Bu Clink

8/12/2019 Bu Clink

1/710

EIGENVALUES OFINHOMOGENEOUS

STRUCTURESUnusual

Closed-Form Solutions

2005 by Issac Elishakoff

8/12/2019 Bu Clink

2/710

CRC PRESS

Boca Raton London New York Washington, D.C.

EIGENVALUES OF

INHOMOGENEOUSSTRUCTURES

Isaac ElishakoffJ. M. Rubin Foundation Distinguished Professor

Florida Atlantic UniversityBoca Raton, Florida

Unusual

Closed-Form Solutions


8/12/2019 Bu Clink

3/710

Library of Congress Cataloging-in-Publication Data

Elishakoff, Isaac.Eigenvalues of inhomogenous structures : unusual closed-form solutions /

Isaac Elishakoff.p. cm.Includes bibliographical references and index.ISBN 0-8493-2892-6 (alk. paper)1. Structural dynamicsMathematical models. 2. Buckling

(Mechanics)Mathematical models. 3. Eigenvalues. I. Title.

TA654.E495 2004624.176015118dc222004051927

This book contains information obtained from authentic and highly regarded sources. Reprintedmaterial is quoted with permission, and sources are indicated. A wide variety of referencesare listed. Reasonable efforts have been made to publish reliable data and information, but theauthor and the publisher cannot assume responsibility for the validity of all materials or for theconsequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means,electronic or mechanical, including photocopying, microfilming, and recording, orby anyinformation storage or retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press does not extend to copying for general distribution, for promotion,for creating new works, or for resale. Specific permission must be obtained in writing from CRCPress for such copying.

Direct all inquiries to CRC Press, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks,

and are used only for identification and explanation, without intent to infringe.


No claim to original U.S. Government worksInternational Standard Book Number 0-8493-2892-6

Library of Congress Card Number 2004051927Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper


Visit the CRC Press Web site atwww.crcpress.com
http://www.crcpress.com/http://www.crcpress.com/

8/12/2019 Bu Clink

4/710

DedicatedtoIFMA,InstituteFranaisdeMcaniqueAvance,Francewhosesuperbly educated engineers have been most instrumental in helping to bringthis book to fruition; without their dedication it would take many more yearsto achieve this humble goal of communicating to the engineers, scientists andstudents the infinite number of closed-form solutions.

Isaac Elishakoff


8/12/2019 Bu Clink

5/710

OtherBooksfromProfessorIsaacElishakoff

TextbookI. Elishakoff,Probabilistic Methods in the Theory of Structures, Wiley-Interscience, New York, 1983(secondedition:DoverPublications,Mineola,NewYork,1999).

Monographs

Y.Ben-HaimandI.Elishakoff,ConvexModelsofUncertaintyinAppliedMechanics,ElsevierSciencePublishers,Amsterdam,1990.G. Cederbaum, I. Elishakoff, J. Aboudi and L. Librescu, Random Vibrations and Reliability ofComposite Structures, Technomic Publishers, Lancaster, PA, 1992.

I. Elishakoff, Y. K. Lin and L. P. Zhu, Probabilistic and Convex Modeling of Acoustically ExcitedStructures,ElsevierSciencePublishers,Amsterdam,1994.I.Elishakoff,TheCouragetoChallenge,1stBooksLibrary,Bloomington,IN,2000.I. Elishakoff, Y. W. Li, and J. H. Starnes,Jr.,Nonclassical Problems in the Theory of Elastic Stability,Cambridge University Press, 2001.I. Elishakoff and Y. J. Ren, Large Variation Finite Element Method for Stochastic Problems, OxfordUniversity Press, 2003.I. Elishakoff, Safety Factors and Reliability :Friends orFoes?, Kluwer Academic Publishers,Dordrecht, 2004.

Edited VolumesI. Elishakoff and H. Lyon, (eds.), Random Vibration - Status and Recent Developments, ElsevierScience Publishers, Amsterdam, 1986.I. Elishakoff and H. Irretier, (eds.),Refined Dynamical Theories of Beams, Plates and Shells, and Their

Applications,SpringerVerlag,Berlin,1987.I.Elishakoff,J.Arbocz,Ch.D.Babcock,Jr.andA.Libai,(eds.),BucklingofStructures-TheoryandExperiment,ElsevierSciencePublishers,Amsterdam,1988.S. T. Ariaratnam, G. Schuller and I. Elishakoff, (eds.), Stochastic Structural Dynamics-Progress inTheoryandApplications,ElsevierAppliedSciencePublishers,London,1988.C. Mei, H. F. Wolfe and I. Elishakoff, (eds.), Vibration and Behavior of Composite Structures, ASMEPress, New York, 1989.

F. Casciati, I. Elishakoff and J. B. Roberts, (eds.), Nonlinear Structural Systems under RandomConditions,ElsevierSciencePublishers,Amsterdam,1990.D. Hui and I. Elishakoff, (eds.),Impact and Buckling of Structures, ASME Press, New York, 1990.A. K. Noor, I. Elishakoff and G. Hulbert, (eds.), Symbolic Computations and Their Impact onMechanics, ASME Press, New York, 1990.Y. K. Lin and I. Elishakoff, (eds.), Stochastic Structural Dynamics-New Theoretical Developments,Springer Verlag, Berlin, 1991.I.ElishakoffandY.K.Lin,(eds.),StochasticStructuralDynamics-NewApplications,Springer,Berlin,1991.I. Elishakoff (ed.),Whys and Hows in Uncertainty Modeling, Springer Verlag, Vienna, 2001.A. P. Seyranian and I. Elishakoff (eds.), Modern Problems of Structural Stability, Springer Verlag,

Vienna, 2002.


8/12/2019 Bu Clink

6/710

Contents

Foreword xv

Prologue 1

Chapter 1 Introduction: Review of Direct, Semi-inverse and InverseEigenvalue Problems 7

1.1 Introductory Remarks 71.2 Vibration of Uniform Homogeneous Beams 81.3 Buckling of Uniform Homogeneous Columns 101.4 Some Exact Solutions for the Vibration of

Non-uniform Beams 191.4.1 The Governing Differential Equation 21

1.5 Exact Solution for Buckling of Non-uniformColumns 24

1.6 Other Direct Methods (FDM, FEM, DQM) 281.7 Eisenbergers Exact Finite Element Method 301.8 Semi-inverse or Semi-direct Methods 351.9 Inverse Eigenvalue Problems 431.10 Connection to the Work by Zyczkowski and Gajewski 501.11 Connection to Functionally Graded Materials 521.12 Scope of the Present Monograph 53

Chapter 2 Unusual Closed-Form Solutions in Column Buckling 552.1 New Closed-Form Solutions for Buckling of a

Variable Flexural Rigidity Column 552.1.1 Introductory Remarks 552.1.2 Formulation of the Problem 562.1.3 Uncovered Closed-Form Solutions 572.1.4 Concluding Remarks 65

2.2 Inverse Buckling Problem for InhomogeneousColumns 65

2.2.1 Introductory Remarks 652.2.2 Formulation of the Problem 652.2.3 Column Pinned at Both Ends 662.2.4 Column Clamped at Both Ends 682.2.5 Column Clamped at One End and Pinned at

the Other 692.2.6 Concluding Remarks 70

vii

http://2892prologue.pdf/http://2892prologue.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch1.pdf/http://2892ch2.pdf/http://2892ch1.pdf/http://2892prologue.pdf/

8/12/2019 Bu Clink

7/710

viii Eigenvalues of Inhomogeneous Structures

2.3 Closed-Form Solution for the Generalized EulerProblem 74

2.3.1 Introductory Remarks 74

2.3.2 Formulation of the Problem 762.3.3 Column Clamped at Both Ends 79

2.3.4 Column Pinned at One End and Clamped atthe Other 79

2.3.5 Column Clamped at One End and Free at theOther 81

2.3.6 Concluding Remarks 83

2.4 Some Closed-Form Solutions for the Buckling of

Inhomogeneous Columns under DistributedVariable Loading 84


2.4.2 Basic Equations 87

2.4.3 Column Pinned at Both Ends 92

2.4.4 Column Clamped at Both Ends 97

2.4.5 Column that is Pinned at One End andClamped at the Other 100


Chapter 3 Unusual Closed-Form Solutions for Rod Vibrations 107

3.1 Reconstructing the Axial Rigidity of a LongitudinallyVibrating Rod by its Fundamental Mode Shape 107


3.1.2 Formulation of the Problem 108

3.1.3 Inhomogeneous Rods with Uniform Density 109

3.1.4 Inhomogeneous Rods with Linearly VaryingDensity 112

3.1.5 Inhomogeneous Rods with ParabolicallyVarying Inertial Coefficient 114

3.1.6 Rod with General Variation of InertialCoefficient (m >2) 115


3.2 The Natural Frequency of an Inhomogeneous Rod

may be Independent of Nodal Parameters 1203.2.1 Introductory Remarks 120

3.2.2 The Nodal Parameters 121

3.2.3 Mode with One Node: Constant InertialCoefficient 124

3.2.4 Mode with Two Nodes: Constant Density 127

3.2.5 Mode with One Node: Linearly VaryingMaterial Coefficient 129

3.3 Concluding Remarks 131

http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch3.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch2.pdf/http://2892ch3.pdf/

