The Rayleigh-Ritz Method for Structural Analysis E-Book

Sinniah Ilanko

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 207

Veröffentlichungsjahr: 2014

Beschreibung

A presentation of the theory behind the Rayleigh-Ritz (R-R) method, as well as a discussion of the choice of admissible functions and the use of penalty methods, including recent developments such as using negative inertia and bi-penalty terms. While presenting the mathematical basis of the R-R method, the authors also give simple explanations and analogies to make it easier to understand. Examples include calculation of natural frequencies and critical loads of structures and structural components, such as beams, plates, shells and solids. MATLAB codes for some common problems are also supplied.

Leseprobe

Contents

Preface

Introduction and Historical Notes

1. Principle of Conservation of Energy and Rayleigh’S Principle

1.1. A simple pendulum

1.2. A spring-mass system

1.3. A two degree of freedom system

2. Rayleigh’s Principle and Its Implications

2.1. Rayleigh’s principle

2.2. Proof

2.3. Example: a simply supported beam

2.4. Admissible functions: examples

3. The Rayleigh–Ritz Method and Simple Applications

3.1. The Rayleigh–Ritz method

3.2. Application of the Rayleigh–Ritz method

4. Lagrangian Multiplier Method

4.1. Handling constraints

4.2. Application to vibration of a constrained cantilever

5. Courant’s Penalty Method Including Negative Stiffness and Mass Terms

5.1. Background

5.2. Penalty method for vibration analysis

5.3. Penalty method with negative stiffness

5.4. Inertial penalty and eigenpenalty methods

5.5. The bipenalty method

6. Some Useful Mathematical Derivations and Applications

6.1. Derivation of stiffness and mass matrix terms

6.2. Frequently used potential and kinetic energy terms

6.3. Rigid body connected to a beam

6.4. Finding the critical loads of a beam

7. The Theorem of Separation and Asymptotic Modeling Theorems

7.1. Rayleigh’s theorem of separation and the basis of the Ritz method

7.2. Proof of convergence in asymptotic modeling

7.3. Applicability of theorems (1) and (2) for continuous systems

8. Admissible Functions

8.1. Choosing the best functions

8.2. Strategy for choosing the functions

8.3. Admissible functions for an Euler–Bernoulli beam

8.4. Proof of convergence

9. Natural Frequencies and Modes of Beams

9.1. Introduction

9.2. Theoretical derivations of the eigenvalue problems

9.3. Derivation of the eigenvalue problem for beams

9.4. Building the stiffness, mass matrices and penalty matrices

9.5. Modes of vibration

9.6. Results

9.7. Modes of vibration

10. Natural Frequencies and Modes of Plates of Rectangular Planform

10.1. Introduction

10.2. Theoretical derivations of the eigenvalue problems

10.3. Derivation of the eigenvalue problem for plates containing classical constraints along its edges

10.4. Modes of vibration

10.5. Results

11. Natural Frequencies and Modes of Shallow Shells of Rectangular Planform

11.1. Theoretical derivations of the eigenvalue problems

11.2. Frequency parameters of constrained shallow shells

11.3. Results and discussion

12. Natural Frequencies and Modes of Three-Dimensional Bodies

12.1. Theoretical derivations of the eigenvalue problems

12.2. Results

13. Vibration of Axially Loaded Beams and Geometric Stiffness

13.1. Introduction

13.2. The potential energy due to a static axial force in a vibrating beam

13.3. Determination of natural frequencies

13.4. Natural frequencies and critical loads of an Euler–Bernoulli beam

13.5. The point of no return: zero natural frequency

14. The Rrm In Finite Elements Method

14.1. Discretization of structures

14.2. Theoretical basis

14.3. Essential conditions at the boundaries and nodes

14.4. Derivation of interpolation functions (shape functions)

14.5. Derivation of element matrix equations using the Rayleigh–Ritz method

14.6. Assembly of element matrices

14.7. Eigenvalue problems: geometric stiffness matrix for calculating critical loads

14.8. Eigenvalue problems: vibration analysis

14.9. Consistent mass matrix for a beam element

14.10. Lumped mass matrix for a beam element

14.11. The Rayleigh–Ritz and the Galerkin methods

Bibliography

Appendix

Index

