Eric Reissner

On Vibrations of Shallow Spherical Shells

The problem of the axi‐symmetrical vibrations of a shallow spherical shell is reduced to two simultaneous differential equations for the tangential and normal components of displacement. The solution of these equations is obtained in terms of Bessel functions. With this solution the third‐order determinant for the frequencies of a shell segment with clamped edge is given but not evaluated. Instead, an approximate value for the lowest frequency is calculated by means of the Rayleigh‐Ritz procedure [Eq. (40)]. It is found that very little curvature is needed to modify appreciably the corresponding flat‐plate frequency. The approximations which are made are those customary for shallow shells: (1) omission of the transverse shear term in the tangential force equilibrium equation, and (2) relations between couples and changes of curvatures as in the theory of flat plates.

On asymptotic solutions for nonsymmetric deformations of shallow shells of revolution

Abstract The paper presents asymptotic solutions of the differential equations for unsymmetrical deformations of such shallow shells of revolution as behave qualitatively similar to shallow spherical shells. Three distinct classes of asymptotic solutions, two of an interior nature, one of a boundary layer nature, and each having its own amplitude factor, are superimposed. Determination of the relative magnitudes of the amplitude factors follows from a consideration of the relevant boundary conditions, as exemplified for two distinct classes of boundary conditions. Determination of the amplitude factors is equivalent to the determination of the relative orders of magnitude of interior bending and direct stresses and of edge zone bending and direct stresses. The general results are illustrated by the explicit solution for a shallow spherical shell with sinusoidally varying meridional edge load, and with vanishing circumferential edge load and meridional bending moment, the fourth condition being that of vanishing transverse shear force or of vanishing transverse deflection.

On the Determination of the Centers of Twist and of Shear for Cylindrical Shell Beams

Abstract : The authors have recently proposed new definitions for the centers of twist and of shear, in terms of influence coefficients for tip-loaded cantilever beams. These definitions are here applied in conjunction with a minimum complementary energy method for the approximate determination of the influence coefficients of the problem of thin-walled open and closed cross section beams, with the possibility of a continuous transition from closed cross section to open cross section. The result is an explicit formula for the coordinate of the centers of twist and of shear of beams the cross sections of which have one axis of symmetry. This formula includes as special cases the known elementary formula for open cross sections, an extension of this formula so as to include the case of flat plates, and a known formula for closed cross section thin-shell beams. (Author)

ON THE NONLINEAR THEORY OF THIN PLATES

ABSTRACT This paper discusses a formulation of the theory of thin elastic plates which is valid for arbitrarily large deformations subject to the assumption of small strain and to the assumption of middle surface normal preservation during deformation. Of primary interest is the formulation of plate theory in such a way that its range of applicability extends beyond that of the small-finite-deflection theory associated with the names of Kirchhoff, A. Foppl and von Karman. This discussion is adapted from a special lecture presented to the Third Southeastern Conference on Theoretical and Applied Mechanics, March 31-April 1, 1966, at the University of South Carolina in Columbia.

A Note on Günther’s Analysis of Couple Stress

The following considerations are concerned with Gunther’s form of couple stress theory [1]. While the observations which follow were made without knowledge of the earlier work1, they are offered here as a supplement to it.

Note on the problem of St. Venant flexure

ZusammenfassungEs wird gezeigt, wie die St. Venantsche Theorie der Querkraftbiegung in einem nichthomogenen transversal isotropen Stabe im allgemeinen mit einem ebenen Dehnungszustand in den Querschnittebenen verbunden ist, und dass dieser ebene Dehnungszustand verschwindet, falls eine gewisse Kombination der Elastizitätszahlen eine Konstante ist.

A note on stress strain relations of the linear theory of shells

ZusammenfassungEs wird gezeigt, wie man gewisse bekannte Elastizitätsgleichungen der linearen Schalentheorie [2] umschreiben kann mit Hilfe von verallgemeinerten Dehnungskomponenten und verallgemeinerten Energiefunktionen, welche von dem erstgenannten Verfasser [3] eingeführt worden sind.

On Stress-Strain Relations and Strain-Energy Expressions in the Theory of Thin Elastic Shells

The stress-strain relations of Flugge and Byrne for thin elastic shells are inverted to express strain quantities, and therewith the strain energy, in terms of stress resultants and couples. In this form, and upon omission of terms which are small of order h2 /R2 , the stress-strain relations and the strain-energy expression are shown to be simply related to corresponding results of Trefftz. The strain-energy formula of Trefftz is generalized to arbitrary orthogonal middle surface co-ordinates.

Elementary Differential Equations

1. Introduction to Differential Equations. 2. First-Order Equations. 3. Second and Higher-Order Linear Differential Equations. 4. Some Physical Applications of Linear Differential Equations. 5. Power Series Solutions of Differential Equations. 6. Laplace Transforms. 7. Introduction to Systems of Linear Differential Equations and Applications. 8. Numerical Methods. 9. Matrix Methods for Systems of Differential Equations. 10. Nonlinear Equations and Stability. 11. Fourier Series and Boundary Value Problems. 12. Partial Differential Equations. Appendices.