8/12/2019 Bu Clink

8/710

Contents ix

Chapter 4 Unusual Closed-Form Solutions for Beam Vibrations 1354.1 Apparently First Closed-Form Solutions for

Frequencies of Deterministically and/orStochastically Inhomogeneous Beams(PinnedPinned Boundary Conditions) 1354.1.1 Introductory Remarks 1354.1.2 Formulation of the Problem 1364.1.3 Boundary Conditions 1374.1.4 Expansion of the Differential Equation 1384.1.5 Compatibility Conditions 1394.1.6 Specified Inertial Coefficient Function 1404.1.7 Specified Flexural Rigidity Function 1414.1.8 Stochastic Analysis 1444.1.9 Nature of Imposed Restrictions 1514.1.10 Concluding Remarks 151

4.2 Apparently First Closed-Form Solutions forInhomogeneous Beams (Other Boundary Conditions) 1524.2.1 Introductory Remarks 1524.2.2 Formulation of the Problem 1534.2.3 Cantilever Beam 1544.2.4 Beam that is Clamped at Both Ends 1634.2.5 Beam Clamped at One End and Pinned at the

Other 1654.2.6 Random Beams with Deterministic Frequencies 168

4.3 Inhomogeneous Beams that may Possess aPrescribed Polynomial Second Mode 1754.3.1 Introductory Remarks 1754.3.2 Basic Equation 1804.3.3 A Beam with Constant Mass Density 1824.3.4 A Beam with Linearly Varying Mass Density 1854.3.5 A Beam with Parabolically Varying Mass

Density 1904.4 Concluding Remarks 199

Chapter 5 Beams and Columns with Higher-Order PolynomialEigenfunctions 203

5.1 Family of Analytical Polynomial Solutions forPinned Inhomogeneous Beams. Part 1: Buckling 2035.1.1 Introductory Remarks 2035.1.2 Choosing a Pre-selected Mode Shape 2045.1.3 Buckling of the Inhomogeneous Column

under an Axial Load 2055.1.4 Buckling of Columns under an Axially

Distributed Load 209


http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch4.pdf/http://2892ch5.pdf/http://2892ch4.pdf/

8/12/2019 Bu Clink

9/710

x Eigenvalues of Inhomogeneous Structures

5.2 Family of Analytical Polynomial Solutions forPinned Inhomogeneous Beams. Part 2: Vibration 2255.2.1 Introductory Comments 2255.2.2 Formulation of the Problem 2265.2.3 Basic Equations 2275.2.4 Constant Inertial Coefficient(m = 0) 2285.2.5 Linearly Varying Inertial Coefficient(m = 1) 2305.2.6 Parabolically Varying Inertial Coefficient

(m = 2) 2315.2.7 Cubic Inertial Coefficient(m = 3) 2365.2.8 Particular Casem = 4 2395.2.9 Concluding Remarks 242

Chapter 6 Influence of Boundary Conditions on Eigenvalues 2496.1 The Remarkable Nature of Effect of Boundary

Conditions on Closed-Form Solutions for VibratingInhomogeneous BernoulliEuler Beams 2496.1.1 Introductory Remarks 2496.1.2 Construction of Postulated Mode Shapes 2506.1.3 Formulation of the Problem 2516.1.4 Closed-Form Solutions for the ClampedFree

Beam 2526.1.5 Closed-Form Solutions for the

PinnedClamped Beam 2716.1.6 Closed-Form Solutions for the

ClampedClamped Beam 2896.1.7 Concluding Remarks 308

Chapter 7 Boundary Conditions Involving Guided Ends 309

7.1 Closed-Form Solutions for the Natural Frequency forInhomogeneous Beams with One Guided Supportand One Pinned Support 3097.1.1 Introductory Remarks 3097.1.2 Formulation of the Problem 3107.1.3 Boundary Conditions 3107.1.4 Solution of the Differential Equation 3117.1.5 The Degree of the Material Density is Less

than Five 312

7.1.6 General Case: Compatibility Conditions 3187.1.7 Concluding Comments 322

7.2 Closed-Form Solutions for the Natural Frequency forInhomogeneous Beams with One Guided Supportand One Clamped Support 3227.2.1 Introductory Remarks 3227.2.2 Formulation of the Problem 3237.2.3 Boundary Conditions 323

7.2.4 Solution of the Differential Equation 324

http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch6.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch5.pdf/http://2892ch7.pdf/http://2892ch6.pdf/

8/12/2019 Bu Clink

10/710

Contents xi

7.2.5 Cases of Uniform and Linear Densities 3257.2.6 General Case: Compatibility Condition 3277.2.7 Concluding Remarks 329

7.3 Class of Analytical Closed-Form PolynomialSolutions for GuidedPinned Inhomogeneous Beams 3307.3.1 Introductory Remarks 3307.3.2 Formulation of the Problem 3307.3.3 Constant Inertial Coefficient(m = 0) 3327.3.4 Linearly Varying Inertial Coefficient(m = 1) 3337.3.5 Parabolically Varying Inertial Coefficient

(m = 2) 3357.3.6 Cubically Varying Inertial Coefficient(m = 3) 3377.3.7 Coefficient Represented by a Quartic

Polynomial(m = 4) 3387.3.8 General Case 3407.3.9 Particular Cases Characterized by the

Inequalityn m + 2 3497.3.10 Concluding Remarks 364

7.4 Class of Analytical Closed-Form PolynomialSolutions for ClampedGuided InhomogeneousBeams 3647.4.1 Introductory Remarks 3647.4.2 Formulation of the Problem 3647.4.3 General Case 3667.4.4 Constant Inertial Coefficient(m = 0) 3767.4.5 Linearly Varying Inertial Coefficient(m = 1) 3777.4.6 Parabolically Varying Inertial Coefficient

(m = 2) 3787.4.7 Cubically Varying Inertial Coefficient(m = 3) 3807.4.8 Inertial Coefficient Represented as a

Quadratic (m = 4) 3857.4.9 Concluding Remarks 392

Chapter 8 Vibration of Beams in the Presence of an Axial Load 3958.1 ClosedForm Solutions for Inhomogeneous

Vibrating Beams under Axially Distributed Loading 395

8.1.1 Introductory Comments 3958.1.2 Basic Equations 3978.1.3 Column that is Clamped at One End and Free

at the Other 3988.1.4 Column that is Pinned at its Ends 4028.1.5 Column that is clamped at its ends 4078.1.6 Column that is Pinned at One End and

Clamped at the Other 411


http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch7.pdf/http://2892ch8.pdf/

8/12/2019 Bu Clink

11/710

xii Eigenvalues of Inhomogeneous Structures

8.2 A Fifth-Order Polynomial that Serves as both theBuckling and Vibration Modes of an InhomogeneousStructure 4178.2.1 Introductory Comments 4178.2.2 Formulation of the Problem 4198.2.3 Basic Equations 4218.2.4 Closed-Form Solution for the Pinned Beam 4228.2.5 Closed-Form Solution for the ClampedFree

Beam 4318.2.6 Closed-Form Solution for the

ClampedClamped Beam 4428.2.7 Closed-Form Solution for the Beam that is

Pinned at One End and Clamped at the Other 4528.2.8 Concluding Remarks 460

Chapter 9 Unexpected Results for a Beam on an Elastic Foundationor with Elastic Support 4619.1 Some Unexpected Results in the Vibration of

Inhomogeneous Beams on an Elastic Foundation 4619.1.1 Introductory Remarks 461

9.1.2 Formulation of the Problem 4629.1.3 Beam with Uniform Inertial Coefficient,Inhomogeneous Elastic Modulus and ElasticFoundation 463

9.1.4 Beams with Linearly Varying Density,Inhomogeneous Modulus and ElasticFoundations 468

9.1.5 Beams with Varying Inertial CoefficientRepresented as anmth Order Polynomial 475

9.1.6 Case of a Beam Pinned at its Ends 4809.1.7 Beam Clamped at the Left End and Free at the

Right End 4869.1.8 Case of a ClampedPinned Beam 4919.1.9 Case of a ClampedClamped Beam 4969.1.10 Case of a GuidedPinned Beam 5019.1.11 Case of a GuidedClamped Beam 5109.1.12 Cases Violated in Eq. (9.99) 515

9.1.13 Does the BoobnovGalerkin MethodCorroborate the Unexpected Exact Results? 517

9.1.14 Concluding Remarks 5219.2 Closed-Form Solution for the Natural Frequency of

an Inhomogeneous Beam with a RotationalSpring 5229.2.1 Introductory Remarks 5229.2.2 Basic Equations 522

9.2.3 Uniform Inertial Coefficient 5239.2.4 Linear Inertial Coefficient 526

http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch8.pdf/http://2892ch9.pdf/

8/12/2019 Bu Clink

12/710

Contents xiii

9.3 Closed-Form Solution for the Natural Frequency ofan Inhomogeneous Beam with a Translational Spring 5289.3.1 Introductory Remarks 5289.3.2 Basic Equations 5299.3.3 Constant Inertial Coefficient 5319.3.4 Linear Inertial Coefficient 533

Chapter 10 Non-Polynomial Expressions for the Beams FlexuralRigidity for Buckling or Vibration 537

10.1 Both the Static Deflection and Vibration Mode ofa Uniform Beam Can Serve as Buckling Modes ofa Non-uniform Column 537