To my late parents Saraswathyppillai and Sinniah, my late brother Senthinathan who encouraged and supported me during my studies at Manchester, my brothers Kumarabharathy, Kathirgamanathan, sister Sooriyakumari, my wife Krshnanandi, daughters Kavitha and Tehnuka, my in-laws, nephews, nieces, my supervisors Emeritus Professor Dickinson, the late Dr Tillman, my teachers from my old schools in Sri Lanka (Veemankamam Mahavithiyalaym, Mahajana College, Tellippalai), and all my lecturers and students, and colleagues both current and past.

Sinniah Ilanko

To my wife and son.Luis Monterrubio

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

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John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

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© ISTE Ltd 2014The rights of Sinniah Ilanko and Luis E. Monterrubio to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014953191

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-638-9

Preface

It is a privilege to have the opportunity to share with you some of our interesting experience in the journey of learning and research about structural analysis. The specific path we are exploring here is a superhighway – a variational technique called the Rayleigh–Ritz method or the Ritz method. I was introduced to this method during my PhD studies by an excellent supervisor, Professor Stuart Dickinson at the University of Western Ontario. Prior to this, another excellent supervisor (my BSc and MSc supervisor), the late Dr Stuart Tillman (University of Manchester), had taught me the Lagrangian multiplier method. Dr Tillman’s lectures were such that one has only to listen once – from then on the material stays crystal clear. So the reason I became interested in these techniques (other than the fact that they are very handy for vibration analysis which was my research area) is perhaps the passion that my teachers showed in the subject.

The thought of writing a book in vibration occurred to me after my first study leave in India where I received some positive feedback after giving a public lecture on “Vibration and Stability of Structures” that was meant to be for a general audience. I tried hard to think of ways of explaining concepts such as natural frequency, stiffness, mass, damping where possible using everyday experiences and analogies. While such analogies may not be accurate, they help to create an image and this technique has since helped me to score some points with my students. For example, I explain natural frequencies as the frequencies at which a structure is easily excitable and give as an example the frequency at which one should meet with a potential partner or friend to sustain the relationship. If we meet too frequently, we may not give our friends enough space and scare them; if do not meet often enough, we may wrongly signal that we are not interested in them. So to get the maximum response, we need to engage at the right frequency. The same goes with structures. I have always wanted to share such thoughts through a book.

The opportunity to write a book has finally come through an invitation from Professor Noël Challamel. The book is about the Rayleigh–Ritz method but as you will see, for historical reasons and for its common potential use, the focus is largely on natural frequencies and modes and the related problem of structural stability. I have tried to think of simple analogies to present this in an interesting way but have only managed to do this in one or two places. The book is a mixture of well-established material (both theory and application) and the result of our own research. An accidental mistake in the sign of an inertia term in an equation in my PhD thesis has led to the discovery that negative values of large magnitude for masses and stiffness can be used in modeling constraints. Dr Luis Monterrubio has utilized this idea for solving a number of different structural problems, and has come up with a nice set of admissible functions to use in the analysis of beams, plates, shells and solids in Cartesian coordinates and has contributed to Chapters 8–12. Dr Yusuke Mochida has helped with proofreading and checking the derivations. I am grateful to Luis, Yusuke, Professor Noël Challamel and ISTE for making this possible.

I must also record my thanks to my family, teachers, students and colleagues for my role as an author would not have been possible without them.

