Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions
The engineering community generally accepts that there exists only a small set of closed-form solutions for simple cases of bars, beams, columns, and plates. Despite the advances in powerful computing and advanced numerical techniques, closed-form solutions remain important for engineering; these include uses for preliminary design, for evaluation of the accuracy of approximate and numerical solutions, and for evaluating the role played by various geometric and loading parameters.

Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions offers the first new treatment of closed-form solutions since the works of Leonhard Euler over two centuries ago. It presents simple solutions for vibrating bars, beams, and plates, as well as solutions that can be used to verify finite element solutions. The closed solutions in this book not only have applications that allow for the design of tailored structures, but also transcend mechanical engineering to generalize into other fields of engineering. Also included are polynomial solutions, non-polynomial solutions, and discussions on axial variability of stiffness that offer the possibility of incorporating axial grading into functionally graded materials.

This single-package treatment of inhomogeneous structures presents the tools for optimization in many applications. Mechanical, aerospace, civil, and marine engineers will find this to be the most comprehensive book on the subject. In addition, senior undergraduate and graduate students and professors will find this to be a good supplement to other structural design texts, as it can be easily incorporated into the classroom.
1120987558
Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions
The engineering community generally accepts that there exists only a small set of closed-form solutions for simple cases of bars, beams, columns, and plates. Despite the advances in powerful computing and advanced numerical techniques, closed-form solutions remain important for engineering; these include uses for preliminary design, for evaluation of the accuracy of approximate and numerical solutions, and for evaluating the role played by various geometric and loading parameters.

Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions offers the first new treatment of closed-form solutions since the works of Leonhard Euler over two centuries ago. It presents simple solutions for vibrating bars, beams, and plates, as well as solutions that can be used to verify finite element solutions. The closed solutions in this book not only have applications that allow for the design of tailored structures, but also transcend mechanical engineering to generalize into other fields of engineering. Also included are polynomial solutions, non-polynomial solutions, and discussions on axial variability of stiffness that offer the possibility of incorporating axial grading into functionally graded materials.

This single-package treatment of inhomogeneous structures presents the tools for optimization in many applications. Mechanical, aerospace, civil, and marine engineers will find this to be the most comprehensive book on the subject. In addition, senior undergraduate and graduate students and professors will find this to be a good supplement to other structural design texts, as it can be easily incorporated into the classroom.
84.99 In Stock
Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions

Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions

by Isaac Elishakoff
Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions
Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions

Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions

by Isaac Elishakoff

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Overview

The engineering community generally accepts that there exists only a small set of closed-form solutions for simple cases of bars, beams, columns, and plates. Despite the advances in powerful computing and advanced numerical techniques, closed-form solutions remain important for engineering; these include uses for preliminary design, for evaluation of the accuracy of approximate and numerical solutions, and for evaluating the role played by various geometric and loading parameters.

Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions offers the first new treatment of closed-form solutions since the works of Leonhard Euler over two centuries ago. It presents simple solutions for vibrating bars, beams, and plates, as well as solutions that can be used to verify finite element solutions. The closed solutions in this book not only have applications that allow for the design of tailored structures, but also transcend mechanical engineering to generalize into other fields of engineering. Also included are polynomial solutions, non-polynomial solutions, and discussions on axial variability of stiffness that offer the possibility of incorporating axial grading into functionally graded materials.

This single-package treatment of inhomogeneous structures presents the tools for optimization in many applications. Mechanical, aerospace, civil, and marine engineers will find this to be the most comprehensive book on the subject. In addition, senior undergraduate and graduate students and professors will find this to be a good supplement to other structural design texts, as it can be easily incorporated into the classroom.

Product Details

ISBN-13: 9780367454272
Publisher: CRC Press
Publication date: 06/30/2020
Pages: 752
Product dimensions: 6.12(w) x 9.19(h) x (d)

