RICE UNIVERSITY FREE VIBRATION OF CIRCULAR ARCHES Carlos Antonio Lopes Pereira A ...

86 Pages·2014·3.78 MB·English

RICE UNIVERSITY
FREE VIBRATION OF CIRCULAR ARCHES
by
Carlos Antonio Lopes Pereira
A THESIS SUBMITTED
IN PARTIAL FULFILLMENT OF THE
REQUIREMENT FOR THE DEGREE OF
MASTER OF SCIENCE
Thesis Director's Signature
CLudu,
uMtpu
Houston, Texas
September, 1968
1
ABSTRACT
FREE VIBRATION OF CIRCULAR ARCHES
by
Carlos Antonio Lopes Pereira
The lower natural frequencies and the associated modes of vibration of
uniform circular arches which are either fixed or simply supported at the ends
are computed and studied in this thesis.
A general theory which considers the effects of rotatory inertia and
shearing deformations as well as extensional and flexural deformations was
used. When both rotatory inertia and shearing deformations are neglected the
general theory reverts to a well-known classical theory which is based upon the
same assumptions as Flugge's theory for the bending of cylindrical shells. The
governing differential equations were solved numerically by a Holzer-type
iterative procedure combined with an initial value integration method. A Runge-
Kutta integration technique was used. The method was programmed for an IBM
7040 computer, and all results were obtained on this computer.
A detailed study has been made of the lowest ten vibration frequencies and
modes for fixed arches for a wide range of slenderness ratios and for angles of
opening of 45° , 90° and 180°, using the classical theory. The effects of rotatory
inertia and shearing deformations have been investigated only for arches with an
angle of opening of 90°.
Simple approximate expressions for estimating the natural frequencies of
fixed-ended arches (classical theory) are developed and their accuracy evaluated
11
by comparing them with the results obtained by exact analysis.
In addition, exact solutions of classical theory were made for hinged
circular arches with angles of opening of 45° and 180° and the accuracy of
approximate expressions for estimating the natural frequencies of hinged arches
proposed by Austin, Veletsos and Wung were evaluated for these angles of open¬
ing.
Ill
CONTENTS
ABSTRACT
LIST OF TABLES
LIST OF FIGURES
I. INTRODUCTION
,1.1 Object and Scope
1. 2 Review of Literature
1.3 Acknowledgement
1. 4 Nomenclature
II. GOVERNING EQUATIONS AND NUMERICAL SOLUTION
2. 1 Fundamental Equations
2. 2 Numerical Procedure
III. PRESENTATION OF RESULTS FOR FIXED ARCH
3.1 Vibrational Behavior of Arch by Classical Theory
3. 2 Approximate Formulas
3. 3 Effects of Rotatory Inertia and Shearing Deformation
IV. PRESENTATION OF RESULTS FOR HINGED ARCH
4. 1 Additional Solutions and Comparison with Approximate Formulas
REFERENCES
TABLES 1 -15
FIGURES 1 -40
iv
TABLES
1. Convergence of the Numerical Method
2. Summary of Solutions of Classical Theory, Fixed-ended Uniform
Circular Arches 0-9O°
o
3. Summary of Solutions of Classical Theory, Fixed-ended Uniform
Circular Arches 0 = 45°
O
4. Summary of Solutions of Classical Theory, Fixed-ended Uniform
Circular Arches 0 =l8O°
o
5. Summary of Solutions of Classical Theory, Hinged-ended Uniform
Circular Arches 00=45°
6. Summary of Solutions of Classical Theory, Hinged-ended Uniform
Circular Arches §,= 180°
7. Summary of Solutions for Theory Including Rotatory Inertia but
Neglecting Shearing Deformation, Fixed-ended Uniform Circular
Arches 0<>=9OO
8. Summary of Solutions for Theory Including Rotatory Inertia and Shear
Deformation, Fixed-ended Uniform Circular Arches 0 = 90°, P=°H
O
9. Summary of Percent Energy, Fixed-ended Uniform Circular Arches
©o = 90°
10. Summary of Percent Energy, Classical Theory, Fixed-ended Uniform
Circular Arches 0 =45°
O
11. Summary of Percent Energy, Classical Theory, Fixed-ended Uniform
Circular Arches 0=l8O°
o
12. Summary of Percent Energy, Classical Theory, Hinged-ended Uniform
Circular Arches 0<,= 45°
13. Summary of Percent Energy, Classical Theory, Hinged-ended Uniform
Circular Arches © =180°
o
14. Summary of Solutions for Theory Including Rotatory Inertia and Shear
Deformation, Fixed-ended Uniform Circular Arches 0,= 9O°, P= 0.3
t
15. Summary of Percent Energy, Theory Including Rotatory Inertia and Shear
Deformation, Fixed-ended Uniform Circular Arches 0 =9O°j 0.3 .