10.1.1 Introductory Remarks 53710.1.2 Basic Equations 53810.1.3 Buckling of Non-uniform Pinned Columns 53910.1.4 Buckling of a Column under its Own Weight 54210.1.5 Vibration Mode of a Uniform Beam as a

Buckling Mode of a Non-uniform Column 54410.1.6 Non-uniform Axially Distributed Load 54510.1.7 Concluding Remarks 547

10.2 Resurrection of the Method of SuccessiveApproximations to Yield Closed-Form Solutions forVibrating Inhomogeneous Beams 54810.2.1 Introductory Comments 54810.2.2 Evaluation of the Example by Birger and

Mavliutov 55110.2.3 Reinterpretation of the Integral Method for

Inhomogeneous Beams 55310.2.4 Uniform Material Density 555

10.2.5 Linearly Varying Density 55710.2.6 Parabolically Varying Density 55910.2.7 Can Successive Approximations Serve as

Mode Shapes? 56310.2.8 Concluding Remarks 563

10.3 Additional Closed-Form Solutions forInhomogeneous Vibrating Beams by the IntegralMethod 566

10.3.1 Introductory Remarks 56610.3.2 PinnedPinned Beam 56710.3.3 GuidedPinned Beam 57510.3.4 FreeFree Beam 58210.3.5 Concluding Remarks 590

Chapter 11 Circular Plates 59111.1 Axisymmetric Vibration of Inhomogeneous Clamped

Circular Plates: an Unusual Closed-Form Solution 591

11.1.1 Introductory Remarks 59111.1.2 Basic Equations 593

http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch10.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch9.pdf/http://2892ch11.pdf/http://2892ch10.pdf/

8/12/2019 Bu Clink

13/710

xiv Eigenvalues of Inhomogeneous Structures

11.1.3 Method of Solution 59411.1.4 Constant Inertial Term(m = 0) 59411.1.5 Linearly Varying Inertial Term(m = 1) 59511.1.6 Parabolically Varying Inertial Term(m = 2) 59611.1.7 Cubic Inertial Term(m = 3) 59811.1.8 General Inertial Term(m 4) 60011.1.9 Alternative Mode Shapes 601

11.2 Axisymmetric Vibration of Inhomogeneous FreeCircular Plates: An Unusual, Exact, Closed-FormSolution 60411.2.1 Introductory Remarks 60411.2.2 Formulation of the Problem 605

11.2.3 Basic Equations 60511.2.4 Concluding Remarks 607

11.3 Axisymmetric Vibration of Inhomogeneous PinnedCircular Plates: An Unusual, Exact, Closed-FormSolution 60711.3.1 Basic Equations 60711.3.2 Constant Inertial Term(m = 0) 60811.3.3 Linearly Varying Inertial Term(m = 1) 609

11.3.4 Parabolically Varying Inertial Term(m = 2) 61011.3.5 Cubic Inertial Term(m = 3) 61211.3.6 General Inertial Term(m 4) 61411.3.7 Concluding Remarks 616

Epilogue 617

Appendices 627

References 653

http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892epilogue.pdf/http://2892epilogue.pdf/http://2892app.pdf/http://2892app.pdf/http://2892ref.pdf/http://2892ref.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ch11.pdf/http://2892ref.pdf/http://2892app.pdf/http://2892epilogue.pdf/

8/12/2019 Bu Clink

14/710

Foreword

This is a most remarkable and thorough review of the efforts that have beenmade to find closed-form solutions in the vibration and buckling of all mannerof elastic rods, beams, columns and plates. The author is particularly, but notexclusively, concerned with variations in the stiffness of structural members.The resulting volume is the culmination of his studies over many years.

What more can be said about this monumental work, other than to express

admiration? The authors solutions to particular problems will be very valu-able for testing the validity and accuracy of various numerical techniques.Moreover, the study is of great academic interest, and is clearly a labor oflove. The author is to be congratulated on this work, which is bound to be ofconsiderable value to all interested in research in this area.

Dr. H.D. ConwayProfessor Emeritus

Department of Theoretical and Applied MechanicsCornell University

It is generally believed that closed-form solutions exist for only a relativelyfew, very simple cases of bars, beams, columns, and plates. This monographis living proof that there are, in fact, not just a few such solutions. Even in thecurrentageofpowerfulnumericaltechniquesandhigh-speed,large-capacitycomputers,thereareanumberofimportantusesforclosed-formsolutions: for preliminary design (often optimal)

as bench-mark solutions for evaluating the accuracy of approximateand numerical solutions

to gain more physical insight into the roles played by the variousgeometric and/or loading parameters

This book is fantastic. Professor Elishakoff is to be congratulated not only

for pulling together a number of solutions from the international literature,but also for contributing a large number of solutions himself. Finally, hehas explained in a very interesting fashion the history behind many of thesolutions.

Dr. Charles W. BertBenjamin H. Perkinson Professor EmeritusAerospace and Mechanical Engineering

The University of Oklahoma xv


8/12/2019 Bu Clink

15/710


8/12/2019 Bu Clink

16/710

1Introduction: Review of Direct, Semi-inverseand Inverse Eigenvalue Problems

This chapter presents a selective review (with emphasis on the word selective)

of the direct, semi-inverse and inverse eigenvalue problem for structuresdescribed by a differential equation with variable coefficients. It gives onlya taste of the extensive research that has been conducted since 1759, whenLeonhard Euler posed, apparently for the first time, a boundary value problem.Since then numerous studies have been conducted for rods, BernoulliEuler

beams, BresseTimoshenko beams, KirchhoffLove and MindlinReissnerplates and shells, and structures analyzed via finer, high-order theories. Thisselective review classifies the solutions as belonging to one of three main classes:(1) direct problems, (2) semi-inverse problems, (3) inverse problems. In addi-

tion, some new closed-form solutions are reported that have been obtained viaposing an inverse vibration problem. Due to the huge body of literature, theauthor limits himself to discussing classic theories of structures.

1.1 Introductory Remarks

The vibration and buckling eigenvalue problems in engineering may beroughly categorized as belonging to one of the following three classes: (1) dir-ect or forward problems, (2) inverse or backward problems, (3) semi-inverseor semi-direct problems. Direct problems are associated with the determin-ation of the vibration frequencies and/or the buckling loads of structureswith specified configuration; namely, in the vibration context, direct prob-lems call for evaluation of the natural frequencies of rods, beam, plates orshells of specified, uniform or non-uniform cross-section (in the case of rodsand beams) or thickness (in the case of plates and shells), for both homogen-eous structures, i.e., those with constant elastic properties and mass density,and inhomogeneous ones, in which the elastic properties and/or the materialdensity vary with the coordinates. There are numerous methods of solvingdirect eigenvalue problems. These methods are conveniently subdivided intothe exact and the approximate ones. Exact solutions may be available forboth uniform and non-uniform homogeneous or inhomogeneous structures.

7


8/12/2019 Bu Clink

17/710

8 Eigenvalues of Inhomogeneous Structures

For completeness, we will recapitulate some simple examples reported in theliterature.

1.2 Vibration of Uniform Homogeneous Beams

Consider first the vibration of the uniform homogeneous BernoulliEulerbeams. The vibration of such beams is governed by the the followingdifferential equation:

D

4w

x4+ A2w

t2 =0 (1.1)where D= EI is the flexural rigidity, E is the modulus of elasticity, I themoment of inertia, the mass density, A the cross-sectional area, w(x, t)the deflection (transverse displacement), x the axial coordinate and t thetime. We are looking for the free harmonic vibrations with frequency ,representing the displacementw(x, t) as

w(x, t)=W(x)eiax

(1.2)

whereW(x)is the mode shape. The differential equation (1.1) becomes

Dd4W

dx4 A2W=0 (1.3)

It is instructive to introduce the parameter

k4 = A2

D(1.4)

so that Eq. (1.3) takes the form

d4W

dx4 k4W= 0 (1.5)

The solutionW (x)can be put in the form

W(x) C1sin(kx) + C2cos(kx) + C3sinh(kx) + C4cosh(kx) (1.6)

The mode shape W(x)must satisfy the associated boundary conditions. Forthe end that is pinned (denoted byP),

W=

0 d2W

dx2 =0 (1.7)


8/12/2019 Bu Clink

18/710

Introduction 9

For the end that is clamped (denoted byC),

W

=0

dW

dx =0 (1.8)

For the free end (denoted by F),

d2W

dx2 =0 d

3W

dx3 =0 (1.9)

whereas for the guided end (denoted byG),

dW

dx=0 d

3W

dx3 =0 (1.10)

Note that the term guided was suggested to the author by Bert (2000b).Usually, textbooks do not discuss the guided end boundary condition. Someof the exceptions are the texts by Dimarogonas (1996) and Inman (1995), who

refer to the end with boundary condition in Eq. (1.10) as the guided end

boundary conditions at both ends yields four homogeneous equations withfour unknownsC1, C2, C3and C4. Requiring that they be non-trivial, i.e.,

C21+ C22+ C23+ C24=0 (1.11)

yields characteristic equations that give, for beams under different boundaryconditions, associated non-trivial fundamental natural frequencies,

PP: sin(kL)=0GP: cos(kL)=0CF: cos(kL) cosh(kL) + 1=0CG: tan(kL)

+tanh(kL)

