Asymptotic analysis of stability for prismatic solids under axial loads

Abstract

This work addresses the stability of axially loaded prismatic beams with any simply connected crosssection. The solids obey a general class of rate-independent constitutive laws, and can sustain finite strains in either compression or tension. The proposed method is based on multiple scale asymptotic analysis, and starts with the full Lagrangian formulation for the three-dimensional stability problem, where the boundary conditions are chosen to avoid the formation of boundary layers. The calculations proceed by taking the limit of the beam's slenderness parameter, ε (ε 2 ≡ area/length 2), going to zero, thus resulting in asymptotic expressions for the critical loads and modes. The analysis presents a consistent and unified treatment for both compressive (buckling) and tensile (necking) instabilities, and is carried out explicitly up to o( ε 4) in each case. The present method circumvents the standard structural mechanics approach for the stability problem of beams which requires the choice of displacement and stress field approximations in order to construct a nonlinear beam theory. Moreover, this work provides a consistent way to calculate the effect of the beam's slenderness on the critical load and mode to any order of accuracy required. In contrast, engineering theories give accurately the lowest order terms ( O( ε 2)—Euler load—in compression or O(1)—maximum load—in tension) but give only approximately the next higher order terms, with the exception of simple section geometries where exact stability results are available. The proposed method is used to calculate the critical loads and eigenmodes for bars of several different cross-sections (circular, square, cruciform and L-shaped). Elastic beams are considered in compression and elastoplastic beams are considered in tension. The O( ε 2) and O( ε 4) asymptotic results are compared to the exact finite element calculations for the corresponding three-dimensional prismatic solids. The O( ε 4) results give significant improvement over the O( ε 2) results, even for extremely stubby beams, and in particular for the case of cross-sections with commensurate dimensions.