Axially Symmetric Motions of Thick Cylindrical Shells

1958 ◽  
Vol 25 (1) ◽  
pp. 97-102
Author(s):  
I. Mirsky ◽  
G. Herrmann
Keyword(s):  
Cylindrical Shells ◽  
Three Dimensional ◽  
Present Theory ◽  
Theory Of Elasticity ◽  
Approximate Theory ◽  
Axially Symmetric ◽  
Dimensional Theory ◽  
Rotatory Inertia ◽  
Transverse Normal Stress ◽  
Wide Range

Abstract An approximate theory of axially symmetric motions of thick, elastic, cylindrical shells, in which the effect of transverse normal stress is retained, is deduced from the three-dimensional theory of elasticity. The present theory contains, in addition to the usual membrane and bending terms, also the influence of rotatory inertia and transverse shear deformation. Thus it may be specialized to a variety of shell, plate, and solid-cylinder equations. The propagation of free harmonic waves in an infinite shell is studied on the basis of the present theory and the three-dimensional theory of elasticity. Excellent agreement is obtained for the phase velocity of the lowest mode of motion for a wide range of the parameters involved.

Dynamic Buckling and Post-buckling of Imperfect Orthotropic Cylindrical Shells Under Mechanical and Thermal Loads, Based on the Three-Dimensional Theory of Elasticity

1999 ◽  
Vol 66 (2) ◽  
pp. 476-484 ◽  
Author(s):  
M. Shariyat ◽  
M. R. Eslami
Keyword(s):  
Circular Cylindrical Shell ◽  
Cylindrical Shells ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Dynamic Buckling ◽  
Curvilinear Coordinates ◽  
Thermal Loads ◽  
Dimensional Theory ◽  
Highly Nonlinear

The three-dimensional theory of elasticity in curvilinear coordinates is employed to investigate the dynamic buckling of an imperfect orthotropic circular cylindrical shell under mechanical and thermal loads. Accurate form of the strain expressions of imperfect cylindrical shells is established through employing the general Green's strain tensor for large deformations and the equations of motion are derived in terms of the second Piola-Kirchhoff stress tensor. Then, the governing equations are properly formulated and solved by means of an efficient and relatively accurate solution procedure proposed to solve the highly nonlinear equations resulting from the above analysis. The proposed formulation is very general as it can include the influence of the initial imperfections, temperature distribution, and temperature dependency of the mechanical properties of materials, effect of various end conditions, possibility of large-deformation occurrence and application of any combination of mechanical and thermal loadings. No simplifications are done when solving the resulting equations. Furthermore, in contrast to the displacement-based layer-wise theories and the three-dimensional approaches proposed so far, the stress, force and moment boundary conditions as well as the displacement type ones, can be incorporated accurately in these formulations. Finally, a few examples of mechanical and thermal buckling of some orthotropic cylindrical shells are considered and results of the present three-dimensional elasticity approach are compared with the buckling loads predicated by the Donnell's equations, some single-layer theories, some available results of the layer-wise theory and the recently published three-dimensional approaches and the accuracy of the later methods are discussed based on the exact method presented in this paper.

Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motions of Cylinders

1956 ◽  
Vol 23 (4) ◽  
pp. 563-568
Author(s):  
George Herrmann ◽  
I. Mirsky
Keyword(s):  
Shear Deformation ◽  
Shell Theory ◽  
Plate Theory ◽  
Wave Length ◽  
Three Dimensional ◽  
Approximate Theory ◽  
Axially Symmetric ◽  
Rotatory Inertia ◽  
Phase Velocities ◽  
Transition Wave

Abstract The frequency (or phase velocity) of axially symmetric free vibrations in an elastic, isotropic, circular cylinder of medium thickness is studied on the basis of the three-dimensional linear theory of elasticity and several different shell theories. To be in good agreement with the solution of the three-dimensional equations for short wave lengths, an approximate theory has to include the influence of rotatory inertia and transverse shear deformation, for example, in a manner similar to Mindlin’s plate theory. A shell theory of this (Timoshenko) type is deduced from the three-dimensional elasticity theory. From a comparison of phase velocities it appears that, to a good approximation, membrane and curvature effects on one hand, and on the other hand, flexural, rotatory-inertia, and shear-deformation effects are mutually exclusive in two ranges of wave lengths, separated by a “transition” wave length. Thus, in the full range of wave lengths, the associated lowest phase velocities may be determined on the basis of the membrane shell theory (for wave lengths larger than the transition wave length) and on the basis of Mindlin’s plate theory (for wave lengths smaller than the transition wave length).

Vibration of Thick Circular Disks and Shells of Revolution

2003 ◽  
Vol 70 (2) ◽  
pp. 292-298 ◽  
Author(s):  
A. V. Singh ◽  
L. Subramaniam
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Published Data ◽  
Axially Symmetric ◽  
Shells Of Revolution ◽  
Displacement Fields ◽  
Wide Range ◽  
Circular Disks

A fully numerical and consistent method using the three-dimensional theory of elasticity is presented in this paper to study the free vibrations of an axially symmetric solid. The solid is defined in the cylindrical coordinates r,θ,z by a quadrilateral cross section in the r-z plane bounded by four straight and/or curved edges. The cross section is then mapped using the natural coordinates (ξ,η) to simplify the mathematics of the problem. The displacement fields are expressed in terms of the product of two simple algebraic polynomials in ξ and η, respectively. Boundary conditions are enforced in the later part of the solution by simply controlling coefficients of the polynomials. The procedure setup in this paper is such that it was possible to investigate the free axisymmetric and asymmetric vibrations of a wide range of problems, namely; circular disks, cylinders, cones, and spheres with considerable success. The numerical cases include circular disks of uniform as well as varying thickness, conical/cylindrical shells and finally a spherical shell of uniform thickness. Convergence study is also done to examine the accuracy of the results rendered by the present method. The results are compared with the finite element method using the eight-node isoparametric element for the solids of revolution and published data by other researchers.

Sign in / Sign up

Export Citation Format

Share Document