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Eigenvalues of inhomogeneous structures : Unusual closed-form solutions
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EIGENVALUES OF
INHOMOGENEOUS
STRUCTURES
Unusual
Closed-Form Solutions
© 2005 by Issac Elishakoff
CRC PRESS
Boca Raton London New York Washington, D.C.
EIGENVALUES OF
INHOMOGENEOUS
STRUCTURES
Isaac Elishakoff
J. M. Rubin Foundation Distinguished Professor
Florida Atlantic University
Boca Raton, Florida
Unusual
Closed-Form Solutions
© 2005 by Issac Elishakoff
EOIS: “2892_fm” — 2004/9/28 — 00:42 — page iv — #4
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 dynamics–Mathematical models. 2. Buckling
(Mechanics)–Mathematical models. 3. Eigenvalues. I. Title.
TA654.E495 2004
624.1’76’015118–dc22
2004051927
This book contains information obtained from authentic and highly regarded sources. Reprinted
material is quoted with permission,and sources are indicated. A wide variety of references
are listed. Reasonable efforts have been made to publish reliable data and information,but the
author and the publisher cannot assume responsibility for the validity of all materials or for the
consequences 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, or by any
information 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 CRC
Press 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.
© 2005 by Issac Elishakoff
No claim to original U.S. Government works
International Standard Book Number 0-8493-2892-6
Library of Congress Card Number 2004051927
Printed in the United States of America 1234567890
Printed on acid-free paper
© 2005 by Issac Elishakoff
Visit the CRC Press Web site at www.crcpress.com
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Dedicated to IFMA, Institute Français de Mécanique Avancée, France whose
superbly educated engineers have been most instrumental in helping to bring
this book to fruition; without their dedication it would take many more years
to achieve this humble goal of communicating to the engineers,scientists and
students the infinite number of closed-form solutions.
Isaac Elishakoff
© 2005 by Issac Elishakoff
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Other Books from Professor Isaac Elishakoff
Textbook
I. Elishakoff, Probabilistic Methods in the Theory of Structures, Wiley-–Interscience,New York,1983
(second edition: Dover Publications, Mineola, New York, 1999).
Monographs
Y. Ben-Haimand I.Elishakoff, Convex Models of Uncertainty in Applied Mechanics, Elsevier Science
Publishers,Amsterdam, 1990.
G. Cederbaum,I. Elishakoff,J. Aboudi and L. Librescu, Random Vibrations and Reliability of
Composite Structures, Technomic Publishers, Lancaster, PA,1992.
I. Elishakoff, Y. K. Lin and L. P. Zhu, Probabilistic and Convex Modeling of Acoustically Excited
Structures, Elsevier Science Publishers, Amsterdam, 1994.
I. Elishakoff, The Courage to Challenge, 1st Books Library, 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, Oxford
University Press,2003.
I. Elishakoff, Safety Factors and Reliability: Friends or Foes?, Kluwer Academic Publishers,
Dordrecht,2004.
Edited Volumes
I. Elishakoff and H. Lyon,(eds.), Random Vibration - Status and Recent Developments,Elsevier
Science Publishers, Amsterdam, 1986.
I. Elishakoff and H. Irretier,(eds.), Refined Dynamical Theories of Beams, Plates and Shells, and Their
Applications, Springer Verlag, Berlin, 1987.
I. Elishakoff, J. Arbocz, Ch. D. Babcock,Jr. and A. Libai,(eds.), Buckling of Structures-Theory and
Experiment, Elsevier Science Publishers, Amsterdam, 1988.
S. T. Ariaratnam,G. Schuëller and I. Elishakoff,(eds.), Stochastic Structural Dynamics-Progress in
Theory and Applications, Elsevier Applied Science Publishers, London, 1988.
C. Mei,H. F. Wolfe and I. Elishakoff,(eds.), Vibration and Behavior of Composite Structures,ASME
Press,New York,1989.
F. Casciati,I. Elishakoff and J. B. Roberts,(eds.), Nonlinear Structural Systems under Random
Conditions, Elsevier Science Publishers, 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 on
Mechanics, ASME Press,New York,1990.
Y. K. Lin and I. Elishakoff,(eds.), Stochastic Structural Dynamics-New Theoretical Developments,
Springer Verlag,Berlin,1991.
I. Elishakoff and Y. K. Lin,(eds.), Stochastic Structural Dynamics-New Applications, 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.
