identically vanish. Hence, after preforming the
integration with respect to (|), equation (3.6.4) can be written as
U =
£
n=0 0
£ E e + Y G Y —n — —n -Ti — --Ti rds
Here
n
277
<
TT
if n=0
if n>0 (3.6.5)
Following the analysis of Chapter II, equation (3.6.5) can be written a;
n
U =
N £
n=0 0
.T -1 B ' E B + B' G B 1 x_ A rds —en en -^n Y^ I —n
(3.6.6)
Denoting,
58
£
r ,n B E B rds = kl""
—cn — —en —
J 0
,n
£ c
B G B rds —Yn — —Yn
= k qYn
J 0
and
j,qen ^ ^qyn ^ ^qn
A-1 k'l^A-l -n — -n (3.6.7)
the strain energy expression is written as
N
"4 X k X Ti -n -n
n=0 (3.6.8)
The matrices B and B are given in Appendices A and B, respectively, -Yn —en
59
3.7 Kinetic Energy
Again assuming that the kinetic energies associated with the
symmetric and antisymmetric deformations are uncoupled, the total
kinetic energy of the element can be written as
T^ = T + T (3.7.1)
Here, T^ is the total kinetic energy, T and T are the kinetic en
ergies associated with the symmetric and antisymmetric deformations.
The kinetic energy of an element is given by (for symmetric part only)
£ 27r h r r r
2
J J J p u u dw rdi ds (3.7.2)
0 0 0
Here p is the density of the material, ii are the displacements at the
nodes, and the "dot" denotes differentiation with respect to time.
Using harmonic dependence of the displacements, and assuming that the
thickness of the element remains constant, equation (3.7.2) can be
written as
N
T =
M £ £
r r
n=0 m=0 T^IJ pu 0 9 u rd<|)ds —n -n —m -m
(3.7.3)
0 0
Here again the terms n j m represent coupling between the hrirmo-
nic terms, and due to the orthogonality of the trignometric functions
in the range of 0 < (}> < 2TT, they will vanish.
Integrating with respect to (|), in the range of 0 £ (f) £ 277, equation
(317:3) can be written as
60
N .n ' :—»
rp __ X • • h pu u rds
n=0
here C = 27r, and C. = TT, (3.7.4)
Differentiation of equation (3.4.3) gives
u —n = $
% (3.7.5)
But from equation (3.4.3),
Sn -n -n (3.7.6)
Substitution of equations (3.7.5) and (3.7.6) into (3.7.4) gives.
N
T = 2
•T -1 h x' A
-il —n
T T -1 =
p$ $ rds A X
n=0 0
(3.7.7)
Defining
£
n
, . p$ ^ rds = m —a
and
A m A = m —n —a —n -n
(3.7.8)
then equation (3.7.7) reduces to.
N
T = •=-X
1 \ 2 /
-1 s -T } X m X
-^1 - n - i l (3.7.9)
n=0
61
In equation (3.7.9), the matrix m is known as the element mass
matrix. Using the definition of the matrix ^ from equation (3.4.3),
equation (3.7.9) can be written as.
£
r m —a G>
J 0
Symmetric rds
Substituting for r = C^ + C^s,
where
, r a b
(3.7.10)
equation (3.7.10) reduces to^
62
- 1
Pi ^2 +z.
8 10-
^3 + z
^1 +z.
^2 + z.
where,
^3 + z
m = C^ h —a 1
Symmetric
^1 + z.
^2
P3 + z
^3 + z
P4 + z
^4
P5 + z.
^5 + z.
'•'6
+ z.
^7 + z
8
^1 + z.
^2 +z.
8
^3 + z
10
P, =
P^ =
P„ =
P. =
P. =
P. =
P7 =
C £
C^£^/2
C^£^/3
4 C £ /4
C^£^/5
C 5, /6
C/ /7
2
3
4
5
6
7
'8
= C^^ /2
= C £^/3 2
= c//4
- c//3
= C / ^ / 6
= C2£'/7
= C2i^/8
z_ =
z„ =
Zo = (3.7.11)
63
The element mass matrix given by equation (3.7.11) is a full 10 x 10
matrix, which is consistent with the assumed displacement field and
the stiffness matrix derived earlier. It is, therefore, known as the
consistent mass matrix.
The mass and stiffness matrices for the starred parameter set
will be the same as those of unstarred parameters.
3.8 Transformation into Global Coordinates
The mass and stiffness matrices derived in the proceeding two sec
tions are in the basic coordinate system (<}>, s, w) of the element.
The local and global coordinate systems are shown in Figure 3,8^1, Be
fore the equation of motion can be derived for the entire structure,
it is necessary that all the element stiffness and mass matrices be
transformed to a single set of reference axes of the entire system.
Based on purely geometrical considerations, the transformation matrix
can be derived as
U n
V n
/ w n L
a n
3. n
cosij; sinifj
•sinij; cosij
cqsi|; sinip
U g
U g zn
I a
m
eg
zn
m loca l a t "a" global a t "a"
64
60 U
4-1 C G)
e cu o
J 1
ex. w
•H «
X I O
iH O
13
r. d
T H d o o hJ
T l P! cd
E QJ 4-> cn >.
CO
<u u rt C
•H Td M o o u
• +J p: <u p: Q)
t H ;xq
T H rH CU
rC CO
i H a o
•H
a o o QJ
x: •M
U-l O
- d
00 »
CO
Q)
fcB •H
d
65
or in matrix notation
x = T U^ -^ n (3.8.1)
Here the superscript g denotes that the displacements are in global
coordinate system, and the matrix T_ is known as the transformation matrix.
Substitution of equation (3.8.1) into the strain energy expression
of equation (3.6.8) yields
N — 1 J
U = > U^ T^ k T U^ 2 / -il — -n n
n=0
N \
L 2/
1
> f
T U8 -n
k? -n
uS - i l
>
n=0
where g T k^ = T k T n — —n —
(3.8.2)
Here k^ is the element stiffness matrix in the global coordinates. -il
Similarly, substituting equation (3.8.1) into the expression for kine
tic energy gives
N
T = ], n=0
^ > U^ T M T Ijg = 7 U^ M^ U^ 2 / — n n n 2 / f—n -n -n
n=0
66
where.
M -n g _ T
T M T, — —n
(3.8.3)
,g Here, m is the element mass matrix in the global coordinate syst em.
3.9 External Work Potential
The work done by the body force x„ and the surface traction T is —B -r
given as
£ • 2 7
r r w =
J 0 0 j
h U X„ r d ({) ds U T^ r d (f) ds
0 0
jj ^ r d ({) ds
0 0
Where
% = ^ ^ 'R
(3.9.1)
Here q_ is the equivalent force vector. If this force vector and the
displacement vector are expanded in a Fourier series then the work
potential can be written as
67
£ 277
u. 6 . e £^ rd(f)ds (3.9.2)
0 0
Here, again due to the orthogonality of the trignometric functions in
the internal of 0 £ ((> £ 277 the double sum for nM vanishes. Integra
ting equation (3.9.2) with respect to (^ we get
W = C ,n
n=0 J 0
} ^ £F ''' '
Where
• 27r
,n c" = <
77
n = 0
n > 0 (3.9.3)
The use of equations (3.4.2) and (3.5.9) will yield
N ^
1
W = C n T -1 T X A ^ Qr, rds —n -n — -4
n- O
0
(3.9.4)
Transforming from local coordinates to global coordinates with
the help of equation (3,8.1). yields.
where.
W = U^ p -n -Hi
68
£n = ll n' J^\-ds (3.9.5)
The above procedure leads to a load vector which is consistent
with the assumed displacement field used in deriving the mass and
stiffness matrices.
3.10 Equation of Motion
Substitution of the strain energy function, kinetic energy func
tion and the function for work potential into the expression for the
-f"Vi
Lagrangian function L, gives the Lagrangian for the j element as
L. = ( T - U - W ) . 3 3
2
T T T .(jg jg yg _ yg jg yg + 2U^ p -il—n-ii -ii-n-n -ii-Hi (3.10.1)
-il
For an assemblage of M finite elements, the total Lagrangian can be
written as (51),
L = 1
•J
T U"' m^ U U^ k^ U^ + 2U^ p
- 1 -Hi -n -n -il -n -il —n
-*1
69
N 1
= k ) \ ^ l \ ^ - X^ K X + 2X' P , 2 /_^ \ -n -n -n -n -n -n ^-n -n I (3.10.2) 'h=0
Here X are the displacements for the entire structure. M and K -n -n
are the total mass and stiffness matrices and P is the total load -n
vector. Now applying variational principles to equation (3.10.2)
we obtain.
<S /L dt = 0 (3.10.3)
Following the analysis of Chapter II, the equation of motion
for the assemblage of M finite elements, for each Fourier harmonic,
can be derived as
M *X + K X = P ,^ ^^ ,^ -n -n -n -n -n (3.10.4)
3.11 Load Vector for Specific Types of Loading
In this section load vectors for certain types of loading are
presented for analyzing the static behavior of the radial tire.
A. Internal Pressure Load
The pressure loads act on the adjacent grid points in such a
manner that the center of the pressure is preserved. The generalized
force at a grid location 'a' due to pressure load on a conical shell
elements between nodal points 'a' and 'b' is given in reference (27)
as:
70
P^ = 277£ I ^ + ^ w
r r
3 • 6 (3.11.1)
This load is in the w coordinate axis of the element and, therefore,
must be transformed into the global coordinate system of the tire.
This load will produce only the axisymmetric deformations, so
harmonic analysis is not required. Also, this load will produce
deflections mainly due to the membrane strains.
B. Centrifugal Force
The effective forces due to the centrifugal or inertial loading
were developed by lumping the components of this load at the nodes,
as in reference (7). For one element, the load due to the centri
fugal force in the radial direction is given as:
p = 7 7 r p t £ r 03 ^r m m
(3.11.2)
where r = / r . r, , and o) is the rotational speed in radians/sec
The effective bending moment due to the centrifugal force at each
node will be
1 2 2 2 m. = ^ 7 r r p t w £ ^ 2 m (3.11.3)
The resulting deformation under the effect of centrifugal loads will
be axisyimnetric, therefore, harmonic analysis is not required.
71
C. Load Vector Due to Ground Contact Loading
In the method of solution selected in this investigation, it is
necessary to assume the footprint area beforehand. Once the footprint
area is decided, some sort of pressure distribution in the footprint
is assumed. For a load acting between the azimuth positions -c|)i<(j)«j>i,
this load is given in reference (8) as:
Dead Load MM
1 n=l
£
r r
J 0
J F' cosip r ds dd) n ^ Y (3.11.4)
Here, F^ is the assumed pressure distribution acting normal to the
surface of the tire, and MM is the number of elements in the trend
region. The pressure F'- will not be axisymmetric and, as a result,
has to be expanded in a Fourier series as:
N
n no + 7 F- cos(n6)
ZLJ nn n=0
The load vector in the r direction, for an element, is:
(3.11.5)
2 r F- (J)i£ no ^
for n=0
p = -^ F^ sin(ncj)i)i' ^r n nn
for n>0 (3.11.6)
72
5.12 Boundary Conditions
Before the governing field equations can be solved to get unique
displacements at the nodes, it is necessary to apply the displacement
boundary conditions.
For the static analysis only half of the tire meridian is consi
dered. The reason for doing this is based on the fact that the tire
is syiranetric with respect to a plane passing through the tire meridian
at the crown and normal to the z axis. Also, if (^ is defined such
that -T7<(f><77, then the loading and deformation are symmetric with res
pect to 4> = 0 . All the nodal displacements and rotations at the bead
are assumed to be zero. In other words, the bead is assumed as a
built-in or fixed end. The displacement in the Z direction, and the
slope in the ({> direction are assumed to vanish at the cro\m. This
follows from the assumed symmetry of the tire meridian at the crowi.
For the dynamic analysis the entire tire meridian is considered.
The reason for this is that if only half of the tire m.eridian is
considered, then the frequencies associated with the unsymmetrical
mode shapes will be omitted. Here again, all the displacements at
the bead are assumed to be zero.
