A Strain-Energy Expression for Thin Elastic Shells

1949 ◽  
Vol 16 (2) ◽  
pp. 183-189
Author(s):  
H. L. Langhaar
Keyword(s):  
Strain Energy ◽  
Elastic Shell ◽  
Strain Tensor ◽  
Middle Surface ◽  
Circular Cylinders ◽  
Flat Plates ◽  
Energy Expression ◽  
Deflection Theory ◽  
Large Deflection Theory ◽  
Elliptical Cylinders

Abstract A derivation is given for the strain energy of an isotropic elastic shell whose radii of curvature are sufficiently large that strains may be assumed to vary linearly throughout the thickness. The work of Love (1) has been the only previous general investigation which expresses the strain energy in terms of the displacements of the middle surface. The effects of the tangential displacements upon the energy due to bending are found to differ appreciably from Love’s results in the first-order terms. As in the classical large-deflection theory of flat plates, quadratic terms in the derivatives of the normal deflection are retained in the strain tensor, but quadratic terms which involve the tangential displacements are neglected. Special forms of the general energy expression derived in this paper are given for shells in the shapes of flat plates, circular cylinders, elliptical cylinders, ellipsoids of revolution, and spheres. These applications, as well as certain intuitive observations, provide checks on the theory.

The Strain-Energy Expression for Thin Elastic Shells

1958 ◽  
Vol 25 (4) ◽  
pp. 546-552
Author(s):  
J. H. Haywood ◽  
L. B. Wilson
Keyword(s):  
Circular Cylinder ◽  
Plane Stress ◽  
Strain Energy ◽  
Middle Surface ◽  
Thin Shells ◽  
Elastic Shells ◽  
Energy Expression ◽  
Small Deflection ◽  
Deflection Theory

Abstract A strain-energy expression is derived for thin isotropic elastic shells in terms of the displacements of the middle surface of the shell. This expression is confined to small-deflection theory, and the condition of plane stress previously used in the theory of thin shells is retained. A simplified expression is also obtained by the introduction of the Kirchhoff-Love hypothesis, and the relative merits of these two expressions are discussed. The strain-energy expression is applied to the thin circular cylinder, and the result is compared with various strain-energy expressions developed by previous authors.

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

1960 ◽  
Vol 27 (1) ◽  
pp. 104-106 ◽  
Author(s):  
J. K. Knowles ◽  
Eric Reissner
Keyword(s):  
Strain Energy ◽  
Middle Surface ◽  
Stress Strain ◽  
Elastic Shells ◽  
Energy Expression ◽  
Energy Formula

The stress-strain relations of Flu¨gge 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.

A new approach to curvature measures in linear shell theories

2020 ◽  
pp. 108128652097275
Author(s):  
Miroslav Šilhavý
Keyword(s):  
Strain Tensor ◽  
Middle Surface ◽  
Surface Strain ◽  
Affine Function ◽  
Shear Deformations ◽  
Strain Measure ◽  
Bending Strain ◽  
Curvature Measures ◽  
Tensor Formula ◽  
Strain Measures

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99

Buckling of Multiple-Bay Ring-Reinforced Cylindrical Shells Subject to Hydrostatic Pressure

1953 ◽  
Vol 20 (4) ◽  
pp. 469-474
Author(s):  
W. A. Nash
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Middle Surface ◽  
Elastic Buckling ◽  
Elastic Instability ◽  
Total Potential ◽  
External Forces ◽  
Work Done

Abstract An analytical solution is presented for the problem of the elastic instability of a multiple-bay ring-reinforced cylindrical shell subject to hydrostatic pressure applied in both the radial and axial directions. The method used is that of minimization of the total potential. Expressions for the elastic strain energy in the shell and also in the rings are written in terms of displacement components of a point in the middle surface of the shell. Expressions for the work done by the external forces acting on the cylinder likewise are written in terms of these displacement components. A displacement configuration for the buckled shell is introduced which is in agreement with experimental evidence, in contrast to the arbitrary patterns assumed by previous investigators. The total potential is expressed in terms of these displacement components and is then minimized. As a result of this minimization a set of linear homogeneous equations is obtained. In order that a nontrivial solution to this system of equations exists, it is necessary that the determinant of the coefficients vanish. This condition determines the critical pressure at which elastic buckling of the cylindrical shell will occur.

Small Rotationally Symmetric Deformations of Shallow Helicoidal Shells

1955 ◽  
Vol 22 (1) ◽  
pp. 31-34
Author(s):  
Eric Reissner
Keyword(s):  
Flat Plate ◽  
Plane Stress ◽  
General Property ◽  
Middle Surface ◽  
Circular Ring ◽  
Sample Problem ◽  
Flat Plates ◽  
Transverse Bending ◽  
Rotationally Symmetric

Abstract Known solutions for transverse bending and plane stress of flat circular ring plates are generalized so as to apply to shallow helicoidal shells. The pitch of the middle surface of the shell is responsible for a coupling of what would be separate problems of plane stress and transverse bending for flat plates. Explicit results are obtained for an important sample problem. A general property of helicoidal cantilever shells is stated. Criteria are obtained indicating, in terms of the parameters of the shell (a) the range of applicability of the results obtained, and (b) the range in which the shell behaves like a flat plate.

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