Gol'Denveizer / Kármán / Dryden Theory of Elastic Thin Shells

Solid and Structural Mechanics
1. Auflage 2014
ISBN: 978-1-4831-6462-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 680 Seiten, Web PDF

Theory of Elastic Thin Shells discusses the mathematical foundations of shell theory and the approximate methods of solution. The present volume was originally published in Russian in 1953, and remains the only text which formulates as completely as possible the different sets of basic equations and various approximate methods of shell analysis emphasizing asymptotic integration. The book is organized into five parts. Part I presents the general formulation and equations of the theory of shells, which are based on the well-known hypothesis of the preservation of the normal element. Part II is devoted to the membrane theory--the most widely used approximate method of analysis of shells that was formulated at approximately the same time as the more general bending theory. In Part III methods of analysis of circular cylindrical shells with the aid of trigonometric series are considered. Part IV is essentially mathematical in character and its purpose is to justify the approximate methods of shell analysis. In Part V approximate methods of analysis of shells are formulated.

Autoren/Hrsg.

Gol'Denveizer, A. L.

Kármán, Th. von

Dryden, H. L.

Inhaltsverzeichnis

1;Front Cover;1
2;Theory of Elastic Thin Shells;6
3;Copyright Page;7
4;Table of Contents;8
5;TRANSLATION EDITOR'S PREFACE TO ENGLISH EDITION;14
6;AUTHOR'S PREFACE TO ENGLISH EDITION;16
7;PREFACE;18
8;PART I: Basic Relations in the Theory of Shells;24
8.1;CHAPTER 1. BRIEF OUTLINE OF THE THEORY OF SURFACES;24
8.1.1;1. Curvilinear Coordinates on a Surface and the First Quadratic Form;24
8.1.2;2. Basic and Auxiliary Trihedra of a Surface. Decomposition of an Arbitrary Vector Along Axes of Basic and Auxiliary Trihedra;27
8.1.3;3. Gauss-Weingarten Derivative Formulas. Codazzi-Gauss Equations;31
8.1.4;4. Resolution of Derivatives of an Arbitrary Vector along the Axes of the Basic and Auxiliary Trihedra;34
8.1.5;5. Second Quadratic Form of a Surface and Dupin's Indicatrix;35
8.1.6;6. Conjugate Lines, Lines of Curvature, Asymptotic Lines;38
8.1.7;7. Gaussian Curvature and Bending of Surfaces;40
8.1.8;8. Fundamental Formulas of the Theory of Surfaces in Orthogonal Coordinates;41
8.2;CHAPTER 2. STATIC AND GEOMETRIC RELATIONS OF THE THEORY OF SHELLS;45
8.2.1;9. Forces and Moments;45
8.2.2;10. Forces and Moments along Oblique Sections;48
8.2.3;11. External Loads;50
8.2.4;12. Equilibrium Equations of the Shell;51
8.2.5;13. Stress Functions;56
8.2.6;14. Vectors of Elastic Displacement and of Elastic Rotation of the Middle Surface;59
8.2.7;15. Components of Tangential Deformation (Strain) of the Middle Surface of the Shell;64
8.2.8;16. Expressions for the Derivatives of the Vector of Elastic Displacement;66
8.2.9;17. Components of Bending Deformation (Strain) of the Middle Surface;68
8.2.10;18. Expressions for Derivatives of the Vector of Elastic Rotation;74
8.2.11;19. Expressions for Components of Deformation and Angles of Rotation in Terms of Displacements;75
8.2.12;20. Determination of Displacements on the Basis of given Components of Deformation. Equations of Compatibility of Strain;78
8.2.13;21. Transformation of Components of Strain;82
8.3;CHAPTER 3. RELATIONS OF ELASTICITY. GENERALTHEOREMS OF THE THEORY CF SHELLS;87
8.3.1;22. Fundamental Hypothesis of the Theory of Shells;87
8.3.2;23. Relations of Elasticity;90
8.3.3;24. Supplementary Equations of the Theory of Shells;97
8.3.4;25. Work of Forces and Moments of Thin Shells;98
8.3.5;26. Strain Energy;101
8.3.6;27. Analysis of Some Variants of Elasticity Relations;105
8.4;CHAPTER 4. FUNDAMENTAL EQUATIONS OF THE THEORY OF SHELLS;109
8.4.1;28. Summary of Fundamental Relations of the Theory of Shells;109
8.4.2;29. Complete System of Equations of the Theory OF Shells ;112
8.4.3;30. Statio-Geometrio Analogy;115
8.4.4;31. Equations of Compatibility in Termsof Forces and Moments;119
8.4.5;32. Equations of Equilibrium in Terms of Displacements;125
8.4.6;33. Boundary Conditions;127
9;PART II: Membrane Theory;133
