ON ACCURATE AND APPROXIMATE SOLUTIONS OF STABILITY PROBLEM OF BAR-STRIP WITH SMALL SHEAR STIFFNESS UNDER UNIFORM AXIAL COMPRESSION

Abstract

A two-dimensional linearized problem of a bar (strip) stability made of a linearly elastic orthotropic material is considered in conditions of the compressive stress which is homogeneous in its length and cross-section.
Its reduction to one-dimensional equations, derived in two variants, is performed without introduction any simplifying assumptions and satisfying the static boundary conditions accurately at the points of the longitudinal edges. The first variant is based on using only trigonometric functions (sine and cosine) at the lateral coordinate (SC-approximation), allowing to obtain the accurate equations, while deriving an equation of the second variant we use as basic a sine, a cosine (even harmonics) and a unit (SC1-approximation). It is shown, that in the pivoting case of the lateral bar edges the derived equations allow to reveal only one practically useful form of stability loss (FSL) which is of shear type.
It is indicated that an application of the shown approximation enables to get one-dimensional equations, allowing to analyze a bending FSL only in case of assuming the absence of normal stresses in the longitudinal sections. Proceeding from the result analysis of the solved equations, obtained for the bar with pivoting lateral edges, a theoretical opportunity of a bar stability loss with a number of half-waves (harmonics in trigonometric functions) along the lateral coordinate morethan a unit, is stated These solutions are used for determining the values of critical loads which are lower than those known in scientific literature and they can't be set proceeding from the existing and more precised theory variants of bars, plates and shells.
Using trigonometric basic functions with an assumption of the normal stress absence in the longitudinal sections, a variant of one-dimensional equations is also derived which allows to perform a limiting process to a bar stability equation according to Kirchhoff theory without any additional transformations. It is shown, that the equations of this type, as for their accuracy, are equal to the equations, derived using the assumption, described above and based on using SCI-approximation of displacements along the lateral coordinate but they differ in pithiness.