VIBRATION OF HINGED CIRCULAR ARCHES

102 Pages·2014·3.71 MB·English

RICE UNIVERSITY
VIBRATION OF HINGED CIRCULAR ARCHES
by
Shyr-jen Wung
A THESIS SUBMITTED
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
Thesis Director's Signature
Houston, Texas
May, 1967
ABSTRACT
The governing differential equations for the vibration of
uniform circular arches have been derived considering in addition
to the usual actions the effect of rotatory inertia and shear de
formation. A trial eigenvalue method is used for determining the
natural frequencies and mode shapes. The Runge-Kutta fifth order
integration technique is used in this method to perform the inte
gration of the differential equations.
A detailed study has been made of the lowest eight vibra
tion frequencies for a hinged arch with an angle of opening equal
to ninety degrees. The effect of slenderness ratio and the in
fluences of the rotatory inertia and shear deformation on frequencies
are presented in detail in tables and curves. It is shown that the
frequencies found neglecting shear and rotatory inertia can be con
sidered in most regions to be either primarily extensional or flex
ural but that a coupling exists in certain transition regions. The
effects of shear deformation and rotatory inertia, which are most
important in and near the transition regions, are to smooth the
frequency curves.
i
CONTENTS
ABSTRACT i
LIST OF TABLES iv
LIST OF FIGURES v
I. INTRODUCTION 1
1.1 Object and Scope 1
1.2 Acknowledgement 2
1.3 Nomenclature 3
II. DERIVATION OF FUNDAMENTAL EQUATIONS 5
2.1 Equations of Equilibrium 5
2.2 Stress Resultants 6
2.3 D'Alembert Reversed Inertia Forces 10
2.4 Governing Equations and Boundary Conditions,Set I 12
2.5 Governing Equations and Boundary Conditions,
Set II 14
2.6 Governing Equations and Boundary Conditions,
Set III 17
2.7 Governing Equations and Boundary Conditions,
Set IV 18
2.8 Energy Formulations 20
III. REVIEW OF THE LITERATURE 24
IV. NUMERICAL METHOD 34
V. PRESENTATION OF RESULTS 44
5.1 Solutions of Classical Theory 44
5.2 Approximate Formulas 44
5.3 Effects of Rotatory Inertia and Shear 51
5.4 Summary 51
ii
-----------------------------~
CONTENTS
REFERENCES 52
TABLES 1 - 5 54
Figures 1 - 20 60
APPENDIX I. FLOW CHART AND FORTRAN PROGRAM 79
iii
TABLES
1. Convergence of the Numerical Method
2. Summary of Solutions of Classical Theory
3. Summary of Solutions For Theory Including Rotatory Inertia
But Neglecting Shear Deformation
4. Summary of Solutions For Theory Including Rotatory Inertia
and Shear Deformation with r = 0.3
5. Summary of Solutions For Theory Including Rotatory Inertia
and Shear Deformation with r = o.l
FIGURES
1. Coordinate and Displacement Notations
2. Stress Resultants and Loads
3. Shear Deformation
4. Typical Plots of Determinant vs. Frequency
5. Natural Frequencies of Symmetrical Vibrations of Hinged-Ended
Uniform Circular Arches (Classical Theory)
6. Natural Frequencies of Anti-Symmetrical Vibrations of Hinged-
Ended Uniform Circular Arches (Classical Theory)
7. Typical Anti-Symmetrical Flexural Modes
B. Typical Symmetrical Flexural Modes
9. Modes on the Plateaus with the Same Height
10. First Extensional Modes of Anti-Symmetrical Vibrations
11. Second Extensional Modes of Anti-Symmetrical Vibrations
iv
12. First Extensional Modes of Symmetrical Vibrations (Breathing
Modes)
13. Second Extensional Modes of Symmetrical Vibrations
14. Change of Mode Shapes in Transition Region
15. Comparison of Approximate Solutions and Exact Solutions For
Anti-Symmetrical Vibrations (Classical Theory)
16. Comparison of Approximate Solutions and Exact Solutions For
Symmetrical Vibrations (Classical Theory)
17. Effect of Shear Deformation and Rotatory Inertia for Symmetri-
cal Vibrations (First, second, and third Mode)
18. Effect of Shear Deformation and Rotatory Inertia for Symmetri-
cal Vibrations (Fourth and Fifth mode)
19. Effect of Shear Deformation and Rotatory Inertia For Anti-
Symmetrical Vibrations (First, second, and third Mode)
20. Effect of Shear Deformation and Rotatory Inertia For Anti
Symmetrical Vibrations (Fourth and Fifth Mode)
v
CHAPTER I.
