Column buckling with shear deformations—A hyperelastic formulation

Column constitutive relationships and buckling equations are derived using a consistent hyperelastic neo-Hookean formulation. It is shown that the Mandel stress tensor provides the most concise representation for stress components. The analogous definitions for uniaxial beam plane stress and plane strain for large deformations are established by examining the virtual work equations. Anticlastic transverse curvature of the beam cross-section is incorporated when plane stress or thick beam dimensions are assumed. Column buckling equations which allow for shear and axial deformations are derived using the positive definiteness of the second order work. The buckling equations agree with the equation derived by Haringx and are extended to incorporate anticlastic transverse curvature which is important for low slenderness, high buckling modes and with increasing width to thickness ratio. The work in this paper does not support the existence of a shear buckling mode for straight prismatic columns made of an isotropic material.

[1]  George A. Kardomateas,et al.  Buckling of moderately thick orthotropic columns: comparison of an elasticity solution with the Euler and Engesser/Haringx/Timoshenko formulae , 1997 .

[2]  James M. Kelly,et al.  The Analysis of Multilayer Elastomeric Bearings , 1984 .

[3]  E. Reissner.  On one-dimensional finite-strain beam theory: The plane problem , 1972 .

[4]  A. Gjelsvik.  Stability of Built-up Columns , 1991 .

[5]  S. Timoshenko.  Theory of Elastic Stability , 1936 .

[6]  J. S. Wilson,et al.  Stability of Structures , 1935, Nature.

[7]  Robert L. Taylor,et al.  Numerical formulations of elasto-viscoplastic response of beams accounting for the effect of shear , 1984 .

[8]  Zdenek P. Bazant,et al.  Shear buckling of sandwich, fiber composite and lattice columns, bearings, and helical springs: Paradox resolved , 2003 .

[9]  Mario M. Attard,et al.  Finite strain––isotropic hyperelasticity , 2003 .

[10]  Norman A. Fleck,et al.  Compressive Kinking of Fiber Composites: A Topical Review , 1994 .

[11]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[12]  J. A. Haringx.  On highly compressible helical springs and rubber rods, and their application for vibration-free mountings , 1950 .

[13]  J. C. Simo,et al.  Remarks on rate constitutive equations for finite deformation problems: computational implications , 1984 .

[14]  Z. Bažant,et al.  Stability and finite strain of homogenized structures soft in shear: Sandwich or fiber composites, and layered bodies , 2006 .

[15]  Goto Yoshiaki,et al.  Elliptic integral solutions of plane elastica with axial and shear deformations , 1990 .

[16]  Engesser Fr.,et al.  Die Knickfestigkeit gerader Stäbe , 1891 .

[17]  E. Reissner.  Some remarks on the problem of column buckling , 1982 .

[18]  S. Mikhlin,et al.  Variational Methods in Mathematical Physics , 1965 .

[19]  Z. Bažant,et al.  Sandwich buckling formulas and applicability of standard computational algorithm for finite strain , 2004 .

[20]  Zdenek P. Bazant,et al.  A correlation study of formulations of incremental deformation and stability of continuous bodies , 1971 .

[21]  Norman A. Fleck,et al.  End compression of sandwich columns , 2002 .

[22]  Mario M. Attard,et al.  Finite strain––beam theory , 2003 .

[23]  B. W. Rosen,et al.  Mechanics of composite strengthening. , 1965 .

[24]  H. Ziegler,et al.  Arguments for and against Engesser's buckling formulas , 1982 .

[25]  Mario M. Attard,et al.  Hyperelastic constitutive modeling under finite strain , 2004 .