O pewnej przybliżonej teorii zginania i skręcania prętów prostych o przekroju pełnym

 An approximate theory of bending and torsion of straight solid bars

An approximate theory of bending, shear and torsion of straight bars of constant arbitrary cross-sections is developed, solid bars being discussed as well as thin-walled bars, independently of the degree of symmetry of the cross-section. The inaccuracies of the theory are chiefly due to the following two simplifying assumptions: (1) the warping of the cross­sections is uniform along the axis of the bar, (2) the cross-sections behave as rigid in their planes. The first assumption signifies that bending by terminal loads is considered instead of pure bending (treated in Strength of Materials). The second is the tacit assumption of Strength of Materials and leads to discrepancies with Saint-Venant's theory of bending, the accordance with his theory of torsion being preserved, however. The assumption that in the theory of shear Poisson' s ratio is equal to zero results, in some cases, in considerable errors, which should be remembered when this theory is being applied.

Assuming the structure to be transversally-isotropic, Eqs. (2.3), we obtain, from the basic relations(2.1), the equations for stresses (2.4) under the assumption of v = v' = 0. The first of these equations can easily be transformed into (2.11). The equation (2.12) for unit angle of twist can be obtained in an equally easy manner. Next, the problem of simple bending (υ = 0) is discussed in the case where the warping function satisfies Poisson's equation (3.2). The coordinates of the centre of shear are determined by the general equations (3.9). It can be verified that for v = 0, the function Φ proposed by Leibenzon, [12], becomes our warping function w_s.

The cases of circular, elliptic and rectangular bars subjected to shear with the force Q_y are considered as well as a bar of narrow, symmetrical cross-section (Fig. 6).