=0

CP: tan(kL)=tanh(kL)CC: cos(kL) cosh(kL) 1=0

(1.12)

The solutions of these transcendental equations are obtainable either analytic-ally or numerically. For the beam pinned at both ends the non-trivial solutionof Eq. (1.12) reads kj= j (j= 1,2, . . .). For the guidedpinned beam weget from Eq. (1.12)kj

=(2j

1)/2. For the other combinations of boundary

conditions the numerical solutions of the transcendental equations (1.12) are


(see alsoNatke, 1989; Korenev and Rabinovich, 1972). Satisfaction of the
http://2892ref.pdf/http://2892ref.pdf/http://2892ref.pdf/http://2892ref.pdf/http://2892ref.pdf/http://2892ref.pdf/

8/12/2019 Bu Clink

19/710


available, and the first three frequency coefficients kjare listed below:

CF: k1L=1.875; k2L=4.694; k3L=7.854

CG: k1L=2.365; k2L=5.498; k3L=8.639CP: k1L=3.927; k2L=7.069; k3L=10.210CP: k1L=4.73; k2L=7.853; k3L=10.996

(1.13)

For higher frequencies (j1), accurate asymptotic expressions can be givenfor the natural frequency coefficientskj:

CF: kj

(j

1

2)

CG: kj (j 14 )CP: kj (j+ 14 )CC: kj (j+ 12 )

(1.14)

Exact solutions for various vibrating uniform beams are given in numerous

1.3 Buckling of Uniform Homogeneous Columns

Buckling of uniform homogeneous columns subjected to compressive load atthe ends is governed by the familiar differential equation

D d4

Wdx4

+ Pd2

Wdx2

=0 (1.15)

One, again, introduces the eigenvalue parameter

4 = PD

(1.16)

leading to the following expression of the buckling modeW (x):

W= C1+ C2x + C3sin(x) + C4cos(x) (1.17)

The boundary conditions for the pinned or clamped ends coincide withtheir counterparts in the vibration case, yet for the free end the boundarycondition (1.9) is replaced by

Dd3W

dx3 +P

dW

dx =0 (1.18)


references (see, e.g.,Gorman,1974).
http://2892ref.pdf/http://2892ref.pdf/http://2892ref.pdf/

8/12/2019 Bu Clink

20/710

Introduction 11

The appropriate characteristic equations are:

PP: sin(L)=0

GP: cos(L)=0CF: cos(L)=0CG: sin(L)=0CP: tan(L)=LCC: cos(L)=1

(1.19)

In the case of thePP, GP, CG, CCandCFcolumns analytical solutions

are obtained:

PP: j= jGP: j= (2j 1)/2CF: j= (2j 1)/2CG: j= jCC: j

=2j

(1.20)

Forj=1, we obtain the buckling loads, denoted by Pcr:

PP: Pcr= 2D/L2

GP: Pcr= 2D/4L2

CF: Pcr= 2D/4L2

CG: Pcr= 2D/L2CC: Pcr=42D/L2

(1.21)

For a PC column, the numerical solution of the transcendental equation(1.19.5) is available. It reads

1L4.49 (1.22)

with the associated buckling load

Pcr20.2D

L2 (1.23)

Exact solutions are also available for specific non-uniform beams and columns,in both the vibration and the buckling contexts.

The governing equations describing buckling of non-uniform columns are

differential equations with variable coefficients. Such equations may arise


8/12/2019 Bu Clink

21/710


even in uniform columns. One such case is the initially vertical uniformcolumn under its own weight with intensity q. We first consider the columnthat is clamped at x= 0 and free at x= L. The equation of bending of thecolumn

D(x)d2W

dx2 =M(x) (1.24)

where M(x), the bending moment, is determined as the sum of elementarymoments of weight intensity, acting on all elements of the part of the columnuntil the cross-sectionx:

M= Lx

q[V(u) W(x)] du=q Lx

V(u)du qW(x)(L x) (1.25)

Substituting Eq. (1.25) into Eq. (1.24) and differentiating with respect to xwe get

d

dx

D(x)

d2W

dx2

+ q(L x) dW

dx=0 (1.26)

If the flexural rigidity D(x) is constant, we get an ordinary differentialequation with variable coefficients

Dd3W

dx3+ q(L x) dW

dx=0 (1.27)

We introduce a new variable

z= 23

q(L x)3

D(1.28)

From this expression we findxand the derivatives ofWwith respect tox:

x=L 3

9Dz2

4q

dW

dx= 3

3qz

2D

dW

dz

d2W

dx2 = 3

9q2

4D2

1

3z3dW

dz+ z2/3 d

2W

dz2

d2W

dx

3

=

3q

2D

1

z

dW

dz

d2W

dz

2

z

d3W

dz

3

(1.29)


8/12/2019 Bu Clink

22/710

Introduction 13

Substitution of Eqs. (1.29) into Eq. (1.27) yields

d3W

dz3

+

1

z

d2W

dz2

+ 1 1

9z2

dW

dz =0 (1.30)

which is the Bessel equation with respect to the function dW/dz, with thesolution

dW

dz=C1J1/3(z) + C2J1/3(z) (1.31)

We will not determine the function Wat this stage since it is sufficient to knowits derivatives. Since at x=0 the column is clamped, we have the followingboundary conditions:

x=0 W= 0 dWdx

=0 or dWdz

=0 (1.32)

The upper end is free, i.e., there

x=L z=L d2

Wdx2

=0 or z1/3 dWdz

+ 3z2/3 d2

Wdz2

=0 (1.33)

Following Dinnik (1912, 1929, 1955a,b) we start from the latter condition. Forextremely smallz, neglecting the terms of higher order, we write

dW

dz=D1z1/3 + D2z1/3 (1.34)

whereD1and D2are new constants, proportional toC1and C2, respectively.Substituting this expression into the boundary condition at the upper end,we get

D1=C1=0 (1.35)

The condition at the lower end yields

C2J1/3

2

3

qL3

D

=0 (1.36)

i.e., eitherC2=0, corresponding to straight equilibrium, or

2

3qL

3

D =n (1.37)


8/12/2019 Bu Clink

23/710


wherenare roots of the transcendental equation J1/3()=0. The first rootis 1.87. This value results in the buckling load

qcr=7.87D

L3 (1.38)

Consider, now, the case in which the column is clamped at x=0, the upperend can move but the slope is zero (guided case). At x= 0 the boundaryconditions specified in Eq. (1.32) hold. At x=L

z

=0

dW

dx =0 or z1/3

dW

dz =0 (1.39)

Consider the latter condition. WetakedW/dzin Eq. (1.31), express the Besselfunction as a series, and then multiply by z1/3:

z1/3dW

dz= C1z

2/3

21/3(4/3)

1 3z

2

16+

+ C22

1/3

(2/3)

1 3z

2

8 +

(1.40)

It is clear that in order for the right-hand side to vanish at z=0, it is necessaryto putC2= 0. The first boundary condition leads to either an uninterestingcase ofC2= 0, signifying the straight form of the equilibrium, or C2= 0,with

J1/3

2

3

qL3

D

=0 (1.41)

with first root equal to 2.90. Accordingly, the buckling intensity equals

qcr=18.9D

L3 (1.42)

Now, consider now the case in which the lower end is clamped but the upperend carries the concentrated compressive load P. The bending moment reads

M= L

x

q[V(u) W(x)] du + P(f W ) (1.43)

where f is the displacement at the upper end. The differential equation reads

Dd3W

dx3 +q(L

x)

dW

dx =0 (1.44)


8/12/2019 Bu Clink

24/710

Introduction 15

Note that Eq. (1.44) coincides with Eq. (1.27), except thatLis replaced byL,where

L=(qL

+P)

q (1.45)

has the dimension of length; Lcan be referred to as an effective lengthof thecolumn. In perfect analogy with the case of the clampedfree column withoutthe concentrated force, we obtain

dW

dz=C1J1/3(z) + C2J1/3(z)

z= 23

q(L x)3

D

(1.46)

The boundary conditions at the lower end read

x=0 z= 23

q(L)3

D=zL

W= 0 dWdx

=0 or dWdz

=0(1.47)

At the upper end

x=L z= 23

q(L L)3

D=zU

d2W

dx2 =0 dW

dz+ 3z d

2W

dz2 =0 (1.48)

wherezLand zUrepresent the values ofztaken at the lower and upper ends,respectively. The boundary conditions lead to the following equations:

C1J1/3(zL) + C2J1/3(zU)=0C1J2/3(zL) C2J2/3(zU)=0

(1.49)

The non-triviality requirementC21

+C22

=0 leads to

J1/3(zL)J2/3(zU+ J1/3(zL)J2/3(zU)=0 (1.50)

In the previous case we dealt with a homogeneous differential equation ofthe third order. If the column is pinned atx=L, then the horizontal forceN,the reaction, should be taken into account. Instead of Eq. (1.27) we get

D(x)d3W

dx3 +q(L

x)

dW

dx =N (1.51)


8/12/2019 Bu Clink

25/710


or with the new variable as in Eq. (1.27),

d3W

dz3 +1

z

d2W

dz2 + 1 1

9z2dW

dz = 2N

3qz = bz (1.52)

Let us first determine dW/dz. The complementary solution is given byEq. (1.31). The particular solution is represented as

dW

dz=A0+ A1z + A2z2 + + Anzn + (1.53)

