Theory of Elastic Thin Shells

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