Modern Trends in Structural and Solid Mechanics 2 E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book comprised of three separate volumes presents the recent developments and research discoveries in structural and solid mechanics; it is dedicated to Professor Isaac Elishakoff. This second volume is devoted to the vibrations of solid and structural members. Modern Trends in Structural and Solid Mechanics 2 has broad scope, covering topics such as: exact and approximate vibration solutions of rods, beams, membranes, plates and three-dimensional elasticity problems, Bolotin s dynamic edge effect, the principles of plate theories in dynamics, nano- and microbeams, nonlinear dynamics of shear extensible beams, the vibration and aeroelastic stability behavior of cellular beams, the dynamic response of elastoplastic softening oscillators, the complex dynamics of hysteretic oscillators, bridging waves, and the three-dimensional propagation of waves. This book is intended for graduate students and researchers in the field of theoretical and applied mechanics.
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Veröffentlichungsjahr: 2021
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Table of Contents
Cover
Title Page
Copyright
Preface: Short Bibliographical Presentation of Prof. Isaac Elishakoff
1 Bolotin’s Dynamic Edge Effect Method Revisited (Review)
1.1. Introduction
1.2. Toy problem: natural beam oscillations
1.3. Linear problems solved
1.4. Generalization for the nonlinear case
1.5. DEEM and variational approaches
1.6. Quasi-separation of variables and normal modes of nonlinear oscillations of continuous systems
1.7. Short-wave (high-frequency) asymptotics. Possible generalizations of DEEM
1.8. Conclusion: DEEM, highly recommended
1.9. Acknowledgments
1.10. Appendix
1.11. References
2 On the Principles to Derive Plate Theories
2.1. Introduction
2.2. Some historical remarks
2.3. Possibilities to formulate plate theories
2.4. Shear correction
2.5. Conclusion
2.6. References
3 A Softening–Hardening Nanomechanics Theory for the Static and Dynamic Analyses of Nanorods and Nanobeams: Doublet Mechanics
3.1. Introduction
3.2. Doublet mechanics formulation
3.3. Governing equations
3.4. Analytical solutions
3.5. Numerical results
3.6. Conclusion
3.7. References
4 Free Vibration of Micro-Beams and Frameworks Using the Dynamic Stiffness Method and Modified Couple Stress Theory
4.1. Introduction
4.2. Formulation of the potential and kinetic energies
4.3. Derivation of the governing differential equations
4.4. Development of the dynamic stiffness matrix
4.5. Application of the Wittrick–Williams algorithm
4.6. Numerical results and discussion
4.7. Conclusion
4.8. Acknowledgments
4.9. References
5 On the Geometric Nonlinearities in the Dynamics of a Planar Timoshenko Beam
5.1. Introduction
5.2. The geometrically exact planar Timoshenko beam
5.3. The asymptotic solution
5.4. The importance of nonlinear terms
5.5. Simplified models
5.6. Conclusion
5.7. References
6 Statics, Dynamics, Buckling and Aeroelastic Stability of Planar Cellular Beams
6.1. Introduction
6.2. Continuous models of planar cellular structures
6.3. The grid beam
6.4. Buckling
6.5. Dynamics
6.6. Aeroelastic stability
6.7. References
7 Collapse Limit of Structures under Impulsive Loading via Double Impulse Input Transformation
7.1. Introduction
7.2. Collapse limit corresponding to the critical timing of second impulse
7.3. Classification of collapse patterns in non-critical case
7.4. Analysis of collapse limit using energy balance law
7.5. Verification of proposed collapse limit via time-history response analysis
7.6. Conclusion
7.7. References
8 Nonlinear Dynamics and Phenomena in Oscillators with Hysteresis
8.1. Introduction
8.2. Hysteresis model and SDOF response to harmonic excitation
8.3. 2DOF hysteretic systems
8.4. Nonlinear modal interactions in 2DOF hysteretic systems
8.5. Conclusion
8.6. Acknowledgments
8.7. Appendix: Mechanical characteristics of SDOF and 2DOF systems
8.8. References
9 Bridging Waves on a Membrane: An Approach to Preserving Wave Patterns
9.1. Introduction
9.2. Problem statement
9.3. Homogenized bridge
9.4. Internal reflections
9.5. Discrete bridge
9.6. Net bridge
9.7. Concluding remarks
9.8. Acknowledgments
9.9. References
10 Dynamic Soil Stiffness of Foundations Supported by Layered Half-Space
10.1. Introduction
10.2. Generation of dynamic soil stiffness
10.3. Numerical examples of the generation of dynamic soil stiffness
10.4. Numerical examples of the generation of FRS
10.5. Conclusion
10.6. References
List of Authors
Index
Summary of Volume 1
Summary of Volume 3
End User License Agreement
List of Illustrations
Preface: Short Bibliographical Presentation of Prof. Isaac Elishakoff
Figure P.1.
Prof. Isaac Elishakoff
Figure P.2.
Elishakoff in middle school in the city of Sukhumi, Georgia
Figure P.3. Elishakoff just before acceptance to university. Photo taken in Sukh...
Figure P.4. Public PhD defense, Moscow Power Engineering Institute and State Uni...
Figure P.5. Elishakoff with Bolotin (middle), member of the Russian Academy of S...
Figure P.6. Prof. Elishakoff presenting a book to Prof. J. Singer, Technion’s Pr...
