First-order generalised beam theory for curved thin-walled members with circular axis

This paper presents a first-order Generalised Beam Theory (GBT) formulation for naturally curved thin-walled members with deformable cross-section, whose undeformed axis is a circular arc with no pre-twist. First, the strain-displacement relations for naturally curved thin-walled members are derived and it is shown how the classic GBT assumptions concerning the strains can be incorporated, namely: (i) Kirchhoff's thin-plate assumption, (ii) Vlasov's null membrane shear strain assumption and (iii) the null membrane transverse extension assumption. The equilibrium equations are obtained in terms of GBT modal matrices and stress resultants. It is demonstrated that, for the so-called “rigid-body” deformation modes (extension, bending and torsion), the GBT equations coincide with those of the Winkler (in-plane case) and Vlasov (out-of-plane case) theories. A standard displacement-based GBT finite element is used to solve a set of representative illustrative examples involving complex local-global deformation. It is shown that the proposed GBT formulation leads to extremely accurate results with a reduced number of DOF and that the GBT modal solution provides an in-depth insight into the structural behaviour of naturally curved members.