Substituting it into Eq. (1.52) and equating the coefficients with equal powersof the variablezwe get

A0=0 A1(9 1)= b A2(9 4 1)= 9A0A3(9 9 1)= 9A1, . . . , An(92n 1)= 9An2

(1.54)

This means that all the coefficients with even index vanish

A0=A2= A4= =A2n=0 (1.55)

while the odd coefficients equal

A1= b

9 12 1 A3=9b

(9 12 1)(9 32 1)

A5= 92b(9 12 1)(9 32 1)(9 52 1) , . . . , A2n+1=

9A2n19(2n + 1)2 1

(1.56)

leading to

dW

dz =C1J1/3(z)

+C2J

1/3(z)

6N

qC(z) (1.57)

where

C(z)=z

1

9 12 1(3z)2

(9 12 1)(9 32 1) +(3z)4

(9 12 1)(9 32 1)(9 52 1)

(3z)2n

(9 12 1)(9 32 1)(9 52 1) [9(2n + 1)2 1] (1.58)


8/12/2019 Bu Clink

26/710

Introduction 17

This series is uniformly convergent for all finite values of the variable z.Integrating Eq. (1.57) we get

W= C1A(z) + C2B(z) (6N/q)D(z) + C3 (1.59)

where

A(z)=

J1/3(z) dz B(z)=

J1/3(z) dz D(z)=

C(z)dz

4

3

A(z)=3 z

24/3 1

23

5 1 4 z

22 +

32

8 1 2 4 7 z

24

33

11 1 2 4 7 10 z

2

6 + +(1)n 3

n

(3n + 2) 1 2 3 n 1 4 7 (3n + 1) z

2

2n

2

35

3B(z)=3 z

22/3

1 34

1

2

z

22 + 3

2

7

1

2

2

5

z

24

33

10 1 2 3 2 5 8 z

2

6 + + (1)n 3

n

(3n + 1) 1 2 3 n 2 5 8 (3n 1) z

2

2n

D(z)= z2

2

1

9 12 1 (3z)2

2(9 12 1)(9 32 1)

+ (3z)43(9 12 1)(9 32 1)(9 52 1)

(1.60)

with Gamma function(x)evaluated atx=4/3:

4

3=0.8910 23 53=1.3541 (1.61)

Consider, now, the boundary conditions. Let the column be clamped at bothends. At the bottom

x=0 z=2

3qL3

D = zL W=dW

dx =dW

dz =0 (1.62)


8/12/2019 Bu Clink

27/710


while at the upper end

x= L z= W= dWdx

= dWdz

=0 (1.63)

In order to satisfy the latter boundary conditions, it is necessary to require that

C2=C3=0 (1.64)

The conditions atx=0 yield

C1A(zL)

(6N/q)D(zL)

=0

C1J1/3(zL) (6N/q)C(zL)=0 (1.65)

resulting in the equation for the determination of the buckling load

A(zL)C(zL) D(zL)J1/3(zL)=0 (1.66)

Dinnik reports 5.72 to be the minimum root of this equation, yielding thecritical, buckling load

qcr=317.15D/L3 (1.67)

Likewise, for the clampedpinned column

zL=4.83 qcr=52.49D/L3 (1.68)

while for the pinned column

zL=2.87 qcr=18.53D/L3 (1.69)

It should be noted that the buckling of uniform columns under their ownweight was revisited by Willers (1941) and Engelhardt (1954). They demon-strated that the general solution for the columns slope can be written in termsof Bessel and Lommel functions:

dW

dz=C1J1/3(z) + C2J1/3(z) + C3s0,1/3(z) (1.70)

Equation (1.70) contains the Lommel function

S,(Z)=

m=0

(1)mz+1+2m

( + 1)2 2

( + 3)2 2

( + 2m + 1)2 2

(1.71)


8/12/2019 Bu Clink

28/710

Introduction 19

with parameters = 0, = 13(Erdelyi, 1953). Lommel functions have also beenutilized in axisymmetric vibrations of a piezoelectric cylinder by Adelmanand Stavsky (1975), and in the impact buckling of a bar in a compressivetesting machine by Elishakoff (1980). Note that without going through theclever transformations of variables, one can solve the problems discussedin this section straightforwardly by the Frobenius (power series) method fordifferentialequationshavingalgebraiccoefficients(Leissa,2000b),asisshownin Section 1.4.

1.4 Some Exact Solutions for the Vibration ofNon-uniform Beams

The literature usually deals with beams of variable cross-sectional area. Thesimplest case of this kind is a beam of constant width and linearly varyingthicknessh:

h= h1+ (h0 h1)(x/L) (1.72)

where h1 is the thickness at the cross-section x= 0 and h0 is the thicknessattained at the cross-section x= L, where Lis the length of the beam. Forthe tapered beam, the moment of inertia and the cross-sectional area areexpressed as

I= bh3/12=(b/12)[h1+ (h0 h1)x/L]3

A

=bh

=b

[h1

+(h0

h1)x/L

]

(1.73)

The governing differential equation is

Ed2

dx2

b

12

h1+ (h0 h1)

x

L

3 d2Wdx2

b2

h1+ (h0 h1)

x

L

W= 0

(1.74)

We introduce a new coordinate

X= h1+ (h0 h1)x/L (1.75)

Equation (1.74) becomes

d2

dX

2 X3d2W

dx

2 =k4XW (1.76)


8/12/2019 Bu Clink

29/710


where

k4 =122L4/E(h0 h1)4 (1.77)

Introducing the linear operatorMsuch that

M= 1X

d

dX

X2

d

dX

(1.78)

Equation (1.76) can be put in the form

(M

+k2)(M

k2)W

=0 (1.79)

which has a solution obtained from the equations

(M+ k2)W= 0(M k2)W= 0

(1.80)

or

d

dX

X2

dW

dX

k2XW= 0 (1.81)

The general solution of Eq. (1.81) is then written as

W(X)=

A1J1

2k

X

+ A2Y1

2k

X

+ A3I1

2k

X

+ A4K1

2k

X

X

(1.82)

whereJ1andY1are first-order Bessel functions of the first and second kind,and I1and K1are first-order modified Bessel functions of the first and secondkind, respectively. The above solution is due to Mabie and Rogers (1964).Satisfying boundary conditions, one gets four algebraic equations for thecoefficients A1,A2,A3andA4.Requirementofnon-trivialityofthecoefficientsyieldatranscendentalequationthatshouldbesolvednumerically.Thesolu-tion in terms of Bessel functions for inhomogeneous beams was pioneeredby Kirchhoff (1879, 1882), who studied the free vibrations of a wedge or conebeams.FurthercontributionswereprovidedbyWard(1913),Nicholson(1920)andWrinch(1922).Considerablesimplificationinderivationsoccurredforthecomplete beams due to the absence of Bessel functions of the second kindin the solution. As of now there is an extensive literature, including the bookby Gorman (1975). The results of numerical evaluation are usually presen-ted in terms of tables of figures, in various references. These were presentedfor the variable cross-section beam that is clamped at one end and pinnedat the other, by Mabie and Rogers (1968). Cantilever beams with constant

width and linearly variable thickness, or beams with constant thickness and


8/12/2019 Bu Clink

30/710

Introduction 21

linearly variable width, with an end mass, have been investigated by Mabieand Rogers (1964). The same authors (Mabie and Rogers, 1974) also studiedtransverse vibrations of double tapered beams. With the solution based onBessel functions, Conway and Dubil (1965) tackled the truncated wedge andcone beams, and presented tables of frequencies for combinations of clamped,pinned and free boundary conditions. Lee (1976) dealt with a cantilever witha mass at one end, while Goel (1976) extended the method to a wedge and acone beam with resilient supports at both ends. Sanger (1968) considered aspecial class of non-uniform beams, with the geometry enabling the expres-sion of the solution in terms of Bessel functions. Solution in terms of Besselfunctions for a beam, part of which is tapered while the other part is uniform,was studied by Auciello and Ercolano (1997).

For the cone and the wedge beams, an exact solution obtained using theFrobenius method was proposed by Naguleswaran (1990, 1992, 1994a,b) andtheresultsweretabulatedfordifferentconstraintconditions.Wang(1967)alsoutilized the method of Frobenius, and considered the following variations ofthe cross-sectional area and the moment of inertia, respectively:

A=A0(x/L)n I= I0(x/L)m (1.83)

where the constants mand nare any two positive numbers, and A0 and I0are the cross-sectional area and moment of inertia, respectively, of the beamat the endx=L.