Figure P.7. Elishakoff having received the William B. Johnson Inter- Professiona...
Figure P.8. Inauguration as the Frank Freimann Visiting Professor of Aerospace a...
Figure P.9. Prof. Elishakoff with Prof. Warner Tjardus Koiter, Delft University ...
Figure P.10. Elishakoff and his colleagues during the AIAA SDM Conference at Pal...
Figure P.11. Elishakoff with his wife, Esther Elisha, M.D., during an ASME award...
Chapter 1
Figure 1.1. Curve 1 corresponds to the rapidly oscillating solution in the inner...
Chapter 3
Figure 3.1.
Arrangement of atoms in a periodic nanostructure
Figure 3.2. Dimensionless displacement of nanorods with dimensionless rod length...
Figure 3.3. Frequency parameter of nanorods with rod length for the first three ...
Figure 3.4. Frequency parameter of nanorods with mode number. For a color versio...
Figure 3.5. Frequency parameters of nanorods with doublet separation distance (η...
Figure 3.6. Wave frequency of nanorods with wavenumber. For a color version of t...
Figure 3.7. Phase velocity of nanorods with wavenumber. For a color version of t...
Figure 3.8. Group velocity of nanorods with wavenumber. For a color version of t...
Figure 3.9. Dimensionless displacement of nanobeams with dimensionless beam leng...
Figure 3.10. Critical buckling load of nanobeams with beam length. For a color v...
Figure 3.11. Frequency parameter of nanobeams with beam length for the first thr...
Figure 3.12.
Wave frequency of nanobeams with wavenumber
Figure 3.13.
Phase velocity of nanobeams with wavenumber
Figure 3.14.
Group velocity of nanobeams with wavenumber
Chapter 4
Figure 4.1.
Coordinate system of a micro-beam
Figure 4.2.
Sign convention for the positive axial force
f,
the shear force
s an...
Figure 4.3.
Boundary conditions for the axial displacement and the axial force
Figure 4.4. Boundary conditions for the bending displacement, the bending rotati...
Figure 4.5.
Amplitudes of displacements and forces at the ends of the micro-beam
Figure 4.6.
Alignment of the local
(OXY)
and global
()
coordinate systems
Figure 4.7. Natural frequencies and mode shapes of a micro-beam with the simply ...
Figure 4.8. Natural frequencies and mode shapes of a micro-beam with the cantile...
Figure 4.9. Natural frequencies and mode shapes of a micro-beam with the clamped...
Figure 4.10.
Effect of the material length scale parameter
l/h on the fundamenta...
Figure 4.11.
An L-shaped micro-frame (
L
=20h
, b
=2h
, h
=17.6 μm)
Figure 4.12.
Mode shapes of an L-shaped micro-frame for
l/h
= 1
Figure 4.13.
A micro portal frame (L=20h, b=2h, h=17.6 μm)
Figure 4.14.
Mode shapes of a micro portal frame for
l/h
= 1
Chapter 5
Figure 5.1. The nonlinear terms in the axial direction. a) Constant terms; b) te...
Figure 5.2. The nonlinear terms in the transversal direction. a) Terms proportio...
Figure 5.3. The nonlinear terms in the rotational direction. a) Terms proportion...
Figure 5.4.
a) ω
0
and b) ω
2
for varying l. n
= 1,
κ
h
= 0
Figure 5.5. The L2 norm percentage of the various nonlinear terms in the constan...
Figure 5.6. The L2 norm percentage of the various nonlinear terms in the part pr...
Figure 5.7. The L2 norm percentage of the various nonlinear terms in the part pr...
Figure 5.8. The L2 norm percentage of the various nonlinear terms in the part pr...
Figure 5.9. The L2 norm percentage of the various nonlinear terms in the part pr...
Figure 5.10. The L2 norm percentage of the various nonlinear terms in the part p...
Figure 5.11. The L2 norm percentage of the various nonlinear terms in a) the con...
Figure 5.12. The L2 norm percentage of the various nonlinear terms in a) the par...
Figure 5.13. The L2 norm percentage of the various nonlinear terms in a) the par...
Figure 5.14. The L2 norm percentage of the various nonlinear terms in a) the con...
Figure 5.15. The L2 norm percentage of the various nonlinear terms in a) the con...
Figure 5.16. The L2 norm percentage of the various nonlinear terms in a) the par...
Figure 5.17. The L2 norm percentage of the various nonlinear terms in a) the par...
Figure 5.18.
The L
2
norm percentage of the various nonlinear terms in a) the par...
Figure 5.19.
The L
2
norm percentage of the various nonlinear terms in a) the par...
Chapter 6
Figure 6.1. Grid beam. For a color version of this figure, see www.iste.co.uk/ch...
Figure 6.2. Single cell of the grid beam and its deformation modes for two diffe...
Figure 6.3. Static response of a grid beam to lateral forces: (a) rigid transver...
Figure 6.4. Buckling of a uniformly compressed grid beam (Ferretti (2018)): (a) ...
Figure 6.5.
Lateral displacements
for the first three modes (n
= 25), continuo...
Figure 6.6. Base-isolated tower building under wind flow: (a) cellular beam, (b)...
Figure 6.7. Critical wind velocity of the base-isolated beam: (a) versus the sti...
Chapter 7
Figure 7.1. Comparison between the conventional approach and the innovative appr...
Figure 7.2. Critical response, derived by the innovative approach, positioned in...
