Statyka i dynamika skręcanego cienkościennego dwuteownika o zmiennym, bisymetrycznym przekroju poprzecznym

The paper is a closure of author's sequence of works [7, 11]. The torsion of an I-beam with variable, bisymmetric section is considered whereby the assumption is made that the section varies slowly and continuously as a consequence of the varying lateral dimensions of its component elements (web and flanges). The theoretical analysis is performed regarding the mass-forces and the dynamic character of the external torsional loading; in particular the static problem and the question of the beam's torsional free vibrations is examined. The theoretical considerations are based on the technical torsion theory of thin-walled beams with constant cross-sections [8].
After a detailed review of the existing research works dealing with the mentioned problem, is given the beam's state of displacements and strains is analysed. The necessity of introducing corrections to the idea of the linear strains in the beam's flanges (δ_p^* instead of δp) and the adequate
normal stresses is settled. Taking these corrections into consideration the problem's fundamental differential equation (3.30) is derived, independently by the energy-method and satisfying the equilibrium condition, and formulas for the internal forces and stresses in the beam's cross-section are determined. Also the proper initial and boundary conditions are discussed and a way of the fundamental equation's solution by means of the finite differences method is suggested. A comparison of the paper's main formulas with these adequate to author's theory of thin-walled cylindrical beam-shells with varying cross-section [9, 10], and those regarding constant sections [8] leads to the conclusion that among them the theory presented here is the most general one.
Two numerical examples demonstrate the practical computation method of the I-beam considered; the first one concerns the static problem, and the second the beam's torsional free vibrations question. Model experiments confirm the correctness of the elaborated theory.