1.4.1 The Governing Differential Equation

The differential equation that governs the free vibrations of the beams that

are inhomogeneous and/or have a variable cross-sectional area reads:

d2

dx2

D()

d2W

dx2

A2W= 0 (1.84)

It can be expressed as

m d4

Wd4

+ 2mm1 d3

Wd3

+ m(m 1)m2 d2

Wd2

2nW= 0

=x/L 2 =2L4A/EI0(1.85)

We multiply Eq. (1.85) by4m and let=4 m + n, yielding

4d4W

d4 +2m3

d3W

d3 +m(m

1)2

d2W

d2 2W

=0 (1.86)


8/12/2019 Bu Clink

31/710


It has a general solution in the form of a linear combination of convergentseries of an infinite form about the origin (Ince, 1956). In this case, the gen-eral solution can be obtained by the method of Frobenius. We introduce thedifferential operator

= dd

(1.87)

which gives the general relations

r dr

dr=( 1)( 2) ( r + 1) (1.88)

Moreover,

( 1)( 2) ( r + 1)xs =s(s 1)(s 2) (s r + 1)xs (1.89)

Equation (1.86) may be rewritten in the form

(

1)(

+m

2)(

+m

3)W

2W

=0 (1.90)

We introduce the following notations

u= 2

4 u=u

d

du

(1.91)

Then, Eq. (1.89) becomes

u

u 1

u 2 m

u 3 m

W uW= 0 (1.92)

which is due to Wang (1967). Equation (1.92) is a type of generalized hyper-geometric equation (Erdelyi, 1953). Its general solution is a linear combinationof the following independent generalized hypergeometric series:

W1

=0F3 (; b1, b2, b3; u)

W2=u1b1 0F3 (; 2 b1, b2 b1+ 1, b3 b1+ 1; u)W3=u1b2 0F3 (; b1 b2+ 1, 2 b2, b3 b2+ 1; u)W4=u1b3 0F3 (; b1 b3+ 1, b2 b3+ 1, 2 b3; u)

(1.93)

where

b1=(3 m + n)/ b2=(2 + n)/ b3=(1 + n)/ (1.94)


8/12/2019 Bu Clink

32/710

Introduction 23

The generalized hypergeometric function is defined as

0F3(a1, a2, . . . , ap; b1, b2, . . . , bq; u)=1 +

n=1 p

i=1(ai)n u

n qj=1

bj

n n!1

(1.95)

The series in Eq. (1.93) are undefined or not linearly independent wheneitherb1, b2orb3is an integer or two of them differ by an integer. For thesecases the logarithmic terms appear in the general solutions. The detailedderivations of the logarithmic solutions by the method of Frobenius were

presented by Wang (1967) and are not recapitulated here. Wang (1967) con-sidered the following four cases: (a) when two bcoefficients are equal (thisoccurs ifm= 1 or m= 2 or m= 4, which yields the coincident values forb1 and b3); (b) when one bvalue equals unity (indeed, when the parametermequals 2 or 3, the value ofb2 or b3 turns out to be unity); (c) when b1 isa negative integer or zero (this happens when is a reciprocal of a positiveinteger); (d) when two bs differ by an integer (this case occurs for certaincombinations of parameters mand n). For example, the combination m= 3andn=1 yieldsb1=

1

2 andb3= 3

2 , which have a difference of unity. Somespecial cases follow from Wangs (1967) general formulation. For uniform

beams, i.e.,m=n=0,=4 and b1= 34 ,b2= 24 ,b3= 14 the hypergeometricfunctions reduce to the familiar solution in Eq. (1.6).

Another special combination of parameters m and n occurs when m n=2or =2, which includes wedge-shaped and cone-shaped beams. The solutionreduces to the Bessel functions on utilizing the relationship between the Besselfunctions and the generalized hypergeometric functions (Rainville, 1960):

Jv(z)=(z/2)v

(1 + v) 0F1(; 1 + v; z2/4)

Iv(z)=(z/2)v

(1 + v) 0F1(; 1 + v; z2/4)

(1.96)

The last special case is for beams with constant thickness and linearly

tapered width. Then, m

=n

=1, yielding b1

= 34 , b2

= 34 , b3

= 12 . Since the

values ofb1andb2are equal, the series solutions for W1andW2in Eq. (1.93)coincide. The fundamental solutionW2is replaced by

W2=W1ln u 4u1/4

44

5 42 3

1

5+ 1

3+ 1

2

u

+

44

9 82

7

44

5 42

3 1

9+

1

7+

1

4+

1

5+

1

3+

1

2u2 +

(1.97)


8/12/2019 Bu Clink

33/710


This form of the solution for the beam with constant thickness and linearlytapered width was given by Ono (1925). Wang (1967) presented numer-ous solutions and attendant numerical evaluations for the fundamental andsecond frequencies.

1.5 Exact Solution for Buckling of Non-uniform Columns

The study of buckling of non-uniform columns was pioneered by Euler (1759).He considered the columns with flexural rigidity given as a polynomial(a

+bx/L)m. Actually, Euler represented the flexural rigidity as a product

of the modulus of elasticity and the positive quantityk2 (denoted by Euler askk), unbeknown to him that the latter quantity was a moment of inertia. Eulerreferred to the product Ek2 as a moment of stiffness (moment du ressort ormoment de roideur). He hit upon the cases in which the buckling modes aregiven by the elementary functions. The analogous flexural rigidity variation,

D(x)=EI(x/b)m (1.98)

in the non-uniform columns was studied by Dinnik (1912). He demonstratedthat for anym, equal to any, positive or negative, integer or decimal number,except 2, the equation

EI(x/b)mW+ PW= 0 (1.99)

is integrable in terms of the Bessel functions of order

n=1/(m 2) (1.100)In these formulas, letting m=0 and using the known relations between theBessel functions of order 12 and trigonometric functions

I1/2(x)=sin(x)

2/x I 1/2(x)=cos(x)

2/x (1.101)

we recover Eulers solution for the uniform column. Atm=4, we again getthe Bessel functions of order

12 , but for an argument other than at m= 0.

Thus, for m=4 the solution can be obtained both by Bessel and trigonometricfunctions. Form=4, Eq. (1.97) is rewritten as

x4W+ U2W= 0 U2 =Pb4/EI (1.102)

with the buckling mode

W(x)=x[Aicos(U/x) + A2sin(U/x)] (1.103)


8/12/2019 Bu Clink

34/710

Introduction 25

whereas form=2, the mode shape reads

W(x)= x{A1cos[ln(x/b)] + A2sin[ ln(x/b)]} (1.104)

with

2 = Pb2

EI 1

4. (1.105)

For the buckling of the inhomogeneous column with the elastic modulusvariation

E(x)=E0[1 k(x/L)]2 (1.106)Freudenthal (1966) introduced a new variable

v2 = [1 k(x/L)]2 (1.107)

and reduced the buckling equation to

v2 d2

Wdv2

+ 0

Lk

W= 0 20= PE0I

(1.108)

For(2L/k)2 >1 he derived a solution analogous to Eq. (1.104):

W= A1v cos(aln v) + A2v sin(aln v)

v= 12|1 (2L/k)2| (1.109)

From the boundary conditions, W= 0 at x= 0 and x= L the followingcritical value follows

Pcr=2EI

L2

k

ln(1 k)

2

k

2

2 (1.110)

where the expression in brackets tends to unity as k0. For

D(x)=D0(x/L) (1.111)

Freudenthal (1966) gave a solution in terms of Bessel functions.Columns with variable cross-section were extensively studied by Dinnik

(1912, 1929, 1955a,b). He considered the columns with the following flexuralrigidity distribution:

D(x)=b(L x)m

(1.112)


8/12/2019 Bu Clink

35/710


The columns were under the axial distributed loading

q=c(L x)n (1.113)

whereb,c,mand nare constants. In the cross-sectionx

M(x)= L

x

c(L u)n[V(u) W(x)] du

=c L

x

(L u)nV (u) du cn + 1 (L x)

n+1W(x) (1.114)

The differential equation reads

W 1 m2

W+ cb(n + 1) (L x)

nm+1W=0 (1.115)

leading to

dW

dx=(L x)(1m)/2

C1J 1m

nm+3(y) + C2J 1m

nm+3(y)

, (1.116)

where

y2 =4c(L x)nm+3/b(n + 1)(n m + 3)2 (1.117)

Letting m= n= 0, b= B and c= q results in the particular case of theuniform column:

dW

dx =(L x)1/2C1J1/323q(L x)

3

D+ C2J1/323q(L x)

3

D

(1.118)

For the cone, whose radius at the base equals Rand whose height equalsL,we get

r=R(L x)/L q=R2k(L x)2/L2 D=EI= R4E(L x)4/4L4

(1.119)

whereris the radius of the cone at distancexfrom the origin. Comparison ofEqs. (1.119) with Eqs. (1.112) and (1.113) results in

m=4 n=2 b=R4E/4L4 c=R2k/L2dW

dx=(L x)3/2[C1J3(z) + C2Y3(z)] (1.120)

z2 =16kL2(L x)/3ER2


8/12/2019 Bu Clink

36/710

Introduction 27

For the clampedfree column, the buckling load is found from the equation

Y3 16kL3/3ER2=0 (1.121)The first root, calculated by Dinnik (1955b) is 4.43. Hence, the critical lengthof the column equals

Lcr=1.5453 3

ER2/K (1.122)

The optimal design problem of columns naturally involved variable cross-sections. The problem was first posed and solved by Lagrange (17701773).He attempted to determine the shape of a column of given length andvolume so that the attendant buckling load attains a maximum. Furthercontributions were made by Clausen (1851), Bairstow and Stedman (1914),Blasius (1914), Darnley (1918), Nikolai (1955), Keller (1960), Tadjbakhsh andKeller (1962), Keller and Niordson (1966), Adali (1979, 1981), Cox (1992),Cox and Overton (1992), Banichuk (1974), Seyranian (1983, 1995), McCarthy