Figure 7.3. SDOF elastic–plastic system with negative post-yield stiffness. For ...
Figure 7.4. Elastic–plastic collapse response of the SDOF system subjected to cr...
Figure 7.5. Energy balance law for the elastic–plastic collapse response of the ...
Figure 7.6. Collapse limit input velocity for various post-yield stiffness ratio...
Figure 7.7. Modeling of the main part of a near-fault ground motion into one-cyc...
Figure 7.8. Displacement and restoring-force responses under Rinaldi Sta. FN wit...
Figure 7.9. Trajectories in restoring-force characteristics for CASEs-A, B, C (H...
Figure 7.10. Three cases (A, B, C) for three input velocity levels and four case...
Figure 7.11. Restoring-force characteristics for Collapse Patterns 1’–4’. For a ...
Figure 7.12. Collapse Pattern 1’ (CASE-A): (a) Restoring-force characteristic an...
Figure 7.13. Collapse limit input velocity for various intervals of two impulses...
Figure 7.14. Response analysis simulation for various combinations of input velo...
Chapter 8
Figure 8.1. (a) SDOF Bouc–Wen oscillator; (b) first loading branch; and (c) rest...
Figure 8.2. SDOF oscillator: frequency response curves for varying forcing ampli...
Figure 8.3.
Response amplitude versus resonance frequency: (a)
and (b)
. For ...
Figure 8.4. 2DOF system: (a) top-hysteresis and (b) base-hysteresis configuratio...
Figure 8.5. 2DOF top configuration close to a (1:1) internal resonance (ω2A/ω1A ...
Figure 8.6. 2DOF top-hysteresis oscillator close to a (3:1) internal resonance ω...
Figure 8.7. 2DOF base-hysteresis configuration close to a (3:1) resonance: (a) F...
Figure 8.8. 2DOF base-hysteresis configuration close to (3:1) resonance for IA1 ...
Figure 8.9. 2DOF base-hysteresis oscillator close to (2:1) internal resonance (ω...
Chapter 9
Figure 9.1. Schematic of an infinite elastic membrane with periodically inserted...
Figure 9.2. Comparisons of the distributed delta function δh(x) for different va...
Figure 9.3. Ratio of the amplitude of the overall transmission to the initial wa...
Figure 9.4. Ratio of the amplitude of the overall transmission to the initial wa...
Figure 9.5. Ratios of the amplitudes of successive reflections [9.37] and transm...
Figure 9.6. Ratios of the amplitudes of the overall reflection [9.41] and transm...
Figure 9.7. Schematic of an infinite elastic membrane with a square lattice of s...
Chapter 10
Figure 10.1. A horizontal soil layer under three-dimensional point loads. For a ...
Figure 10.2. Layered half-space under three-dimensional forces. For a color vers...
Figure 10.3. Foundation-site model and discretization of foundation. For a color...
Figure 10.4. Flexible foundation nodes and discretization of foundation. For a c...
Figure 10.5. Foundation-site models and discretization of foundations. For a col...
Figure 10.6. Dynamic soil stiffness of the square foundation on a layer and half...
Figure 10.7. Dynamic soil stiffness of the circular foundation on a layer and ha...
Figure 10.8. Dynamic soil stiffness of the circular foundation on half-space. Fo...
Figure 10.9. Dynamic soil stiffness of circular foundation on a layer and half-s...
Figure 10.10. Primary and secondary systems in a nuclear power plant. For a colo...
Figure 10.11. Dynamic soil stiffness of a reactor building. For a color version ...
Figure 10.12.
Horizontal transfer matrix of a reactor building
Figure 10.13.
FLIRS of a reactor building
Figure 10.14. Comparison of FRS at node 4 at the reactor building. For a color v...
List of Tables
Chapter 1
Table 1.1. Comparison of frequencies obtained using various approximation method...
Chapter 4
Table 4.1.
Natural frequencies of a simply supported (S–S) micro-beam
Table 4.2.
Natural frequencies of a cantilever (C–F) micro-beam
Table 4.3.
Natural frequencies of a clamped–clamped (C–C) micro-beam
Table 4.4.
Natural frequencies of an L-shaped micro-frame
Table 4.5.
Natural frequencies of a micro portal frame
Chapter 5
Table 5.1. The L2 norm percentage of the various nonlinear terms in the axial eq...
Table 5.2.
The L
2
norm percentage of the various nonlinear terms in the transver...
Table 5.3. The L2 norm percentage of the various nonlinear terms in the rotation...
Chapter 8
Table 8.1.
Mechanical characteristics of SDOF systems
Table 8.2.
Mechanical characteristics of 2DOF systems
Chapter 9
Table 9.1. Values of ϵ [9.65] for a Jacobi theta forcing [9.51] on a bridge with...
Chapter 10
Table 10.1.
Rocking and torsional stiffness in the reactor building case
Guide
Cover
Table of Contents
Title Page
Copyright
Preface: Short Bibliographical Presentation of Prof. Isaac Elishakoff
Begin Reading
List of Authors
Index
Summary of Volume 1
Summary of Volume 3
Other titles from ISTE in Health Engineering and Society
End User License Agreement
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Series Editor
Noël Challamel
Modern Trends in Structural and Solid Mechanics 2
Vibrations
Edited by
Noël Challamel
Julius Kaplunov
Izuru Takewaki
First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2021
The rights of Noël Challamel, Julius Kaplunov and Izuru Takewaki to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2021932076
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-715-6
PrefaceShort Bibliographical Presentation of Prof. Isaac Elishakoff
This book is dedicated to Prof. Isaac Elishakoff by his colleagues, friends and former students, on the occasion of his seventy-fifth birthday.