© 2005 by Issac Elishakoff
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Contents
Foreword xv
Prologue 1
Chapter 1 Introduction: Review of Direct, Semi-inverse and Inverse
Eigenvalue Problems 7
1.1 Introductory Remarks 7
1.2 Vibration of Uniform Homogeneous Beams 8
1.3 Buckling of Uniform Homogeneous Columns 10
1.4 Some Exact Solutions for the Vibration of
Non-uniform Beams 19
1.4.1 The Governing Differential Equation 21
1.5 Exact Solution for Buckling of Non-uniform
Columns 24
1.6 Other Direct Methods (FDM,FEM,DQM) 28
1.7 Eisenberger’s Exact Finite Element Method 30
1.8 Semi-inverse or Semi-direct Methods 35
1.9 Inverse Eigenvalue Problems 43
1.10 Connection to the Work by Zyczkowski and Gajewski 50 ˙
1.11 Connection to Functionally Graded Materials 52
1.12 Scope of the Present Monograph 53
Chapter 2 Unusual Closed-Form Solutions in Column Buckling 55
2.1 New Closed-Form Solutions for Buckling of a
Variable Flexural Rigidity Column 55
2.1.1 Introductory Remarks 55
2.1.2 Formulation of the Problem 56
2.1.3 Uncovered Closed-Form Solutions 57
2.1.4 Concluding Remarks 65
2.2 Inverse Buckling Problem for Inhomogeneous
Columns 65
2.2.1 Introductory Remarks 65
2.2.2 Formulation of the Problem 65
2.2.3 Column Pinned at Both Ends 66
2.2.4 Column Clamped at Both Ends 68
2.2.5 Column Clamped at One End and Pinned at
the Other 69
2.2.6 Concluding Remarks 70
vii
© 2005 by Issac Elishakoff
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viii Eigenvalues of Inhomogeneous Structures
2.3 Closed-Form Solution for the Generalized Euler
Problem 74
2.3.1 Introductory Remarks 74
2.3.2 Formulation of the Problem 76
2.3.3 Column Clamped at Both Ends 79
2.3.4 Column Pinned at One End and Clamped at
the Other 79
2.3.5 Column Clamped at One End and Free at the
Other 81
2.3.6 Concluding Remarks 83
2.4 Some Closed-Form Solutions for the Buckling of
Inhomogeneous Columns under Distributed
Variable Loading 84
2.4.1 Introductory Remarks 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 and
Clamped at the Other 100
2.4.6 Concluding Remarks 105
Chapter 3 Unusual Closed-Form Solutions for Rod Vibrations 107
3.1 Reconstructing the Axial Rigidity of a Longitudinally
Vibrating Rod by its Fundamental Mode Shape 107
3.1.1 Introductory Remarks 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 Varying
Density 112
3.1.5 Inhomogeneous Rods with Parabolically
Varying Inertial Coefficient 114
3.1.6 Rod with General Variation of Inertial
Coefficient (m > 2) 115
3.1.7 Concluding Remarks 118
3.2 The Natural Frequency of an Inhomogeneous Rod
may be Independent of Nodal Parameters 120
3.2.1 Introductory Remarks 120
3.2.2 The Nodal Parameters 121
3.2.3 Mode with One Node: Constant Inertial
Coefficient 124
3.2.4 Mode with Two Nodes: Constant Density 127
3.2.5 Mode with One Node: Linearly Varying
Material Coefficient 129
3.3 Concluding Remarks 131
© 2005 by Issac Elishakoff
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Contents ix
Chapter 4 Unusual Closed-Form Solutions for Beam Vibrations 135
4.1 Apparently First Closed-Form Solutions for
Frequencies of Deterministically and/or
Stochastically Inhomogeneous Beams
(Pinned–Pinned Boundary Conditions) 135
4.1.1 Introductory Remarks 135
4.1.2 Formulation of the Problem 136
4.1.3 Boundary Conditions 137
4.1.4 Expansion of the Differential Equation 138
4.1.5 Compatibility Conditions 139
4.1.6 Specified Inertial Coefficient Function 140
4.1.7 Specified Flexural Rigidity Function 141
4.1.8 Stochastic Analysis 144
4.1.9 Nature of Imposed Restrictions 151
4.1.10 Concluding Remarks 151
4.2 Apparently First Closed-Form Solutions for
Inhomogeneous Beams (Other Boundary Conditions) 152
4.2.1 Introductory Remarks 152