Some of our explanations, for example the use of analogies, may not be based on principles of science, or may be in subject areas such as management in which we do not have any expertize. While we have tried to eliminate errors, our attempts to continuously improve the manuscript with new examples and explanations may have led to some mistakes. We would be grateful to receive any comments, criticisms or suggestions for improvements regarding the contents of this book.

Sinniah ILANKOOctober 2014

Introduction and Historical Notes

In many practical engineering problems, it is not possible or convenient to develop exact solutions. A convenient method for solving such problems originated from attempts to calculate natural frequencies and modes of structures. This method is known as the Rayleigh–Ritz method or the Ritz method [RAY 45a, RAY 45b, RIT 08, RIT 09, LEI 05, ILA 09, YOU 50]. In this book we will see how to apply this method for solving a variety of common problems engineers and scientists encounter. We will first provide some historical notes on the development of the method and show how the principle of conservation of energy leads to this procedure. Those who are keen to get on with the application may want to proceed to Chapter 3.

Chapter 1 starts with application of the principle of conservation of energy for a simple pendulum showing how the natural frequency can be found by applying this principle for a system that can vibrate only in one mode or shape. Such systems that can only vibrate in one mode are called single degree of freedom systems, as a single coordinate is sufficient to describe the actual shape of natural vibration and such a natural vibration without any external dynamic force takes place only at one frequency which is its natural frequency. Then we consider a spring-mass vibratory system which has two independent coordinates. It can be easily shown, by applying Newton’s second law of motion, that the system has two natural frequencies with associated modes. Application of the conservation of energy for this system requires an assumption about the shape of vibration. Although the natural frequencies and modes can be calculated conveniently by solving the equations of motion derived from Newton’s second law, application of the principle of conservation energy shows that in this case this the application leads to one value for the frequency which depends on the assumed mode (the assumed ratio of the displacement of two masses) and takes minimum and maximum values when the assumed modes correspond to the actual first and second modes, yielding the respective natural frequencies. In Chapter 2, we proceed to show how this illustrates Rayleigh’s Principle which in Lord Rayleigh’s own words is stated as follows:

The period of a conservative system vibrating in a constrained type about a position of stable equilibrium is stationary in value when the type is normal.

Chapter 2 also presents a well-known proof that for a system with a finite number of degrees of freedom, the frequency obtained by applying the principle of conservation of energy is an upper bound to the fundamental natural frequency, and a lower bound to the highest natural frequency, provided no essential (geometric) constraints are violated. The requirement that the essential conditions are not violated leads to the notion of admissible forms (these could be vector or functions) of displacements. This chapter also deals with what is meant by admissibility.

Lord Rayleigh has shown in several of his papers and books, how a good estimate for the fundamental natural frequency may be obtained by adjusting the shape of a chosen function to seek the lowest possible values for the frequency (or highest value for the period). The expression for the frequency is a quotient with potential energy being the numerator and a kinetic energy function being the denominator. This quotient is called the Rayleigh quotient. However, the credit for introducing a systematic method for performing this minimization should be given to Walter Ritz. For this reason there are some who argue that the method should be called the Ritz method. The arguments and counterarguments for the name are available in literature [LEI 05, ILA 09]. So we will not focus on it here except to say that to be inclusive we are using the name Rayleigh–Ritz method which gives credit to both Rayleigh and Ritz.

Thus, Chapter 3 takes us from the implication of Rayleigh’s principle to the Rayleigh–Ritz method. Typical minimization equations are formulated for a conservative structural system possessing potential and kinetic energy, to the point of developing the eigenvalue equations. A cantilever beam is used as an illustrative example showing how the method is applied to obtain the natural frequencies and modes. The effect of adding partial restraints and rigid body attachments are also explained in this chapter. In addition to natural frequency calculations, static analysis is also demonstrated as a special case. It may be worth noting here that while the origin of the Rayleigh–Ritz method can be traced back to problems of finding natural frequencies and modes, it can also be used to solve boundary value problems. In structural analysis, this corresponds to calculation of displacements using the minimum total potential energy theorem, but the procedure for minimization is the same as the one used in the Rayleigh–Ritz method for vibration analysis.

