## 1 Introduction  
Thin-walled structures are commonly used in civil, aeronautical, and mechanical engineering structures. For simplicity and computational efficiency, one-dimensional theories are preferred to two- and three-dimensional models. However, the development of an efficient one-dimensional theory faces a fundamental challenge of finding a general and simple procedure to determine the cross-section deformation modes capable of describing cross-section in-plane (distortion) and out-of-plane (warping) deformations [1]. Moreover, the set of deformation modes should have a hierarchic capability, to obtain a reduced model and a clear physical interpretation of structural behaviors. To support this, several trends of enhancing beam theories have been developed in the last decades. [PERSON] and [PERSON] [2; 3] developed an asymptotic method and contributed much to variational asymptotic beam sectional analyses. Expansion of the beam displacement field through Taylor or Maclaurin series is a general approach to model both thin-walled and solid structures, and has branched out to composite structures, due to the work of [PERSON] and [PERSON] [4; 5]. Saint-Venant-driven models have been widely studied from the beginning, and new developments can be referred to [PERSON] et al. [6], and [PERSON] and [PERSON] [7]. [PERSON] and [PERSON] were the pioneers in the research of thin-walled structures. [PERSON] [8] established a general thin-walled beam theory with warping of open cross-sections considered. Warping of closed cross-sections was first described by [PERSON] [9] in his study of non-uniformed torsion of thin-walled beams. Since then, refinement of their thin-walled theories has been extensively studied and generated many achievements, such as the recent work of [PERSON] et al. [10] and [PERSON] et al. [11]. Some special structural behaviors, exhibiting their importance in engineering, were also studied where shear deformation was considered. These include the studies of multi-cell distortion ([PERSON] et al. [12]), warping due to shear-lag effects ([PERSON] et al. [13]) and other secondary effects ([PERSON] [14], [PERSON] et al. [15], [PERSON] et al. [16]). Additionally, there are also some studies focused on the physical interpretation of cross-section deformation modes [17; 18]. Recently, the generalized beam theory (GBT) has been the subject of intense scholarly debate, which can accurately handle cross-section deformation stemming from the so-called GBT cross-section analysis. Since the groundbreaking work of [PERSON] [19], GBT has been developed to account for both warping and distortion with shear deformation [20] and transverse extension included; it has also been applied to multi-cell cross-sections [21], composite materials [22] and curved members [23]. As a comparable theory, the beam model, based on generalized eigenvectors [24; 25] has been obtained by considering a mixed Hellinger-Reissner variational formulation. Its sectional deformation modes have been defined through a cross-section discretization, using two-dimensional elements. [PERSON] et al. [26; 27] have also recently proposed a remarkable higher-order one-dimensional model, which allows one to accurately reproduce three-dimensional behaviors of thin-walled structures. The key point is the application of a criterion to uncouple governing equations which have been established based on the solution of a generalized eigenvalue problem. In all fairness, the three theories mentioned above are powerful enough to deal with almost any arbitrary thin-walled cross-sections and any mechanical analyses (including static, dynamic, bulking, and so on), with an acceptable and optional precision. However, one common problem is the solution of the generalized eigenvalue problem, which is quite involved and demanding in uncoupling cross-sectional deformation modes. Actually, a simplified approach that does not include all the features but has a high precision, is more suitable, in some cases.