In this post, we will see the book Theory Of Anisotropic Shells by V. A. Ambartsumyan.

# About the book

Shells are widely used as structural elements in modem construction engineering, aircraft construction, shipbuilding, rocket construction, etc. A careful study of the shells used in engineering leads to the conclusion that they are most often anisotropic (naturally or structurally) and in many cases are anisotropic and laminar.

Despite the large number of articles appearing in journals, there is as yet not one book devoted to the theory of anisotropic laminar shells. In the present book the author partially fills this gap. The text is based on the author’s investigations over the last few years. It consists of the following divisions: (a) fundamental equations of the theory of elasticity of an anisotropic body in curvilinear coordinates; (b) general theory of anisotropic laminar shells; (c) membrane theory of anisotropic shells; (d) theory of symmetrically loaded anisotropic shells of revolution; (e) anisotropic cylindrical shells; (f) shallow anisotropic shells; (g) new theories of anisotropic shells and plates.

The book does not deal with the undeniably important problems of non linear theory, the theories of stability and vibration, as well as temperature problems of anisotropic laminar shells. Nor does it deal with problems associated with plastic and elastic-plastic deformations of the material of the shell layers, since these problems have not been adequately investigated.

Within each chapter the formulas have a two-part enumeration. Where reference is made to the formulas of preceding chapters a three-part enumeration is used (the first digit referring to the chapter).

The book was translated from Russian by was published under NASA Technical Translation series.

Credits to original uploader.

You can get the book here.

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# Contents

## CHAPTER I

FUNDAMENTAL EQUATIONS OF THE THEORY OF ELASTICITY OF AN ANISOTROPIC BODY IN CURVILINEAR COORDINATES

Section 1. Some Remarks on Curvilinear Coordinates in Space 1

Section 2. Deformation Components and Differential Equations of Equilibrium in the Triorthogonal System of Curvilinear Coordinates 4

Section 3. Curvilinear Anisotropy. Generalized Hooke’s Law 7

Section 4. Transformation of Elastic Constants with Rotation of the Coordinate System 12

Section 5. Elastic Constants for Certain Anisotropic Materials 14

## CHAPTER II

FUNDAMENTAL EQUATIONS OF THE THEORY OF SHELLS CONSISTING OF AN ARBITRARY NUMBER OF ANISOTROPIC LAYERS

Section 1. Basic Concepts, Initial Relationships and Hypotheses 18

Section 2. Displacements and Deformations 22

Section 3. Equations of Continuity of Deformations of the Coordinate

Surface 25

Section 4. Stressesin Layers 26

Section 5. Conditions of Contact of Adjacent Layers 28

Section 6. Internal Forces and Moments 29

Section 7. Equilibrium Equations 31

Section 8. Potential Energy of Deformation 33

Section 9. Elasticity Relationships 35

Section 10. Boundary Conditions 38

Section 11. Additional Remarks Concerning the Conditions of Contact

of Adjacent Layers and the Conditions at the Outer Surfaces of a Shell 40

Section 12. Special Cases of Anisotropy of the Material of the Shell Layers 43

Section 13. Shells Consisting of an Odd Number of Layers Symmetrically Arranged Relative in the Coordinate Surface 45

Section 14. Single-Layer Anisotropic Shells 50

Section 15. Further Remarks Concerning Elasticity Relationships 55

Section 16. Calculation of Stiffnesses for Arbitrary Directions 58

## CHAPTER III

MEMBRANE THEORY OF ANISOTROPIC SHELLS

Section 1. General I’remises and Initial Relationships in the Membrane Theory of Single-Layer Isotropic Shells 61

Section 2. Boundary Conditions 64

Section 3. Area of Applicability of the Membrane Theory 65

Section 4. Fundamental Equations of the Membrane Theory of Symmetrically loaded Shells of Revolution 66

Section 5. Examples of Calculation of Symmetrically Loaded Shells

Of Revolution 73

Section 6. Evaluation of Results Obtained in the Preceding Section 86

Section 7. Continuation of Section 5 87

Section 8. An Arbitrarily Loaded Cylindrical Shell of Arbitrary Shape 95

Section 9. Some Remarks Concerning the Membrane Theory of Anisotropic Laminar Shells 104

