ELASTIC BENDING OF A STRIP WITH A SIGNIFICANT DEFLECTION UNDER THE ACTION OF APPLIED FORCES AND MOMENT

Abstract

The elastic bending of a strip of rectangular cross-section in the form of a cantilevered rod is considered. In the course of resistance of materials problems of deformation of beams under the influence of the applied forces and the moments allowing insignificant deflections are usually considered. Due to the small angular displacements of the elastic line, a simplified expression of its curvature is adopted, and the theory of bending in this case is linear. At considerable deflections of cores the linear theory of a bend cannot be used, it is necessary to use exact expression of curvature of a longitudinal elastic axis of a core.
 The article considers the folding of a strip with its small thickness in comparison with the width, which can be exemplified by a ruler. It is assumed that the strip material works resiliently according to Hooke's law and after the cessation of the deforming forces completely restores its shape. Due to the small thickness of the strip in the plane of the bend in comparison with the length of this strip, the displacement of the end of the point of the strip are significant. At the same time, in any small volume of this band with linear dimensions of the order of its thickness, all deformations remain small. Thus, with small internal elastic deformations, large displacements of the end of the cantilever strip are achieved during its bending.
 Under the action of applied forces and moments, the elastic axis of the strip acquires a curvilinear shape, which is determined by the dependence of its curvature on the length of the arc. According to the theory of elastic bending, the curvature of the strip is directly proportional to the applied moment and inversely proportional to its stiffness. This dependence of the curvature is a natural equation of the curve. In differential geometry, there are formulas for the transition from the natural equation of the curve to the parametric equations on which the curve can be constructed. However, this transition to the final result is almost impossible due to the double integration of expressions, even for the simplest cases of band load. For example, under the action of a concentrated force at the end of the console, the moment has a linear relationship. The curve with a linear natural equation is known and is called clothoid. However, its construction must be carried out by numerical methods. The paper considers this case in combination with the action of the applied moment and constructs the elastic axis of the strip.