ON PERTURBATIONS OF BUCKLING MODES OF ROD SYSTEMS CORRESPONDING TO MULTIPLE CRITICAL FORCES WHEN THE POSITION OF CONSTRAINTS CHANGES

Abstract

The article is devoted to the study of the influence of the position of supports of rod systems containing longitudinally compressed elements on their critical forces and the corresponding forms of buckling. Many issues related to the design and operation of such systems, in particular ensuring their stability, require taking into account the features of these forms, in particular the location of their nodes, extreme points, etc. Of special complexity is the case of a multiple critical force, for which the buckling mode is not uniquely determined, since an infinitely many buckling modes correspond to a multiple critical force. In the proposed work, for the case of a concentrated deformable or absolutely rigid hinged support, it is studied how, with a small displacement of the support, two simple critical forces are formed from a multiple critical force, and two uniquely determined buckling forms are formed from the corresponding infinite set of forms. In this case, significant use is made of analytical and qualitative methods of the theory of stability of rod systems, in particular, well-known theorems on the influence of imposing constraints on their critical forces, as well as previously established relationships determining the derivatives of the critical forces with respect to the coordinates determining the positions of the moving supports. Analytical expressions are proposed that allow one to describe the buckling modes formed after a small shift of the support in one direction or another, from which, in particular, it follows that on a moving support the angles of slope of the rod axis for these forms at the same value of the support reaction are numerically equal, but opposite in direction. The conclusions of the article are demonstrated on specific examples of two-span prismatic rods compressed by a longitudinal force constant along the length. In one of them, the position of the deformable intermediate support varies with absolutely rigid end supports. In the other, the intermediate absolutely rigid support moves when one of the end supports has a finite rigidity. In both examples, at a certain value of the rigidity of the deformable support, the main critical force becomes twofold and the rod can lose stability in an infinite number of configurations. Direct calculations performed for these cases show that the shift of the intermediate support leads to the effect described in the article and confirm its results.