TRANSVERSE VIBRATIONS

OF BELLO\VS EXPANSION JOINTS

by

VAIDUTIS FELIKSAS JAKUBAUSKAS, B.Sc., M.Sc., M. Eng.

A Thesis

Submitted to the School of Graduate Studies

In Partial Fulfillment of the Requirements

For the Degree

Doctor of Philosophy

McMaster University

(c) Copyright by Vaidutis Feliksas Jakubauskas, June 1995

TRANSVERSE VIBRATIONS

OF BELLO\VS EXPANSION JOINTS

DOCTOR OF PHILOSOPHY (1995) (Mechanical Engineering)

TITLE:

McMaster University Hamilton. Ontario

Investigations of Transverse Vibrations of Bellows Expansion Joints

AUTHOR: Vaidutis Feliksas Jak"Ubauskas,

B.Sc. (Kaunas Politechnical Institute, Lithuania)

M.Sc. (Odessa Civil Engineering Institute, Ukraine)

M.Eng. ( McMaster University, Canada)

SUPERVISOR: Professor D.S. Weaver

NUMBER OF PAGES: xv, 245

II

ABSTRACl

Bellows expansion joints are used in piping systtms to absorb significant axial

and/or transverse motions. Unfortunately, their flexibility also makes them susceptible to

vibration. This thesis presents a detailed analysis of the transverse vibrations of single and

double bellows expansion joints, including the effects of internal fluid.

A differential equation of motion is developed which treats transverse bellows

vibrations including the effects of fluid added mass, rotary inertia and internal pressure.

The added mass is determined from potential flow theory and provided in the form of a

mode dependent added mass coefficient. The equation of motion is solved for the first four

transverse modes and comparison with experiments shows excellent agreement. The

neglect of rotary inertia and the effect of convolution distortion on fluid added mass in the

EJMA Standard makes the latter's preC:ictions for natural frequency significantly higher

than those measured , especially for transverse modes above the fundamental.

The equation of motion is also solved approximately to provide an analytical

expression for transverse natural frequencies. The results are presenteC: in a form which

makes hand calcuiations possible for the first four modes of single and double bellows

expansion joints. Experiments in still fluid as well as flow-induced motion show excellent

agreement with predicted frequencies.

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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my supervisor Dr. D.S. Weaver for

suggesting the problem. His constant encouragement and advice, often extending beyond

the s~ope of this work, are greatly appreciated.

I am grateful to the late Dr. F.A. Mirza for the use of his computer software.

I also am grateful to Dr. M.A. Dokainish for making it possible to use his

computing facility.

I am thankful for the assistance offered by Mr. L.Baksys in the preparation of

drawings and the final copy of the manuscript.

I acknowledge the financial support of the Natural Sciences and Engineering Re-

search Council of Canada.

Lastly, I would like to thank my parents for their unending support and encourage-

ment.

IV

TABLE OF CONTENTS

Page CHAPTERl INTRODUCTION 1

CHAPTER2 REVIEW OF INVESTIGATIONS 7 OF TRANSVERSE VIBRATIONS IN CORRUGATED PIPE EXPANSION JOINTS

2.1 General Information about Flow- 7 Induced Vibrations

2.2 The Survey of Flow Induced Vibrations 10 of Corrugated Pipe Expansion Joints

2.3 The Survey oflnvestigations 15 of Axial Stiffness of Bellows

2.4 Investigations of Axial 19 Vibrations in Bellows

2.5 The Survey of Investigations of Transverse Vibrations 29 of Corrugated Pipe Expansion Joints

CHAPTER3 STIFFNESS AND INERTIAL PROPERTIES 35 IN CASE OF TRANSVERSE VIBRATION OF BELLOWS

3.1 The Bending Stiffhess of Bellows 35 3.2 Bellows as a Shell of Revolution 37 3.3 Bellows as a Solid of Revolution 42 3.4 Peculiarities and Comparison of 47

Results of Calculation of Axial Stiffness of Bellows 3.5 Bellows Mass 50 3.6 Additional Fluid Mass of Bellows 51 3.7 FE Formulation of Laplace Equation 58

in 30 Domain with Mixed Boundary Conditions 3.o Added Fluid Mass, i. 70 3.9 Second Moment oflnertia of 75