We have noted that a requirement of the Rayleigh–Ritz method is that the choice of displacement functions for formulating the energy terms is subject to the requirement that they satisfy all geometric conditions. In actual fact, it is not necessary for each function to satisfy the constraints but the series as a whole does need to. A way to relax this requirement is to use the Lagrangian Multiplier method where each function is allowed to violate the geometric constraints but then these constraints are enforced by including additional constraint equations which are associated with undetermined coefficients called the Lagrangian Multipliers. Chapter 4 deals with this approach and demonstrates the method through a propped cantilever.

Chapter 5 presents some mathematical derivations and formulas for computing the terms in the eigenvalue matrix equations. For example, integral expressions for stiffness and mass matrices are presented in Chapter 5.

Chapter 6 introduces the penalty method. While the Lagrangian Multiplier method helps to relax the limitations on the choice of admissible displacement functions or vectors, it introduces extra equations that need to be solved together with a set of minimization equations. There is another clever way to achieve the enforcement of geometric conditions without increasing the number of equations. This involves a gemoetric constraint with an artificial spring of very high stiffness and including the strain energy associated with any violation of the constraint. This idea was introduced by Richard Courant in [COU 43] and has since then become very popular and widely accepted. This is known as the penalty method. The penalty parameter corresponds to the stiffness of the artificial spring and serves as a penalty against any constraint violation. There have been two criticisms about this approach. One is that while high stiffness may minimize any constraint violation, it is not possible to determine the effect it has on the accuracy of the results. Furthermore, choosing a stiffness that is large enough to prevent any constraint violation, but not too large as to cause any numerical problems due to round-off errors, can be challenging. In the case of frequency calculations, the approximation of a rigid boundary condition with a less than ideally rigid condition relaxes the structure and could result in lower estimates for the natural frequencies. This means that the Rayleigh–Ritz method would then yield an upper bound solution to a lower bound model. However, recent advances in the penalty method where stiffness parameters of positive and negative values had been used were found to give bounded results for frequencies, as far as the constraint violation is concerned. This is explained in Chapter 6.

Although it is well known that the Rayleigh–Ritz method gives upper bound to the fundamental natural frequencies, it is not well known that the method actually gives upper bound to all but the highest of the natural frequencies. The proof of boundedness of the Rayleigh–Ritz method, which comes in the form of Theorem of Separation, is presented in Chapter 7. This chapter also gives rigorous mathematical proof of theorems that justify the use of negative stiffness parameters, which were derived by one of the authors.

The fact that the admissible functions do not have to satisfy all geometric constraints is great news for the Rayleigh–Ritz method fans because this removes the restrictions on the choice of admissible functions. However, with the use of penalty terms, some functions are known to cause numerical problems. So, are there any well-behaved functions? This question is answered in Chapter 8, where a recipe for formulating shape functions can be found, presenting a specific set for many common structural elements including beams, rectangular plates, shells of rectangular planform and solids – all are shapes which can be described conveniently in Cartesian coordinates. It is a convenient set consisting of a cosine series, and linear and quadratic functions. These functions are easy to work with and have shown to be well-behaved even under challenging conditions such as high penalty terms and higher modes where most common admissible functions cause numerical problems.

Chapters 9 deals with application of the special set of functions for beams, which is then extended to rectangular plates in Chapter 10, shells of rectangular planform in Chapter 11 and solid bodies in Chapter 12. In all these cases, it has been shown that the set of admissible functions presented in Chapter 7 can be used to find the natural frequencies and modes of the totally unconstrained system, without causing any numerical problems. This is observed consistently for beams, plates, shells and solids, irrespective of the number of terms used or the number of natural frequencies and modes obtained. For any other set of boundary conditions, the penalty method has been employed to obtain the natural frequencies and modes.