9.1;CHAPTER 5. MEMBRANE THEORY Of SHELLS OF ARBITRARY SHAPE;133
9.1.1;1. General Assumptions of Membrane Theory;133
9.1.2;2. Static, Geometric and Mixed Problems of Membrane Theory;135
9.1.3;3. Boundary Conditions in Membrane Theory;139
9.1.4;4. Three Classes of Membrane Shells;140
9.1.5;5. Relation Between Membrane Theory and the Theory of Infinitesimal Flexure of Surfaces;142
9.1.6;6. Conjugate Geometrio and Static Problems of Membrane Theory;145
9.1.7;7. Membrane Shell of Positive Curvature with One Geometric Condition;150
9.2;CHAPTER 6. MEMBRANE THEORY OF SHELLS OF ZERO CURVATURE;154
9.2.1;8. Curvilinear Coordinates on Cylindrical and Conical Surfaces;154
9.2.2;9. General Integral of Equations of Membrane Theory of Shells of Zero Curvature;158
9.2.3;10. Boundary Conditions;162
9.2.4;11. Examination of the State of Stress ofa Cylindrical Membrane Shell;164
9.2.5;12. Examples of Analysis of Cylindrical Membrane Shells;167
9.2.6;13. Examples of Analysis of Cylindrical Membrane Shells, Continued;172
9.3;CHAPTER 7. MEMBRANE THEORY OF SPHERICAL SHELLS;183
9.3.1;14. Transformation of Membrane Equations of a Spherical Shell;183
9.3.2;15. Integration Methods for the Equations of Membrane Theory of Spherical Shells;186
9.3.3;16. Application of the Methods of the Theory of Functions of a Complex Variable to the Analysis of Spherical Membrane Shells;190
9.3.4;17. Integral Equations of Equilibrium;193
9.3.5;18. Static Meaning of Poles of the Complex Stress Function;196
9.4;CHAPTER 8. ANALYSIS OF CLOSED SPHERICAL MEMBRANE SHELLS;203
9.4.1;19. Analysis of Closed Spherical Membrane Shells Under the Action of Concentrated Forces and Moments;203
9.4.2;20. Example;205
9.4.3;21. Displacements of a Closed Spherical Shell Subjected to Concentrated Forces and Moments;208
9.4.4;22. Analysis of Closed Spherical Membrane Shells Subjected to Distributed Loads;213
9.4.5;23. Generalizations;220
9.5;CHAPTER 9. ANALYSIS OF MEMBRANE SHELLS TAKING BOUNDARY CONDITIONS INTO ACCOUNT;228
9.5.1;24. The Simplest Problems in Which Account Must be Taken of Boundary Conditions;228
9.5.2;25. Examples;235
9.5.3;26. Number of Solutions of Static and Geometric Problems for Membrane Shells of Positive Curvature;242
9.5.4;27. Examples of Statically Determinate and Geometrically Variable Membrane Shells;247
10;PART III: Circular Cylindrical Shells;251
10.1;CHAPTER 10. METHOD OF EXPANSION IN TRIGONOMETRIC SERIES;251
10.1.1;1. Basic Equations of the Theory of Cylindrical Shells;251
10.1.2;2. The Solving Equation of Circular Cylindrical Shells;255
10.1.3;3. Application of Trigonometric Series to the Analysis of Circular Cylindrical Shells;258
10.2;CHAPTER 11. ANALYSIS OF CLOSED CYLINDRICAL SHELLS;267
10.2.1;4. Basic Formulas for Analysis;267
10.2.2;5. Properties of Roots of the Characteristic Equation. Simplification of the Characteristic Equation;272
10.2.3;6. Physical Meaning of Zero Roots of the Characteristic Equation;280
10.2.4;7. Analysis of the State of Stress of Closed Cylindrical Shells;282
10.2.5;8. Approximate Methods of Analysis of the Basic State of Stress of Circular Cylindrical Shells;290
10.2.6;9. Approximate Methods of Analysis of Edge Effects;296
10.2.7;10. States of Stress Corresponding to Large Values of m;299
10.2.8;11. Imposition of Boundary Conditions;304
10.3;CHAPTER 12. ANALYSIS OF OPEN CYLINDRICAL SHELLS;314
10.3.1;12. Basic Formulas for Analysis;314
10.3.2;13. Properties of Roots of Characteristic Equation;319
10.3.3;14. Analysis of the State of Stress in Open Cylindrical Shells;323
10.3.4;15. Approximate Methods of Analysis of Open Cylindrical Shells;327
10.3.5;16. Imposition of Boundary Conditions;333
11;PART IV: Analysis of the State of Stress in an Arbitrary Shell;337
11.1;CHAPTER 13. ASYMPTOTIC INTEGRATION OF PARTIAL DIFFERENTIAL EQUATIONS;337
11.1.1;1. Classification of Linear Differential Operators with Partial Derivatives;337
11.1.2;2. Nomenclature and Notations;342
11.1.3;3. Asymptotic Expansion of Integrals of a Homogeneous Differential Equation;346
11.1.4;4. Three Fundamental Cases;349
11.1.5;5. Construction of Functions of Variation;353
11.1.6;6. Integrals with Given Non-Characteristic Supporting Contour;356
11.1.7;7. Case of Multiple Characteristics;359
