Thin Circular Conical Shells Under Arbitrary Loads

1955 ◽  
Vol 22 (4) ◽  
pp. 557-562
Author(s):  
N. J. Hoff
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conical Shell ◽  
Strain Energy ◽  
Variational Calculus ◽  
Radius Of Curvature ◽  
Median Surface ◽  
Conical Shells ◽  
Arbitrary Loading ◽  
Circular Conical Shells

Abstract Equations defining the displacements of the median surface of a conical shell under arbitrary loads are developed. In their derivation only the essential parts of the strain energy are considered and three simultaneous partial differential equations are obtained through the use of the variational calculus. When the minimum radius of curvature of the median surface of the cone is made to approach a constant value, the cone goes over into a cylinder. At the same time the equations here developed for the cone are transformed into the Donnell equations of the theory of cylindrical shells. It is shown how eigenfunctions of the homogeneous equations can be constructed and how particular solutions can be found for any arbitrary loading.

A Donnell Type Theory for Asymmetrical Bending and Buckling of Thin Conical Shells

1957 ◽  
Vol 24 (4) ◽  
pp. 547-552
Author(s):  
Paul Seide
Keyword(s):  
Type Theory ◽  
Variable Coefficients ◽  
Radius Of Curvature ◽  
Median Surface ◽  
Order Partial Differential Equation ◽  
Circular Plates ◽  
Conical Shells ◽  
Arbitrary Loading ◽  
Circular Conical Shells ◽  
Asymmetrical Bending

Abstract Equations, somewhat more accurate than those recently presented by N. J. Hoff, are derived for bending and buckling of thin circular conical shells under arbitrary loading. These equations reduce to Donnell’s equations for thin cylindrical shells when the cone semivertex angle becomes very small and the minimum radius of curvature of the median surface approaches a constant value. At the other end of the scale the equations reduce to the well-known equations for flat circular plates when the cone semivertex angle approaches a right angle. In addition, for the entire range of cone semivertex angles the equations reduce to the known equations for axisymmetrical bending when variations of the displacements around the circumference vanish. The problem of bending is reduced to the solution of a single fourth-order partial differential equation with variable coefficients.

scholarly journals Generalized shear deformations for isotropic incompressible hyperelastic materials

1977 ◽  
Vol 20 (2) ◽  
pp. 129-141 ◽  
Author(s):  
James M. Hill
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Shear Deformations ◽  
Partial Differential ◽  
Hyperelastic Materials ◽  
Restricted Form ◽  
Consistent Solution

AbstractFor isotropic incompressible hyperelastic materials the single function characterizing generalized shear deformations or as they are sometimes called anti-plane strain deformations must satisfy two distinct partial differential equations. Knowles [5] has recently given a necessary and sufficient condition for the strain–energy function of the material which if satisfied ensures that the two equations have consistent solutions. It is shown here for the general material not satisfying Knowles' criterion that the only possible consistent solution of the two partial differential equations are those which are already known to exist for all strain–energy functions. More general types of generalized shear deformations for such meterials are shown to exist only for special or restricted form ot the strain-energy function. In derving these results we also obtain an alternative derivation of Knowles' criterion.

scholarly journals STRESS-STRAIN STATE OF THE CONICAL SHELL OF VARIABLE THICKNESS BASED ON THREE-DIMENSIONAL EQUATIONS OF ELASTICITY THEORY

2020 ◽  
Vol 82 (2) ◽  
pp. 189-200
Author(s):  
Val.V. Firsanov ◽  
V.T. Pham
Keyword(s):  
Differential Equations ◽  
Classical Theory ◽  
Variable Thickness ◽  
Conical Shell ◽  
Strain State ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Median Surface ◽  
Stress Strain ◽  
Stress Strain State

The results of a study of the stress-strain state of a conical shell of variable thickness based on a non-classical theory are presented. The sought-for displacements of the shell are approximated by polynomials in the normal coordinate to the median surface two degrees higher in relation to the classical theory of the Kirchhoff-Love type. When developing the theory, the three-dimensional equations of the theory of elasticity, as well as Lagrange variational principle are used as the equation of the shell state. As the result of minimizing the specified value of the total energy of the shell, a mathematical model is constructed, which is a system of differential equations of equilibrium in the displacements with variable coefficients and the corresponding boundary conditions. Two cases are considered: the shell is under the action of symmetric and asymmetric loads. Two-dimensional equations are transformed to the system of ordinary differential equations by means of trigonometric sequences as per circumferential coordinate. To solve the formulated boundary value problem, finite difference and matrix sweep methods are applied. The calculations have been made by means of a computer program. After having determined the displacements, shell deformations and tangential stresses are found from geometric and physical equations, transverse stresses - from the equilibrium equations of the three-dimensional theory of elasticity. As an example, a conical shell rigidly restrained at the two edges, with asymmetrically varying thickness is considered. Compared are the results of the VAT calculations obtained as per the improved and classical theories. The significant contribution of additional stresses in the boundary zone to the total stress state of the shell is shown. The received results can be used in the strength and durability calculations and tests of machine-building facilities of various purposes.

Dynamic Axisymmetric Buckling of Shallow Conical Shells Subjected to Impulsive Loads

1965 ◽  
Vol 32 (1) ◽  
pp. 129-134 ◽  
Author(s):  
R. E. Fulton
Keyword(s):  
Conical Shell ◽  
Strain Energy ◽  
Maximum Displacement ◽  
Spherical Cap ◽  
Single Degree Of Freedom ◽  
Conical Shells ◽  
Simply Supported ◽  
Single Degree ◽  
Impulsive Loads ◽  
Axisymmetric Buckling

A theoretical investigation is made of the axisymmetric snap-through buckling of a shallow conical shell subjected to an idealized impulse applied uniformly over the surface of the shell. The shell is assumed to behave as a single-degree-of-freedom system, and a study is made of the strain energy at maximum displacement: i.e., zero velocity. Under certain conditions this equilibrium position becomes unstable and the shell can snap through (or buckle). Nonlinear strain displacement equations are used and solutions are obtained for clamped and simply supported boundaries at the edge of the shell. Results for the cone are compared with similar results for a shallow spherical cap having the same rise as the cone. This comparison indicates that the spherical shell can resist a larger impulse than the conical shell before buckling.

Plastic Analysis of Circular Conical Shells

1960 ◽  
Vol 27 (4) ◽  
pp. 696-700 ◽  
Author(s):  
P. G. Hodge
Keyword(s):  
Conical Shell ◽  
Collapse Load ◽  
Finite Area ◽  
Concentrated Load ◽  
Conical Shells ◽  
Edge Angle ◽  
Rotationally Symmetric ◽  
Plastic Analysis ◽  
Circular Conical Shells ◽  
Support Conditions

A right circular conical shell of edge angle α is subjected to a concentrated load Q directed along the axis. The collapse load is found to be Q = 2πM0 cos2 α, independently of the size or support conditions of the shell. Some solutions are also obtained for the case where the load is distributed over a finite area. Bounds are found on the collapse load of a general rotationally symmetric shell under a concentrated load.

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