Influence Coefficients for Thin Smooth Shells of Revolution Subjected to Symmetric Loads

B. R. Baker; G. B. Cline

doi:10.1115/1.3640551

Influence Coefficients for Thin Smooth Shells of Revolution Subjected to Symmetric Loads

Journal of Applied Mechanics ◽

10.1115/1.3640551 ◽

1962 ◽

Vol 29 (2) ◽

pp. 335-339 ◽

Cited By ~ 7

Author(s):

B. R. Baker ◽

G. B. Cline

Keyword(s):

Differential Equations ◽

Asymptotic Integration ◽

Asymptotic Solutions ◽

Thin Shells ◽

Uniform Thickness ◽

Shells Of Revolution ◽

Influence Coefficients ◽

Independent Variable

The differential equations governing the deformation of shells of revolution of uniform thickness subjected to axisymmetric self-equilibrating edge loads are transformed into a form suitable for asymptotic integration. Asymptotic solutions are obtained for all sufficiently thin shells that possess a smooth meridian curve and that are spherical in the neighborhood of the apex. For design use, influence coefficients are derived and presented graphically as functions of the transformed independent variable ξ. The variation of ξ with the meridional tangent angle φ is given analytically and graphically for several common meridian curves—the parabola, the ellipse, and the sphere.

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On the Accuracy of Some Shell Solutions

Journal of Applied Mechanics ◽

10.1115/1.4012115 ◽

1959 ◽

Vol 26 (4) ◽

pp. 577-583

Author(s):

G. D. Galletly ◽

J. R. M. Radok

Keyword(s):

Negative Curvature ◽

Numerical Solutions ◽

Asymptotic Solutions ◽

Thin Shells ◽

Shells Of Revolution ◽

Equilibrium Equations ◽

Influence Coefficients ◽

Bending Loads ◽

Edge Influence

Abstract R. B. Dingle’s method [1] for finding asymptotic solutions of ordinary differential equations of a type such as occur in the bending theory of thin shells of revolution is presented in outline. This method leads to the same results as R. E. Langer’s method [2], recently used for problems of this kind, and permits a simple analytical and less formal interpretation of the asymptotic treatment of such equations. A comparison is given of edge influence coefficients due to bending loads, obtained by use of these asymptotic solutions and numerical integration of the equilibrium equations, respectively. The particular shells investigated are of the open-crown, ellipsoidal, and negative-curvature toroidal types. The results indicate that the agreement between these solutions is satisfactory. In the presence of uniform pressure, the use of the membrane solutions for the determination of the particular integrals appears to lead to acceptable results in the case of ellipsoidal shells. However, in the case of toroidal shells, the membrane and the numerical solutions disagree significantly.

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On Finite Symmetrical Deflections of Thin Shells of Revolution

Journal of Applied Mechanics ◽

10.1115/1.3564619 ◽

1969 ◽

Vol 36 (2) ◽

pp. 267-270 ◽

Cited By ~ 44

Author(s):

Eric Reissner

Keyword(s):

Differential Equations ◽

Nonlinear Theory ◽

Transverse Shear ◽

Shell Theory ◽

Middle Surface ◽

Thin Shells ◽

Shells Of Revolution ◽

Shear Deformations ◽

Linear Shell Theory ◽

Classical Subject

Recent simplifications of linear shell theory through consideration of transverse shear deformations and stress moments with axes normal to the shell middle surface suggest analogous approaches to the corresponding problem of nonlinear theory. As a first step in this direction consideration is given here to the classical subject of finite symmetrical deformations of shells of revolution. The principal new results of the present analysis concern the form of strain-displacement and compatibility differential equations.