11.1.8;8. Integrals with Given Characteristic Supporting Contour;367
11.1.9;9. Asymptotic Expansion of Particular Solution of a Nonhomogeneous Partial Differential Equation;373
11.1.10;10. Example;381
11.2;CHAPTER 14. ASIMPTOTIC INTEGRATION OF EQUATIONS OF THE THEORY OF SHELLS;389
11.2.1;11. Asymptotic Integration of a System of equation;389
11.2.2;12. Non-Contradictory Values of Indices of Intensity;396
11.2.3;13. Construction of Functions of Variation;399
11.2.4;14. Determination of Coefficients of Asymptotic Expansion of Functions of Intensity for Fundamental Integrals;403
11.2.5;15. Construction of Approximate Equations of the Theory of Shells;405
11.2.6;16. Asymptotic Error of Equations of Membrane Theory;408
11.2.7;17. Elementary States of Stress in an Arbitrary Shell;412
11.2.8;18. The Complete State of Stress in an Arbitrary Shell;415
11.3;CHAPTER 15. ELEMENTARY STATES OF STRESS;419
11.3.1;19. Fundamental State of Stress. Membrane and Pure Bending States of Stress;419
11.3.2;20. Approximate Equations for States of Stress with Large Indices of Variation;424
11.3.3;21. Region of Applicability of Equation (20.11);429
11.3.4;22. Simple Edge Effect;436
11.3.5;23. Integration of the Solving Equation of Simple Edge Effect;441
11.3.6;24. Solving Equations of Non-Degenerate Generalized Edge Effects;446
11.3.7;25. Solving Equation of Generalized Edge Effect in a Shell of Zero Curvature;451
11.3.8;26. Range of Applicability of Solving Equations (25.5);455
11.3.9;27. Further Simplification of Solving Equations (25.5);458
11.3.10;28. Range of Applicability of Membrane Theory in the Analysis of Shells of Zero Curvature;465
11.3.11;29. Estimating the Aoouracy of Construction of a Complete State of Stress;471
12;PART V: Approximate Methods of Analysis of Shells;476
12.1;CHAPTER 16. APPLICATION OF EXPANSIONS IN ORTHOGONAL FUNCTIONS TO THE ANALYSIS OF SHELLS;476
12.1.1;1. Expansion of Functions in Fourier Series;476
12.1.2;2. Methods of Construction of Closed Orthogonal Systems of Functions;478
12.1.3;3. Continuation;481
12.1.4;4. Index of Variation of the State of Stress and of External Loading;486
12.2;CHAPTER 17. GENERAL APPROXIMATE METHODS;494
12.2.1;5. Membrane Theory;494
12.2.2;6. Region of Applicability of Membrane Theory;497
12.2.3;7. Properties of the Simple Edge Effect;507
12.2.4;8. Approximate Theory of the Simple Edge Effect;509
12.2.5;9. Analysis of Shells by the Membrane Theory with Consideration of Edge Effects;517
12.2.6;10. Particular Cases;518
12.2.7;11. Example;525
12.2.8;12. Approximate Methods of Analysis of Shells with Large Indices of Variation;531
12.2.9;13. Example;538
12.2.10;14. Shells with Non-Rigidly Supported Edges;545
12.3;CHAPTER 18. CYLINDRICAL AND CONICAL SHELLS;550
12.3.1;15. The Generalized Edge Effect in a Shell of Zero Curvature;550
12.3.2;16. The Solving Equations of the Generalized Edge Effect in Shells of Zero Curvature;553
12.3.3;17. Integration of the Solving Equations of the Generalized Edge Effect for Cylindrical Shells;557
12.3.4;18. Imposition of Boundary Conditions;560
12.3.5;19. Integration of Equations of the Generalize dEdge Effect for Conical Shells
;565
12.3.6;20. Analysis of the State of Stress of Shells of Zero Curvature;571
12.3.7;21. Approximate Theory of the Non-Degenerate Edge Effect;577
12.3.8;22. Integration of Equations of the Non-Degenerate Edge Effect for Cylindrical and Conical Shells;580
12.3.9;23. Integration of System (22.9);587
12.3.10;24. Continuation;593
12.3.11;25. Tables of Elastic Reactions and of Elastic Displacements of Cylindrical Shells of Medium Reduced Length;598
12.3.12;26. Example;607
12.3.13;27. Analysis of Cylindrical Shells of Medium Reduced Length Subjected to Loads Distributed along a Generator;612
12.3.14;28. Example;617
12.3.15;29. Analysis of Conical Shells;626
13;Author's Addendum to English Edition SOME MATHEMATICAL PROBLEMS OF THE LINEAR THEORY OF ELASTIC THIN SHELLS;630
13.1;I. Asymptotio Methods of Integration of Partial Differential Equations;630
13.2;II. Imposition of Boundary Conditions;647
13.3;III. The Influence of the Conditions of Edge Constrainton the State of Stress in the Shell;663
14;AUTHOR'S AMENDMENTS;670
15;AUTHOR INDEX;677
16;SUBJECT INDEX;678