Three-Dimensional Elasticity Solution for the Buckling of Sandwich Columns

2001 ◽  
Author(s):  
George A. Kardomateas
Keyword(s):  
Transverse Shear ◽  
Sandwich Structure ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Elasticity Solution ◽  
Dimensional Theory ◽  
Column Buckling ◽  
The Core ◽  
Timoshenko Column ◽  
Three Dimensional Elasticity

Abstract A study of the buckling of a sandwich column, based on the three dimensional theory of elasticity, and a comparison with the simple Euler or transverse shear correction Engesser / Haringx / Timoshenko column buckling formulas, is presented. All three phases of the sandwich structure (two face sheets and the core) are assumed to be orthotropic and the column is in the form of a hollow circular cylinder. The Euler or Engesser / Haringx / Timoshenko loads are based on the equivalent axial modulus. Representative results show the significance of the effect of transverse shear in these sandwich structures.

Three-Dimensional Elasticity Solution of a Composite Beam

1967 ◽  
Vol 1 (2) ◽  
pp. 122-135 ◽  
Author(s):  
Staley F. Adams ◽  
M. Maiti ◽  
Richard E. Mark
Keyword(s):  
Three Dimensional ◽  
Solid Wood ◽  
Theory Of Elasticity ◽  
Orthotropic Materials ◽  
Stress And Strain ◽  
Cantilever Beams ◽  
Dimensional Theory ◽  
Reinforced Plastics ◽  
Three Dimensional Elasticity ◽  
Moduli Of Elasticity

This investigation was undertaken to develop a rigorous mathe matical solution of stress and strain for a composite pole con sisting of a reinforced plastics jacket laminated on a solid wood core. The wood and plastics are treated as orthotropic materials. The problem of bending of such poles as cantilever beams has been determined by the application of the principles of three- dimensional theory of elasticity. Values of all components of the stress tensor in cylindrical coordinates are given for the core and jacket. Exact values for the stresses have been obtained from computer results, using the basic elastic constants—Poisson's ratios, moduli of elasticity and moduli of rigidity—for each ma terial. A comparison of the numerical results of the exact solu tion with strength of materials solutions has been completed.

Bifurcation of Equilibrium in Thick Orthotropic Cylindrical Shells Under Axial Compression

1995 ◽  
Vol 62 (1) ◽  
pp. 43-52 ◽  
Author(s):  
G. A. Kardomateas
Keyword(s):  
Axial Compression ◽  
Orthotropic Material ◽  
Shell Theory ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Elasticity Solution ◽  
Timoshenko Theory ◽  
Cylinder Axis ◽  
Dimensional Theory ◽  
Nonshallow Shell

The bifurcation of equilibrium of an orthotropic thick cylindrical shell under axial compression is studied by an appropriate formulation based on the three-dimensional theory of elasticity. The results from this elasticity solution are compared with the critical loads predicted by the orthotropic Donnell and Timoshenko nonshallow shell formulations. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. The bifurcation points from the Timoshenko formulation are always found to be closer to the elasticity predictions than the ones from the Donnell formulation. For both the orthotropic material cases and the isotropic one, the Timoshenko bifurcation point is lower than the elasticity one, which means that the Timoshenko formulation is conservative. The opposite is true for the Donnell shell theory, i.e., it predicts a critical load higher than the elasticity solution and therefore it is nonconservative. The degree of conservatism of the Timoshenko theory generally increases for thicker shells. Likewise, the Donnell theory becomes in general more nonconservative with thicker construction.

Three-Dimensional Elasticity Solution and Edge Effects in a Spherical Dome

1977 ◽  
Vol 44 (4) ◽  
pp. 599-603 ◽  
Author(s):  
Shun Cheng ◽  
T. Angsirikul
Keyword(s):  
Edge Effects ◽  
Thin Shell ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Elasticity Solution ◽  
Uniform Thickness ◽  
Spherical Dome ◽  
Elasticity Solutions ◽  
The Subject ◽  
Three Dimensional Elasticity

The subject of this analysis is a homogeneous, isotropic, and elastic spherical dome of uniform thickness subjected to prescribed edge stresses at the end surface. Starting from three-dimensional equations of theory of elasticity, solutions of Navier’s equations and the characteristic equation are obtained. Eigenvalues are computed for various values of the thickness and radius ratio and their special features are analyzed. Coefficients of the nonorthogonal eigenfunction expansions are then determined through the use of a least-squares technique. Many numerical results are obtained and illustrated by figures. These results show that the method presented herein yields very satisfactory solutions. These solutions are fundamental to the understanding of thin shell theories.

Three-Dimensional Elasticity Solution for the Buckling of Transversely Isotropic Rods: The Euler Load Revisited

1995 ◽  
Vol 62 (2) ◽  
pp. 346-355 ◽  
Author(s):  
G. A. Kardomateas
Keyword(s):  
Critical Load ◽  
Transverse Shear ◽  
Isotropic Material ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Radius Ratio ◽  
Isotropic Case ◽  
Elasticity Solution ◽  
Three Dimensional Elasticity ◽  
Isotropic Bodies

The bifurcation of equilibrium of a compressed transversely isotropic bar is investigated by using a three-dimensional elasticity formulation. In this manner, an assessment of the thickness effects can be accurately performed. For isotropic rods of circular cross-section, the bifurcation value of the compressive force turns out to coincide with the Euler critical load for values of the length-over-radius ratio approximately greater than 15. The elasticity approach predicts always a lower (than the Euler value) critical load for isotropic bodies; the two examples of transversely isotropic bodies considered show also a lower critical load in comparison with the Euler value based on the axial modulus, and the reduction is larger than the one corresponding to isotropic rods with the same length over radius ratio. However, for the isotropic material, both Timoshenko’s formulas for transverse shear correction are conservative; i.e., they predict a lower critical load than the elasticity solution. For a generally transversely isotropic material only the first Timoshenko shear correction formula proved to be a conservative estimate in all cases considered. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one and therefore, a more precise estimate of the transverse shear effects. Furthermore, by performing a series expansion of the terms of the resulting characteristic equation from the elasticity formulation for the isotropic case, the Euler load is proven to be the solution in the first approximation; consideration of the second approximation gives a direct expression for the correction to the Euler load, therefore defining a new, revised, yet simple formula for column buckling. Finally, the examination of a rod with different end conditions, namely a pinned-pinned rod, shows that the thickness effects depend also on the end fixity.

A Refined Theory of Elastic, Orthotropic Plates

1958 ◽  
Vol 25 (4) ◽  
pp. 437-443 ◽  
Author(s):  
S. J. Medwadowski
Keyword(s):  
Body Force ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Refined Theory ◽  
System Of Equations ◽  
Order Partial Differential Equation ◽  
Type Solution ◽  
Orthotropic Plates ◽  
Dimensional Theory ◽  
Nonlinear System Of Equations

Abstract A refined theory of elastic, orthotropic plates is presented. The theory includes the effect of transverse shear deformation and normal stress and may be considered a generalization of the classical theory of von Karman modified by the refinements of the Levy-Reissner-Mindlin theories. A nonlinear system of equations is derived directly from the corresponding equations of the three-dimensional theory of elasticity in which body-force terms have been retained. Next, the system of equations is linearized and reduced to a single sixth-order partial differential equation in a stress function. A Levy-type solution of this equation is discussed.

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