NASA TECHNI CAL NASA TT F-118 T R A N S L A T I ON CO <c CO THEORY OF ANISOTROPIC SHELLS by S. A. Ambartsumyan State Publishing House for Physical and Mathematical Literature Moscow, 1961 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. MAY 1964 THEORY OF ANISOTROPIC SHELLS By S. A. Ambartsumyan Translation of "Teoriya anizotropnykh obolochek" State Publishing House for Physical and Mathematical Literature Moscow, 1961 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Office of Technical Services, Department of Commerce, Washington, D. C. 20230 -- Price $6.09 CONTENTS Foreword ............................................................................ v 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. Stresses in L ay ers................................................................................ 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 Deform ation..................................................... 33 Section 9. Elasticity Relationships........................................................................ 35 i 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 Sym­ metrically Arranged Relative to the Coordinate Surface . . . 46 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 m MEMBRANE THEORY uF ANISOTROPIC SHELLS Section 1. General Premises 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................................................................................. 13 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 Form ulas........................... 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 11 Section 7. Internal Forces, Moments, Stresses and Displacements.............................................................................. 132 Section 8. Edge Effect in Anisotropic Shel'c ............................. 136 Section 9. Long Shells of Revolution........................................... 139 Section 10. Examples of Calculation of Long Shells of Revolution . . . . 143 Section 11. Solution of a Few Problems of Shells of Revolution of Zero 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 Displace­ ments ........................................................................................... 198 Section 3. Cylindrical Shells Consisting of an Arbiti'ary Number of Orthotropic L ayers.............................................................. 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 Shell by the Method of Double Trigomeir-'c Series................................................................... 240 Section 10. Integration of Equations in the Engineering Theory of Cylindrical Shells by the Method of Single Trigono­ metric Series.............................................................................. 265 CHAPTER VI SHALLOW ANISOTROPIC SHELLS Section 1. Basic Premises. Initial Relationships and Equations. . . . 277 Section 2. Equations of Solution and Design Form ulas.......................... 280 Section 3. Shells Consisting of an Odd Number of Homogeneous Anisotropic Layers Symmetrically Arranged Relative to the Median Surface .............................................................. 285 ill 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 Orthotropic Shells................... 356 Section 8. Approximate Theory of an Extremely Shallow Shell Considering Transverse Shear ...................................... 365 Section 9. ^mother 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 lv FOREWORD Shells are widely used as structural elements in modem construction engineering, aircraft construction, shipbuilding, rocket cons—uction, 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. The last few years have seen the appearance of a number of thorough monographs: V.Z. Vlasov, Obshchaya teoriya obolochek (General Theory of Shells); A.S. Vol'mir, Gibkiye plastinki i obolochki (Elastic plates and shells); A. L. Gol'denveyzer, Teoriya uprugikh tonkikh obolochek (Theory of thin elastic shells); A.I. Lur'ye, Statika tonkostennykh uprugikh obolochek (Statics of thin- walled elastic shells); Kh. M. Mushtar' and K.Z. Galimov, Nelineynaya teoriya uprugikh obolochek (Nonlinear theory of elastic shells); V.V. Novozhilov, Teoriya tonkikh obolochek (Theory of thin shells). These monographs shed light on many fundamental problems in the theory of shells in general. However, there is almost nothing therein dealing with the theory of anisotropic laminar shells. 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 aniso­ tropic shells and plates. In distinction from the first divisions of the book, which are based on the hypothesis of nondeformable normals a6 given for the stack of the shell as a v whole, the last chapter attempts to construct a theory of essentially anisotropic shells and plates without the hypothesis of nondeformable normals. 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 associ­ ated 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 enumer­ ation is used (the first digit referring to the chapter). In conclusion I wish to express my deep gratitude to my colleagues at the Institute of Mathematics and Mechanics AN of the Armenian SSR, D.V. Peshtmaldzhyan, A. A. Khachatryan and L.A. Movsesyan, who were of great assistance in the preparation of this book. S.A. Ambartsumyan May 1959, Yerevan vl 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 Here we shall discuss in brief and without proof those aspects of the theory of curvilinear coordinates in space which will be used in subsequent dis­ cussion. As is known, the position of any point M in space may be uniquely defined by its radius vector r relative to a certain fixed point O. In rectangular Cartesian coordinates for r we have r = xl + yj + zk. (1.1) where, as usual, /, j, k represent the corresponding unit vectors. In problems of the theory of shells the location of any point M with radius - vector r is conveniently defined not by three cartesian coordinates x, y and z but by any three other numbers a, p, 7. The quantities a. p, 7,which uniquely de­ fine the location of point M in space are known as the curvilinear coordinates of point M. Each of these coordinates is a function of radius vector r or a function of the components x, y and z of this radius-vector in Cartesian coordinates: a(/-) = a(jc. y. z), p(r) = p(jc. y, z), (1.2) l(r) = 1 (x, y, z). Conversely, since the radius vector r of any point M in space is completely de­ fined when we are given or, 8 and y, it is a function of these independent 2 variables, and, consequently, the components of this radius vector will be func­ tions of curvilinear coordinates * — jc<a. ,i. 7). y^y(a, -I, 7), z -=-• z (a, fl, 7). (1.3) In (1.2), assuming (o -const, (!(/•)— const, and 7 (/■; — const, we obtain three fam­ ilies of surfaces. Through each point M in space there passes a surface of each of these families. These surfaces are known as the coordinate surfaces. The lines of intersection of the coordinate surfaces are referred io as coordin­ ate lines. It is evident that in our case there will be three such coordinate lines: coordinate line a (with current coordinate a), which is formed by the intersec­ tion of coordinate surfaces [5 const and 7 const; coordinate line /? (with current coordinate 8), which is formed by the intersection of coordinate surfaces a = constand 7 -- const; coordinate line y (with current coordinate y), which is formed by the intersection of coordinate surfaces p — constand 2— const (Figure 1). Figure 1 In subsequent discussion we shall be interested only in orthogonal c *rvi- linear coordinates, that is, those curvilinear coordinates of which all coordinate lines o:, /5 and y are mutually perpendicular at each point M in space. In the given triorthogonal system of curvilinear coordinates we have for the square of a linear element in space ds2 = H] rfa* + Hi dtf 4 Hi d^, (1 • 4) where HX = HX (a, ft, 7), //.* = //,(a, p, 7), H3 = H3{a. p, 7),