(1999) and others. The finite element method in buckling optimization wasemployed by Simitses et al. (1973), Manickarajah et al. (2000) and others.Xie and Steven (1993) proposed a simple evolutionary method for the topo-logy, shape and layout optimization of structures. In this method, a structureor the design domain is divided into a fine mesh of elements, and inefficientmaterial is gradually removed according to the design criteria, and the resid-ual structure evolves towards the desired optimum. Keller (1960) proved thatthe optimal solid convex cross-section against buckling failure has the form

of an equivalent triangle; moreover, the dimension of that triangle shouldchange along the axis according to the law found earlier for the optimalshape of a circular column. Szyszkowski and Watson (1988) proposed thatthe optimum shape of a structure with respect to buckling should have theconfiguration for which the specific bending energy due to the fundamentalbuckling mode is uniform. Gajewski andZyczkowski (1988) write: . . .Tad-jbakhsh and Keller (1962) derived the optimal solutions for columns clampedat one end and pinned at the other, and for clampedclamped columns. Thesolution in the latter case, however is incorrect; it was obtained with respectto the first buckling mode, whereas a bimodal solution is here necessary(Olhoff and Rasmussen, 1977). This sentiment seems to be shared by Cox(1992) and Cox and Overton (1992). Yet, Spillers and Meyers (1997) arguedthat the work of Cox and Overton (1992), which claims to correct errors inthe classic solution of Tadjbakhsh and Keller (1962) (for the optimal design ofa column), itself contains an error and actually only serves to further estab-lish the validity of the TadjbakhshKeller solution. They also note that itwas... a surprise to many to see a rather robust discussion of the validity

of the TadjbakhshKeller solution . . . running from the late 1970s into the


8/12/2019 Bu Clink

37/710


mid-1980s. It appears that a detailed and careful literature review of the cal-culations and/or miscalculations in the optimum design of columns is calledfor, for the benefit of the general audience. The reader may consult reviewson selected topics by Ashley (1982), Banichuk (1982) and others. It is worthnoting that the optimum design of vibrating structures was conducted byNiordson (1965, 1970), Karihaloo and Niordson (1973), Olhoff (1970, 1976a,b,

Exact methods of calculation of buckling loads and natural frequencies wereextensively reviewed by Williams and Wittrick (1983).

1.6 Other Direct Methods (FDM, FEM, DQM)

In the earlier sections we considered exact solutions of differential equationsdescribing buckling or vibration of structures. As Fried (1979) writes: thetheoretical limiting process in the mathematical formulation createsdifferen-tial equations to describe physical phenomena or engineering processes. Asymbolic solution to these equations in terms of some elementary functions,

even if existing, is, as a rule, very hard to come by and the programmingnecessary to obtain such a symbolic solution impossible. To overcome this dif-ficulty, numerical techniques of great generality have been invented, based ondiscretizationof the problem, on the division of the continuous flow of eventsor continuous change of state into a series of discrete states formulated algeb-raically, with the limiting process on convergence deferred to the numericalstage of the solution.

Many symbolic algebraic or numerical methods have been developed in

the past few decades that make it possible to carry out the vibration andbuckling analysis of beams with arbitrarily varying cross-sectional areas orinhomogeneities. The class of approximate methods can be compared, inthe terminology of Leissa (2000a) to the limitless Pandoras box. In thesecircumstances, we confine ourselves to presenting a superficial, yet hopefullystill meaningful, overview of them.

The method of successive approximations, discussed by Engesser (1893)and Vianello (1898), was extensively employed in the pre-FEM literat-ure. Sekhniashvili (1966) utilizes this method, as discussed by Bernshtein(1941) with attendant theoretical justification provided by Sushenkov (see

1933), who showed that in the limit the method leads tothe fundamental frequency and the corresponding mode shape. Later,the method was utilized by Ratzensdorfer (1943), Popovich (1962),Mitelman (1970), Hodges (1997) and others. The numerical integra-tion method was utilized by Newmark (1943), Lu et al. (1983) andSakiyama (1986).

Special approximate methods for columns of varying flexural rigidity,

including lower bound approximations, were developed by Miesse (1949),


1977) and others (see alsoSection 1.9).

Papkovich,

8/12/2019 Bu Clink

38/710

Introduction 29

Silver (1951), Abbassi (1958), Appl and Zorowski (1959), Fadle (1962, 1963),Mazurkiewicz (1964, 1965), Pnueli (1972ac, 1999) and others.

Some authors (Thomson, 1949; Ram and Rao, 1951; Fadle, 1962; Ram, 1963;Bert, 1984a,b, 1987a,b) dealt with the so-called n-section column or steppedcolumn which is a column consisting ofnsections, each having a differentbut constant flexural rigidity. The exact solutions were obtained by solvingsimultaneous differential equations.

The basic idea of the finite-difference method (FDM) is the approximationof a derivative of a function at a point by an algebraic expression containingthe value of the function at that point and at several nearby points. Thus,the governing differential equation is replaced by an algebraic equation. Thereplacement of a continuous function by an algebraic expression composed

of the values of the function at several discrete points is equivalent to repla-cing an original distributed system by one with several lumped masses, i.e.,reduction of the continuous system to a n-section column. The use of theFDM for columns with varying flexural rigidity and pinned ends was usedby Salvadori (1951), Srinivasan (1964), Szidarovski (1964), Girijavallabhan(1969) and Iremonger (1980).

The differential quadrature method (DQM) was proposed Bellman andCasti (1971). In the DQM the discretization is accomplished by expressing

at each grid point the calculus operator value of a function with respect to acoordinate direction at any discrete point as the weighted linear sum of thevalues of the function at allthe discrete points chosen in that direction. Therth partial derivative of a function is expressed as

r

xr

x=xi=

Nxk=1

Arikki i=1, 2, . . . , Nx (1.123)

Likewise,

s

ys

y=yj

=Nyl=1

Bsjljl j=1, 2, . . . , Ny (1.124)

where A(r)ik

and B(s)jl

are the respective weighting coefficients. Also ij=(xi yj). Bellman et al. (1972) presented an idea of using the polynomial testfunctions for the determination of the weighting coefficients. DQM was util-ized for vibration problems by Bert et al. (1988, 1994), whereas Jang et al.(1989), and Sherbourne and Pandey (1991) employed it for buckling analysis.An extensive review of this method was provided by Bert and Malik (1996).

The finite element method (FEM) is a numerical technique that realizesthe system as an assembly connected to one another at points called nodes.Each element is associated with generalized displacements and generalizedforces, that are internal forces as far as overall structure is concerned, but

they represent external forces when individual elements are involved. The


8/12/2019 Bu Clink

39/710

8/12/2019 Bu Clink

40/710

Introduction 31

with boundary conditions

W (0)=W(L)=0 (1.129)

Substituting Eq. (1.127) into Eq. (1.128) leads to

1 2a2x0 + 2 3a3x1 + 3 4a4x2 + + 2a0x0 + 2a1x1 + 2a2x2 + =0(1.130)

We order the terms as

x

0 1 2a2+

2

a0+ x1 2 3a3+ 2a1+ x2 3 4a4+ 2a2+ =0

(1.131)

Since this expression must vanish for anyx, we get

a2= 2a0

1 2 a3= 2a1

1 2 3 a4=4a0

1 2 3 4 a5=4a1

5! a6= 6a0

6!(1.132)

and so on. Thus, the displacement becomes

W(x)=a0+ a1x 2a0

2! x2

2a1

3! x3

2a0

4! x4 +

4a1

5! x5 (1.133)

atx=L, we get

a1

L

2L3

3! +4L5

5! 6L7

7! + + (1)n

2nL2n+1

(2n + 1)!

=0 (1.134)

For the solution to be non-trivial we set a1 = 0, and the expression inparentheses must vanish. This leads, once we substitute L= , to

1

2/3! +

4/5!

6/7! +

8/9! + =

0 (1.135)

Taking into account that

sin = 3/3! + 5/5! 7/7! + sin n= 3/3! + 55! 7/7! +

(1.136)

we conclude that Eq. (1.134) is equivalent to Eq. (1.19.1) with

= n, and

thus we obtain the minimal eigenvalue given in Eq. (1.20.1).


Duetotheboundaryconditionatx = 0, a0 = 0.Due totheboundarycondition

8/12/2019 Bu Clink

41/710


By taking only two terms we get the following approximation

2 =3!/L2 Pcr=6EI/L2 (1.137)

which constitutes a 39% underestimation of the buckling load; but, with threeterms

2 =9.48/L2 Pcr=9.48EI/L2 (1.138)

and this yields a 3.94% error. With more terms the error diminishes rapidly.We followed Brgermeister and Steup (1957) here in exposing the direct series

representation method.The method was greatly expanded and re-interpreted by Eisenberger(1991ad). He considered bars, columns and beams of variable properties.To illustrate his approach, consider the behavior of the beam in the presenceof variable axial forces

d2

dx2

D(x)

d2W

dx2

d

dx

N(x)

dW

dx

=P(x) (1.139)

where Wis the transverse displacement, N(x) the axial force, P(x)the dis-tributed transverse load along the member and D(x)= EI(x)is the flexuralrigidity. As Eisenberger (1991) notes,

The solution for the general case of polynomial variation of I(x), N(x)and P(x)along the beam is not generally available. Using the finite ele-ment technique, it is possible to derive the terms in the stiffness matrix.We assume that the shape functions for the element are polynomials andwe have to find the appropriate coefficients. It is widely known thatexact terms will result, and if one uses the solution of the differentialequation as the shape functions, for the derivation of the terms in thestiffness matrix. In this work exact shape functions are used, to derivethe exact flexural rigidity coefficients. These shape functions are exactup to the accuracy of the computer, or up to a preset value set by theanalyst.

