Reissner’s New Mixed Variational Principle Applied to Laminated Cylindrical Shells

1992 ◽  
Vol 114 (1) ◽  
pp. 115-119 ◽  
Author(s):  
K. Bhaskar ◽  
T. K. Varadan
Keyword(s):  
Cylindrical Shells ◽  
Three Dimensional ◽  
Order Theory ◽  
Equilibrium Equations ◽  
Mixed Variational Principle ◽  
Elasticity Solutions ◽  
Transverse Stresses ◽  
Higher Order Theory ◽  
Displacement Approach ◽  
Three Dimensional Elasticity

Reissner’s new mixed variational theorem, which allows independent interpolation, through the thickness, of the three transverse stresses besides that of the three displacements, is applied here to derive a higher-order theory of laminated orthotropic cylindrical shells. The accuracy of the theory is verified by comparison with three-dimensional elasticity solutions. It is shown that Reissner’s principle does not directly lead to accurate transverse shear stress predictions, but requires the use of the equilibrium equations of three-dimensional elasticity as is common in the conventional displacement approach.

A Higher-Order Theory for Plane Stress Conditions of Laminates Consisting of Isotropic Layers

1999 ◽  
Vol 66 (1) ◽  
pp. 95-100 ◽  
Author(s):  
X. J. Wu ◽  
S. M. Cheng
Keyword(s):  
Three Dimensional ◽  
Order Theory ◽  
Higher Order ◽  
Shear Stresses ◽  
Stress Equilibrium ◽  
Material Discontinuities ◽  
Interlaminar Shear ◽  
Higher Order Theory ◽  
Three Dimensional Elasticity ◽  
Higher Order Functions

In this paper, a higher-order theory is derived for laminates consisting of isotropic layers, on the basis of three-dimensional elasticity with displacements as higher-order functions of z in the thickness direction. The theory employs three stress potentials, Ψ (an Airy function), p (a harmonic function), and its conjugate q, to satisfy all conditions of stress equilibrium and compatibility. Interlaminar shear stresses, i.e., antiplane stresses, are shown to be present at the interfaces, especially near material discontinuities where gradients of in-plane stresses are usually high. For illustrating its practical application, the problem of a plate containing a hole patched with an intact plate is solved.

Buckling of Thick Orthotropic Cylindrical Shells Under Torsion

1999 ◽  
Vol 66 (1) ◽  
pp. 41-50 ◽  
Author(s):  
Y. S. Kim ◽  
G. A. Kardomateas ◽  
A. Zureick
Keyword(s):  
Critical Loads ◽  
Shell Theory ◽  
Cylindrical Shells ◽  
Three Dimensional ◽  
Wide Range ◽  
Classical Shell Theory ◽  
Torsional Loads ◽  
Elasticity Solutions ◽  
The Galerkin Method ◽  
Three Dimensional Elasticity

A three-dimensional elasticity solution to the problem of buckling of orthotropic cylindrical shells under torsion is presented. A mixed form of the Galerkin method with a series of Legendre polynomials in the thickness coordinate has been applied to solve the governing differential equations. The accuracy of existing shell theory solutions has been assessed through a comparison study for both isotropic and orthotropic cylinders. For isotropic cylinders the solutions based on the Donnell shell theory were found to predict nonconservative values for the critical loads. As the circumferential wave numbers increase, shell theory solutions provide more accurate values. For orthotropic cylinders, the classical shell theory predicts much higher critical loads for a relatively short and thick cylinder, while the shear deformation theories provide results reasonably close to the elasticity solutions. Detailed data are also presented for the critical torsional loads over a wide range of length ratios and radius ratios for isotropic, glass/epoxy, and graphite/epoxy cylinders.

Conservation laws of linear elasticity in stress formulations

2005 ◽  
Vol 461 (2053) ◽  
pp. 99-116 ◽  
Author(s):  
Shaofan Li ◽  
Anurag Gupta ◽  
Xanthippi Markenscoff
Keyword(s):  
Conservation Laws ◽  
Linear Elasticity ◽  
Three Dimensional ◽  
Cauchy Stress ◽  
General Boundary Conditions ◽  
Cauchy Stress Tensor ◽  
Equilibrium Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Three Dimensional Elasticity

In this paper, we present new conservation laws of linear elasticity which have been discovered. These newly discovered conservation laws are expressed solely in terms of the Cauchy stress tensor, and they are genuine, non–trivial conservation laws that are intrinsically different from the displacement conservation laws previously known. They represent the variational symmetry conditions of combined Beltrami–Michell compatibility equations and the equilibrium equations. To derive these conservation laws, Noether's theorem is extended to partial differential equations of a tensorial field with general boundary conditions. By applying the tensorial version of Noether's theorem to Pobedrja's stress formulation of three–dimensional elasticity, a class of new conservation laws in terms of stresses has been obtained.

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