4.2.2 Formulation of the Problem 153
4.2.3 Cantilever Beam 154
4.2.4 Beam that is Clamped at Both Ends 163
4.2.5 Beam Clamped at One End and Pinned at the
Other 165
4.2.6 Random Beams with Deterministic Frequencies 168
4.3 Inhomogeneous Beams that may Possess a
Prescribed Polynomial Second Mode 175
4.3.1 Introductory Remarks 175
4.3.2 Basic Equation 180
4.3.3 A Beam with Constant Mass Density 182
4.3.4 A Beam with Linearly Varying Mass Density 185
4.3.5 A Beam with Parabolically Varying Mass
Density 190
4.4 Concluding Remarks 199
Chapter 5 Beams and Columns with Higher-Order Polynomial
Eigenfunctions 203
5.1 Family of Analytical Polynomial Solutions for
Pinned Inhomogeneous Beams. Part 1: Buckling 203
5.1.1 Introductory Remarks 203
5.1.2 Choosing a Pre-selected Mode Shape 204
5.1.3 Buckling of the Inhomogeneous Column
under an Axial Load 205
5.1.4 Buckling of Columns under an Axially
Distributed Load 209
5.1.5 Concluding Remarks 224
© 2005 by Issac Elishakoff
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x Eigenvalues of Inhomogeneous Structures
5.2 Family of Analytical Polynomial Solutions for
Pinned Inhomogeneous Beams. Part 2: Vibration 225
5.2.1 Introductory Comments 225
5.2.2 Formulation of the Problem 226
5.2.3 Basic Equations 227
5.2.4 Constant Inertial Coefficient (m = 0) 228
5.2.5 Linearly Varying Inertial Coefficient (m = 1) 230
5.2.6 Parabolically Varying Inertial Coefficient
(m = 2) 231
5.2.7 Cubic Inertial Coefficient (m = 3) 236
5.2.8 Particular Case m = 4 239
5.2.9 Concluding Remarks 242
Chapter 6 Influence of Boundary Conditions on Eigenvalues 249
6.1 The Remarkable Nature of Effect of Boundary
Conditions on Closed-Form Solutions for Vibrating
Inhomogeneous Bernoulli–Euler Beams 249
6.1.1 Introductory Remarks 249
6.1.2 Construction of Postulated Mode Shapes 250
6.1.3 Formulation of the Problem 251
6.1.4 Closed-Form Solutions for the Clamped–Free
Beam 252
6.1.5 Closed-Form Solutions for the
Pinned–Clamped Beam 271
6.1.6 Closed-Form Solutions for the
Clamped–Clamped Beam 289
6.1.7 Concluding Remarks 308
Chapter 7 Boundary Conditions Involving Guided Ends 309
7.1 Closed-Form Solutions for the Natural Frequency for
Inhomogeneous Beams with One Guided Support
and One Pinned Support 309
7.1.1 Introductory Remarks 309
7.1.2 Formulation of the Problem 310
7.1.3 Boundary Conditions 310
7.1.4 Solution of the Differential Equation 311
7.1.5 The Degree of the Material Density is Less
than Five 312
7.1.6 General Case: Compatibility Conditions 318
7.1.7 Concluding Comments 322
7.2 Closed-Form Solutions for the Natural Frequency for
Inhomogeneous Beams with One Guided Support
and One Clamped Support 322
7.2.1 Introductory Remarks 322
7.2.2 Formulation of the Problem 323
7.2.3 Boundary Conditions 323
7.2.4 Solution of the Differential Equation 324
© 2005 by Issac Elishakoff
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Contents xi
7.2.5 Cases of Uniform and Linear Densities 325
7.2.6 General Case: Compatibility Condition 327
7.2.7 Concluding Remarks 329
7.3 Class of Analytical Closed-Form Polynomial
Solutions for Guided–Pinned Inhomogeneous Beams 330
7.3.1 Introductory Remarks 330
7.3.2 Formulation of the Problem 330
7.3.3 Constant Inertial Coefficient (m = 0) 332
7.3.4 Linearly Varying Inertial Coefficient (m = 1) 333
7.3.5 Parabolically Varying Inertial Coefficient
(m = 2) 335
7.3.6 Cubically Varying Inertial Coefficient (m = 3) 337
7.3.7 Coefficient Represented by a Quartic
Polynomial (m = 4) 338
7.3.8 General Case 340
7.3.9 Particular Cases Characterized by the
Inequality n ≥ m + 2 349
7.3.10 Concluding Remarks 364
7.4 Class of Analytical Closed-Form Polynomial
Solutions for Clamped–Guided Inhomogeneous
