# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## 1 Introduction
The presented discussion concerned the influence of member length on the critical stress magnitude and corresponding buckling mode shape. Comparison of the analytical GBT results to FE simulations by shell elements revealed very good consistency with respect to both local and global structural behavior. Afterwards, the author extended his research to account for elliptical cylindrical shells and tubes under compression [27]. The problem of variable curvature along the cross-section mid-line was solved by introducing parametric functions of tangential angle and their Fourier series expansions to approximate the local arch radius. The proposed formulation proved to be very effective since only three deformation modes (one local-shell, one distortional and one global) were required to accurately evaluate the buckling behavior of elliptical members for a wide range of specimen slenderness. The structural behavior of circular cross-section members was studied also by [PERSON] [28]. The author considered conical shells under three different boundary conditions and subjected to buckling load. In the proposed approach the member deformed configuration was approximated by a combination of the predetermined shell-type deformation modes. Thereafter, [PERSON] and [PERSON] [29] adopted a GBT framework to analyse the ovalization of a tubular cross-section beam when subjected to bending. In the kinematic analysis of the member cross section, distortional and bi-distortional strains were introduced in addition to the regular strain measures of rigid cross-section beams. Several cases of static loadings applied at the free end were investigated followed by large-amplitude free vibrations analysis. This research was continued to account for tubular beams with a soft core, possibly made of foam materials [30]. The findings showed the inclusion of a foam core could improve performance of the pipes due to the shifted structural instability limit load. In the subsequent paper [31], [PERSON] adopted this model to capture the double-layered pipe designs. New terms representing longitudinal slipping of the two layers were added to the kinematic description of the structure profile. Next, a homogenization procedure was applied to obtain the nonlinear, coupled, elastic response function of the beam-like member. An efficient method to model beams with arbitrary but open cross section shapes within Generalized Beam Theory was presented by [PERSON] and [PERSON] [32]. The core of this approach was a modified two-stage cross sectional analysis procedure. Within the first step the curved mid-line of the cross-section was finely approximated by a series of straight segments constituting the polyline. The second stage consisted in selection just a few of natural nodes of this polyline to generate the independent DOFs for warping analysis while the intermediate nodes of the profile were treated like in classic procedure. By this approach it was possible to describe the cross-section geometry accurately, without generating an excessive number of deformation modes. The method was applied to several cross-sections with curved walls and it was shown it assured sufficiently accurate results even with a rough selection of cross-section DOFs. The method proved to be particularly efficient for polygonal sections with rounded corners. The presented above discussion and review of thin-walled structures bibliography demonstrates that the problem of structural behavior of thin-walled beams with curvilinear open profiles has not been comprehensively studied within the GBT framework. The main purpose of present contribution is to fill this gap and to derive the analytical model of a generic thin-walled curvilinear cross-sections beam within the GBT modelling framework. Therefore the outline of the paper is as follows. In Section 2.1 the analysis of the profile kinematics is presented and expressions for strains in curvilinear reference system are given.