In this chapter, the mathematical model for a radial tire, based
on the finite element analysis, is derived. To obtain the solution
for a particular tire it is only necessary to furnish the geometrical
data and the constructional parameters. Four automobile tires were
considered for the purpose of the analysis and their results are
presented in Chapter V.
CHAPTER IV
THE DYNAMIC ANALYSIS
4_._1_ n-Step Iteration Method
The eigenvalue problem is stated by equation (2.7,1), and is
K Y = A Y (4.1.1)'
Here 1 and M are n order symmetric positive definite banded
ji;atrices. The to's are the n distinct real eigenvalues and the Y*s
are the corresponding orthogonal eigenvectors. ( The stiffness and
the mass matrices for the tire vvere derived in Chapter III ) Iy\
most cases of practcal interest, the stiffness matrix will be posi
tive definite. However, if the stiffness matrix is positive semi-
definite a small shift parameter should be chosen to obtain a
decomposable matrix. Also, if a consistent formulation based
upon variational priciples is used, the resulting mass matrix will
also be positive definite. The stiffness matrix in equation (4.1.1)
is decomposed into a product of a lower triangular matrix and its
transpose by the well known Cholesky's decomposition method, such
that
K = L L^ (4.1.2)
Here, L is the lower triangular matrix, including the diagonal terms
Now, equation (4.1.1) can be written as.
- Y = ]^"Vl (4.1.3) 0)
73
T Mul t ip ly ing equat ion (4 .1 .4) by L
0)
T Let L Y = 0 , then we have.
- 1 . . . - 1 ^ 2
CO
and t h i s can be w r i t t e n a s .
where,
^ 2 0)
and
74
T -1 -1 -1
S u b s t i t u t i n g K_ = L L in equat ion (4 .1 .3 ) we g e t ,
T - 2 l = k^ k \ l . (4 .1 .4 ) CO
I T -1 -^ L 'Y = L M Y .
2 (4 .1 .5)
e = I ^ M ] L " 0 . (4 .1 .6)
X ^ = Z^^ . (4 .1 .7)
_1 _iT (4 .1 .8) Z = L -M L
75
In actual computation the inversion of the stiffness matrix is
not necessary, forward and backward substitution can be employed to
obtain the required result. The reduction of the n order eigenvalue
problem to a reduced set of generalized coordinates is achieved by
using the nodal displacements as unknowns and through the transforma
tion (33).
9 = V S_ (4.1.9) nxl nxm mxl
The transformation matrix V relates the discrete displacements represen
ted by 9 to some generalized displacements S. This transformation is
orthogonal such that
v' V = I (4.1.10)
The Raleigh quotient of the system is
^R = i 1 ^ / e G_ (4.1.11)
where 9 is the approximation to 0. The accuracy of the method based
upon this formulation is due to the fact that X ^ is stationary wit'
respect to Z and the eigenvectors 9. Substituting equation (4.1.9)
into equation (4.1.11), we get
X„ = s V z V S / s V v S (4.1.12) R —
76
Making \^ stationary with respect to arbitrary variations in the •f-V.
elements of S yields the reduced m order eigenvalue problem as.
R S = \S_ (4.1.13)
where
R = V^ Z V (4.1.14)
If the reduction from the original system's n-coordinates to m-
generalized coordinates is viewed as imposing n-m constraints upon
the original system, then Raleigh's principle states that.
X3 < XJ < XJ*"-" j <m (4.1.15) - R -
Thus, a l l t he ?tp's a re contained between X and X , and the approxi-R 3 - .1 mation becomes exact for n = m, i.e., when X^ = X . Thus, if the
transformation matrix V_ is an approximation of the exact nodal vectors,
X will be an approximation to the unreduced system. In this analysis K
the matrix V is built, vector by vector, by using Lanczos' algorithm
(30), such that the resulting m x m matrix is a symmetric tridiagonal
matrix. The eigenvalues of this matrix approximate the higher modes
of the matrix, and consequently, the lower natural frequencies of
the structure. The matrices R and V are of the following form:
77
R =
^11 ^2
2 22 S
S 33 4
d r m mm m X m
(4.1.16)
V = v. V, V, m
n x m (4.1.17)
4.2 Determination of the Matrix R and Vectors V
The original eigenvalue problem is
Z 0 = X9 (4.2.1)
First, assume an arbitrary vector y . The choice of this vector should
be such that it should contain all the eigenvectors of the system. That
is if the eigenvectors were known the vector y_ would be a linear com
bination of all of them. If this vector is deficient in some eigenvector,
convergence to that particular eigenvalue is almost impossible. One way
to be almost sure that this assumed eigenvector contains all the eigen
vectors is to generate it by random numbers. This random vector should
be normalized with respect to the Z matrix. The second vector y_ can
be obtained by ordinary iteration as.
78
^2 = ^ 1 (4.2.2)
Now instead of using y * as the first iterated vector, it should be
made orthogonal to the vector y_ by the Gram-Schimidt orthogonalization
process.
2 = 2 - ''lA f"- -^
T Multiply equation (4.2.3) by v to get
•-r 'T' * T
Vl V2 = v ^ l 2 - ^ 1 ^ l A ^"-^-^^
Now a s A , i s o r t h o g o n a l t o v^y
T T V v^ = 0 and yjx YJI = 1 ( 4 . 2 . 5 )
Substituting these results in equation (4.2.4), and rearranging the
terms yields
r = V v^ (4.2.6) 1 1 - ^ 1 - 2 ^
* Substitution for y_^ yields,
^n = I-I^ILI-I (4.2.7)
and
V = Z v - r V (4.2.8) X.2 ±1 1 ^11-1
79
Normal iz ing y^^, v>?e ge t
1
^^2 = ^-2/{^-h^ ^ ( 4 -2 .9 )
similarly.
*
IJS ^ --2" ( 4 .2 .10 )
Making y„ orthogonal to y and v_
•k
^3 = y-3 - ^22^2 - ^2^1 (4.2.11)
T r..- can be found by multiplying equation (4.2.11) by v^ , and d^ can
T be found by multiplying the same equation by y^ ,
^22= 1.2^ 1-2
^2 = ^l'?^ ^2
with
V3 = i l 2 - = 22 2 - ''2 1 ^^-2-12)
The recurrence relations to find the matrix coefficients are given by
(38) as.
r.. = V. Z V. , 11 —1 1
T d. = V. , Z V., 1 —1-1 1
V. , = Z v . -r..v. -d.v. , —1+1 1 11—1 1—1-1
80
and
(4.2.13)
For large system of matrices the vectors generated by equation
(4.2.12) should be theoretically orthogonal to one another, but in
practice, this need not be the case. As the computation proceeds,
the round-off error propagation quickly destroys the orthogonality
between the vectors (52) . This is also caused by repeated multipli
cation and divisions ..involved in generating these vectors. Ojalvo
and Newman (53) noted that this drifting away from the orthogonal
set of vectors results in false "bunching" of the eigenvalues at the
higher end. Lanczos (54) suggested a reorthogonalization correction
of the following form.
i-1
V. = V. —1 —1
> I V. . Z V. V. . (4.2.14)
Here v. is the value of v. computed from equation (4.2.13). However,
this correction procedure is not adequate to produce good results.
Ojalvo and Newman (53) suggested the following iteration loop based
on the Lanczos' reothogonalization criterion, and this procedure is
used in this investigation. Iterate the equation (4.2.14) "s"
times such that
±
s+1 s \ T s V. , = v . T - / v . v . T j -1.1 -1.1 ^ [-3 -1.1 J V
j= l
satisfies the condition.
81
(4.2.15)
max i<j<i
T s V. V. --1 -1+1
< Tol (4.2.16)
Here, y. is the starting vector obtained from using the equation
(4.2.13), and Tol is some small value. After this reorthogonaliza
tion criterion is satisfied, the vector generated from equation
(4.2.15) is normalized and the computation is carried in the usual
manner. Usually only about two to three iterations are necessary
to completely reorthogonalize the vector.
4.5 Size Criteria
Althogh there is no definite criterion for the required size of
the reduced matrix for a finite number of eigenvalues to be deter^
mined, at least theroetically n degrees of freedom system can be
reduced to a very small system. However, it is found that when
m«n, then the m/2 eigenvalues are in exact agreement with the
true natural frequencies of the unreduced system, while the re
maining m/2 frequencies lie in the spectrum of the remaining true
frequencies of the unreduced system (55). Thus, if "q" lower
82
frequencies are desired, then the size of the reduced problem should
be,
i(2q+l) q > 3
min (7,n) <1 £ 5
The eigenvalues and the associated set of the orthogonal eigen-
vectors of the reduced m order matrix can be found by using methods
such as the LR or QR algorithm, Jacobi method, inverse iteration
method, etc. In this thesis, however, the QR algorithm as given in
Reference (36) is employed to compute the eigenvalues of the reduced
tridiagonal matrix. The reason for employing this method is due to
the fact that when the matrix is symmetric and tridiagonal, the QR
algorithm is perhaps the fastest of all the methods available. The
associated eogenvectors are computed by inverse iteration (57) after
the eigenvalues have been evaluated.
CHAPTER V
NUMERICAL RESULTS
5.1 General
This section presents the numerical results of the method
developed in previous chapters. In the remainder of this chapter
the finite element method used will be referred to as the FEM. The
aim in this chapter is to present the main characteristics of the
method of solution, the FEM, as applied to the analysis of pneumatic
tires* primarily radial tires, and to compare these results vi ith the
experimental data discussed in the literature. Three radial tires
and one bias ply tire are considered in this analysis.
Two digital computer programs were written in the Fortran IV
language and executed on the Texas Tech IBM 570 computer. Program
one performs the axisymmetric and non-axisyimTietric (which will be
called asyimnetric) static analysis and has the capability of gener
ating the mass matrix to be used in the dynamic analysis. Program,
two is essentially an eigenvalue and eigenvector extraction routine
which takes the stiffness and mass matrix as input data. This fea
ture allows this program to be a very general purpose eigenvalue
extraction routine. In the eigenvalue program, Guyan's (42) reduc
tion was employed to rem.ove the degrees of freedom associated with
the nodes where the displacement specified has zero value. In order
to increase the degree of precision, both programs were written using
double precision variables.
The results in this chapter are presented in three different
sections. Section 5.2 deals with the asymmetric deformation of the
83
84
tire under the effect of static ground contact loading. In section
5.3 the axisymmetric deflection patterns for the crown and sidewall
of the radial tire under the effect of changes in the internal infla
tion pressure are presented. Also presented'is'the crovm'deflection
pattern of a radial tire subjected to various rotational speeds.
Section 5.4 is devoted to the results of the dynamic analysis of the
Firestone HR78-15 tire. Natural frequencies and some of the mode
shapes associated with these frequencies are given in this section.
The important constructional parameters needed in this analysis
to generate the elastic stiffness matrices for the four tires studied
are given in Table 5.1.1. Figures 5.1.1 through 5.1.4 show the unde
formed meridian geometry of these tires.
The detailed analytical data of the graphs presented in this
chapter are given in Appendix C. The figures follow the terminology
and the coordinate system defined in Figures 1.2.1 and 1.2.2. The
term "crown deflection" is used to indicate the changes in the radial
dimension r of the crown from some reference position. Unless other
wise specified this reference position is the undeformed geometry of
the tire at the indicated inflation pressure.
Figure 5.1.5 is presented in this chapter to visualize the terms
"axisymmetric" and "asynmietric." The term "axisymmetric" is used
whenever the loading or the resulting deflection is syTmnetric with
respect to a lateral axis passing through the center of rotation of
the tire and which is independent of angle <|). Examples of such types
of loading are internal pressure and centrifugal force. The term
"asymmetric" is used to indicate the exactly opposite situation.
