13. Transforming Deformation Model of Flat Beams with Finite Length Fastening Areas Located on One of the Front-Face Surfaces

On the example of a plane problem of the mechanics of a beam with finite length fastening areas located on one of the front-face surfaces, it is shown that in the study of deformation processes with additional account of the fixed area compliance requires the introduction of the concept of the stress–strain state type transformation at the transition across the boundary from an unfastened to a fastened section. Appropriate mathematical models are also required to describe such a phenomenon. Within the framework of the classical Kirchhoff–Love model it is impossible to take into account the presence of such fixed sections. At the same time, within the framework of the simplest S.P. Timoshenko shear model, this transformation is possible if the section is fixed only on one of the front-face surfaces. The kinematic and force conditions for coupling of fastened and unfastened beam sections have been formulated. Based on the derived relations, an exact analytical solution to the problem of static bending of a cantilevered beam under the action of a constant surface load has been found. This solution is in good agreement with the results obtained by modeling of a beam using rectangular finite elements in a plane stress state, as well as using the ANSYS software package based on the equations of a plane problem of elasticity theory. An exact analytical solution has been obtained for the problem of transverse bending vibrations of a flat beam with two cantilevered parts and a finite length section between them under vibration loading by a transverse force acting on one of the unfastened section.