Figure P.1.Prof. Isaac Elishakoff
Prof. Isaac Elishakoff is an international leading authority across a broad area of structural mechanics, including dynamics and stability, optimization and anti-optimization, probabilistic methods, analysis of structures with uncertainty, refined theories, functionally graded material structures, and nanostructures. He was born in Kutaisi, Republic of Georgia, on February 9, 1944.
Figure P.2.Elishakoff in middle school in the city of Sukhumi, Georgia
Elishakoff holds a PhD in Dynamics and Strength of Machines from the Power Engineering Institute and Technical University in Moscow, Russia (Figure P.4 depicts the PhD defense of Prof. Isaac Elishakoff).
Figure P.3.Elishakoff just before acceptance to university. Photo taken in Sukhumi, Georgia
Figure P.4.Public PhD defense, Moscow Power Engineering Institute and State University; topic “Vibrational and Acoustical Fields in the Circular Cylindrical Shells Excited by Random Loadings”, and dedicated to the evaluation of noise levels in TU-144 supersonic aircraft
His supervisor was Prof. V. V. Bolotin (1926–2008), a member of the Russian Academy of Sciences (Figure P.5 shows Elishakoff with Bolotin some years later).
Figure P.5.Elishakoff with Bolotin (middle), member of the Russian Academy of Sciences, and Prof. Yukweng (Mike) Lin (left), member of the US National Academy of Engineering. Photo taken at Florida Atlantic University during a visit from Bolotin
Currently, Elishakoff is a Distinguished Research Professor in the Department of Ocean and Mechanical Engineering at Florida Atlantic University. Before joining the university, he taught for one year at Abkhazian University, Sukhumi in the Republic of Georgia, and 18 years at the Technion – Israel Institute of Technology in Haifa, where he became the youngest full professor at the time of his promotion (Figure P.6 shows Elishakoff presenting a book to Prof. Josef Singer, Technion’s former president).
Figure P.6.Prof. Elishakoff presenting a book to Prof. J. Singer, Technion’s President; right: Prof. A. Libai, Aerospace Engineering Department, Technion
Elishakoff has lectured at about 200 meetings and seminars, including about 60 invited, plenary or keynote lectures, across Europe, North and South America, the Middle East and the Far East.
Prof. Elishakoff has made vital and outstanding contributions in a number of areas in structural mechanics. In particular, he has analyzed random vibrations of homogeneous and composite beams, plates and shells, with special emphasis on the effects of refinements in structural theories and cross-correlations. Free structural vibrations have been tackled using a non-trivial generalization of Bolotin’s dynamic edge effect method. Nonlinear buckling has been investigated using a novel method, incorporating experimental analysis of imperfections. As a result, the fundamental concept of closing the gap – spanning the entire 20th century – between theory and practice in imperfection-sensitive structures has been proposed. Novel methods of evaluating structural reliability have been proposed, taking into account the error associated with various low-order approximations, as well as human error; innovative generalization of the stochastic linearization method has been advanced. A non-probabilistic theory for treating uncertainty in structural mechanics has been established. Dynamic stability of elastic and viscoelastic structures with imperfections has been studied. An improved, non-perturbative stochastic finite element method for structures has been developed. The list of Elishakoff’s remarkable research achievements goes on.
His research has been acknowledged by many awards and prizes. He is a member of the European Academy of Sciences and Arts, a Fellow of the American Academy of Mechanics and ASME, and a Foreign Member of the Georgian National Academy of Sciences. Elishakoff is also a recipient of the Bathsheva de Rothschild prize (1973) and the Worcester Reed Warner Medal of the American Society of Mechanical Engineers (2016).
Figure P.7.Elishakoff having received the William B. Johnson Inter- Professional Founders Award
Elishakoff is directly involved in numerous editorial activities. He serves as the book review editor of the “Journal of Shock and Vibration” and is currently, or has previously been an associate editor of the International Journal of Mechanics of Machines and Structures, Applied Mechanics Reviews, and Chaos, Solitons & Fractals. In addition, he is or has been on the editorial boards of numerous journals, for example Journal of Sound and Vibration, International Journal of Structural Stability and Dynamics, International Applied Mechanics and Computers & Structures. He also acts as a book series editor for Elsevier, Springer and Wiley.
Figure P.8.Inauguration as the Frank Freimann Visiting Professor of Aerospace and Mechanical Engineering; left: Rev. Theodore M. Hesburgh, President of the University of Notre Dame; right: Prof. Timothy O’Meara, Provost
Prof. Elishakoff has held prestigious visiting positions at top universities all over the world. Among them are Stanford University (S. P. Timoshenko Scholar); University of Notre Dame, USA (Frank M. Freimann Chair Professorship of Aerospace and Mechanical Engineering and Henry J. Massman, Jr. Chair Professorship of Civil Engineering); University of Palermo, Italy (Visiting Castigliano Distinguished Professor); Delft University of Technology, Netherlands (multiple appointments, including the W. T. Koiter Chair Professorship of the Mechanical Engineering Department – see Figure P.9); Universities of Tokyo and Kyoto, Japan (Fellow of the Japan Society for the Promotion of Science); Beijing University of Aeronautics and Astronautics, People’s Republic of China (Visiting Eminent Scholar); Technion, Haifa, Israel (Visiting Distinguished Professor); University of Southampton, UK (Distinguished Visiting Fellow of the Royal Academy of Engineering).