Cross-section of Bellows 3.10 Mass Moment of Inertia of Connecting 78

Pipe of Universal Expansion Joint

CHAPTER4 ASSUMPTIONS AND DIFFERENTIAL EQUATION 81 OF TRANSVERSE VIBRATION OF BELLOWS

4.1 The Modes ofNatural Transverse 81 4.2 The Influence of Shear and Inertia 84

of Rotation of Cross-section on Vibrating Bellows

v

4.3 The Influence of Coriolis Forces 89 on Natural Vibrations of Bellows

4.4 The Influence oflnside Pressure and the Centrifugal 97 Force of the Flow on Transverse Vibration of Bellows

4.5 Assumptions and Derivation of Differential Equation of 104 Transverse Vibration of Bellows Using Newtonian Approach

CHAPTERS THEORETICAL INVESTIGATION OF NATURAL 107 TRANSVERSE VIBRATIONS OF SINGLE BELLO\VS EXPANSION JOINT

5.1 Solution of Differential Equation 107 5.2 Single Bellows Type Expansion Joint I I 1

Natural Frequencies 5.3 The Exact Solution of Single Bellows Expansion 114

Joint Natural Frequencies and its Comparison with Rayleigh Quotient Solution

5.4 Instability condition for Single 1 I 7 Bellows expansion Jo:nt

CHAPTER6 THEORETICAL INVESTIGATION OF I 19 NATURAL VIBRATIONS OF UNIVERSAL EXPANSION JOINT LATERAL MODE

6.1 Derivation of Boundary Conditions for Vibration I 19 of Universal Expansion Joint in Lateral Mode

6.2 Solution of Differential equation 122 6.3 Solution of Bernoulli-Euler eq1iation 125 6.4 General Expression for Universal 129

Expansion Joint Lateral Modes Natural Frequencies 6.5 First Lateral Mode Natural Frequency 130

of Universal Expansion Joint without Lateral Supports 6.6 Second and Third Lateral Modes Natural Frequencies 136

of Universal Expansion Joint without Lateral Supports 6.7 First Lateral Mode Natural Frequency 140

of Universal Expansion Joint with Lateral Supports 6.8 The Exact Solution of Universal Expansion Joint 145

Lateral Mode Natural Frequency and Its Comparison with the Rayleigh Quotient Solution

6.9 Instability Condition for Universal 150 Expansion Joint Lateral Mode

CHAPTER 7 THEORETICAL INVESTIGATION OF I 5 I NATURAL VIBRATIONS OF UNIVERSAL EXPANSION JOINT ROCKING MODE

7.1 Derivation of Boundary Conditions for Vibration 151 of Universal Expansion Joint Rocking Mode

VI

7.2 Derivation of Differential Equation 155 and Boundary Conditions Using Hamilton's Principle

7.3 Solution of Differential Equation 160 7.4 Solution of Bernoulli-Euler Equation 163 7.5 General Expression for Universal Expansion Joint 168

Rocking Modes Natural Frequencies 7.6 First Rocking Mode Natural Frequency 169

of Universal Expansion Joint without Lateral Supports 7.7 Second and Third Rocking Mode Natural Frequency 177

of Universal Expansion Joint without Lateral Supports 7.8 First Rocking Mode Natural Frequency 180

of Universal Epansion Joint with Lateral Supports 7.9 The Exact Solution of Universal Expansion Joint 186

Rocking Mode Natural Frequency and Its Comparison with Rayleigh Quotient Solution

7.10 Instability Condition for Universal 191 Expansion Joint Rocking Mode

CHAPTERS EXPERil\IENTAL INVESTIGATION OF NATURAL 193 TRANSVERSE VIBRATIONS OF BELLO\VS

8.1 Apparatus for Investigation of Natural Transverse 193 Vibrations of Fixed-Fixed Bellows

8.2 The Method and Results of Experimental Investigation 197 of Natural Vibrations of Bellows Expansion Joint

CHAPTER9 EXPERIMENTAL INVESTIGATION OF FLOW- 216 INDUCED TRANSVERSE VIBRATIONS OF BELLOWS EXPANSION JOINTS

9.1 General Information about Water/Wind Tunnels 216 9.2 Water Tunnel Design 220 9.3 Experimental Results 222

CHAPTER 10 CONCLUSIONS 236

REFERENCES 240

Vll

LIST OF FIGURES

Figure Page

2.1 The Geometry of bellows 11

2.2 Various types of expansion joints 12

2.3 The kinds of bellows vibration modes 13

2.4 Cr for convoluted bellows, EJMA Std. (1980) 17

2.5 Finite element discretisation of bellows and fluid domains 24

2.6 Bellows added mass coeffi~ient and bellows 26 frequency f (Hz) as functions of mode number for five convolution bellows, Jakubauskas and Weaver ( 1992)

2.7 First six axisym netric modes of five 27 convolution bellows, Jakubauskas and Weaver (1992)

2.8 Installation P.xamplt:s of bellows 31

3.1 Two node 6 DOF axisymmetric element 38

3.2 Half-convolution domain for shell element 40

3.3 8-noded parent and isoparametric elements 41

3.4 Half-convolution domain for solid element 43

3.5 Distribution of added mass, m12 (x), over the 52 length of fixed-fixed bellows (first mode)

3.6 Initiai and deformed shapes of bellows convolution 53

3.7 Division of convolution by planes of symmetry 60

3.8 The solution domain with boundary conditions 62

3.9 20-noded parent and isoparametric elements 65

3.10 Spatial view of3D mesh of 1/8 convolution space 69

3.11 Added mass,, versus h!Rm 75

3.12 Rocking mode of universal expansion joint 79

4.1 Universal expansion joint as elastic system 82

vm

4.2 Lateral modes: a) first, b) second 82

4.3 Rocking modes: a) first, b) second 83

4.4 Distribution of Coriolis for simply supported 94 ends - 1, and fixed-fixed ends - 2

4.5 Differential element of bellows 97

5.1 Mathematical model for single bellows 10