# Theory of Elastic Thin Shells

### Solid and Structural Mechanics

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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.

## Table of Contents

Translation Editor's Preface to English Edition

Author's Preface to English Edition

Preface

Part I Basic Relations in the Theory of Shells

Chapter 1. Brief Outline of the Theory of Surfaces

1. Curvilinear Coordinates on a Surface and the First Quadratic Form

2. Basic and Auxiliary Trihedra of a Surface. Decomposition of an Arbitrary Vector Along Axes of Basic and Auxiliary Trihedra

3. Gauss-Weingarten Derivative Formulas. Codazzi-Gauss Equations

4. Resolution of Derivatives of an Arbitrary Vector along the Axes of the Basic and Auxiliary Trihedra

5. Second Quadratic Form of a Surface and Dupin's Indicatrix

6. Conjugate Lines, Lines of Curvature, Asymptotic Lines

7. Gaussian Curvature and Bending of Surfaces

8. Fundamental Formulas of the Theory of Surfaces in Orthogonal Coordinates

Chapter 2. Static and Geometric Relations of the Theory of Shells

9. Forces and Moments

10. Forces and Moments along Oblique Sections

11. External Loads

12. Equilibrium Equations of the Shell

13. Stress Functions

14. Vectors of Elastic Displacement and of Elastic Rotation of the Middle Surface

15· Components of Tangential Deformation (Strain) of the Middle Surface of the Shell

16. Expansions for the Derivatives of the Vector of Elastic Displacement

17· Components of Bending Deformation (Strain) of the Middle Surface

18. Expressions for Derivatives of the Vector of Elastic Rotation

19· Expressions for Components of Deformation and Angles of Rotation in Terms of Displacements

20· Determination of Displacements on the Basis of Given Components of Deformation. Equations of Compatibility of Strain

21. Transformation of Components of Strain

Chapter 5. Relations of Elasticity. General Theorems of the Theory of Shells

22. Fundamental Hypothesis of the Theory of Shells

23. Relations of Elasticity

24. Supplementary Equations of the Theory of Shells

25. Work of Forces and Moments, of Thin Shells

26. Strain Energy

27· Analysis of Some Variants of Elasticity Relations

Chapter 4. Fundamental Equations of the Theory of Shells

28. Summary of Fundamental Relations of the Theory of Shells

29· Complete System of Equations of the Theory of Shells

30. Static-Geometric Analogy

31. Equations of Compatibility in Terms of Forces and Moments

32. Equations of Equilibrium in Terms of Displacements

33· Boundary Conditions

Part II Membrane Theory

Chapter 5· Membrane Theory of Shells of Arbitrary Shape

1. General Assumptions of Membrane Theory

2. Static, Geometric and Mixed Problems of Membrane Theory

3. Boundary Conditions in Membrane Theory

4· Three Classes of Membrane Shells

5· Relations between Membrane Theory and the Theory of Infinitesimal Flexures of Surfaces

6. Conjugate Geometric and Static Problems of Membrane Theory

7· Membrane Shell of Positive Curvature with One Geometric Condition

Chapter 6. Membrane Theory of Shells of Zero Curvature

8. Curvilinear Coordinates on Cylindrical and Conical Surfaces

9. General Integral of Equations of Membrane Theory of Shells of Zero Curvature

10. Boundary Conditions

11. Examination of the State of Stress of a Cylindrical Membrane Shell

12. Examples of Analysis of Cylindrical Membrane Shells

13. Examples of Analysis of Cylindrical Membrane Shells, Continued

Chapter 7. Membrane Theory of Spherical Shells

14. Transformation of Membrane Equations of a Spherical Shell

15. Integration Methods for the Equations of Membrane Theory of Spherical Shells

16. Application of the Methods of the Theory of Functions of a Complex Variable to the Analysis of Spherical Membrane Shells