## CHAPTER IV

SYMMETRICALLY LOADED ANISOTROPIC SHELLS OF REVOLUTION

Section 1. Basic Premises. Initial Relationships and Equations 109

Section 2. Equations of Solution and Design Formulas 113

Section 3. Shells of Revolution Consisting of an Odd Number of Layers Symmetrically Arranged Relative to the Median Surface of the Shell 117

Section 4. Single-Layer Shells of Revolution 120

Section 5. Reduction of the System of Equations in (3.16) and

(3.17) to a Single Equation. A Particular Solution of the

Inhomogeneous Equation 121

Section 6. Asymptotic Integration of the Equation of Solution (5.9) 124

Section 7. Internal Forces, Moments, Stresses and Displacements 132

Section 8. Edge Effect in Anisotropic Shell 136

Section 9. Long Shells of Revolution 26 139

Section 10. Examples of Calculation of Long Shells of Revolution 143

Section 11. Solution of a Few Problems of Shells of Revolution of Zere Gaussian Curvature Consisting of an Arbitrary Number of Layers 166

Section 12. Anisotropic Cylindrical Shells of Revolution Reinforced

by Lateral Ribs 177

## CHAPTER V

ANISOTROPIC CYLINDRICAL SHELLS

Section 1. Basic Premises. Initial Relationships and Equations 195

Section 2. System of Differential Equations of Solution in Displacements 198

Section 3. Cylindrical Shells Consisting of an Arbitrary Number of Orthotropic Layers 201

Section 4. Engineering Theory of Cylindrical Shells Consisting of an Arbitrary Number of Anisotropic Layers 208

Section 5. Continuation of Section 4. 214

Section 6. Cylindrical Shells Consisting of an Odd Number of Homogeneous Anisotropic Layers Symmetrically Arranged Relative to the Median Surface 224

Section 7. Engineering Theory of Cylindrical Shells Consisting of an Odd Number of Homogeneous Anisotropic Layers Symmetrically Arranged Relative to the Median Surface 228

Section 8. Continuation of Section 7. 232

Section 9. Integration of the Equations of Engineering Theory of a Cylindrical Sne!l by the Method of Double Trigonometric Series 240

Section 10. Integration of Equations in the Engineering Theory of Cylindrical Shells by the Method of Single Trigonometric 265

## CHAPTER VI

SHALLOW ANISOTROPIC SHELLS

Section 1. Basic Premises. Initial Relationships and Equations 277

Section 2. Equations of Solution and Design Formulas 280

Section 3. Shells Consisting of an Odd Number of Homogeneous Anisotropic Layers Symmetrically Arranged Relative to the Median Surface 285

Section 4. Extremely Shallow Shells. Basic Premises, Initial Relationships and Equations 288

Section 5. Equations of Solution and Design Formulas in the Theory of Extremely Shallow Shells Consisting of an Arbitrary Number of Anisotropic Layers 292

Section 6. Equations of Solution and Design Formulas in the Theory of Extremely Shallow Shells Consisting of an Odd Number of Homogeneous Anisotropic Layers Symmetrically Arranged Relative to the Median Surface 300

Section 7. Integration of the Equations of Solution in the Theory of Extremely Shallow Orthotropic Shells 306

## CHAPTER VII

NEW THEORIES OF ANISOTROPIC SHELLS AND PLATES

Section 1. Basic Premises and Hypotheses 315

Section 2. Theory of Orthotropic Plates 316

Section 3. Theory of Bending of a Plate Possessing Cylindrical Anisotropy 328

Section 4. Approximate Theory of an Anisotropic Plate Considering Transverse Shear 332

Section 5. Another Approximate Theory of Anisotropic Plates 336

Section 6. Examples of Plate Calculations 342

Section 7. Theory of Extremely Shallow Anisotropic Shells 358

Section 8. Approximate Theory of an Extremely Shallow Shell Considering Transverse Shear 365

Section 9. Another Approximate Theory for an Extremely Shallow Shell 368

Section 10. Example of Calculation of a Shell 372

Section 11. Theory of Extremely Shallow Laminar Orthotropic Shells. 376

Section 12. Examples of Calculation of Laminar Shells and Plates 386