About the Author

Elishakoff, Isaac

Table of Contents

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Forewordxv
Prologue1
Chapter 1Introduction: Review of Direct, Semi-inverse and Inverse Eigenvalue Problems7
1.1Introductory Remarks7
1.2Vibration of Uniform Homogeneous Beams8
1.3Buckling of Uniform Homogeneous Columns10
1.4Some Exact Solutions for the Vibration of Non-uniform Beams19
1.4.1The Governing Differential Equation21
1.5Exact Solution for Buckling of Non-uniform Columns24
1.6Other Direct Methods (FDM, FEM, DQM)28
1.7Eisenberger's Exact Finite Element Method30
1.8Semi-inverse or Semi-direct Methods35
1.9Inverse Eigenvalue Problems43
1.10Connection to the Work by Zyczkowski and Gajewski50
1.11Connection to Functionally Graded Materials52
1.12Scope of the Present Monograph53
Chapter 2Unusual Closed-Form Solutions in Column Buckling55
2.1New Closed-Form Solutions for Buckling of a Variable Flexural Rigidity Column55
2.1.1Introductory Remarks55
2.1.2Formulation of the Problem56
2.1.3Uncovered Closed-Form Solutions57
2.1.4Concluding Remarks65
2.2Inverse Buckling Problem for Inhomogeneous Columns65
2.2.1Introductory Remarks65
2.2.2Formulation of the Problem65
2.2.3Column Pinned at Both Ends66
2.2.4Column Clamped at Both Ends68
2.2.5Column Clamped at One End and Pinned at the Other69
2.2.6Concluding Remarks70
2.3Closed-Form Solution for the Generalized Euler Problem74
2.3.1Introductory Remarks74
2.3.2Formulation of the Problem76
2.3.3Column Clamped at Both Ends79
2.3.4Column Pinned at One End and Clamped at the Other79
2.3.5Column Clamped at One End and Free at the Other81
2.3.6Concluding Remarks83
2.4Some Closed-Form Solutions for the Buckling of Inhomogeneous Columns under Distributed Variable Loading84
2.4.1Introductory Remarks84
2.4.2Basic Equations87
2.4.3Column Pinned at Both Ends92
2.4.4Column Clamped at Both Ends97
2.4.5Column that is Pinned at One End and Clamped at the Other100
2.4.6Concluding Remarks105
Chapter 3Unusual Closed-Form Solutions for Rod Vibrations107
3.1Reconstructing the Axial Rigidity of a Longitudinally Vibrating Rod by its Fundamental Mode Shape107
3.1.1Introductory Remarks107
3.1.2Formulation of the Problem108
3.1.3Inhomogeneous Rods with Uniform Density109
3.1.4Inhomogeneous Rods with Linearly Varying Density112
3.1.5Inhomogeneous Rods with Parabolically Varying Inertial Coefficient114
3.1.6Rod with General Variation of Inertial Coefficient (m [greater than sign] 2)115
3.1.7Concluding Remarks118
3.2The Natural Frequency of an Inhomogeneous Rod may be Independent of Nodal Parameters120
3.2.1Introductory Remarks120
3.2.2The Nodal Parameters121
3.2.3Mode with One Node: Constant Inertial Coefficient124
3.2.4Mode with Two Nodes: Constant Density127
3.2.5Mode with One Node: Linearly Varying Material Coefficient129
3.3Concluding Remarks131
Chapter 4Unusual Closed-Form Solutions for Beam Vibrations135
4.1Apparently First Closed-Form Solutions for Frequencies of Deterministically and/or Stochastically Inhomogeneous Beams (Pinned-Pinned Boundary Conditions)135
4.1.1Introductory Remarks135
4.1.2Formulation of the Problem136
4.1.3Boundary Conditions137
4.1.4Expansion of the Differential Equation138
4.1.5Compatibility Conditions139
4.1.6Specified Inertial Coefficient Function140
4.1.7Specified Flexural Rigidity Function141
4.1.8Stochastic Analysis144
4.1.9Nature of Imposed Restrictions151
4.1.10Concluding Remarks151
4.2Apparently First Closed-Form Solutions for Inhomogeneous Beams (Other Boundary Conditions)152
4.2.1Introductory Remarks152
4.2.2Formulation of the Problem153
4.2.3Cantilever Beam154
4.2.4Beam that is Clamped at Both Ends163
4.2.5Beam Clamped at One End and Pinned at the Other165
4.2.6Random Beams with Deterministic Frequencies168
4.3Inhomogeneous Beams that may Possess a Prescribed Polynomial Second Mode175