17. Integral Equations of Equilibrium

18. Static Meaning of Poles of the Complex Stress Function

Chapter 8. Analysis of Closed Spherical Membrane Shells

19. Analysis of Closed Spherical Membrane Shells under the Action of Concentrated. Forces and Moments

20. Example

21. Displacements of a Closed Spherical Shell Subjected to Concentrated forces and Moments

22. Analysis of Closed Spherical Membrane Shells Subjected to Distributed Loads

23. Generalizations

Chapter 9. Analysis of Membrane Shells Taking Boundary Conditions into Account

24. The Simplest Problems in Which Account Must be Taken of Boundary Conditions

25. Examples

26. Number of Solutions of Static and Geometric Problems for Membrane Shells of Positive Curvature

27. Examples of Statically Determinate and Geometrically Variable Membrane Shells

Part III Circular Cylindrical Shells

Chapter 10. Method of Expansion in Trigonometric Series

1. Basic Equations of the Theory of Cylindrical Shells

2. The Solving Equation of Circular Cylindrical Shells

3. Application of Trigonometric Series to the Analysis of Circular Cylindrical Shells

Chapter 11. Analysis of Closed Cylindrical Shells

4. Basic Formulas for Analysis

5. Properties of Roots of the Characteristic Equation. Simplification of the Characteristic Equation

6. Physical Meaning of Zero Roots of the Characteristic Equation

7. Analysis of the State of Stress of Closed Cylindrical Shells

8. Approximate Methods of Analysis of the Basic State of Stress of Circular Cylindrical Shells

9. Approximate Methods of Analysis of Edge Effects

10. States of Stress Corresponding to Large Values of m

11. Imposition of Boundary Conditions

Chapter 12. Analysis of Open Cylindrical Shells

12. Basic Formulas for Analysis

13. Properties of Roots of Characteristic Equation

14. Analysis of the State of Stress in Open Cylindrical Shells

15. Approximate Methods of Analysis of Open Cylindrical Shells

16. Imposition of Boundary Conditions

Part IV Analysis of the State of Stress in an Arbitrary Shell

Chapter 15. Asymptotic Integrations of Partial Differential Equations

1. Classification of Linear Differential Operators with Partial Derivatives

2. Nomenclature and Notations

3. Asymptotic Expansion of Integrals of a Homogeneous Differential Equation

4. Three Fundamental Cases

5. Construction of Functions of Variation

6. Integrals with Given Non-Characteristic Supporting Contour

7. Case of Multiple Characteristics

8. Integrals with Given Characteristic Supporting Contour

9. Asymptotic Expansion of Particular Solution of a Nonhomogeneous Partial Differential Equation

10. Example

Chapter 14. Asymptotic Integration of Equations of the Theory of Shells

11. Asymptotic Integration of a System of Equations

12. Non-Contradictory values of Indices of Intensity

13. Construction of Functions of Variation

14. Determination of Coefficients of Asymptotic Expansion of Functions of Intensity for Fundamental Integrals

15. Construction of Approximate Equations of the Theory of Shells

16. Asymptotic Error of Equations of Membrane Theory

17· Elementary States of Stress in an Arbitrary Shell

18. The Complete State of Stress in an Arbitrary Shell

Chapter 15. Elementary States of Stress

19. Fundamental State of Stress. Membrane and Pure Bending States of Stress

20. Approximate Equations for States of Stress with Large Indices of Variation

21. Region of Applicability of Equation (20.ll)

22. Simple Edge Effect

23. Integration of the Solving Equation of Simple Edge Effect

24. Solving Equations of Non-degenerate Generalized Edge Effects

25. Solving Equations of Generalized Edge Effect in a Shell of Zero Curvature

26. Range of Applicability of Solving Equations (25.5)

27. Further Simplification of Solving Equations (25.5)

28. Range of Applicability of Membrane Theory in the Analysis of Shells of Zero Curvature

29. Estimating the Accuracy of Construction of a Complete State of Stress

Part V Approximate Methods of Analysis of Shells

Chapter 16. Application of Expansions in Orthogonal Functions to the Analysis of Shells

1. Expansion of Functions in Fourier Series

2. Methods of Construction of Closed Orthogonal Systems of Functions

3. Continuation

4. Index of Variation of the State of Stress and of External Loading

Chapter 17. General Approximate Methods

5. Membrane Theory

6. Region of Applicability of Membrane Theory

7. Properties of the Simple Edge Effect

8. Approximate Theory of the Simple Edge Effect

9. Analysis of Shells by the Membrane Theory with Consideration of Edge Effects

10. Particular Cases

11. Example

12. Approximate Methods of Analysis of Shells with Large Indices of Variation

13. Example

14. Shells with Non-rigidly Supported Edges

Chapter 18. Cylindrical and Conical Shells

15. The Generalized Edge Effect in a Shell of Zero Curvature

16. The Solving Equations of the Generalized Edge Effect in Shells of Zero Curvature

17. Integration of the Solving Equations of the Generalized Edge Effect for Cylindrical Shells

18. Imposition of Boundary Conditions .

19. Integration of Equations of the Generalized Edge Effect for Conical Shells

20. Analysis of the State of Stress of Shells of Zero Curvature

21. Approximate Theory of the Non-degenerate Edge Effect

22. Integration of Equations of the Non-degenerate Edge Effect for Cylindrical and Conical Shells

23. Integration of System (22.9)

24. Continuation

25. Tables of Elastic Reactions and of Elastic Displacements of Cylindrical Shells of Medium Reduced Length

26. Example

27. Analysis of Cylindrical Shells of Medium Reduced Length Subjected to Loads Distributed along a Generator

28. Example

29. Analysis of Conical Shells

Author's Addendum to English Edition (Some Mathematical Problems of the Linear Theory of Elastic Thin Shells)

Author's Amendments

Author Index

Subject Index

## Product details

- No. of pages: 680
- Language: English
- Copyright: © Pergamon 1961
- Published: January 1, 1961
- Imprint: Pergamon
- eBook ISBN: 9781483164625

## About the Author

### A. L. Gol'Denveizer

## About the Editors

### Th. Von Kármán

#### Affiliations and Expertise

Case Institute of Technology, Cleveland, Ohio

### H. L. Dryden

#### Affiliations and Expertise

Case Institute of Technology, Cleveland, Ohio