Skręcanie Mostowych i Hydrotechnicznych Konstrukcji Cienkościennych o Przekroju Zamkniętym - Dąbrowski, R. (Engineering Transactions, 1958)
In the first (introductory) and second sections, types of thin-walled structures of closed cross-sections are described as applied to bridge or water gate structures. In the third article, the equations of the engineer's theory of torsion of thin-walled bars with closed cross-sections are, as a point of departure for the subsequent considerations, described briefly in the more accurate form proposed by Umansky. This theory is based on the assumption that the geometric form of the cross-section does not change during torsion and that the longitudinal normal stresses due to the fact that the warping is not free are in a form similar to that of the longitudinal displacements appearing for free warping. Besides the angle of torsion ϕ, a certain warping function 6 is introduced, see the Eq. (3.9). In the fourth section, the above solution is generalized to cross-sections of the open-closed type which are the most frequent in bridge and hydraulic structures. Next, for a series of bridge cross-sections (closed, open and open-closed), transversal influence lines are represented for normal stresses at certain specific points of the cross-section in the case of a concentrated force acting at the middle section of the span, with variable eccentricity. The normal stresses are due to bending and torsion. The influence lines illustrate the advantages of a closed cross-section which enables the structure to carry transversal loads of considerable eccentricity and, if rigid diaphragms are spaced sufficiently closely, with relatively small increase of normal stresses. In the next part of the fourth section, a numerical-graphical method for determining the quantities appearing in the theory of hampered torsion (assuming undeformable cross-sections) in presented in the case of torsion of a lens shaped flap where the axis of torsion is given beforehand. In the last part of this section, the equilibrium equations and the relations presented above are used to derive differential equations of torsion of a curved bridge of bisymmetric cross-section with the simplifying assumption that the functions representing the warping and the angle of twist are identical. The fifth section is devoted to the influence of deformation of the cross-section for hampered torsion. This deformation occurs in the case where the distances between diaphragms are considerable (or where the diaphragms are arranged over the supports only or are not present at all) and the load may act between diaphragms. The solution is of an approximate character. This is because during the calculation of the normal stress due to the so-called deforming load of shear forces acting in a cross-section), the (a self-equilibrating system influence of shearing strain in the plane of each wall is disregarded. The equation of the problem shows an analogy to that of a beam resting on an elastic foundation, and the solution may be reduced to that of the latter after introducing a substitute system with suitable boundary conditions. Theoretical considerations of this paper refer to load carrying structures with constant cross-section over the entire length.