85
V) 4-i • H
3
CM
• H
CM
• H
(N
• H • H • H •H
ex
to to c c:
•H - H
to o u • H E -
O
W U o 4-> CD
e
ex
i n X^°o 1 i ^
00 I ^ ^ Cd o S:: ci i
r—« rt
• H -o rt a:
- =-- OO
o o . t o
e x . E - Cri CQ
V) .-<1> to
'O o
CJ
r-. ~;
o u
• H H
O LO " -
• ^U o
px
t—1 rt
• H n3 Cj
c i
o 00
o o o
o o vO
VO CM
VO vO
o o
i n
i n
i n
o o o o in
o o o o o vO
o o o o
vO vO
vO
i n vo
I—)
CM
o
in in to o
o CNl
to CN O
OO
o vO vC
o
in CM vO o
o
o to
o to
c o o o c o
vl v |
v | v l 4-
CM
o • H +-> O
O
«/) vO \ 0 - ::s LO r- l (U ^ - ^ CJ bO <t) • - H
•XJ f-i M-l - a O - H O c3
O E- Cii C i
• i n CM vO
V' 00 O CNJ
i n
o o o LO
OO
in CN o
in o 00 CNJ
in
<
tf) on c« o fH r i
U
en to cJ (J f-i c j O
V) 3
I—I
rJ -a o E
f-i (U
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cr: 1
-M I—1 <D
.o
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cr: LU
tn to a o u o to p
1—(
:3 TD O
e fn rt <U r-;
en 1
ci u
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o • H 4-> a u c o to to
• H o
C-, 1
c^ :3.
to to CJ o -!
c i o 1
to D
r - <
:3 -d o E
x> f-i O
CJ 1
o t u
-o u o o
o • H 4-> a u c o to to
• H O
CL, 1
o a
1 o u o *-> a c
• H " . to -o c w
1 o
t—t r^ UJ
to to C3 o f-l CTJ
o •
rt • H -d
o >
• H •M O O
t4-4 MH
tu 1
O -o
to to c i o ^ c i o 1
to to o c
^ o
• H f
•M
X r-H cx
1
o ^
o S-i 3 to to o u c-
c o
• H +-> rt
1—1
t+-t c
r—
1
r^.
X +-> •H to (— o d.
1
G.
86
to +-> •H
c
CM
• H
CM
• H
CM
• H • H • H
•H • H • H
t N
C O
U
to o f-l
• H t - i
CD A 4->
t H O
to
u o +-> (D E CIj ^ cd
Cl,
I—1 OS
c o • H •P O
tru
to
c o u
to
.-H o •M c3
(X
to
<D t o
- d o
Q
to
0)
i n f—<
1 00 r-g -^
CM
O JH
• H E-
r -.*. o u
•j-K t—t
vO
t o o ' d o
Q
JH • H E--
/ — < 00 i n •-V ^ 0)
c^
r—-i 00 V -<
. U-^ o
c :
/ — V
\ D LO V •>
. t H O
r~^
r~^
(56
. t H 0)
/-v^
1—4
rt i H
" d rt /--
to c:
. —J
CO
I - H
rt • H - d rt
c i
1—1
• H " d r;
r V
vO O r-i X cr» CM
vO <o f—1
X cr> CM
4
O O O O o LO OO
t o •
t o
•
r--
• o o o CM
• lO o r--
50.
r
r--CM O
•
t o CM o
•
t o t o o
•
r-o
•
'^ o
«
LO o
•
o CN
o CM
CN CM
O . CO vO
o . LO LO
o LO
vO r--
CN
• o o o CN
o o o rH
o cr» \ 0
o C7^ ' ^ i "
l O •
LO
LO
LO
"^ •
625
LO UO
CM l O
i n •
t ^
595
r--
-en LO
r
vO o t ^
t o OO
• t o .—f
OO
CM 1 — •
r^
r—i f—<
CM CTv
CM
LO
CO
<
0) CO
to to to to to w C3 XI
•c •>• o u o +J ca o
r—1 t o C C\J
CO - d »-H f-H
W CJ
f-l o x^ jn d <
" d a o ?H P
t H o to d
rH d
- d o
u <D
X X d ^
- d rt <D ^1 P
t H O
O • H •P CTJ
u c; o to to
• H
o a.
^-1
o . D ^ p—*
< - d CTJ (D fH P
t H O
to to CD C ^ o
•H
to d
•H - d rt f-i
- d Cj o
c r
c ^
to d
•H - d lA
u p 3: O
u CJ
c>
•H-
o •H to
c CD
E •H Q •H -d
87
Tire Half Width, in
Figure 5.1.1 Meridian Profile of Dodge's Tire No. 6 (radial)
88
12 .
o • H to
CD E
• H Q
03 • H
c3
11 . -
10. -
9 .
8. -
7.
Ti re Half Width, ; in^
Figure 5.1.2 Meridian Profile of Dodge's Tire No. 7 (radial)
89
^ 15,
_ 12
-. 11
- 10
o •H to
c CD
E •H Q cd
•H
CTJ cd
- 9 .
8
Tire Half Width in
Figure 5.1.5 Meridian Profile for Patel's Tire No. 2 (bias)
90
p: •H
o •H to
CD E
•H Q
ccS • H
zi
Tire Half Width, in
Figure 5.1.4 Meridian Profile of Firestone HR78-15 Tire (Radial)
91
Internal Pressure Load (axisymmetric)
Contact Patch Length
Static Ground Contact Load (asymmetric) f
Figure 5.1.5 The Axisymmetric and Asymmetric Loading
92
The best example of this type of loading is the load on the tire in
the footprint region. This load can be made symmetric with respect to
the vertical axis r, which passes through the center of the footprint,
but can not be made symmetric with any other r axis.
5.2 The Asymmetric Deformation Under Static Ground Contact Loading
The tires originally investigated by Dodge (56) and Patel (8)
for the effect of static ground contact loading were selected to examine
the accuracy of the finite element method of this analysis, the FEM.
The investigation reported by Dodge covered several aspects of the
problem associated with the pneumatic tire. Dodge presented analytical
results based on a cylindrical shell model of the tire, as well as ex
perimental results for five bias ply and two radial tires. Dodge's
experimental data for the crown deflection of the two radial tires
under the effects of various ground contact loads were used in this
investigation for comparison purposes. Figures 5.2.1 and 5.2.2 show
the experimental crown deflection obtained by Dodge and the analyti
cally obtained crown deflections by the FEM. In analyzing Dodge's
tire No. 6 by the FEM, the tire was subjected to the static ground
contact loads of 700 lbs and 1000 lbs and was subdivided into 17
finite elements. The internal pressure was 28 psig. The numerical
results are given in Table 5.2.1. While analyzing Dodge's tire No. 7
by the FEM, 15 elements were used to represent the tire meridian and
the applied static loads were 580 lbs and 900 lbs. The inflation
pressure for this tire was 20 psig. The numerical results are given
in Table 5.2.2. For both of the tires, good agreement with the
1.75
93
1.5 Analytical (FEM)
o - -. Experimental Reference (56)
1.25--p = 28 psig i
c o • H +->
o CD r—I tH CD Q
1.0
/
A -4
o u u
.75-
.5
/
/
/ 0
y y
/
y
/
(3
.25,
0, 200
"~T T" 400 . 600
Load Ibf
— ,
800 . 1000
Figure 5.2.1 Crown Deflection for Dodge's Tire No. 6 Under Static Ground Contact Loads
94
1.75 ^ TT
1.50 Ana ly t i ca l (FEM)
/
/ 0
1.25 -.
•H
O •H
o 1.0 CD tH CD Q
O ft
^ .75
O — « Experimental Reference (5 6)
p = 20 psig
/
0
/
^ /
/
/
/
/
P' /
.5 -
/
O
25
0 200.
-T r 400. 600
Load Ibf
— f -
800 1000
Figure 5.2.2 Crown Deflection for Dodge's Tire No Under Static Ground Contact Loads
95
experimental data is obtained, be using the present FEM.
In Figures 5.2.3, 5.2.5, 5.2.6 and 5.2.7, the shapes of the tire
meridian which have been deformed due to the load in the footprint
region, are shown. For consistency with other published data, some
of the tire meridians are sho\ m in inverted position. That is, the
flattened footprint regions are shown at the top rather than at the
bottom of the figures.
The plot of meridian profile (for (|) = 0°, or the center of the
footprint) for the bias ply tire, as given in Figure 5.2.5, compares
very closely with that obtained by Patel (8). In this case, half of
the tire meridian was represented by 16 finite elements. For both
the FEM and Reference (8), the applied dead load was 700 lbs and the
internal pressure was 30 psig. Table 5.2.3 gives the numerical re
sults of the crown and sidewall deflection for this tire.
In all of the three tires discussed so far, constant pressure
distribution in the footprint area was assumed. Fifteen Fourier har
monics, as discussed in Chapter 3, were found to be adequate. In these
analyses, the tires were stationary (not spinning).
In addition to the tires discussed so far, a Firestone HR78-15 tire
was also analyzed by the FEM method. In analyzing this tire, three
footprint pressure distributions, as shown in Figure 5.2.4, were used
to simulate the actual pressure distribution in the footprint area.
The plot of meridian profiles (for ({) = 0°, 45° and 180°) for all three
pressure distributions for this tire under the static ground contact
load of 1000 lbs are given in Figures 5.2.5 through 5.2.7. Figure
5.2.8 shows a portion of the circumferential profile of the crown for
96
13
•H
O •H to
CD
E •H
a •H
12.»
11.-
p = 30 psig
Load = 700 lbs.
10. -
9. -
8. -
7.
0 1 r~ 2. 3.
Tire Half Width, in,
Figure 5.2.3 Meridian Profile for Patel's Tire No. 2 Under the Effect of Static Ground Contact Load
Table 5.2.1 Crown Deflection for Dodge's Tire No. 6
97
Load Ibf.
700
1000
Deflections, in.
Analytical(FEM)
-.940
-1.202
Dodge(57)
-.951
-1.244
Table 5.2.2 Crown Deflection for Dodge's Tire No. 7
Load Ibf.
580
900
.-, , , ,,
Deflections, in.
Analytical(FEM)
- 1.052
-1.549
Dodge (57)
-1.0
-1.464
Table 5.2.5 Deflections for Patel's Tire No. 2 Load = 700 Ibf, p = 30 psig
Location
Crown
Sidewall
Analytical(FEM)
-0.351
.104
Patel (8)
-0.4
-.12
98
Constant Pressure Distribution
iiji ii iiJ^ j ^ j ^ n ji J ]k >r h 4i 4i .^ ,\, > i
Ffi
i t* Contact Patch Length i h Tire Widt H
Trapezoidal Pressure D i s t r i b u t i o n
/
Ji h Ji ii ii J Ffi
L
>k i\. ii y k >*•
[-» Contact Patch Length-^ [-Tire Width"}
Cosine Pressure Distribution
T"
Fn
K Contact Patch Length-^
n T\
L_UL
Tire Width
Figure 5.2.4 Different Types of Pressure Distributions in the Footprint Area.
99
o • H to CD
E •H
a CCJ
• H TJ
14. f 13 .
12.
11
10
9. -
8. ,
0.
A • ^A..
O 9
@ <l> = 0 °
A * = 45°
<}) = 180 o
p = 30 p s ig
Load = 1000. lbs
N A
®
^A
® ^
\9 Ji \
I
A / /
/ A /
/
.. / /
A ®
4
—T
3. 4.
A ,
TT
5
Tire Half Width, in.
Figure 5.2.5 Meridian Profiles for HR78-15 Radial Tire for Constant Pressure Distribution in Footprint.
14 A - A-
100
•H
o •H to c <D
E •H Q
zi
• H
CT3
13 . -0 ^
1 2 . .
1 1 . -
a cl) = o"
^ = 45
180 o
10 . -
9 . -
8. -
0.
p = 50 p s i g
Load = 1000. l b s
A.
N \
\
\
A I \ \ I
A / / /
A /
.A /
/
/4
/
/ /
1. "T" 2.
©
5.
Tire Half Width, in
Figure 5.2.6 Meridian Profiles for HR78-15 Radial Tire for Trapezoidal Pressure Distribution in Footprint
101
14.
•H
O •H to
c (D E
•H
a cd • H
A----A--A A A-^-
13 . > \
12. -
1 1 . -
10. -
9. -
8. -
7.
9 O
0.
d d ' \
9 (}) = 0
A cf> = 45^
ct> = 180
p = 30 ps ig
Load = 1000 lbs
1. 2.
