ASYMPTOMATIC ANALYSIS OF STABILITY FOR PRISMATIC SOLIDS UNDER AXIAL LOADS

This work addresses the stability of axially loaded prismatic beams wi th any simply connected cross-section. 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 mul tiple scale asymptotic analysis, and starts with the full Lagrangian f ormulation for the three-dimensional stability problem, where the boun dary conditions are chosen to avoid the formation of boundary layers. The calculations proceed by taking the limit of the beam's slenderness parameter, epsilon (epsilon(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 com pressive (buckling) and tensile (necking) instabilities, and is carrie d out explicitly up to O(epsilon(4)) in each case. The present method circumvents the standard structural mechanics approach for the stabili ty problem of beams which requires the choice of displacement and stre ss 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 accuratel y the lowest order terms (O(epsilon(2))-Euler load-in compression or O (1)-maximum load-in tension) but give only approximately the next high er 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 differ ent cross-sections (circular, square, cruciform and L-shaped). Elastic beams are considered in compression and elastoplastic beams are consi dered in tension. The O(epsilon(2)) and O(epsilon(4)) asymptotic resul ts are compared to the tract finite element calculations for the corre sponding three-dimensional prismatic solids. The O(epsilon(4)) results give significant improvement over the O(epsilon(2)) results, even for extremely stubby beams, and in particular for the case of cross-secti ons with commensurate dimensions. (C) 1998 Elsevier Science Ltd. All r ights reserved.