# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## 1 Introduction
Since the mid 1990s, [PERSON]'s original theory received significant attention from the research community and was extended to take into account other cases and effects. [PERSON] [16] proposed to use the Generalized Beam Theory to estimate the shear stiffnesses of various cross-sections as particular cases of a general analysis embracing torsion, bending, extension and shear of regular prismatic systems. [PERSON] and [PERSON] [17] developed a geometrically nonlinear Generalized Beam Theory to consider large structural deformations. The proposed formulation was used to study the post-buckling behavior of steel columns with cold-formed thin-walled profiles. The presented analysis showed, among other things, the need to include shear and transverse extension modes to properly capture system kinematics. The relevant shear-deformable GBT structural model was elaborated and presented in authors later research [18; 19]. [PERSON] and [PERSON] [20] adopted the GBT to study the response of prismatic thin-walled members stiffened at different cross-sections along their span. Next, [PERSON] and [PERSON] [21] revisited the standard GBT algorithm for the determination of the conventional modes, in which bending, shear and local modes are evaluated separately. Authors proposed to use the dynamic modes of an unconstrained planar frame similar in shape to the member profile to calculate cross-section in-plane deformation modes. Next, based on the enforcement of shear conditions, the out-of-plane warping component was evaluated. [PERSON] et al. [22; 23; 24] proposed the new method to evaluate a suitable basis of modes for the elastic analysis of thin-walled members. The suggested treatment relied on the solution of two distinct eigenvalue problems governing the in-plane and the out-of-plane free oscillations of a thin-walled beam cross-section. The first eigenvalue problem was solved posing an inextensibility condition for the planar frame beam having the shape of the TWB middle line. In the second eigenvalue analysis the transverse extension and membrane shear strain of the plate elements forming the cross-section were accounted for and modes orthogonal to the inextensional ones were calculated. The efficiency of the proposed method was confirmed by the outcomes of numerical examples. The research was continued later by the authors to extend the theory to account for nonlinear effects exhibited by elastic thin-walled beams [25]. To this aim, both linear and nonlinear functionswere used to describe the displacement field of the member cross-section. The admissible set of trial functions was determined requiring that the classic [PERSON]'s kinematic hypotheses of the linear theory (i.e., (a) transverse inextensibility and (b) unshearability) were satisfied also in the nonlinear sense. An illustrative example of C-lipped cross-section beam with uniform thickness subjected to a uniform vertical pressure load was presented. The accuracy of GBT analytical results was confirmed by the outcomes of FE simulations. Studying the invoked above references and other GBT related literature one observes the vast majority of research deals with polygon-like cross sections while the cases of non-prismatic open or closed profiles are very scarce. This results from the fact that curved geometries make the problem to find the cross-section deformation modes quite complex. The exception might be tubular beam designs studied by [PERSON] in [14], who presented solutions to the problem of uniform bending of a closed cylinder specimen. The analysis was enhanced by discussion of solution accuracy when approximating the cylinder profile by a regular polygon shape. In recent years, [PERSON] [26] investigated the buckling behavior of circular hollow section (CHS) members. He postulated to extend the cross section analysis and include axisymmetric and torsional modes apart from the set of classical orthogonal shell-type deformations of the profile.