\ A
I I I
A I
A
/
A
/
A /
/
/m
3. T 4
"T 5
Tire Half Width, in
Figure 5.2.7 Meridian Profiles for HR78-15 Radial Tire for Coine Pressure Distribution in Footprint.
102
this tire. From the results obtained, it can be concluded that a flat
footprint region is generated in all three cases. Also, the resulting
meridian profiles for all three pressure distributions are rather simi
lar in shape. From Figure 5.2.8, it is evident that the asymmetric
phenomenon vanishes not far from the footprint area, and from Figures
5.1.4 and 5.2.5 through 5.2.7, it can be concluded that the meridian
profiles for the cases, ({> = 45° and 180° do not differ greatly from
the undeformed tire meridian. This tire was subdivided into 18 finite
elements. The inflation pressure was 50 psig. Fifteen Fourier har
monics were again found to be adequate and the tire was stationary.
In the analysis of the Firestone HR78-15 tires, only the analyti
cal results are presented. Comparison with experimental results was
not made because of the lack of suitably documented experimental data.
In Table 5.2.4 the load input data for all the tires studied are
presented; c}) and F were defined in Chapter 3, and the pressure dis
tributions are described in Figure 5.2.4.
5.3 Axisymmetric Deformation
For the purpose of investigating the axisymmetric deformation
pattern of the tire, loads due to inflation pressure and rotational
speeds were considered. A Firestone GR70-15 radial tire was inflated
and the deformations at the crown and sidewall were measured with a
dial guage as the pressure was varied. The tire was already mounted
and inflated with air. To begin the experiment the tire was deflated
to 0 psig and the crown and sidewall locations at this pressure vv'ere
taken as the reference points. The tire did not carry any axial load.
103
Pressure Distribution in Footprint
Undeformed Geometr
Deflection
Figure 5.2.8 Circumferential Profile of the Crown Under The Static Ground Contact Load, HR78-15 Radial Tire.
Table 5.2.4 Static Ground Contact Loading Data for the Tires
104
Tire
.Dodge's No. 6
Dodge's No. 7
Patel's No. 2
HR78-15
Loading From Element #
to Element #
— . 1 ,.,•
13 - 17
12 - 15
12 - 16
13 - 18
Static Load Ibf.
700.
1000.
580.
900.
700.
1000.
1
1000.
1000.
Pressure Distri
bution
Const.
Const.
Const.
Const.
Const.
Const.
Co sin
Trapez.
*i
13.24°
16.38°
19.30°
28.03°
12.02°
11.61
11.61
11.61
FA psi. n
28.11
28.45
20.18
21.20
28.67
30.94
48.14
32.04
105
The results of this experiment, along with the analytical results ob
tained for the Firestone HR78-15 tire, are presented in Figure 5.3.1.
Although these tires are different in size, the construction types are
identical. The deflection pattern at the crov\m is similar, with the
smaller tire showing more deflection. The deflection pattern at the
sidewall, however, shows substantial difference. The reason for not
conducting a similar experiment on the HR78-15 tire was the unavaila
bility of this tire. The numerical data of the deflections for these
two tires are given in Table 5.3.1.
Figure 5.3.2 shows the analytical results of the deflection at the
crown when the HR78-15 tire was subjected to various rotational speeds.
In this case also, no external load was applied on the tire.
The axisymmetric behavior of a bias ply tire was also analyzed.
For this purpose an experimental treadless tire, constructed and studied
by Walter (5), was selected for the analysis. The tire was a two ply
bias tire. The analysis was performed to study the effect of infla
tion pressure and inertial loading on the meridian geometry. The re
sults obtained by the FEM analysis were, however, far removed from the
actual experimental data. The reason for this discrepancy is due to
the fact that, under the effect of axisymmetric loads due to inflation
pressure and inertial loading, the deflections are primarily caused by
the membrane strains, and a bias ply tire, under such conditions, ex
hibits a highly nonlinear load-deformation behavior. The method pre
sented in this analysis which uses the initial strain conditions to
derive the elastic stiffness matrices of the tire, does not have the
capability of handling such a highly nonlinear behavior.
106
- . 0 1 4 -
- . 0 1 2 -
- . 0 1 -
•H
O •H •P O <D
I—t
tH CD
Q
o u
CJ
- . 0 0 8 -
- .006 .
- . 0 0 4 -
- . 0 0 2
.0
0
Analytical (FEM) HR78-15 /
^ 0
O Experimental GR70-15
: .12
/
- • o ^0-0--"'-0 o
"~r" 5.
"^T—
10. 15 •nr— 20
.0
25 30.
Inflation Pressure, psig
Figure 5.3.1 Axisymmetric Deflection of Two radial Tires Under the Effect of Inflation Pressure.
Table 5.3.1 Axisymmetric Deflections at the Crown and Sidewall of the Two Radial Tires Under the Effects of Internal Pressure
107
TIRE:
Experiment 1
pre.psig.
6
10
12
18
26
defl.in.
-.0034
-.0053
-.0066
-.0087
-.0106
GR70-15 TIRE: HR78-15
LOCATION: CKOm
Experiment 2
pre.psig
7
12
20
50
defl.in.
-.0041
-.0063
-.0091
-.0124
Analytical
pre.psig.
6
12
18
24
50
defl.in.
-.0019
-.0038
-.0058
-.0077
-.0096
LOCATION: SIDEWALL
pre.psig.
6
. 10
17
23
30
defl.in.
.009
.011
.014
.017
.022
pre.psig
8
11
18
25
30
defl.in
.008
.012
.015
.018
.019
pre.psig.
6
12
18
24
30
defl.in.
.011
.022
.032
.043
.054
108
•H
to
O •H •P O <D
rH tH
CD Q
o U
35. .
30. .
p = 30 ps ig
^ FEM >
25. ^
20. „
1 5 . -
10,
i > . ••
0.
0
R P M
800.
1200
1600
Centrifuga force, Ibf
IJDeflection at crown, in
511
1151.
2047
8 X 10
19 X 10
53 X 10
-3
-3
-3
1000.
Centrifugal Force Ibf
1 2000 .
Figure 5.5.2 Axisymmetric Deflection at the Crown for HR78-15 Under the Effect of Centrifugal Loads
109
5.4 Natural Frequencies and Mode Shapes
The natural frequencies and the associated mode shapes were eval
uated for the Firestone HR78-15 radial tire only. The reason for con
sidering only one tire was the lack of experimental information for
comparison purposes. Most information on the experimental frequencies
reported in the literature do not give the details of the specific mass
and stiffness data needed to conduct an analysis. The analytically ob
tained natural frequencies and the experimental natural frequencies
reported by Potts (57) are presented in Table 5.4.1. In this table, n
denotes the modal number of radial modes, as idealized in the illus
trations in Figure 5.4.1. m is a modal number for noncircumferential
modes, consisting of vibrating displacements in and perpendicular to
the meridial plane. Some of these modes are shown in Figures 5.4.2
through 5.4.4. Any single mode is specified by both a value of n and
m and consists of a superposition of the appropriate modal displace
ment patterns corresponding to these values of n and m. The placement
of the experimental values of the frequencies within their respective
position in the table was done by the author of this work after careful
study of the mode shapes given by Potts. No attempt was made to plot
these superimposed mode shapes because of their complexity, but modal
data is presented in Appendix C.
110
CO vO LO
CO LO
o CM CN
00 vO CM to
o LO
I-H LO CtJ ^ -^
.-H -P CCJ C CD O CD O
•H E C +-> - H CD >^ H ^
t—I <D Oi a fXit+H C X 0 < w oi
< CQ
Oi to
VD vO CM
O t o
LO lO
00
o LO
CM
to CM
OO t o
t o
vO LO LO
o LO
CD
ZH • H
-d
LO rH
I 00
oi
t H O
to CD
• H O
<D d cr CD fH
C^J .
d x: +->
LO
CD I-H
x zi
H
tn
LO
o o
CO LO 00
CM
t o vO
t o to
CO
vO to
to r- l
CN
CM O^
O CO
O CO
vO LO LO
.—( CN
00 LO t o
o rH CN
Oi
t o
to o CM t o
vO O to to
00
CO
CQ CQ CQ
to CN
< CQ
to LO
CM LO
LO
t o
o r H t o
LO to CN
CQ
to CD
-d o E
ctJ • H nd cd U I
o
II
E
to CD
nd o E c3
• H
^ (
II
CM t o
Ill
n = 0
n = 1 n = 2
n = 3 n = 4
\^ ^
n = 5 n = 6
Figure 5.4.1 Radial Modes of the Radial Tire
112
1 4 . - .
• H
. 1 2 . - .
p: o
•H to C <D E
•H Q
ctJ •H
1 0 . . , .
8 . H.
p = 50 p s i g n = 1, m = 1 0) = 80 Hz.
-4 - 2 . r 0 .
T i r e Width, i n .
F i g u r e 5 . 4 . 2 Mode Shape of HR78-15 Rad ia l T i r e fo r t h e Frequency of 80 Hz.
113
• H
o •H to
<D
E •H Q
cd •H
zi a:
14. -
12.
10. .
8. -
• ~ T —
- 4 .
p n CO
=
=
=
30 ps ig 1, m = 2 181 Hz
T 1 r-2. 0. 2.
T i re Width, i n .
F igure 5 .4 .5 Mode Shape of HR78-15 Radial T i re for the Frequency of 181 Hz.
114
15
1 3 . -
• H
c l l o •H to CD
E •H
a CtJ
• H 9. _
• 4 .
30 p s i g 1, m = 3 297 Hz
- T 1 - 2 . 0.
T i r e Width, i n .
T" 4
F i g u r e 5 . 4 . 4 Mode Shape of HR78-15 Radia l T i r e f o r t h e Frequency of 297 Hz.
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
6.1 General
A mathematical model for the static and dynamic analyses of radial
automobile tires was presented. The direct stiffness finite element
method using a conical shell element was employed. The displacements
were approximated in polynomial form along the meridian direction and,
along the circumferential direction, Fourier expansion was used. In
the case of static ground contact loading, the Fourier series was
trucated after fifteen terms as the displacements for the higher order
terms were of much smaller magnitude.
The tire was considered to be a thin shell of revolution with
membrane and bending stiffnesses. The material properties of the tire
were derived by considering the tire to be composed of thin layers of
composite materials linearly orthotropic in nature. In the static
analysis the resulting simultaneous linear equations were solved by
the Gauss elimination method, while in dynamic analysis the natural
frequencies and the mode shapes were evaluated by a method based
on Lanczos' n-step iteration procedure.
6.2 Conclusions
From the results discussed in the proceeding Chapter the following
can be concluded:
1. For a radial ply tire, the method has the capability of
predicting the deformed shape of the tire meridian under
115
116
the effect of static axisymmetrical and/or asymmetrical
loading conditions.
2. The displacements along the meridian under static ground
contact loading for bias ply tire were also found to be
very accurate with respect to the data available in the
literature.
3. The present method of analysis did not give accurate
results for bias ply tires when subjected to the axi
symmetrical loadings of inflation pressure and centri
fugal force.
4. The method also had the ability to evaluate the natural
frequencies and the associated mode shapes for the
radial tire.
5. The method did not require large amounts of computer
time, thus making the method more attractive for the
future use. The total computational time associated with
the axisymmetric analysis was found to be about thirty
seconds. The total computational time for the asymmetric
analysis for each of the tires studied was about four
minutes. Each natural frequency of vibration and the
mode shape were evaluated in about eight seconds. The
computational time will vary according to the number of
elements selected and the number of Fourier harmonics
used.
117
6.3 Recommendations
Following recommendations are made for the solution of the overall
tire problem:
1. A problem associated with static ground contact loading
was the assumption of the footprint area and the foot
print pressure distribution. One recommended way of
handling this problem would be to consider this to be a
displacement boundary value problem and solve for the
load. This would have to be done in several increments
until the required load is reached.
2. Attention should also be focused in the direction of
evaluating the cord forces under dynamic conditions.
3. It is recommended that the transient behavior of the
tire should also be analyzed, as it plays an important
role in the overall performance of the tire.
4. It is also recommended that attention should be given
to collecting experim.ental data because, at present,
not enough mrormation for comparison of analysis with
experiment is available.