Beams 364
7.4.1 Introductory Remarks 364
7.4.2 Formulation of the Problem 364
7.4.3 General Case 366
7.4.4 Constant Inertial Coefficient (m = 0) 376
7.4.5 Linearly Varying Inertial Coefficient (m = 1) 377
7.4.6 Parabolically Varying Inertial Coefficient
(m = 2) 378
7.4.7 Cubically Varying Inertial Coefficient (m = 3) 380
7.4.8 Inertial Coefficient Represented as a
Quadratic (m = 4) 385
7.4.9 Concluding Remarks 392
Chapter 8 Vibration of Beams in the Presence of an Axial Load 395
8.1 Closed–Form Solutions for Inhomogeneous
Vibrating Beams under Axially Distributed Loading 395
8.1.1 Introductory Comments 395
8.1.2 Basic Equations 397
8.1.3 Column that is Clamped at One End and Free
at the Other 398
8.1.4 Column that is Pinned at its Ends 402
8.1.5 Column that is clamped at its ends 407
8.1.6 Column that is Pinned at One End and
Clamped at the Other 411
8.1.7 Concluding Remarks 416
© 2005 by Issac Elishakoff
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xii Eigenvalues of Inhomogeneous Structures
8.2 A Fifth-Order Polynomial that Serves as both the
Buckling and Vibration Modes of an Inhomogeneous
Structure 417
8.2.1 Introductory Comments 417
8.2.2 Formulation of the Problem 419
8.2.3 Basic Equations 421
8.2.4 Closed-Form Solution for the Pinned Beam 422
8.2.5 Closed-Form Solution for the Clamped–Free
Beam 431
8.2.6 Closed-Form Solution for the
Clamped–Clamped Beam 442
8.2.7 Closed-Form Solution for the Beam that is
Pinned at One End and Clamped at the Other 452
8.2.8 Concluding Remarks 460
Chapter 9 Unexpected Results for a Beam on an Elastic Foundation
or with Elastic Support 461
9.1 Some Unexpected Results in the Vibration of
Inhomogeneous Beams on an Elastic Foundation 461
9.1.1 Introductory Remarks 461
9.1.2 Formulation of the Problem 462
9.1.3 Beam with Uniform Inertial Coefficient,
Inhomogeneous Elastic Modulus and Elastic
Foundation 463
9.1.4 Beams with Linearly Varying Density,
Inhomogeneous Modulus and Elastic
Foundations 468
9.1.5 Beams with Varying Inertial Coefficient
Represented as an mth Order Polynomial 475
9.1.6 Case of a Beam Pinned at its Ends 480
9.1.7 Beam Clamped at the Left End and Free at the
Right End 486
9.1.8 Case of a Clamped–Pinned Beam 491
9.1.9 Case of a Clamped–Clamped Beam 496
9.1.10 Case of a Guided–Pinned Beam 501
9.1.11 Case of a Guided–Clamped Beam 510
9.1.12 Cases Violated in Eq. (9.99) 515
9.1.13 Does the Boobnov–Galerkin Method
Corroborate the Unexpected Exact Results? 517
9.1.14 Concluding Remarks 521
9.2 Closed-Form Solution for the Natural Frequency of
an Inhomogeneous Beam with a Rotational
Spring 522
9.2.1 Introductory Remarks 522
9.2.2 Basic Equations 522
9.2.3 Uniform Inertial Coefficient 523
9.2.4 Linear Inertial Coefficient 526
© 2005 by Issac Elishakoff
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Contents xiii
9.3 Closed-Form Solution for the Natural Frequency of
an Inhomogeneous Beam with a Translational Spring 528
9.3.1 Introductory Remarks 528
9.3.2 Basic Equations 529
9.3.3 Constant Inertial Coefficient 531
9.3.4 Linear Inertial Coefficient 533
Chapter 10 Non-Polynomial Expressions for the Beam’s Flexural
Rigidity for Buckling or Vibration 537
10.1 Both the Static Deflection and Vibration Mode of
a Uniform Beam Can Serve as Buckling Modes of
a Non-uniform Column 537
10.1.1 Introductory Remarks 537
10.1.2 Basic Equations 538
10.1.3 Buckling of Non-uniform Pinned Columns 539
10.1.4 Buckling of a Column under its Own Weight 542
10.1.5 Vibration Mode of a Uniform Beam as a
Buckling Mode of a Non-uniform Column 544
10.1.6 Non-uniform Axially Distributed Load 545
10.1.7 Concluding Remarks 547
10.2 Resurrection of the Method of Successive
Approximations to Yield Closed-Form Solutions for