o
V
FIGURES
1. Coordinate and Displacement Notations
2. Typical Plots of Determinant vs. Frequency - Fixed Ended Arches
3. Natural Frequencies for Fixed Ended Arches - Classical Theory
4. Predominantly Flexural Modes of Vibration - Fixed Ended Arches
5. Frequencies Associated with Predominantly Extensional Modes of
Vibration - Fixed Ended Arches
6. a, b Predominantly Extensional Antisymmetrical Modes of Vibration -
Fixed Ended Arches
7. a, b Predominantly Extensional Symmetrical Modes of Vibration - Fixed
Ended Arches
8. Modes Corresponding to Frequencies of Nearly Equal Values - Fixed
Ended Arches
9. a, b Antisymmetrical Modes Corresponding to Frequencies of Nearly
Equal Values - Fixed Ended Arches
10. a,b Symmetrical Modes Corresponding to Frequencies of Nearly Equal
Values - Fixed Ended Arches
11. Comparison of Approximate and Exact Solutions - Fixed Ended Arches
with 0 = 45°
O
12. Comparison of Approximate and Exact Solutions - Fixed Ended Arches
with 0-9O°
o
13. Comparison of Approximate and Exact Solutions - Fixed Ended Arches
with 9=l80°
o
14. Effects of Shear Deformation and Rotatory Inertia for Antisymmetric
Modes - Fixed Ended Arches
15. Effects of Shear Deformation and Rotatory Inertia for Symmetric Modes
Fixed Ended Arches
VI
16. a, bEffects of Shear Deformation and Rotatory Inertia for Antisymmetric
Modes - Fixed Ended Arches
17. a, bEffects of Shear Deformation and Rotatory Inertia for Symmetric Modes -
Fixed Ended Arches
18. Frequencies Associated with Predominantly Extensional Modes -
Hinged Ended Arches
19. a, bPredominantly Extensional Antisymmetrical Modes of Vibration - Hinged
Ended Arches
20. a, b Predominantly Extensional Symmetrical Modes of Vibration - Hinged
Ended Arches
21. Comparison of Approximate and Exact Solutions - Hinged Ended with
9 = 45°
0
22. Comparison of Approximate and Exact Solutions - Hinged Ended with
Q = 180°
0
4
CHAPTER I.
INTRODUCTION
1.1. Object and Scope
This study is concerned with the computation of the natural frequencies
and the associated modes of vibration of uniform circular arches which are
either fixed or simply supported at the ends.
Although the theory of vibrating arches is well established, only a limited
number of numerical solutions are available in the literature. The most compre¬
hensive numerical study of the natural frequencies of circular arches appears to
have been the one conducted recently by Wung (1) at Rice University. This study
was concerned with hinged arches and involved the evaluation of the first eight
natural frequencies of vibration using both the classical theory and a more
general theory which considers the effects of shearing deformations and rotatory
inertia. On the basis of these data and by application of Rayleigh's method,
simple approximate expressions were proposed in Ref. (2) for estimating the
frequency values obtained from the classical theory. Wung’s numerical study
was limited to an arch with an angle of opening of 90°, and the accuracy of the
approximate expressions proposed in Ref. (2) could be evaluated only for this
particular arch.
The object of the present study is to extend the previous investigation
along the following lines:
1. To investigate the response of fixed ended arches using both the
2
classical and the more general theory,
2. To develop simple approximate expressions for estimating these
frequencies, and
3. To obtain additional data for simply-supported arches and to check
the accuracy of the approximate expressions proposed over a wider
range of parameters than considered before.
The nth circular frequency of an arch, p , may conveniently be expressed
in the form
(1)
In this expression S denotes the curved length of the arch axis, El the
flexural rigidity of its cross section, m the mass per unit of length, and C a
n
dimensionless coefficient.
When the effects of shearing deformations are neglected, C is a function
n
of the angle of opening of the arch, 0 , and its slenderness ratio, S/r, where r
O
denotes the radius of gyration of the arch cross section.
In Wung's study the first eight natural frequencies of hinged-ended arches
with 0 = 90° were evaluated for a range of S/r from 0 to 360. In the present
O
study the corresponding frequencies were also evaluated for hinged arches with
0 = 450 and 0 = 180° using the classical theory.
O O
A detailed study has been made of the lowest ten vibration frequencies and
modes for a fixed arch with S/r varying from 0 to 500, and for angles of opening
of 45°, 90° and 180°, using the classical theory. The effects of rotatory in¬
ertia and shearing deformation have been investigated only for arches with0 = 90°.
o
3
These solutions were obtained by applications of the numerical procedure and the
computer program used in Wung's study. This computer program was extended
to make possible the computation of the proportions of extensional and flexural
energies associated with the various modes of vibration. This additional infor¬
mation proved very helpful in the interpretation of the numerical data.
A major part of this investigation has been concerned with the develop¬
ment of simple approximate expressions for estimating the natural frequencies
of fixed-ended arches. In this effort the ordinary theory was used. These
equations are presented in the text and their accuracy is evaluated by comparing
them with the results obtained by exact analysis. In addition, the accuracy of
the approximate expressions proposed in Ref. (2) for hinged-ended arches is re¬
evaluated in the light of the additional data that have been obtained, and certain
modifications are proposed.
1.2. Review of Literature
A brief resume of two papers which are concerned with the vibrations of
fixed-ended circular arches is presented below. A more complete review is
presented by Wung (1).
Den Hartog (3) derived approximate formulas for the lowest extensional
and inextensional modes for hinged and fixed-ended circular arches, by using
the Rayleigh-Ritz method and by neglecting the rotatory inertia and shearing
deformation effects. For the first antisymmetrical inextensional mode for
fixed-ended arches, the following series were used,