Figure P.9.Prof. Elishakoff with Prof. Warner Tjardus Koiter, Delft University of Technology (center), and Dr. V. Grishchak, of Ukraine (right)
Figure P.10.Elishakoff and his colleagues during the AIAA SDM Conference at Palm Springs, California in 2004; Standing, from right to left, are Prof. Elishakoff, the late Prof. Josef Singer and Dr. Giora Maymon of RAFAEL. Sitting is the late Prof. Avinoam Libai
Elishakoff has made a substantial contribution to conference organization. In particular, he participated in the organization of the Euro-Mech Colloquium on “Refined Dynamical Theories of Beams, Plates and Shells, and Their Applications” in Kassel, Germany (1986); the Second International Conference on Stochastic Structural Dynamics, in Boca Raton, USA (1990); “International Conference on Uncertain Structures” in Miami, USA and Western Caribbean (1996). He also coordinated four special courses at the International Centre for Mechanical Sciences (CISM), in Udine, Italy (1997, 2001, 2005, 2011).
Prof. Elishakoff has published over 540 original papers in leading journals and conference proceedings. He championed authoring, co-authoring or editing of 31 influential and extremely well-received books and edited volumes.
Here follows some praise of his work and books:
– “It was not until 1979, when Elishakoff published his reliability study … that a method has been proposed, which made it possible to introduce the results of imperfection surveys … into the analysis …” (Prof. Johann Arbocz, Delft University of Technology, The Netherlands,
Zeitschrift für Flugwissenschaften und Weltraumforschung
).
– “He has achieved world renown … His research is characterized by its originality and a combination of mathematical maturity and physical understanding which is reminiscent of von Kármán …” (Prof. Charles W. Bert, University of Oklahoma).
– “It is clear that Elishakoff is a world leader in his field … His outstanding reputation is very well deserved …” (Prof. Bernard Budiansky, Harvard University).
– “Professor Isaac Elishakoff … is subject-wise very much an all-round vibrationalist” (P. E. Doak, Editor in Chief,
Journal of Sound and Vibration
, University of Southampton, UK).
– “This is a beautiful book …” (Dr. Stephen H. Crandall, Ford Professor of Engineering, M.I.T.).
– “Das Buch ist in seiner Aufmachunghervorragendgestaltet und kannalsäusserstwertvolleErganzung … wäzmstensempfohlenwerden …” [The book’s appearance is perfectly designed and can be highly recommended as a valuable addition.] (Prof. Horst Försching, Institute of Aeroelasticity, Federal Republic of Germany,
Zeitschrift für Flugwissenschaften und Weltraumforschung
).
– “Because of you, Notre Dame is an even better place, a more distinguished University” (Prof. Rev. Theodore M. Hesburgh, President, University of Notre Dame).
– “It is an impressive volume …” (Prof. Warner T. Koiter, Delft University of Technology, The Netherlands).
– “This extremely well-written text, authored by one of the leaders in the field, incorporates many of these new applications … Professor Elishakoff’s techniques for developing the material are accomplished in a way that illustrates his deep insight into the topic as well as his expertise as an educator … Clearly, the second half of the text provides the basis for an excellent graduate course in random vibrations and buckling … Professor Elishakoff has presented us with an outstanding instrument for teaching” (Prof. Frank Kozin, Polytechnic Institute of New York,
American Institute of Aeronautics and Astronautics Journal
).
– “By far the best book on the market today …” (Prof. Niels C. Lind, University of Waterloo, Canada).
– “The book develops a novel idea … Elegant, exhaustive discussion … The study can be an inspiration for further research, and provides excellent applications in design …” (Prof. G. A. Nariboli,
Applied Mechanics Reviews
).
– “This volume is regarded as an advanced encyclopedia on random vibration and serves aeronautical, civil and mechanical engineers …” (Prof. Rauf Ibrahim, Wayne State University,
Shock and Vibration Digest
).
– “The book deals with a fundamental problem in Applied Mechanics and in Engineering Sciences: How the uncertainties of the data of a problem influence its solution. The authors follow a novel approach for the treatment of these problems … The book is written with clarity and contains original and important results for the engineering sciences …” (Prof. P. D. Panagiotopoulos, University of Thessaloniki, Greece and University of Aachen, Germany,
SIAM Review
).
– “The content should be of great interest to all engineers involved with vibration problems, placing the book well and truly in the category of an essential reference book …” (Prof. I. Pole,
Journal of the British Society for Strain Measurement
).
– “A good book; a different book … It is hoped that the success of this book will encourage the author to provide a sequel in due course …” (Prof. John D. Robson, University of Glasgow, Scotland, UK,
Journal of Sound and Vibration
).
– “The book certainly satisfies the need that now exists for a readable textbook and reference book …” (Prof. Masanobu Shinozuka, Columbia University).
– “[the] author ties together reliability, random vibration and random buckling … Well written … useful book …” (Dr. H. Saunders,
Shock and Vibration Digest
).
– “A very useful text that includes a broad spectrum of theory and application” (
Mechanical Vibration
, Prof. Haym Benaroya, Rutgers University).
