Sectional Shape Functions for a Box Beam Under Torsion: Wall-Bending Field

In Chap. 4, the sectional shape functions (ψzW, ψsχ) for a box beam under torsion, which correspond to the wall-membrane field, were derived. This section is devoted to the derivation of the sectional shape functions (ψnχ, ψnη¯, ψnη^) corresponding to the wall-bending field. (See Ferradi and Cespedes (2014), Bebiano et al. (2015), and Choi et al. (2017) for earlier developments.) To argue for the co-existence of ψnχ with ψsχ, we observe that if the distortion mode χ has a non-zero ψsχ (the s-directional displacement component) only (see Fig. 5.1a) without its n-directional counterpart, ψnχ, two adjacent sectional edges cannot remain connected at the corners. Therefore, ψnχ cannot be zero. It was shown in Chap. 2 that the zeroth-order distortion mode χ₀ has a non-zero ψnχ0, as given by Eq. (2.50). If ψnχ(z,s) does not vanish, it will induce a non-zero u~_s(z, n, s) for n≠ 0, as expressed by Eq. (3.3c) and thus causes the bending of cross-sectional walls. The sectional shape functions ψnχk for k≥ 1 will be derived in Sect. 5.3 identically to how ψsχk was derived in Chap. 4; ψnχk can be obtained as the secondary deformation of the axial stress through Poisson’s effect.