LIST OF REFERENCES
1. Purdy, J. F., Goodyear Tire and Rubber Research Report, 1928.
2. Biderman, V. L., "Calculation of the Profile and Stresses in Elements of Pneumatic Tires Under Inflation Pressure," (Russian) Transactions, The Research Institute (Nil ShP) State Scientific and Technical Publishing House, Moscow, 1957.
3. Walter, J. D., "Centrifugal Effects in Inflated, Roteating Bias-Ply Tires," Textile Research Journal, Vol. 40, No. 1, pp. 1-7, 1970.
4. Clark, S. K., "An Analog for the Static Loading of a Pneumatic Tire Model," Office of the Research Administration, University of Michigan, Ann Arbor, Report No. 02957-19-7, 1964.
5. Clark, S. K., "An Analog for the Rolling Pneumatic Tire Under Load," Office of Research Administration, University of Michigan, Ann Arbor, Michigan, 1965.
6. Brewer, H. K., "Prediction of Tire Stresses and Deformation from Composite Theory,"Tire Science and Technology, Vol. 1, No. 1, 1975.
7. Dunn, S. E., and Zorowosky, C. F., "A Mathematical Model for the Pneumatic Tires," Report to Office of Vehicle System Research, Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, North Carolina, 1970.
8. Patel, H. P., and Zorowosky, C. F., "Mathematical Analysis of a Statically Loaded Pneumatic Tire," Ph.D. Disseration, Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, North Carolina, 1975.
9. Tielking, J. T., "Plane Vibration Characteristics of a Pneumatic Tire Model," SAE Mid-Year Meeting, Paper No. 650492, 1965.
10. Fiala, "Radial Vibrations of Belted Radial Tire," Translation, University of Michigan.
11. Bohm, "Mechanics of Belted Tire," Translation No. 5, University of Michigan, Ann Arbor, 1967.
12. Dodge, R. N., "The Dynamic Stiffness of Pneumatic Tire Model," SAE Mid-Year Meeting, Paper No. 650491, 1965
13. Turner, Clough, Martin, and Topp, "Stiffness and Deflection Analysis of Complex Structures, Journal of Aerospace Science, Vol. 23, No. 9, 1956.
118
119
14. Johnson, W. M., and McLay, R. W., "Convergence on the Finite Element Method in Theo-ry of Elas-'-icity," Transaction of ASCE, Vol. 35, 1968 ~
15. Key, S. W., "A Convergence Investigation of the Direct Stiffness Method," Ph.D. Dissertation, University of Washington, Seattle, Washington, 1966.
16. Tong, P., and Pian, T. H. H., "The Convergence of Finite Element Method in Solving Linear Elastic Problems," International Journal of Solids and Structures, Vol. 3, 1967. ~~
17. Desai, C. S., and Abel, J. F., "Introduction to the Finite Element Method," Van Nostrand-Reinhold Company, 1972.
18. Zinckiewicz, 0. C , and Cheung, Y. K., "The Finite Element Method in Structural and Continuum Mechanics," McGraw-Hill Book Company, London, 1967.
19. Argyris, J. H., "Continua and Discontinua," Proceedings, First Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Ohio, 1965.
20. Jones, R. E., and Strome, D. R., "A Survey of Analysis of Shells by the Displacement Method," Proceedings, Second Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Ohio, 1966.
21. Gallagher, R. H., "Analysis of Plate and Shell Structures," Proceedings of the Symposium on Application of Finite Element Methods in Civil Engineering, Vanderbilt University, Nashville, Tennessee, 1969.
22. Iyer, M. S., "Analysis of a Pressure Vessel Junction by the Finite Element Method," Ph.D. Dissertation, Texas Tech University, Lubbock, Texas, 1972.
23. Grafton, P. E., and Strome, D. R., "Analysis of Axisymmetrical Shell by the Direct Stiffness Method," AIAA Journal, Vol. 1, No. 4, 1963.
24. Popov, E. P., Penzien, J., and Lu, Z. A., "Finite Element Solution for Axisymmetric Shells," ASCE Engineering Mechanics Division Journal, 1964.
25. Percy, J. H., Pian, T. H., Klein, S., and Navratna, D. R., "Application of the Displacement Method to Linear Elastic Analysis of Shells of Revolution," AIAA Paper No. 65-142, 1965.
120
26. Jones, R. E., and Strome, D. R., "Direct Stiffness Method Analysis of Shells of Revolution Using Curved Elements," AIAA Journal No. 3, 1965.
27. The Nastran Theoretical Manual, Edited by MacNeal, R. H., Office of Technology Utilization, National Aeronautics and Space Administration, Washington, D. C , 1969.
28. Oliveira, E. R., de A., "Theoretical Foundations of the Finite Element Method," International Journal of Solids and Structure," Vol. 4, 1968.
29. Irons, B. M. R., and Draper, K. J., "Inadequacy of Nodal Connections in a Stiffness Solution of Plate Bending," AIAA Journal, Vol. 3, No. 5, 1965.
30. Crandall, S. H., "Engineering Analysis," McGraw-Hill Book Company, New York, 1976.
31. Ralston, A., "A First Course in Numerical Analysis," McGraw-Hill Book Company, New York.
32. Hildbrand, F. B., "Introduction to Numerical Analysis," McGraw-Hill Book Company, New York.
33. Ketter, R. L., and Prawel, S. P., Jr., "Modem Methods in Engineering Computation," McGraw-Hill Book Company, New York, 1969.
34. Bathe, K. J., "Solution Methods for Large Generalized Eigenvalue Problems in Structural Engineering," Ph.D. Dissertation, University of California, Berkeley, California, 1971.
35. Schwarz, H. R., "The Method of Coordinate Overrelaxation," Numerical Methods in Engineering, Vol. 23, 1974.
36. Wilkinson, J. H., "The Algebraic Eignevalue Problem," Clarendon Press, Oxford, 1965.
37 Vann, W. P., Vallabhan, C. V. G., and Iyer, S. M., "Eigenvalue Analysis of Large Structural Systems," Report, Texas Tech University, Lubbock, Texas.
38 Newman, M., Pipano, A., "Fast Modal Extraction in NASTRAN via the FEER Computer Program," NASA Technical Memorandum, National Aeronautics and Space Administration, Washington D. C , 1973.
39. Hestness, M. R., and Krdush, W., Journal of Research, National Bureau of Standards, Vol. 47, 1951.
121
40. Jennings, A., Orr, D. R. L., "Application of Simultaneous Iteration Method to Undamped Vibration Problems," International Journal of Numerical Methods in Engineering, Vol. 3, No. 1, 1971.
41. Dong, S. B., Wolf, J. A., and Petterson, F. E., "On a Direct It-terative Eigensolution Technique," International Journal of Numerical Methods in Engineering, Vol. 4, No. 2, 1972.
42. Guyan, R. J., "Reduction of Stiffness and Mass Matrices," AIAA Journal, Vol. 3, No. 2, 1965.
43. Oden, J. T., "Calculation of Stiffness Matrices for Finite Elements of Thin Shells of Arbitrary Shape," AIAA Journal, Vol. 6, 1968.
44. Krauss, "Theory of Thin Shell," John Wiley, International.
45. Mang, H. A., "Analysis of Doubly Corrugated Shell Structures by Finite Element Method," Ph.D. Dissertation, Texas Tech University, Lubbock, Texas, 1974.
46. Saada, A. S., "Elasticity: Theory and Application," Pergamon Unified Engineering Series, New York, 1974.
47. Sokolinkoff, I. S., "Tensor Analysis Theory and Application to Geometry and Mechanics of Continua," John Wiley and Sons, New York, 1964.
48. Walter, J. D., "Advances in Tire Composite Theory," Akron Rubber Group Series, Akron, Ohio, 1972.
49. Calcote, L. R., "The Analysis of Laminated Composite Structure," Van Nostrand-Reinhold Company, New York, 1969.
50. Ashton, J. E., Halpain, J. C , and Petit, P. H., "Primer on Composite Materials Analysis," Technomic Publishing Company, Inc., Connecticut, 1969.
51 Ghosh, S. K., and Wilson, E., "Dynamic Stress Analysis of Axisymmetric Structures Under Arbitrary Loading," Report No. EERC 69-10, University of California, Berkeley, California, 1969
52 Causey, R. L., and Gregory, R. T., "On Lanczos' Algorithm for Tridiagonalizing Matrices," Soc. Ind. and Applied Math Review, Vol 3, 1961.
122
53. Ojalvo, I. U., and Newman, M., "Vibration Modes for Large Structures by an Automatic Matrix Reduction Method," AIAA Journal Vol. 8, No. 7, 1970
54 Lanczos , C , "An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators," Jounal of Research, National Bureau of Standards, Vol. 45, 1950.
55. Potts, J. S., Newman, M., and Wang, S. L., "HABEAS - A Structural Dynamics Analysis System," Proceedings, 24th National Conference of Association of Computer Mechanics, ACM Publication, 1969.
56. Dodge, R. N., "Prediction of Pneumatic Tire Characteristics from a Cylindrical Shell Model," Technical Report No. 24, The University of Michigan, Ann Arbor, Michigan, 1966.
57. Potts, G. R., "Application of Holography to the Study of Tire Vibrations," Tire Science and Technology, Vol. 1, No. 3, 1973.
58. Personal Communication with J. R. Potts and H. P. Patel of Firestone Tire and Rubber Company, Akron, Ohio, November 1975.
APPENDIX A
The matrix B is given in Reference (27) as follows. Here, the
subscript n which denotes the harmonic dependence of the B matrix
is dropped for convinience, and the subscript y is made super-
rd script. All the terms in the 3 row of the B matrix will be
^ -Yn
zero. The non-zero elements in the second row will be.
• 2,9 = 1 ' and b^^jQ = s
The elements of the first row are.
1,1
1,2
1 Q
l^ Q
' 1 1 n 0"^ ' ' D , ^ncosY( - - >- ) - ^ o s H ' s i n ^ - ^ ( D , , + 2D „ ^)
1,2 ^ r r, 2 ir 6 ,6 2 ,2 ' a b
D ' n l E ^ ^ - i n s i n ^ c o s V - i ^ ( 3 D ' ^ + D* ) 1,2 r, 2 71 6 ,6 2 ,2
b ^ I
5 ' 02 + :rnD , -cos^*
2 6 , 6 IT
J
1,5 1 . 1 2 ^ ' ^ 03 . Q( 2 ^ ^ 6 , 6 ^ ° ^ ^ — ^ '
1,4
1 1 9 ' 13
1,5
1,6
1_ Q
1 Q
n ^ D ' . . i h - h ) - n ' s i n . ^ ( 2 D ' ^ ^ ^ . D ' 2 ^ , ) 1'2^ ^b ^a
D ' . ( ^ ^ ) - n'^sinH' - ^ ( 2D , , + D , , ) i-,^ ^b
I,
1_5 IT 6,6 2,2
n? 9 ' 2 ' + - ^ ( 2n^D ^ . + sin^VD „ ^ )
IT 6 , 6 ^, ^
b^ . i , /
l_ Q
2D l , l ^ ^ a - ^ b ^ ^ ^ l , 2 ^ r
2,2 ( ^ ) -
21 12
. < 2n D ^^^
I . s i n ^ ^ o ' ^ ^ ^ ) - n ' ^ i ^ ^ ^ ( 2 D ^ ^ ^ . D ^^^ )
123
1,8 Q 6D , , r , l - D , , ( ^ ) ^ - ^ ( 2n2D ^ ^ + 1,1 b 1,2*^ r .
sin^YD* ^ -2__._.„ 33 2 , 2 ^ - n s i n . - ^ ( 2D ^^^ . D ^^^ )
1,9 Q
1,10 Q
- - ^ - ^ ( D ' 2 , 2 ^ ^ ' 6 , 6 ) IT
" ^ °'i.2 *'''e.e 5 " ^ - ^ ^ ( D ' I , 2 * V e ^ < t /
124
Here ,
Q = I t G . r 1,1 av
1 + \ "'P'6,6 ^ ^^^'^P'2,2 ^ TT ItG, r 1,1 av
r = 7r( r + r^ ) av 2 a b " (A.l)
The integral I is given as mn *'
I = TT / s r ds , mn '
a D where r = C + C s; C = r , and C = -:J
1 ^ 1 a z 1 (A. 2)
The integral I is evaluated from the following formula.