– “A treatise on random vibration and buckling … The reviewer wishes to compliment the author for the completion of a difficult task in preparing this book on a subject matter, which is still developing on many fronts …” (Prof. James T. P. Yao, Texas A&M University,
Journal of Applied Mechanics
).
– “It seems to me a hard work with great result …” (Prof. Hans G. Natke, University of Hannover, Federal Republic of Germany).
– “The approach is novel and could dominate the future practice of engineering” (
The Structural Engineer
).
– “An excellent presentation … well written … all readers, students, and certainly reviewers should read this preface for its excellent presentation of the philosophy and raison d’être for this book. It is well written, with the material presented in an informational fashion as well as to raise questions related to unresolved … challenges; in the vernacular of film critics, ‘thumbs up’” (Dr. R. L. Sierakowski, U.S. Air Force Research Laboratory,
AIAA Journal
).
– “This substantial and attractive volume is a well-organized and superbly written one that should be warmly welcomed by both theorists and practitioners … Prof. Elishakoff, Li, and Starnes, Jr. have given us a jewel of a book, one done with care and understanding of a complex and essential subject and one that seems to have ably filled a gap existing in the present-day literature and practice” (
Current Engineering Practice
).
– “Most of the subjects covered in this outstanding book have never been discussed exclusively in the existing treatises … (
Ocean Engineering
).
– “The treatment is scholarly, having about 900 items in the bibliography and additional contributors in the writing of almost every chapter … This reviewer believes that
Non-Classical Problems in the Theory of Elastic Stability
should be a useful reference for researchers, engineers, and graduate students in aeronautical, mechanical, civil, nuclear, and marine engineering, and in applied mechanics” (
Applied Mechanics Reviews
).
– “What more can be said about this monumental work, other than to express admiration? … The study is of great academic interest, and is clearly a labor of love. The author is to be congratulated on this work …” (Prof. H. D. Conway, Department of Theoretical and Applied Mechanics, Cornell University).
– “This book … is prepared by Isaac Elishakoff, one of the eminent solid mechanics experts of the 20th century and the present one, and his distinguished coauthors, will be of enormous use to researchers, graduate students and professionals in the fields of ocean, naval, aerospace and mechanical engineers as well as other fields” (Prof. Patricio A. A. Laura, Prof. Carlos A. Rossit, Prof. Diana V. Bambill, Universidad Nacional del Sur, Argentina,
Ocean Engineering
).
– “This book is an outstanding research monograph … extremely well written, informative, highly original … great scholarly contribution …. There is no comparable book discussing the combination of optimization and anti-optimization … magnificent monograph …. This book, which certainly is written with love and passion, is the first of its kind in applied mechanics literature, and has the potential of having a revolutionary impact on both uncertainty analysis and optimization” (Prof. Izuru Takewaki, Kyoto University,
Engineering Structures
).
– “This book is a collection of a surprisingly large number of closed form solutions, by the author and by others, involving the buckling of columns and beams, and the vibration of rods, beams and circular plates. The structures are, in general, inhomogeneous. Many solutions are published here for the first time. The text starts with an instructive review of direct, semi-inverse, and inverse eigenvalue problems. Unusual closed form solutions of column buckling are presented first, followed by closed form solutions of the vibrations of rods. Unusual closed form solutions for vibrating beams follow. The influence of boundary conditions on eigenvalues is discussed. An entire chapter is devoted to boundary conditions involving guided ends. Effects of axial loads and of elastic foundations are presented in two separate chapters. The closed form solutions of circular plates concentrate on axisymmetric vibrations. The scholarly effort that produced this book is remarkable” (Prof. Werner Soedel, then Editor-in-Chief of
Journal Sound and Vibration
).
– “The field has been brilliantly presented in book form …” (Prof. Luis A. Godoy
et al
., Institute of Advanced Studies in Engineering and Technology, Science Research Council of Argentina and National University of Cordoba, Argentina,
Thin-Walled Structures
).
– “Elishakoff is one of the pioneers in the use of the probabilistic approach for studying imperfection-sensitive structures” (Prof. Chiara Bisagni and Dr. Michela Alfano, Delft University of Technology;
AIAA Journal
).
– “Recently, Elishakoff
et al
. presented an excellent literature review on the historical development of Timoshenko’s beam theory” (Prof. Zhenlei Chen
et al
.,
Journal of Building Engineering
).
Professor Isaac Elishakoff is the author or co-author of an impressive list of seminal books in the field of deterministic and non-deterministic mechanics, presented below.
Books by Elishakoff
Ben-Haim, Y. and Elishakoff, I. (1990). Convex Models of Uncertainty in Applied Mechanics. Elsevier, Amsterdam.
Cederbaum, G., Elishakoff, I., Aboudi, J., Librescu, L. (n.d.). Random Vibration and Reliability of Composite Structures. Technomic, Lancaster.
Elishakoff, I. (1983). Probabilistic Methods in the Theory of Structures. Wiley, New York.
Elishakoff, I. (1999). Probabilistic Theory of Structures. Dover Publications, New York.
Elishakoff, I. (2004). Safety Factors and Reliability: Friends or Foes? Kluwer Academic Publishers, Dordrecht.
Elishakoff, I. (2005). Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions of Semi-Inverse Problems. CRC Press, Boca Raton.
Elishakoff, I. (2014). Resolution of the Twentieth Century Conundrum in Elastic Stability. World Scientific/Imperial College Press, Singapore.