4.3.1Introductory Remarks175
4.3.2Basic Equation180
4.3.3A Beam with Constant Mass Density182
4.3.4A Beam with Linearly Varying Mass Density185
4.3.5A Beam with Parabolically Varying Mass Density190
4.4Concluding Remarks199
Chapter 5Beams and Columns with Higher-Order Polynomial Eigenfunctions203
5.1Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 1: Buckling203
5.1.1Introductory Remarks203
5.1.2Choosing a Pre-selected Mode Shape204
5.1.3Buckling of the Inhomogeneous Column under an Axial Load205
5.1.4Buckling of Columns under an Axially Distributed Load209
5.1.5Concluding Remarks224
5.2Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 2: Vibration225
5.2.1Introductory Comments225
5.2.2Formulation of the Problem226
5.2.3Basic Equations227
5.2.4Constant Inertial Coefficient (m = 0)228
5.2.5Linearly Varying Inertial Coefficient (m = 1)230
5.2.6Parabolically Varying Inertial Coefficient (m = 2)231
5.2.7Cubic Inertial Coefficient (m = 3)236
5.2.8Particular Case m = 4239
5.2.9Concluding Remarks242
Chapter 6Influence of Boundary Conditions on Eigenvalues249
6.1The Remarkable Nature of Effect of Boundary Conditions on Closed-Form Solutions for Vibrating Inhomogeneous Bernoulli-Euler Beams249
6.1.1Introductory Remarks249
6.1.2Construction of Postulated Mode Shapes250
6.1.3Formulation of the Problem251
6.1.4Closed-Form Solutions for the Clamped-Free Beam252
6.1.5Closed-Form Solutions for the Pinned-Clamped Beam271
6.1.6Closed-Form Solutions for the Clamped-Clamped Beam289
6.1.7Concluding Remarks308
Chapter 7Boundary Conditions Involving Guided Ends309
7.1Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Pinned Support309
7.1.1Introductory Remarks309
7.1.2Formulation of the Problem310
7.1.3Boundary Conditions310
7.1.4Solution of the Differential Equation311
7.1.5The Degree of the Material Density is Less than Five312
7.1.6General Case: Compatibility Conditions318
7.1.7Concluding Comments322
7.2Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Clamped Support322
7.2.1Introductory Remarks322
7.2.2Formulation of the Problem323
7.2.3Boundary Conditions323
7.2.4Solution of the Differential Equation324
7.2.5Cases of Uniform and Linear Densities325
7.2.6General Case: Compatibility Condition327
7.2.7Concluding Remarks329
7.3Class of Analytical Closed-Form Polynomial Solutions for Guided-Pinned Inhomogeneous Beams330
7.3.1Introductory Remarks330
7.3.2Formulation of the Problem330
7.3.3Constant Inertial Coefficient (m = 0)332
7.3.4Linearly Varying Inertial Coefficient (m = 1)333
7.3.5Parabolically Varying Inertial Coefficient (m = 2)335
7.3.6Cubically Varying Inertial Coefficient (m = 3)337
7.3.7Coefficient Represented by a Quartic Polynomial (m = 4)338
7.3.8General Case340
7.3.9Particular Cases Characterized by the Inequality n [greater than or equal] m + 2349
7.3.10Concluding Remarks364
7.4Class of Analytical Closed-Form Polynomial Solutions for Clamped-Guided Inhomogeneous Beams364
7.4.1Introductory Remarks364
7.4.2Formulation of the Problem364
7.4.3General Case366
7.4.4Constant Inertial Coefficient (m = 0)376
7.4.5Linearly Varying Inertial Coefficient (m = 1)377
7.4.6Parabolically Varying Inertial Coefficient (m = 2)378
7.4.7Cubically Varying Inertial Coefficient (m = 3)380
7.4.8Inertial Coefficient Represented as a Quadratic (m = 4)385
7.4.9Concluding Remarks392
Chapter 8Vibration of Beams in the Presence of an Axial Load395
8.1Closed-Form Solutions for Inhomogeneous Vibrating Beams under Axially Distributed Loading395
8.1.1Introductory Comments395
8.1.2Basic Equations397
8.1.3Column that is Clamped at One End and Free at the Other398
8.1.4Column that is Pinned at its Ends402
8.1.5Column that is clamped at its ends407
8.1.6Column that is Pinned at One End and Clamped at the Other411