INTRODUCTION
1.1. Object and Scope
The object of this study is to obtain information on the
free vibration characteristics of circular arches. Despite the fact
that the theory has been available for many years thre is a paucity
of information on the natural frequencies and modes of vibration of
arches, due primarily to the labor required to extract numerical in
formation from the theory.
This thesis contains a derivation of the governing differ
ential equations, the development of a numerical procedure for the
solution of these equations, and a detailed study of the lowest
eight vibration frequencies for a circular hinged arch with an in
cluded angle of 90 degrees.
The governing differential equations consider in addition to
the usual actions the effects of rotatory inertia and shear deforma
tion. When rotatory inertia and shear deformation are neglected the
equations are consistent with Flugge's theory of cylindrical shells.
The equations are derived by both the customary displacement method
and also by an energy method. The equations are presented in such a
wayfuat it is easy to either consider or neglect rotatory inertia and
shearing deformations independently.
A trial eigenvalue method is used for solving the differential
equations and determining the frequencies. The Runge-Kutta fifth
order integration technique is used in the method to perform the re
quired integrations. The method has been programmed for an IBM 7040
2
computer. All solutions were obtained on this computer.
A detailed study has been made of the lowest eight vibra
tion frequencies for a hinged arch with an angle of opening of
ninety degrees. The effect of slenderness ratio and the influences
of rotatory inertia and shear deformation on the frequencies are
presented in detail in tables and curves. It is shown that the
frequencies found neglecting shear and rotatory inertia can be
considered in most regions to be either primarily extensional or
primarily flexural, but that a coupling exists in certain transi
tion regions. The effects of rotatory inertia and shear deforma
tion seem to smooth the frequency curves. These effects are most
important in and near the transition regions. Their effect is
negligible when the slenderness ration is very large. As has been
reported previously, the effects of shear deformation far exceed the
effects of rotatory inertia.
1.2. Acknowledgements
I am grateful to Professor W. J. Austin for his help and
s.
guidance of this work. I wish to thank Professor A. Veletsos
for his help and guidance. The valuable technical assistance of
Professor W. P. Vann, Professor C. Oran and Professor E. C. Holt
is gratefully acknowledged.
3
1.3. Nomenclature
Latin Symbols.
A cross section area of arch
a radius of centroidal axis of arch
potential energy due to bending and stretching,
see Eq. (39.a)
potential energy due to shear, see Eq. (39.b)
translatory kinetic energy, see Kq. (39.c)
rotatory kinetic energy, see Eq. (39.d)
determinant, see Eqs. (66) or (72)
E Young's modulus
2 2
F non-dimensional frequency parameter equal to mp a /EA
G shearing modulus
g integrand of the Lagrangian, see Eq. (40)
I moment of inertia of cross section of arch
2
r
k non-dimensional dimension parameter, equal to 2
a
L length of arch
Lagrangian of the dynamic system, see Eq. (40)
bending moment, see Fig. 2
m mass per unit length
N normal force, see Fig. 2
p angular frequency
translatory inertia force per unit length in circum-
p~
ferential direction, see Eq. (18)
translatory inertia force per unit length in radial
direction, see Eq. (19)
4
r radius of gyration of cross section of arch
R rotatory inertia index, equals to unity if rotatory
inertia is included, equals to zero if not
s shear index; equals to unity if shear deformation is
included; equals to zero if not
T rotatory inertia force, see Eq. (17)
t time variable
u shear variable, equal to aS
v circumferential displacement
w radial displacement
z radial coordinate
Greek S~bols
a angle of opening of arch, see Fig. 1
s
shear deformation, see Fig. 1 and Eq. (7.a)
r non-dimensional shear parameter, equal toEAG:
y mass per unit volume
extensional strain
€
A constant of proportionality, equal to Q/G A ~
normal stress on cross-section
0
T shear stress on cross-section
angular coordinate, see Fig. 1
~
rotation of the tangent to the centroidal axis, see
xl
Fig. 3
rotation of cross-section, see Fig. 3