Vibrating Inhomogeneous Beams 548
10.2.1 Introductory Comments 548
10.2.2 Evaluation of the Example by Birger and
Mavliutov 551
10.2.3 Reinterpretation of the Integral Method for
Inhomogeneous Beams 553
10.2.4 Uniform Material Density 555
10.2.5 Linearly Varying Density 557
10.2.6 Parabolically Varying Density 559
10.2.7 Can Successive Approximations Serve as
Mode Shapes? 563
10.2.8 Concluding Remarks 563
10.3 Additional Closed-Form Solutions for
Inhomogeneous Vibrating Beams by the Integral
Method 566
10.3.1 Introductory Remarks 566
10.3.2 Pinned–Pinned Beam 567
10.3.3 Guided–Pinned Beam 575
10.3.4 Free–Free Beam 582
10.3.5 Concluding Remarks 590
Chapter 11 Circular Plates 591
11.1 Axisymmetric Vibration of Inhomogeneous Clamped
Circular Plates: an Unusual Closed-Form Solution 591
11.1.1 Introductory Remarks 591
11.1.2 Basic Equations 593
© 2005 by Issac Elishakoff
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xiv Eigenvalues of Inhomogeneous Structures
11.1.3 Method of Solution 594
11.1.4 Constant Inertial Term (m = 0) 594
11.1.5 Linearly Varying Inertial Term (m = 1) 595
11.1.6 Parabolically Varying Inertial Term (m = 2) 596
11.1.7 Cubic Inertial Term (m = 3) 598
11.1.8 General Inertial Term (m ≥ 4) 600
11.1.9 Alternative Mode Shapes 601
11.2 Axisymmetric Vibration of Inhomogeneous Free
Circular Plates: An Unusual,Exact,Closed-Form
Solution 604
11.2.1 Introductory Remarks 604
11.2.2 Formulation of the Problem 605
11.2.3 Basic Equations 605
11.2.4 Concluding Remarks 607
11.3 Axisymmetric Vibration of Inhomogeneous Pinned
Circular Plates: An Unusual,Exact,Closed-Form
Solution 607
11.3.1 Basic Equations 607
11.3.2 Constant Inertial Term (m = 0) 608
11.3.3 Linearly Varying Inertial Term (m = 1) 609
11.3.4 Parabolically Varying Inertial Term (m = 2) 610
11.3.5 Cubic Inertial Term (m = 3) 612
11.3.6 General Inertial Term (m ≥ 4) 614
11.3.7 Concluding Remarks 616
Epilogue 617
Appendices 627
References 653
© 2005 by Issac Elishakoff
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Foreword
This is a most remarkable and thorough review of the efforts that have been
made to find closed-form solutions in the vibration and buckling of all manner
of elastic rods,beams,columns and plates. The author is particularly,but not
exclusively,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 author’s solutions to particular problems will be very valuable for testing the validity and accuracy of various numerical techniques.
Moreover,the study is of great academic interest,and is clearly a labor of
love. The author is to be congratulated on this work,which is bound to be of
considerable value to all interested in research in this area.
Dr. H.D. Conway
Professor Emeritus
Department of Theoretical and Applied Mechanics
Cornell University
It is generally believed that closed-form solutions exist for only a relatively
few,very simple cases of bars,beams,columns,and plates. This monograph
is living proof that there are,in fact,not just a few such solutions. Even in the
current age of powerful numerical techniques and high-speed, large-capacity
computers, there are a number of important uses for closed-form solutions:
• for preliminary design (often optimal)
• as bench-mark solutions for evaluating the accuracy of approximate
and numerical solutions
• to gain more physical insight into the roles played by the various
geometric 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,he
has explained in a very interesting fashion the history behind many of the
solutions.
Dr. Charles W. Bert
Benjamin H. Perkinson Professor Emeritus
Aerospace and Mechanical Engineering
The University of Oklahoma xv
© 2005 by Issac Elishakoff