8.1.7Concluding Remarks416
8.2A Fifth-Order Polynomial that Serves as both the Buckling and Vibration Modes of an Inhomogeneous Structure417
8.2.1Introductory Comments417
8.2.2Formulation of the Problem419
8.2.3Basic Equations421
8.2.4Closed-Form Solution for the Pinned Beam422
8.2.5Closed-Form Solution for the Clamped-Free Beam431
8.2.6Closed-Form Solution for the Clamped-Clamped Beam442
8.2.7Closed-Form Solution for the Beam that is Pinned at One End and Clamped at the Other452
8.2.8Concluding Remarks460
Chapter 9Unexpected Results for a Beam on an Elastic Foundation or with Elastic Support461
9.1Some Unexpected Results in the Vibration of Inhomogeneous Beams on an Elastic Foundation461
9.1.1Introductory Remarks461
9.1.2Formulation of the Problem462
9.1.3Beam with Uniform Inertial Coefficient, Inhomogeneous Elastic Modulus and Elastic Foundation463
9.1.4Beams with Linearly Varying Density, Inhomogeneous Modulus and Elastic Foundations468
9.1.5Beams with Varying Inertial Coefficient Represented as an mth Order Polynomial475
9.1.6Case of a Beam Pinned at its Ends480
9.1.7Beam Clamped at the Left End and Free at the Right End486
9.1.8Case of a Clamped-Pinned Beam491
9.1.9Case of a Clamped-Clamped Beam496
9.1.10Case of a Guided-Pinned Beam501
9.1.11Case of a Guided-Clamped Beam510
9.1.12Cases Violated in Eq. (9.99)515
9.1.13Does the Boobnov-Galerkin Method Corroborate the Unexpected Exact Results?517
9.1.14Concluding Remarks521
9.2Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Rotational Spring522
9.2.1Introductory Remarks522
9.2.2Basic Equations522
9.2.3Uniform Inertial Coefficient523
9.2.4Linear Inertial Coefficient526
9.3Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Translational Spring528
9.3.1Introductory Remarks528
9.3.2Basic Equations529
9.3.3Constant Inertial Coefficient531
9.3.4Linear Inertial Coefficient533
Chapter 10Non-Polynomial Expressions for the Beam's Flexural Rigidity for Buckling or Vibration537
10.1Both the Static Deflection and Vibration Mode of a Uniform Beam Can Serve as Buckling Modes of a Non-uniform Column537
10.1.1Introductory Remarks537
10.1.2Basic Equations538
10.1.3Buckling of Non-uniform Pinned Columns539
10.1.4Buckling of a Column under its Own Weight542
10.1.5Vibration Mode of a Uniform Beam as a Buckling Mode of a Non-uniform Column544
10.1.6Non-uniform Axially Distributed Load545
10.1.7Concluding Remarks547
10.2Resurrection of the Method of Successive Approximations to Yield Closed-Form Solutions for Vibrating Inhomogeneous Beams548
10.2.1Introductory Comments548
10.2.2Evaluation of the Example by Birger and Mavliutov551
10.2.3Reinterpretation of the Integral Method for Inhomogeneous Beams553
10.2.4Uniform Material Density555
10.2.5Linearly Varying Density557
10.2.6Parabolically Varying Density559
10.2.7Can Successive Approximations Serve as Mode Shapes?563
10.2.8Concluding Remarks563
10.3Additional Closed-Form Solutions for Inhomogeneous Vibrating Beams by the Integral Method566
10.3.1Introductory Remarks566
10.3.2Pinned-Pinned Beam567
10.3.3Guided-Pinned Beam575
10.3.4Free-Free Beam582
10.3.5Concluding Remarks590
Chapter 11Circular Plates591
11.1Axisymmetric Vibration of Inhomogeneous Clamped Circular Plates: an Unusual Closed-Form Solution591
11.1.1Introductory Remarks591
11.1.2Basic Equations593
11.1.3Method of Solution594
11.1.4Constant Inertial Term (m = 0)594
11.1.5Linearly Varying Inertial Term (m = 1)595
11.1.6Parabolically Varying Inertial Term (m = 2)596
11.1.7Cubic Inertial Term (m = 3)598
11.1.8General Inertial Term (m [greater than or equal] 4)600
11.1.9Alternative Mode Shapes601
11.2Axisymmetric Vibration of Inhomogeneous Free Circular Plates: An Unusual, Exact, Closed-Form Solution604
11.2.1Introductory Remarks604
11.2.2Formulation of the Problem605
11.2.3Basic Equations