Elishakoff, I. (2017). Probabilistic Methods in the Theory of Structures: Random Strength of Materials, Random Vibration, and Buckling. World Scientific, Singapore.
Elishakoff, I. (2018). Probabilistic Methods in the Theory of Structures: Solution Manual to Accompany Probabilistic Methods in the Theory of Structures: Problems with Complete, Worked Through Solutions. World Scientific, Singapore.
Elishakoff, I. (2020). Dramatic Effect of Cross-Correlations in Random Vibrations of Discrete Systems, Beams, Plates, and Shells. Springer Nature, Switzerland.
Elishakoff, I. (2020). Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories. World Scientific, Singapore.
Elishakoff, I. and Ohsaki, M. (2010). Optimization and Anti-Optimization of Structures under Uncertainty. Imperial College Press, London.
Elishakoff, I. and Ren, Y. (2003). Finite Element Methods for Structures with Large Stochastic Variations. Oxford University Press, Oxford.
Elishakoff, I., Lin, Y.K., Zhu, L.P. (1994). Probabilistic and Convex Modeling of Acoustically Excited Structures. Elsevier, Amsterdam.
Elishakoff, I., Li, Y., Starnes Jr., J.H. (2001). Non-Classical Problems in the Theory of Elastic Stability. Cambridge University Press, Cambridge.
Elishakoff, I., Pentaras, D., Dujat, K., Versaci, C., Muscolino, G., Storch, J., Bucas, S., Challamel, N., Natsuki, T., Zhang, Y., Ming Wang, C., Ghyselinck, G. (2012). Carbon Nanotubes and Nano Sensors: Vibrations, Buckling, and Ballistic Impact. ISTE Ltd, London, and John Wiley & Sons, New York.
Elishakoff, I., Pentaras, D., Gentilini, C., Cristina, G. (2015). Mechanics of Functionally Graded Material Structures. World Scientific/Imperial College Press, Singapore.
Books edited or co-edited by Elishakoff
Ariaratnam, S.T., Schuëller, G.I., Elishakoff, I. (1988). Stochastic Structural Dynamics – Progress in Theory and Applications. Elsevier, London.
Casciati, F., Elishakoff, I., Roberts, J.B. (1990). Nonlinear Structural Systems under Random Conditions. Elsevier, Amsterdam.
Chuh, M., Wolfe, H.F., Elishakoff, I. (1989). Vibration and Behavior of Composite Structures. ASME Press, New York.
David, H. and Elishakoff, I. (1990). Impact and Buckling of Structures. ASME Press, New York.
Elishakoff, I. (1999). Whys and Hows in Uncertainty Modeling. Springer, Vienna.
Elishakoff, I. (2007). Mechanical Vibration: Where Do We Stand? Springer, Vienna.
Elishakoff, I. and Horst, I. (1987). Refined Dynamical Theories of Beams, Plates and Shells and Their Applications. Springer Verlag, Berlin.
Elishakoff, I. and Lin, Y.K. (1991). Stochastic Structural Dynamics 2 – New Applications. Springer, Berlin.
Elishakoff, I. and Lyon, R.H. (1986), Random Vibration-Status and Recent Developments. Elsevier, Amsterdam.
Elishakoff, I. and Seyranian, A.P. (2002). Modern Problems of Structural Stability. Springer, Vienna.
Elishakoff, I. and Soize, C. (2012). Non-Deterministic Mechanics. Springer, Vienna.
Elishakoff, I., Arbocz, J., Babcock Jr., C.D., Libai, A. (1988). Buckling of Structures: Theory and Experiment. Elsevier, Amsterdam.
Lin, Y.K. and Elishakoff, I. (1991). Stochastic Structural Dynamics 1 – New Theoretical Developments. Springer, Berlin.
Noor, A.K., Elishakoff, I., Hulbert, G. (1990). Symbolic Computations and Their Impact on Mechanics. ASME Press, New York.
Figure P.11.Elishakoff with his wife, Esther Elisha, M.D., during an ASME awards ceremony
On behalf of all the authors of this book, including those friends who were unable to contribute, we wish Prof. Isaac Elishakoff many more decades of fruitful works and collaborations for the benefit of world mechanics, in particular.
Modern Trends in Structural and Solid Mechanics 1 – the first of three separate volumes that comprise this book – presents recent developments and research discoveries in structural and solid mechanics, with a focus on the statics and stability of solid and structural members.
The book is centered around theoretical analysis and numerical phenomena and has broad scope, covering topics such as: buckling of discrete systems (elastic chains, lattices with short and long range interactions, and discrete arches), buckling of continuous structural elements including beams, arches and plates, static investigation of composite plates, exact solutions of plate problems, elastic and inelastic buckling, dynamic buckling under impulsive loading, buckling and post-buckling investigations, buckling of conservative and non-conservative systems, buckling of micro and macro-systems. The engineering applications concern both small-scale phenomena with micro and nano-buckling up to large-scale structures, including the buckling of drillstring systems.
Each of the three volumes is intended for graduate students and researchers in the field of theoretical and applied mechanics.
Prof. Noël CHALLAMEL
Lorient, France
Prof. Julius KAPLUNOV
Keele, UK
Prof. Izuru TAKEWAKI
Kyoto, Japan
February 2021
For a color version of all the figures in this chapter, see www.iste.co.uk/challamel/mechanics2.zip.
