Torsional Warping and Torsional Distortion as Fundamental Deformable Section Modes

As can be observed from the T-joint problem considered in Fig. 1.8, the effects of section deformations, i.e., torsional warping and torsional distortion, on the overall stiffness of a thin-walled beam are significant due to their coupling behavior with torsional rotation when the beam is subjected to a torsional load or a more general load. The torsional warping and torsional distortion modes involve deformation of the cross-section of a beam, but they will be treated as modes belonging to the fundamental mode set in the HoBT; these two modes as well as the six rigid-body section modes are treated as fundamental modes in this book. As shall be shown later, none of the higher-order section-deformable modes, such as torsional warping and torsional distortion modes (see Table 1.1), produce net non-zero resultant force or moment. Therefore, the stress and displacement of these modes should decay along the axis of a beam under a static load. However, their decay rates can be quite different depending on the mode type and order. For instance, the stress and displacement of torsional warping and torsional distortion modes can survive even several times longer than the cross-sectional width, while those of other higher-order modes decay rapidly. It should also be noted that due to geometric characteristics, the effects of the torsional warping mode for open-section beams generally survive longer than those for closed-sectioned beams. Because the warping effect in open-section beams is generally significant, a structural analysis of such beams using field variables having non-zero resultant forces and moments only produces unacceptably inaccurate results. For this reason, Vlasov (1961) mostly focused his analyses on the torsional warping mode in open-section beams in his beam theory. In the case of closed thin-walled cross-sections, the torsional distortion mode can be induced by torsional warping (Kim and Kim 1999a, b, 2000, 2003; Choi and Kim 2021; Camotim et al. 2010; Goncalves et al. 2010; Yu et al. 2012). Therefore, both torsional warping and torsional distortion modes should be used simultaneously to yield accurate results. (Note that torsion warping is generally coupled with torsion in thin-walled closed-section beams.)