I = TT mn
m! m + 1
jj^m-n+2
(-C )Jr " - n - j - 2
'( m - j )! j! ( m - n - j + 2 )
n b , ^ . m - n + 2 T . ^ (-CJ ) log^( r )
• ( m - n + 2 ) ! ( n - 2 ) !
j=m-n+2
(A.3)
125
The second term in the equation (A.3) replaces the term in the
series for j=m-n+2, this term should be included only if j=m-n+2.
The matrix B appearing in the equation (3.5.7) is the first Y -
row of the matrix B -yn
APPENDIX B
The non-zero elements of the B matrix are given in reference -€n ^
(27) as follows. Here, the subscript n which denotes the harmonic
dependence of the B matrix and the subscript e which denotes the
membrane and bending strains are dropped for convinience.
^,4 = 1 ' n
5,5
n 2,1 r *
ns 5,4
, ns ^2,2 = T '
sinY ^2,3 =-F- *
_ ssin'y ^2,4 ~ r *
\ . l - ^ >
^,8 = ^ '
5,1 ncosV sin7 , y — 2 F-^1,1
, cosY ^2,5 = T " '
_ scosH ^2,6 " r '
5,2
5,3
ncos^ sin^ , Y —2 F - ^ , 2
sm^ , Y b' _ r 1,3
2 m
s cos^ ^ 2,7 " r '
5,4 sm^ , Y
b- . r 1,4
_ s cos'y ^2,8 " r *
n" siiW , Y •5,5 = - ; 2 - - F - h , 5
- sin'y ^3,1 " r *
b o = 1 - -^in^ » 3,2 r
5,6 r
sinY n s sin^ Y 2 r 1,6
2 2 2ssinH' n s sinY , Y
b = — - — T - ° 5,7 r 2
r 1,7
126
2 2 3 V, _ 5s sin'y n s sin'i' , Y ° 5 , 8 T -—2 - " T ~ ^ l , 8 '
127
,9 - - 7- -T-^,9 '
ns sm'y , Y 5,10 • r " r ^1,10 '
, sinH'cos'i' n ,y
. 1 ,ssin"l'cos"t' 3C0S*, n , Y ^6 ,2 = 2 ^ — — 2 —^ * 7 h , 2 '
, _ ncos'y ^ n , Y "6,3 ^ ^ F h , 3 •
ncos^ n , Y ^i^ A - ~ n~ + — b., . ,
6,4 2r^ ^ ^ "
2ns m'i' n , Y b . _ = X— + — b , ^ , 6,5 2 r 1,5
2nssin^ 2n . n , Y ^6 ,6 = 2 T - ' r ^1 ,6 '
6,7
2 2ns sin'F 4ns . n , Y
2 " - ^ 7 ^ , 7
6,8 2ns^sin^ 6ns^ ^ n , Y
2 " T - • ' 7 ^ 1 , 8 '
sinH' n , Y ^6,9 = - T - ^ 7 ^ , 9 >
b ^ = - 1 + ^ i n ^ + - bT , ^ . 6,10 r r 1,10 (B.l)
APPENDIX C
to CD U
• H
CD
tH O
to CD •P ctJ
• H 'd f-i o o
•H -d • H J-i (D
CD
cd
8-15
HR7
Patel's Tire #2
Dodge's Tire #7
Dodge's Tire #6
Y-Cor in
.
X-Cor in.
Y-Cor in
.
X-Cor in.
Y-Cor in.
X-Cor in.
Y-Cor in
.
X-Cor in.
o
L O C M r H t ^ ^ L O C M L O LO >—• C M ' ^ C N v O L O O O O O v O C M t O O - — I L O - — ( 0 0 < — i C N C O
o t o o o t — i t ^ t o o o c M C ^ ' ^ c O ' — ( L o o r ^ r ^ o o o c o o
O O O O O O C T ^ C T l O O i — I r H C M C M t O t O t O t O t O C O t O t O I — t r H r H i — ( r — I r H i — t i — t r H r H i — ( i — t i — I r H
CM t—ILOtO ' s f t O O O 1—IvO • ^ LO t o v o r ^ o ^ t ^ o o o o o o o o o o c r > L n r ^ O L o r - « r H " < ; t " ^ ' ^ t O t - ) o o L o c r > t o c r k ' ^ r ^ L o o
C M t O t O t O • 5 ; ^ ' ^ ' * T ; t ^ ' s f t O t O C N C N t — t i - H O O O
lO cr> L O ' v f vo o c o o o o v o L O O L O o r ^ O L o o L o t ^ o c M t O L o o r ^ o o
r - . 0 0 0 0 C T i C r > O O > — l - - t - - H C M r M C M f M f M C M C N 1 — I r H t — ( r H i — ( . — I l — I r H . — I l — ( r — t r H
lO OO O O v O - ^ O O O O ' ^ t CM CM L O C ^ l \ O O O O C r > O O v O C M O v O t O O L O O L O O
C N t O t O t O ' s t t O t O t O t O t O C M C N C M ^ r H O O
1
LO CT) LO LO LO ';^ r— L O O O O t O C O C M L O O L O O C N t O L O O O I ^
r ^ t ^ O O O O O O C r . C T i O O r H i — I T — 1 . — I r H r H r H t — I r H r H i — I r H i — l i — I r H r H
LO r~ LO LO LO r^ LO " * \ O O C M L O r - - C O O O O O v O t O O O O " - O O L , 0 0
r H C M C M C M C M C M C M C N C M C M C M r H r H ' H O O
LO O O t O i — 1 1 — l O O O O C M O ' ^ r ^ O L O O L O O L O O L O o o o c M L o r - ^ c o c T i
t ^ r - > i ^ o o o o c n c r > o O r H . - H r H C M C N r s i r M C M C M r H i — l r H i — I I — I r H r H r H r H r H r - i
LO LO t o LO t O r - ^ O C N - ^ L O L O L O ' ^ r H O O L O t O O L O O L O O
C M C N t O t O t O t O t O t O t O t O C M C N C N C N - — I I — t O
r H C M t O ' ^ L O v D t - - C O O % O r H C M t O ' ^ L O O t - - C X ) C 7 » , — I r H r H r H i — I r H . — I r H r H r — 1
128
129
TABLE C . l MERIDIAN! COORD r i A T E S : H R 7 8 - 1 5
STATIC lO^D: 1"!C0 LBF
CGNST PRESSURE D I S TR I BUT K l \' I \ FCGTPRrJT AREA
4> = 0 "
N X-COR Y-COR
>
2 . 7 5 2
3 . J 0 4
3 . 5 7 3
3 . B 3 0
4 . 4 9 6
4 , ^ 2 5
14 2 . 3 2 1
1 .9 58
16 1 . 4 7 7
17 0 . 7 42
18 D . 4 9 3
\-i C^. )
8 . : 0 8
8 . 3 1 0
8 . 7 7 3
8 . 9 5 2
9 . 430
9 . 96 I
5 . 0 9 9 1 0 . 5 1 4
8 4 . 9 B8 1 1 . 19 4
4 . 7 6 8 1 1 . 6 2 2
I. 4 . 4 1 8 1.-^.023
11 3 . 9 71 1 2 . 3 1 1
12 3 . 5 0 3 1 2 , 5 0 7
13 2 . 3 49 1 2 . 6 8 5
1 2 . 7 5 1
1 2 . 7 8 7
1 2 . 8 3 8
1 2 . 8 6 3
12 .^^69
1 2 . 8 8 1
(I) = 45 "
X-CGK Y-CDR
2 . 7 5 2
3 . 0 ^ 0
3 . 5 2 8
3 . 7 59
4 . 1 5 1
2 . 9 . 4
2 . 3 5 3
1 .9 7'
1 . 4 « 3
0 . 7 47
0 . ^ 9 6
8 . 0 0 8
8 . 3 2 6
3 . 8 4 7
9 . 1 3 3
9 . 7 9 0
4 . 3 5 6 1 0 . 3 7 5
A . 4 5 7 1 C . 9 0 6
4 . 4 2 3 1 1 . 5 4 9
A . 3 4 5 1 1 . 9 8 7
4 . 1 7 0 1 2 . 4 5 4
3 . 8 8") 1 2 . 8 7 0
3 . 5 1 5 1 3 . 2 1 1
1 3 . 5 5 7
1 3 . 6 9 5
1 3 . 7 4 5
1 3 . 7 9 9
1 3 . 8 1 8
1 3 . 3 2 3
1 3 . 8 3 3
({) = 180
X-COR Y-COR
2 . 7 5 2
3 . 0 0 0
3 . 5 2 5
3 . 7 5 7
4 . 1 3 5
8 . 0 0 8
8 . 3 2 7
8 . 8 5 0
9 . 1 4 L
9 . 8 0 6
4 . 3 4 0 1 0 . 3 9 5
4 . 4 2 4 1 0 . 9 2 6
4 . 3 3 6 1 1 . 5 6 7
4.3>J6 1 2 , 0 0 4
4 . 1 H 5 1 2 . 4 7 1
3 . 8 6 3 1 2 . 8 9 J
3 . 5 0 0 1 3 . 2 3 8
2 . 9 0 2 1 3 . 6 0 3
2 . 3 5 7 1 3 . 7 6 1
1 . 9 8 4 1 3 . 8 2 4
1 . 4 9 4 1 3 . 8 9 3
. 7 5 0 1 3 . 9 2 5
0 . 4 9 8 1 3 . 9 3 3
0 . '. 1 3.9 ^o
130 --
TABLE C,3 Wr-:IDIAN COOKHINATtS: h^78-15
STATIC, L 0 A C : 1 (»0 0 L. B P
. IP.AP."Z ' ' !AAkJ: i I ST^_LBufinv TN F C U T P F I N T fl?EA
(J, = 0 ' (|) = 45 ^ = 180
...
- - - • -
N
i""" 2
_.,... 3
4
5
f,
7
8
Q
. I J -
1 1
1 2
1 3 ,
1 4
1? , '
1 6
1 7
1 p
^
X-CQR
•••2; i s 7
3 . 0 0 3
_ 3 . 5 7 2
3 . 8 3 7
4 . 4 8 6
4 , 9 > 7
_ 5<, i 7 7
4 . 9 0fl
2 - . 7 0 1
„._._^^o4.i5.„..
3 . ^ ^ ^ 1
> » — -
„. 2 , 13 9
2 . 3 1 1
7'" l- ' ^ "
1 , 4 - 9
n . 7 3 8
" " " ' t ' . ^ :^l
) 3 ')
Y-COP
8 . J :8
3 . 3 1 1
8 . 7 7 5
8 . 9 5 7
9 . ' T ' » 1
" " • o . - ; - 7 4
...AO* ""'-"
l i . 2 0 < ^
1 1 i '< ^ i L • .. ' '
... .1 « J 3 ^
1 2 . 3 2 7
1 -; C 0 C i ;! • .• r
_ 1 2 , L ; ' ^
1 2 . 7 t > 8
" "1 9 . q<;]
12c - 4 9
12 . ' " i.
1 2 . F 7 7 •'
10 .^ '^^8
X—c n R
2 . 7 5 2
3 . !''U"'
3 . 5 27
.._- ' . 7 S 8
4 , r t 5
4 . 3 5 5
4 , 4-v4
A- .4 i ' ^
^ " 4 . 3 ^2
. 4 » : 7 .
3 , 8 ' » 3
3 , 5 1 2
2 3^' >0,
2 , ^ ^ - 2
' 1 . 0 7S
l.^""'"'
: } . 7 - 7
' """'j.^y'^^
) o !
Y-CO?