1Bolotin’s Dynamic Edge Effect Method Revisited (Review)
A comprehensive review of Bolotin’s edge effect method is presented. This chapter begins with a toy problem and is concluded by nonlinear considerations that have not been developed by Bolotin himself. Various generalizations and modifications of the method are described, along with a variety of solved problems for which a wide list of references is provided. Attempts are also made to frame the method among the known methods for finding rapidly oscillating solutions.
1.1. Introduction
Professor Isaac E. Elishakoff was a doctoral student of the world-renowned scientist V.V. Bolotin (March 29, 1926 to May 28, 2008) (Bolotin 2006). The first research works of I. Elishakoff and his PhD thesis were devoted to the application and development of the dynamic edge effect (EE) method proposed by V.V. Bolotin. After moving from the Soviet Union to the Western world, Prof. Elishakoff made great efforts to popularize the dynamic EE method in the Western scientific community (Elishakoff 1974, 1976; Elishakoff and Wiener 1976).
Therefore, the appearance of a review of papers related to Bolotin’s method in the volume devoted to Prof. Elishakoff’s 75th birthday seems quite reasonable. Moreover, the previous comprehensive reviews of the subject were published in 1976 (Elishakoff 1976) and 1984 (Bolotin 1984).
In the early 1960s, V.V. Bolotin put forward an asymptotic method for studying natural oscillations of plates and shells, which used the inverse of the dimensionless vibration frequency as a small parameter (Bolotin 1960a, 1960b). In a more general formulation, it is a method for solving self-adjoint eigenvalue problems defined in a rectangular domain, called the boundary value problems with quasi-separable variables, according to Bolotin’s terminology. For this reason, the method is referred to as Bolotin’s method or the dynamic edge effect method (DEEM). And despite the fact that 60 years have passed since the method creation, it is still relevant. The purpose of this review is describing various generalizations and modifications of DEEM, the problems solved with the use of this method and also trying to determine the place of DEEM among the known methods for finding rapidly oscillating solutions. Thus, we demonstrate that DEEM can be broadly applied for solving modern problems.
1.2. Toy problem: natural beam oscillations
Demonstrate the main idea of DEEM on a spatially 1D problem, which can be reduced to a transcendental equation and solved numerically with any degree of accuracy (Weaver et al. 1990). Consider the natural oscillations of a beam of length L, described by the following PDE:
Here, w is the normal displacement, E is the Young modulus, F is the cross-sectional area of the beam, I is the axial inertia moment of the beam cross-section, and ρ is the density of the beam material.
Let us compare two versions of boundary conditions:
We use the following ansatz:
where ω is the eigenfrequency and W (x) is the eigenfunction.
The equation for eigenfunction W (x) has the form
The solution of the eigenvalue problem [1.4], [1.2] is given by
The eigenvalue problem [1.4], [1.3] does not allow separation of variables. However, if the eigenfunction oscillates rapidly along x (i.e. a rather high form of oscillations is considered), then we can hope that in this case a solution of the form [1.5] is also valid for the inner domain sufficiently distant from the boundaries (Figure 1.1). Such an expression does not satisfy the boundary conditions. However, if the solution that compensates the residuals at the boundary conditions and decays rapidly, then the approximate expressions for the eigenfunctions and eigenfrequencies can be obtained.
Figure 1.1.Curve 1 corresponds to the rapidly oscillating solution in the inner domain, and curve 2 corresponds to the sum of DEE and the rapidly oscillating solution
Let us suppose the solution of equation [1.4] in the form
The oscillation frequency ω is
Factorization of ODE [1.4] is (Vakhromeev and Kornev 1972)
The general solution of ODE [1.9] is given by
where functions W0 and W1,2 are the general solutions of the following equations:
For large frequencies (aω ≫1), the following estimates for the derivatives of functions W0 and W1,2 are obtained:
The behavior of these solution components is different: W1 is the rapidly oscillating function, and W2 is the sum of exponentials rapidly decreasing from the edges of the beam.
Therefore, the situation under consideration is fundamentally different from the case when the characteristic equation has small and large modulo roots, which is typical for boundary layer theory. In our case, we are talking about the separation of solutions, one of which oscillates at the same rate as the EE decays (i.e. the characteristic equation has large real and imaginary roots with moduli of the same order). The self-adjoint eigenvalue problem [1.1], [1.2] can be referred to as the boundary value problem with quasi-separable variables (Bolotin 1960a, 1960b, 1961a, 1961b, 1961c; Bolotin et al. 1950, 1961).
We proceed to the construction of the EE described by equation [1.11]. Taking into account the expression for the natural frequency [1.8], we obtain the following relations for EEs localized in the vicinity of the edges x = 0 and x = L, respectively:
We assume that the beam is so long such that EEs do not affect each other, i.e. exp(−πλ-1L) ≪ 1.
Now, it remains to find the quantities x0, λ and constants C1, C2 from the boundary conditions to determine the eigenmodes and eigenfrequencies:
Note that in original works by Bolotin, a slightly different matching procedure is used. Namely, the matching conditions [1.14], [1.15] are not given at the domain boundaries. They are set at some points, which are then determined from the solution of the system of transcendental equations along with other unknown constants.
Substituting expressions [1.7] and [1.12] into conditions [1.14], and expressions [1.7] and [1.13] into [1.15], we obtain
For λ and x0, we have the following expressions:
