# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## 1 Introduction  
Thin-walled beams are structural elements with three characteristic dimensions of different orders of magnitude: the thickness is small when compared to the dimensions of the cross-section, which in turn are small when compared to the beam length. Starting from the second half of the 19 th century, slender thin-walled members have found applications in civil and naval engineering as beams, columns, frame-works etc. Later they have been adopted to the needs of aeronautics and aerospace design, as well as, e.g., turbomachinery industry. They offer a number of advantages over the classical compact section beams. The most important ones are lightweight combined with strong rigidity and loads resistance, lower manufacturing costs due to reduced resource consumption and labor-saving design, lower transport and maintenance expenses etc.. Moreover, thanks to their specific layout they leave to the designer much more flexibility regarding the choice of the material and cross-section shape to meet any specific design requirements. The slenderer thin-walled elements exhibit significantly different kinematic behavior when compared to solid section beams. In particular the profile warping occurs combined with elastic couplings involving bending and twisting; other local effects arise as well. Therefore, the recognized classical beam theories like Euler-Bernoulli and Timoshenko cannot be trustworthy to analyse thin-walled members. The first successful attempt to model the kinematics of thin-walled beams was done by [PERSON] [1]. He developed the Theory of Sectorial Area postulating the non-uniform torsion along the beam axis contributing to out-of-plane warping of the cross-section. This hypothesis was accompaniedby further assumptions regarding the cross-section being rigid in its own plane and no shear deformability along the profile mid-line. [PERSON]'s analytical model was widely adopted by the research community and extended over the years to account for other effects like geometric non-linearities in longitudinal deformations caused by large cross sectional rotation [2], shear deformability along the wall thickness [3], variable cross section specimens (stepped or tapered) [4; 5], curved axis beams [6; 7] or nonlinear warping effects [8]. A further generalization of [PERSON]'s model was proposed by [PERSON] [9]. He revisited the original assumptions to include shear deformation over the cross-section midline by generalizing the description of warping. Later, significant research was focused on modern materials with directional properties like composites and laminates. The works by [PERSON] and [PERSON] [10] and Variational Asymptotic Beam Sectional (VABS) theory developed by [PERSON] and [PERSON] [11] extending the original [PERSON] model are particularly worth noting. Research on the dynamics of composite thin-walled beams with closed sections was also carried out by [PERSON] et al. [12; 13]. According to the original assumptions on beam profile kinematics the Vlasov theory fails to take into account the effects of cross-section distortion and any local in-plane deformations of the walls. In this respect, The Generalized Beam Theory (GBT) is an alternative modeling concept that adds to the [PERSON]'s model the distortion of the cross-section. The method was originally developed by [PERSON] [14; 15] to deal with buckling stability of prismatic cross-section columns. In its original formulation the slender thin-walled element was considered as an assembly of plate segments which are free to bend in the plane of the cross-section. The consequence of this assumption were in-plane deformations of the profile. The latter were assumed to be expressed as the superposition of a series of cross-sectional natural modes whose magnitudes varied along the member span. As a result the two-dimensional (2D) analysis within the shell framework theory is condensed into one-dimensional (1D) beam theory.