8 . 0 0 8
8 a ? 2 6
P.. c 4 9
9 . 1 3 5
9 . 79 6
1 7 7 2 8 3 "
1 > . > ) } ^
1 1 . ^ ^ ^ 5
' l l . c q 3
1 2 . ^ ^ . 1
1 2 . . ^ 7 9
1 3 . 2 23 "
1 0 . ^ 7 2
1 3 . 7 n
1 3 . 7 d 5
i 3 . . 4 2
i 3 7 e ^ ^ 7
1 -^-858
X-COR
2 . 7 5 2
S^ri^o
3 . 5 2 6 _
._J.*J'o.
4 . 132
' T . ^ 2 32 ''
- .- •l 'ri
4 , 3 7 6
V,~2'^>^
4 . . 1 J 8 . .
3 . 6 58
" ' " 3 7 4 ^
2 ,R'^8
2 . 3 ^ 5
1 . 9 « 1
1,4^^2
0 , 7 A 9
) , " - 7
) -, '
_ Y - r o 5 ___
8Vooa
8 , 3 2 7
J . 85 I
9 . 1 ^ 7
9 .31 ' .1
10.40^7
7-2.^1L
1 1 . 5 7 ?
1 ' 2 . 0 0 8 '
1 2 . ^ 7 ^ _ .
1 2 . 8 9 ^
1 3 . 7 4 6
. _ 1 3 , 6 1 2 . _
1 3 , 7 7 0
1 3 . 8 3 4
1 3 , 9 ^ 4
1 3 . ^ 3 7
1 3 . 9 4 5 ' "
1 3 , 9 5 7
—
—
.— -..
. . . .
— " " • " • — — ~ — . - -
- — - • - - • - •
- • • -
151
T ^ H L F r ^^^^^l ••) IN c^":'0^n !r4iT = s • H K 7 8 - I D
C'OSP! :^^--OS'|C'E O I O ^ ^ I O M T i n o IN - ' X T P^O NT A-^ A
<}) = 0 (|) = 45 (|) = 180
\ v-C'^'< Y-CC'c X-C''R Y - r n q x-Cr*^ Y - f Q '
1 2 . 1 ^ . » IH 2 , 7 5? 0 . 00 8 2 . 7 - 2 a.nO"^
2 3 . 0 '4 0 . 3 1' ^ 3 . . ' . ^ 0 3 o 2 2 6 3 ,C0r« 8 , 3 2 7
^ 3 . S ' 5
_^7_ 3o ;-- 2
5 - . 5 " 7
7 5 . r ^ )
P, : ' • • >
8 , 9 , A
9 . ^ 1 ^ ;
- . f 0 ^ •
1 ; . 5 0^
M . ^ M -
3 . 5 2 7 g . ' ^AP
3 . 7 ^ ^ 9 . 1 3 5
4 , 1 4 7 C . 7 9 3
"ATO^ 0 1 >. 3 79
^ , ^ ^ 7 i l , ^ ' ^ 2
3 , 5 26 8 . 8 5 '
3 , 7 ^ 7 q . l ' ' - l
4 . 1 '5 5 9 . ^ . •> ^
4 , 3'^9 1 0 . 3 ^ 5
'. ,. ', >^ l f ^ . 9 2 6
4 , 3 - ^ 1 : > ^ o 7
1 1 . 0 1 A , 3 3 11 . c c :•) -t . 3 .' ^ 2,':= JO
0 , 4 ^ 1 1 ^ - M o . 1 7 o 1 3 . ^ : 5 8 1 2 , - 7
1 1 3 . 9 A \ l . l ^ 3 , 8 ^ ^ 1 2 . ^ 7 c 3 , ^ i : 4 1 2 , 8 0 - ^
\ /. , ' . 5 3 1 3 . ' > 7 9 3 , 5 1 7 1 3 . 2 1 ^ 3 . ^ 0 1 1 3 . 2 3 ^
2 , - ^ 7 i 3 . 5 0 H ^ O^ ' -^ '^^ . ^ . ^ • O " ' '
1 0 . ^ '-^ ? a O 1 3 . 7 . « * 3 ^^^ 1 3 , 7 c "
1 '-" \ '? , ?'-^o 1 ' - • ; ^ l 3 . 7 f l U 0 F 5 1 3 . > ^ 0 ^
U-
17
] 8
1 ' • ;
' ( 7 7 t;
0,:- n
! 0 J " " '-
1 -3 . '' '••''
1 % / 0 7
1 - , 0 ': .1 O ' i-
, i . 7 ' • 8
I 3 » - 1 8 l.O-^-^ 1 3 . ^ 1 9 7
1 0 . '^^ "^^ • : , H 9 S 1 3 . ^ 3
1 0 , ^ 5 3 J ,.,- 1^ , 9 5 »
152
T a b l e C.5 Modal Data fo r HR78-15 Rad ia l T i r e f o r n = 1,
and m = 1 , 2 , and 5 .
0 . 0 1 . 8 2 6 6 0 -
a . 2 4 23-D-0 , 3 215r>-0 , 4 7^ R o l l , ^ ! U n -•^ .732 0 ^ -n , 7 7 3 9")_
0>. ^ 5 2 0 0 -n , 9 R O 0 P -0 . 1 0 6 o n
._0^ X! 790 0 . 1 3 5 1 0 1^, 1483*^
U
n = 1, m = 1 Frequency = 80 Hz
U
0 2 0 1 ^ 1 0 1 (*1
. 156-5n_ 1
57
•J ^ . 1 6 5 6 n K 1 7 5 i n
D.-I76<^n n . 1 7 T 4 n 0 . 1 7 - ^ n 0 . 1752'"^ 0 . 1659^1
0 . 1 ^ 7 0 0 /\ 1 ,^.9 o n
^ . 175A^^ 0 . 1 1 8 4 0
0 . 1 ^6 4 ^ •>. c > 5 3 i n -
0 . 3 5^7 ' ^ -:^. 7 7 6 10-0 . 7 - ^ - ^ ' ' ^ -0 . 6 7 4 A ' v r», 4 7 9 9 - -^ . 3 248'3-
- . 8 ' ' 5 2 n
01
0 1
0 1 •^1 p->
JO. A M
o n
o r
OQ
0 0
0 0 ( ^ r • < • ^
r-. ,'»
o n
; • ) • )
•':n
. - 1
- < • ' 1
- )^
- 0 1 -02
.-) ^ -
J . 3 3 4 3 0 -0 . Z01PD.-• ^ • 3 2 • * 2 0 -0 . ~ 8 2 ^ - n -
.0 . 8 -24 1 0 -». 0 22 2 0 -
0 , 7 9 7 ; ) n -
^ . ^ 1 1 8 ^ -0 . - 7 1 / . n -
••'. '+17 6 0 -.) .20i6 2 Q -' ' , 1 1 1 4 0 -'>, '5''>^7 0 -
0 . 2 8 ^ 3 0 -0 . i^"^*'"^-
'). ' so in-0 . 2 6^^50-0 . 2 1 2 3 0 -
. •- J I. /
0 . ^ ^ 6 9 V - 0 . >n, . 9 0 -_ n ^ ' ^ 4 2 ' ) ^ -- - ,45>3 5!^-_ A , T 0 0 "^'^-
- 0 . 2 62'TO-- ) . ^ 0 ) ^ 10-- 0 . 0 6 ' ^ 0 0 -
- ) . 7 i ; ) R n -- ^ , 7 9 7 3 ^ -
_?> , 8 2 ' ' 2 - -
- i , 3 -5620-- " . , 58 5 ^ 0 -- f ' . : ^ 2 ? 4 ' V _{>, 7 0 5 / 1 0 -
- 0 . 3 3 7 1^-
1. ^
)2 0-1 01 01 •n »i
0 1 > L
01 )1
i l •n t..«2 -10
. •» 7
• )3 •03 -.^3
-n - 9 ^ - ) 3 -"'2 -02 -'•n -^01 -11 - -1 - M
.1 i .
- •.' L - > " <
-:n - u - 9 1
- 0 2
o.i^ - ) . 2 9 7 3 D -- 0 . . 2 0 8 8 ^ -- 0 . 3 i • ^ 9 0 --{J,6.spAl^-
-0. . .5 5 0 4 n -- n . 5 6 : ^ 1 0 -- 0 . 5 6 7 ! ) 0 -')»'=ySy'^D-
- 0 , 6 i - ^ 5 0 -- ^ ) . 7 6 ' ' 9 n -- 0 ,.9 2 93 0 -- 0 . 12 H D » 0 , 1 4 : ^ 2 0 - - 0 . 1 5 7(-n^
- 0 . 1 7 3 1 0 - U 1 8 7 6 n - 0 . 1 ^ ->60 - 0 » 19 ' U 0 - 0 . 1 9 ' ^ 3 D - ' : ^ . 1 3 7 9 0 - , ) • 17 7 5 0 - 0 . 1 5 7 5 n - o , i 4 4 0 D - ^ ' ' . 1 2 1 0 0 -O .OO-^SD-- 9 . 7h~^2 0-- 0 . 6 4'-^00-- 0 , 5 9 ' 5 n -- 0 . 5 6 ^ 7 ^ -- ' ^ . 9 6 ^ 7 0 -- ) , OA'^'sn-- 9 , 48 5 30-- ' ' , 3 1 3 10-- 0 . 2 1 ^ ' - 9 - 0 . 7 Q 9 7 n .
U )
0 2 QI -0 1
0 1 0 1 . M 0 1
\ 1
0 1 •0 1 • j l
9:)
CO VJ (-0 CO C'" Ot.' 0 0 0 0
OD 0 0
- L ' l - 0 1 - 0 1
- u - 0 1 - 0 1 - ) 1 _ r 1
- i l
- 0 1 - 0 2
• 0 . 2 1 0 2 0 -•0 . 4.L4 5 0 -
• 0 . 43 6 5 C -• 0 . 3 5 8 5 0 -:Q«_56Q-4Q-^ . 6 5 2 2 0 -0 . 1 3 7 3 9 -. ) . 2 5 2 8 0 -; ' . 3 5 4 9 n -
9 , 4 2 37 '^ -O-* .-4.5 4-10-
O . 4 t 0 5 0 -
0 1 . .Dl-_ • U
0 1
02^-P 7
0 1 >L-0 1
01
01 : ' . 3 9 2 1 D - 0 1 9 . 3 4 5 ^ 0 . - 0 1 ^ 27 2 0 0 - 0 1
0 . 1 ' - 2 7 n - D l 0 . 9 5 9 5 0 - 0 2 -•;. 1 6 9 1 0 - 3 3
- 0 . 9 7 6 8 * ' ' - 0 2 - M - 3 9 5 n - / U _ - 0 . 2 6 9 6 0 - 0 1 - 9 . 3 4 3 9 9 - 0 1 - 0 . 3 9 0 7 n - ; a - 0 . 4 4 1 1 0 - 0 1 - 0 „ 4 5 6 0 n - n
-^ .-^-2 6 2 0 - 0 1 - ^ . 3 5 7 8 n - 0 1
- j , 2 5 5 9 D - n - 0 . 140 4-0-01 - • ^ . 6 8 0 9 n - 0 2
D . 5 7 6 6 0 - 0 2 0 . 3 5 9 0 0 - 9 1
' ) . 43 8 9 9 - 0 1
0.^-1 73 f^-'^l
''^. 2 1 1 9 . 0 - 0 1
Table C.5 ( c o n t ' d ) n = 1 , m = 3
Frequency = 297 Hz.