Following Eisenberger, the functionsD(x),N(x)and P(x)are written as

D(x)=j

i=0Dix

i N(x)=l

i=0Nix

i P(x)=m

i=0Pix

i (1.140)

where j,land m are integers representing the number of terms in each series.Introducing a non-dimensional variable=x/L, where L is the length of the


8/12/2019 Bu Clink

42/710

Introduction 33

beam Eq. (1.139) is rewritten as

d2

d2 d() d

2W

d2

d

dn() dW

d=p() (1.141)

where

d()=j

i=0DiL

ii =j

i=0di

i

n()=

l

i=0

NiLi

+2i

=

l

i=0

nii

(1.142)

p()=m

i=0PiL

i+4i =m

i=0pi

i

The solutionW()is chosen as an infinite series, as in Eq. (1.127);

W()=

i=0wi

i

(1.143)

Substitution of Eq. (1.143) into Eq. (1.141) bearing in mind Eq. (1.142)results in

i=0i

k=0(k + 1)(i k + 1)nk+1wik+1i

i=0

ik=0

(i k + 1)(i k + 2)nkwik+2i

+

i=0

ik=0

(k + 1)(k + 2)(i k + 1)dk+2wik+2wik+2i

+

i=0

i

k=0

2(k+

1)(i

k+

1)(i

k+

2)(i

k+

3)dk+

1wi

k+

3i

+

i=0

ik=0

(i k + 1)(i k + 2)(i k + 3)(i k + 4)dkwik+4i

=m

i=0pi

i (1.144)


8/12/2019 Bu Clink

43/710


This equation must be satisfied for every value of. Hence, the followingequality must hold

ik=0

(k + 1)(i k + 1)nk+1wik+1 ik=0

(i k + 1)(i k + 2)nkwik+2

+i

k=0(k + 1)(k + 2)(i k + 1)dk+2wik+2

+i

k=0

2(k + 1)(i k + 1)(i k + 2)(i k + 3)dk+1wik+3

+i

k=1(i k + 1)(i k + 2)(i k + 3)(i k + 4)dk wik+4=pi (1.145)

allowing one to expresswi+4as follows:

wi+4= 1d0(i + 1)(i + 2)(i + 3)(i + 4)

pi+

ik=0

(k + 1)(i k + 1)nk+1wik+1

+i

k=0(i k + 1)(i k + 2)nkwik+2

i

k=0(k + 1)(k + 2)(i k + 1)dk+2wik+2

i

k=02(k + 1)(i k + 1)(i k + 2)(i k + 3)dk+1wik+3

i

k=0(i k + 1)(i k + 2)(i k + 3)(i k + 4)dkwik+4

(1.146)

The missing first four coefficients are found by imposing boundary condi-tions; the next step consists in solving the problem numerically. For detailsone should consult the study by Eisenberger (1991ad). The important con-tribution by Eisenberger (1991ad) lies in his re-interpretation of the aboveseries method. He proposed to use the above solution to form the stiffnessmatrixSin the context of the finite element method:

S

= 1

0 [F()

]EI()F()d (1.147)


8/12/2019 Bu Clink

44/710

8/12/2019 Bu Clink

45/710


process:

W(x)

=

n

i=0

cii (1.148)

where i(x)are coordinate functions that satisfy all or part of the boundaryconditions pertinent to the problem. The functions i(x) are chosen so asto represent in an appropriate manner the anticipated mode shape W(x).Most convenient are coordinate functions drawn from the complete set offunctions. The parameters ci are determined from the condition of the bestapproximation ofW(x)by the series (1.148). These methods are referred to as

semi-inverseones

by

Grigoliuk

and

Selezov

(1969)

and

appears

to

best

repres-ent their essence. Indeed, the unknown solution is postulated to be represented

byknownfunctionsi(x), whereas the coefficientsciare determined by somecondition of best approximation; it is as if we guess the solutions but notcompletely. The coefficients of the guessed solutions should still be evaluated.These methods can also be referred assemi-directproblems.

The first method of this kind is the method of Rayleigh (1873) sugges-ted for the determination of the fundamental frequency of the string. Hethen generalized it for higher modes (Rayleigh, 1899a,b, 1911) [the impact ofLord Rayleigh on engineering vibration theory is described in an interestingpaper by Crandall (1995)]. Ritz (1908) provided the mathematical foundationof the method. This method in its single-term application is usually calledRayleighs method, while in the multi-term form is usually referred to as theRayleighRitzmethod (or sometimes simply as the Ritz method). Usually, thepolynomial or trigonometric expressions are utilized as the coordinate func-tions, which should satisfy the geometric boundary conditions. Rayleighsmethod is universally used, in almost all textbooks on both vibration and

buckling, due to its simplicity. In the buckling context recent applicationsinclude those by Manicka Selvam (1997, 1998). Numerous applications ofboth the Rayleigh and the RayleighRitz method were given by Laura andCortinez (1985, 1986a,b, 1987) and others. They published numerous stud-ies in which the coordinate functions were polynomial functions, for beams,circular plates and rectangular plates for a wide range of practical prob-lems. Thirty-six terms in conjunction with the RayleighRitz method wereemployed by Leissa (1973) to study the vibrations of plates under various

boundary conditions, when the exact solution is unavailable. As Timoshenko(1953) wrote:

The idea of calculating frequencies directly from an energy considera-tion,withoutsolvingdifferentialequations,was...elaboratedbyWalterRitz (1909) and the RayleighRitz method is now widely used not only instudying vibration but in solving problems in elasticity, theory of struc-tures, nonlinear mechanics, and other branches of physics. Perhaps noother single mathematical tool has led to as much research in the strength

of materials and theory of elasticity.


8/12/2019 Bu Clink

46/710

Introduction 37

The first application of Rayleighs method to the buckling problem wasperformed by Bryan (1888). In 1910 Timoshenko utilized the Rayleigh andRayleighRitz methods extensively for buckling problems (Timoshenko,1910). His book was submitted for the Zhuravsky prize. Boobnov (1913)provided one of the recommendation letters. In his overly positive review,Boobnov (1913), however, made an extremely constructive criticism ofTimoshenkos work (may all criticisms turn out to be as efficient asBoobnovs!). He stated that the final equations can be obtained, more straight-forwardly, by substituting into the differential equation the series of type(1.148) with each coordinate function satisfying all boundary conditions. Theresultofsuchasubstitutionshouldbemadeorthogonaltoeachofthecoordin-ate functions. Galerkin (1915) wrote an elegant paper in which he provided

several evaluated examples of the above idea. Since then the method has beenintensively utilized in the East. In the Russian literature it is mostly referredto as the BoobnovGalerkin or the Galerkin method, although Grigoliuk(1975, 1996) maintains that it should be referred to as the Boobnov method.One can visualize that Galerkins (1915a,b) work was a wonderful vehiclethrough which the idea exposed by Boobnov (1913) was not lost. Well writ-ten, supplemented with many examples, executed in detail, this paper [byGalerkin (1915)] should have attracted an interest, according to Grigoliuk

(1975). Indeed, clear exposition of the 1915 paper attracted Biezeno (1924),Henky(1927),Duncan(1937),andotherWesternscientiststothismethod.IntheWestitismostoftencalledtheGalerkinmethod.Todigress, notethatCran-dall (1956), when describing the Ritz method wrote: The same ideas hadearlier been applied to eigenvalue problems by Lord Rayleigh. . .We call it theRitz method when it is applied to equilibrium problems and the RayleighRitz method when applied to eigenvalue problems. Presently, the uniformlyaccepted term is the RayleighRitz method forboth equilibrium and eigenvalue

problems. Returning tothe methodadvancedbyBoobnov (1913)and Galerkin(1915a,b), it appears, as in the case of the RayleighRitz method, that the mostappropriatenameofthemethodwouldbetheBoobnovGalerkin method,asitiscalledintheextensivereviewofit,inthevolumededicatedtoGalerkins100thanniversary of birth, by Vorovich (1975). Hestressed that in his paperGalerkin(1915a,b), in addition to utilizing Boobnovs idea, or proposing it independ-ently, also provided a mechanical treatment of the method as a procedureof choosing parameters of approximating the solution, during which on thewidening classes of the possible displacements the total work of internal andexternal forces of the system turns out to be zero. Another widely usedname for this technique is theweighted residual method(Finlayson, 1972; Meir-ovitch, 1997). As Schmidt (1990) wrote: According to Finlayson (1972), the(Boobnov) Galrkin method may be regarded as a special case of the method

A general method for the analytical solution via the BoobnovGalerkinmethod for the problem of buckling of a column with varying flexural rigidityand non-ideal boundary conditions was introduced by Durban and Baruch

(1972), whereas the non-uniform beams of variable mass density were treated


of weighted residuals. (see alsoPomraning,1966).

8/12/2019 Bu Clink

47/710


by Sekhniashvili (1950a,b, 1966), Lashenkov (1961) and others. It should benoted that coordinate functions utilized in the Boobno

Bu Clink

Documents

Transcript of Bu Clink