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Author's Addendum to English Edition: SOME MATHEMATICAL PROBLEMS OF THE LINEAR THEORY OF ELASTIC THIN SHELLS**The material presented comprises three chapters from an article of the same title published in the Soviet journal Uspekhi matematicheskikh nauk (Progress in Mathematical Sciences) [vol. 15 (1960), No. 5, page 95]. The article is here condensed to less than half at the expense of the chapters dealing with the mechanical significance of problems in shell theory. Those chapters are retained in which the connection is discussed between the methods of asymptotic integration of partial differential equations which have recently appeared in the mathematical literature and the methods used in this book. The question of boundary conditions is discussed in much greater detail than in the book.

Theory of Elastic Thin Shells ◽

10.1016/b978-0-08-009561-5.50027-1 ◽

1961 ◽

pp. 607-645

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Boundary Conditions ◽

Linear Theory ◽

Asymptotic Integration ◽

Thin Shells ◽

English Edition ◽

Mathematical Problems ◽

Mathematical Sciences ◽

Mathematical Literature

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The Stokes phenomenon and certain nth-order differential equations I. Preliminary investigation of the equations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100032400 ◽

1957 ◽

Vol 53 (2) ◽

pp. 399-418 ◽

Cited By ~ 21

Author(s):

J. Heading

Keyword(s):

Differential Equations ◽

Preliminary Investigation ◽

Asymptotic Solutions ◽

Stokes Phenomenon ◽

Physical Problems ◽

Power Series Solutions ◽

Independent Variable ◽

Linear Differential ◽

The Relationship ◽

Integral Solutions

1. Introduction. The object of this investigation is to obtain by means of contour integrals exact solutions of certain nth-order differential equations, together with their n independent power-series solutions, their asymptotic solutions and the relationship between these two types of solution. The Stokes phenomenon associated with the changing constants in these asymptotic solutions will then be investigated by various methods. Paper I is concerned with the properties of the various contour-integral solutions of the equations under consideration. The manner in which the arbitrary constants in the general asymptotic solution must change as the argument of the independent variable z varies is dealt with in paper II. This phenomenon is considered in a way that has proved profitable for the approximate solution of more general linear differential equations of the nth order that are approximately identical with those considered here hi certain regions of the complex z–plane. In later publications, these approximations will be described together with their application to explicit physical problems.

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Effect of Internal Pressure on the Influence Coefficients of Spherical Shells

Journal of Applied Mechanics ◽

10.1115/1.3630112 ◽

1963 ◽

Vol 30 (1) ◽

pp. 91-97 ◽

Cited By ~ 12

Author(s):

G. B. Cline

Keyword(s):

Differential Equations ◽

Internal Pressure ◽

Asymptotic Solutions ◽

Constant Thickness ◽

Nonlinear Coupling ◽

Spherical Shells ◽

Spherical Cap ◽

Influence Coefficients

Uniform asymptotic solutions are found for the differential equations governing the symmetric bending of spherical shells of constant thickness subjected to self-equilibrating edge loads and internal pressure. The solutions include the effect of nonlinear coupling of the stresses and edge loads. These solutions are used to obtain expressions for the influence coefficients of both a spherical cap and the complementary portion of the sphere. The effect of the pressure coupling on these influence coefficients is presented graphically.

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On Influence Coefficients and Nonlinearity for Thin Shells of Revolution

Journal of Applied Mechanics ◽

10.1115/1.4011924 ◽

1959 ◽

Vol 26 (1) ◽

pp. 69-72

Author(s):

Eric Reissner

Keyword(s):

Linear Theory ◽

Thin Shells ◽

Shells Of Revolution ◽

Nonlinear Formulation ◽

Elastic Shells ◽

Edge Zone ◽

Influence Coefficients ◽

Rotationally Symmetric

Abstract The paper is concerned with a nonlinear formulation of the problem of rotationally symmetric deformations of thin elastic shells of revolution, which are acted upon by edge forces and moments. Determined are, in particular, nonlinear corrections to the known results of the linear theory, for edge displacements and rotations. The calculations are for cases for which thickness and curvature of the shell are such as to insure that stresses and deformations are effectively contained within a narrow edge zone of the shell.

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