133
U
37
a . 2 5 4 1 0 - 0 1 ! 0 . . 6 S A j n - i / i _ 0 . a 8 6 9 n - Q i
0 0
00 }0 00 00 00
• 0 . 1 2 1 5 0 •xj, 15^?D - 0 . 1652D • 0 , 1701^^ •o,X7-5_9.::> - 0 . 1 7 6 2 0 - 0 . 1 6 7 1 0
- 0 . 9 3 7 1 0 - v ) l - 0 . 3 7 5 1 0 - 0 1
r> ^770n- f^7 9 . 5 4 7 C 0 - 0 1 0 . 1 1 5 3 0 0 0
.^„0. 1^R2.0_ DO 0 . 1 3 9 2 0 0 0 0 . I 3 0 4 D DO
- .0-..X1-8-60 00 A . 6 0 4 Q 0 - 0 1 0 . 1 0 1 ^ 0 - - a
_^a..J2OiiAD-01 - 0 . 8 6 0 2 0 - 0 1 - : ) . 1 3 73f^ }u - ' » . 1 5 ^ 0 0 - 0 . 1 6 ^ 1 0 - 0 . 1 7 0 4 0 - 0 . 1 6 7 1 ' : ^
00 00 nn •\fi
09 / s /•>
00
- " M 6 4 0n - 0 . 1 5 8 30 - 0 . - 1 3 QIO - 0 . 9 6 9 ^ 0 - 0 1 - 0 . 7 6 7 2 0 - 0 1 - 1 . 2 7 7 1 0 - 0 1
U — z
0 . 0 • 0 . ^ 0 3 0 0 -- 1 . 3 . 9 8 6 Q -• 0 . 5 9 1 7 D -• 0 . 0 7 6 1 0 -. 9 . 65.810^ - 0 . 4 1 6 3 0 -• 0 . 7 4 4 2 0 -0 . 5 . 8 1 4 0 -'^'.^2 2 8 0 -0 . 4^Q60-
_D.4346Q.-0 ,7 65 80 -0 . 1 ^ 0 8 0 -
-0. -32320-) . 3 3 ^ 5 0 -
- 0 . 3 6 6 7 0 --i:?.Z43.3iO.-
02 0 1 0 1 0 1 01 0 1 01 02
• 0 . 2 9 1 00 ' - 0 . 1 3 ^ 4 0 •0»-335-7a - 0 . 3 6 9 7 0 ' • ) . 8 5 6 ^ 0 •0.144-6.0 - 0 . 2 7 1 4 0 - • ) .^A5 30 -• ) , /164 9 0 • } , 3 4 ! 7 0 - 0 . 7 7 1 1 0 0'^. 7 7 0 5 0
) . ^ ) 6 30 ^••.6 5 ^ 5 0 0 . 9 0 1 3 0 D. '^ l ' ^on r" , 4 1 S 9 0 ).7^.9 60
. ' •V ,'":
- 0 1 - 0 1 -Ql_ - 0 1 - 0 1 -iL2 - ) 7 -04 -03 -03
-.03 - 0 2 - 1 2 - 0 1 - 0 1 - 0 1 - 0 1 - ^ 1 -^^2 - 9 1 - 0 1 - 0 1 _ n
_ n i - 17
U
0 . 0 0 . 7 1 4 9 0 - J 2
._1.A3.8.2J0.-_I<JL_ 0.61340-01 0.81750-01
- . D , 7 a 2 a o - a i . _ . . 0 , -30 - ' ' 50 -01 • : ) . 3 1 9 3 0 - J 1
--Q-.8-8.64D.r01._-O . 9 9 0 0 0 - 0 1 0 . 1 0 3 0 0 0 0
__.D*J.D58D—aO— -). 7 4 7 9 0 - 0 1 0 . 7 6- ' ' -90-01
- 0 . 1 2 1 8 0 - 0 - 1 -- 0 . 6 2 ^ 3 0 - 0 1 - 0 , 1 2 4 1 0 0 0 -0 , .13-71J )_00 - 0 . 1 ^ 7 8 0 00 - 0 . 1 3 7 8 0 CO - O . 1 2 52Q-0O -- 0 . A 4 5 9 D - 0 1 - 0 . 1 3 7 7 0 - 3 1
0 . 2 4 5 3 0 - 0 1 0 . 7 4 1 8 0 - 0 1 0 . 1 0 6 1 0 DO
_0-..L08.9Q_ Q-C 0.1^^02 0 00 0 . 8 9 ^ 1 0 - 0 1
. O . 8 0 1BO-Ol._ 0 . 8 1 3 0 0 - 0 1 0 . 7 9 4 6 0 - 0 1 3 . 8 3 9 I D - 0 1 0 . 6 3 ^ ^ 4 9 - 0 1 0 . 4 5 4 4 0 - 0 1 J . - 7 4 1 6 0 - 0 2
0 . 0
. . - _ e ^ 0 . 0 . . -0 . 4 5 9 6 0 - 0 1 J . , JA5.1D-^JL 0 . 6 6 1 7 0 - 0 1 0 . 2 0 8 2 0 - 0 1
-CL._53JL3O-0L. - 0 , 6 7 2 0 0 - 0 1 - 0 . 7 0 6 0 0 - 0 1
. - 0 . J 6 . £ I J L M 1 - 0 1 _
- 0 . 4 6 3 7 0 - 0 1 - 0 . 1 3 7 6 0 - 0 1 ^0,..2^tBitD-0 1_
J . 7 4 3 7 0 - J l 0 .9 '=>770-01
. O.-lOL^J'a-OQ-C.OSO^D-Ol 0 . 6 0 5 4 0 - 0 1
-._0!.AlB£O-j..L-0 . S 0 9 3 0 - 0 3
- 0 . 4 0 4 4 0 - 0 1 -D...39.32n--0L -- 0 . 9 8 7 2 0 - 0 1 - 0 , 1 0 630 0 0 -0^ lOQCQ 0 0 - 0 . 7 5 8 2 0 - 0 1 - 0 . 2 6 3 9 0 - ' > l - 0 . 12-4^0-01 _
' 0 « 4 5 8 2 D - 0 1 0 . 6 5 6 2 0 - 0 1 0 . 7 2 1 0 0 - 0 1 „ 0 . 6 9 7A.n«01 0 . 5 6 0 1 0 - 0 1
- - . 1 , 1 9 6 9 0 - 0 1 - 0 . 6 7 8 ^ 0 - 0 1 - > . 7 7 4 4 n - 0 1 - 0 . - ^ 7 9 8 0 - 0 1 .
0 . 0
I
Table C.5 ( c o n t ' d )
n = 1, m = 2
Frequency = 181 Hz
134
U U
0 . 0 - 0 . 1 5 ' ^ 4 0 -
. — _ - O . . A 2 9 . 2 0 -^ , 1 , 5 5 9 3 0 -- >. 7 9 9 7 0 -
--0... . iC_83n . . - 0 . 1 1 7 0 0 - 0 . 1 2 2 8r> - 0 . 1 3 3 0 0 -OU 144^^r> - 9 . 1 5 3 7 0 - 0 . 1 5 9 80 - ^ • . 1 5 8 3 0 _ n , 1^ .600 - 0 . , 1 3 3 49 - 0 . 10 9 19 - i ^ , ^ • ' ^ 9 Q n -
~ < ^ - ^ 1 1 7 0 -- 0 . 2 0 8 7 0 -
9 . 3 7 1 \ n .
- 0 . 5 6 1 7 D -0 . 1 0 5<^P 0 , 1 3 r 9 n
- 0 . 1 4 4 7 D 0 . 1 5 6 89 0 . 1 5 ^ 1 ^ 0 . 1 ^ 7 9 9 ">, 1^7'^n 0 . 1 3 1 7 9
) . 1 2 ^ 0 0 0 . 1 1 ^ : 6 0 0 . l f ' 8 5 0 ) . 81 5 9 n
f , 5 7 s 3 ^ • > , 4 4 1 ' S ^ r-, 15-^'^n • "> - • ' \
0 1 0 1__ •01 01 0 0 -0 ) 0 0 0 ''.V-
• • ' ^
OO .0 0 no 0 9 '' .'r -o n
- 9 1
- 0 1 - 0 2 - 0"' - 0 . 1 -
Of) 9 0 r-0
) 0
0 n '1 ^ i\ t'^
^Cj
• V )
" 0 . • * r ^
- 0 1 -•^'1 - 9 1 ~ '> 1
O.Q - 0 . 9 6 7 8 0 - 0 2 .- 0- ..3 2 ^ 70 -^21- -- 0 . 50 ' ^^>0-01 - ) . 8 3 1 8 0 - 0 1
- ._-0 , .-830aD--OL - " ^ , 7 3 3 3 0 - 0 1 - 0 . 6'^5 5 D - 0 1 -•" . -333 .9 .D-01 - 9 . 1 1 4 5 0 - . ; 2
0 , ? 6 6 4 0 - 0 1 • ' ^ , 4 5 5 i n - 0 X •^'. ~66 3 0 - ' ' U ;'. , 5 6 ^ - 6 0 - " 1 U ' ^ 5 I 9 0 - - U l
0 , 5 2 ^ 6 0 - 0 1 , , 4 9 4 8 0 - ? > l 0 , --8 3 7 0 - 9 1
.. 0 . ^ - 7 7 7 ^ - 0 1 , , 4 8 0 8 0 - )1
0 . ^ 9 4 ^ 5 0 - 9 1 A , r 7 Q 3 r ) > . - » ]
r . - S I / i n - 0 1 ' ' ^ . ^ 6 ' ^ 2 D - M 1 ) . S 7 i 7 0 - ' l
'^ . ^^60 6 ^ - 0 1 0 , 2 7 7 1 0 - 0 1 0 . ^-r^.^- 7^'-'"' '-
- • ' ' , 3 27 4 9 - 0 1
- U '"^^^ . ^ n - J l - 0 , ^ 2 5 6 ^ - 0 1 -'•• . '* ^ 6 0 9 - 0 1 - 0 . 8 3 7 . 5 0 - 0 1 - 0 . 9 1 1 « 0 - ' U - ; ^ ^ " - 7 4 l O - 0 1 - ' " ' . ^ "72 1 0 - " ' 2
, -^ . -
J 2
0 1 0 1 0 1 0 1 . — J 1 . . - .__.
V 1
01 0 1
. . • " >
. n .
0 1--, ' • ' ' • • ;
0 0 • • ' : )
oc ) 1
l - j l . - 0 2
) - . > 1
.' w
0 0 /'• r,
•.. 0
CO CO
•9 1 • 0 1 • n - . • 0 1 • 0 1 -ol -01 - n -02
J
e
0 . 0 ._-9 , 3 6 1 6 D - U 0.-64^880^-01.-) . 62 8 5 0 - 0 1
' ^ . ^ 5 9 1 0 - 0 1 - 9 . 2 7 0 20 . -01 - 9 . ^ 6 4 j 0 - n . -- 0 . 5 6 0 8 0 - 0 1 - 3 . 67-6 .5^-01 . - 0 . 7 1 1 2 0-0 1 - 0 . 6 4 1 9 0 - 0 1 - . ) . A 7 9 A D - 0 - l -- 0 . l ' - -740-r ; l
0. l59fP-^n ^ ' ^ . - ^ 6 1 3 0 - 0 1
0 . 5 ^ 7 1 ^ - 0 1 : ) . 8 5 4 8 9 - 0 1 ^*',°0 6 3 9 - 0 1 -0 . 9 5 0 4 0 - n , » . 9 1 2 8 0 - 0 1
-__a..8 6 3 5 0 - 9 l -0 . 6 1 2 ) 0 - 0 1 '">.^7 78 0- ' . ' l 9.017.8'^ 0 - 0 1
- 1 . 1 7 ^ . - 0 - 9 1 - 9 . 4 7 1 2 0 - 0 1 - 0 , 6 3 6 8 0 - n - 0 . 7 0 99 '^ ' -01 - 0 . 67939- '">l
- ) . 5 6 7 3 0 - n -•">. 4""2 3 0 - 0 1 - G . 2894^^-01
r\ , 3 5 •-5 0 - 9 1 9 . 6 3 2 3 0 - 0 1 9 . f 5 5 2 0 - 0 1 0 . 3 6 6 7 : ^ - 0 1 0 , 0
^ 1
APPENDIX D
The steps involved in incorporating zero transverse shear strains
are as follows.
1. In the element stiffness matrix k (equation 3.6.7), the fifth
and tenth rows and columns should be deleted.
2. In the transformation matrix A (equation 3.5.7), the fifth
and tenth rows and the ninth and tenth coluirmis are delted. Also A = A , —n —n
3. In the stiffness matrix k^ (equation 3.6.7) the ninth and
tenth rows and columns are ommited. Also kO (equation 3.6.7) should
be equal to zero, and the matrix B is set equal to zero.
4. In the element mass matrix m (equations.7.10) the ninth -ct •
and tenth rows and columns should be deleted.
a 5. While applying the boundary conditions the rotations 6 and
0^ should be zero rn
135
